## AP 9th Class Maths Bits 10th Lesson Heron’s Formula

Multiple Choice Questions (MCQs) :

Question 1.
Area of what type triangle can be solved using Heron’s formula ?
A) Right triangle
B) Isosceles triangle
C) Equilateral triangle
D) Any triangle
(OR)
Heron has derived the formula for the area of _________
A) equilateral triangle
B) scalene triangle
C) isosceles triangle
D) any triangle
D) any triangle

Question 2.
Which of the following formulas can be used to find the area of a triangle using Heron’s formula ?
A) A = $$\frac{1}{2}$$bh
B) A = bh
C) A = $$\frac{1}{2}$$ab sin C
D) A = $$\sqrt{(s(s-a)(s-b)(s-c)}$$
D) A = $$\sqrt{(s(s-a)(s-b)(s-c)}$$

Question 3.
If a triangle has sides of lengths 5 cm, 12 cm and 13 cm, what is its area using Heron’s formula ?
A) 20 cm2
B) 30 cm2
C) 24 cm2
D) 60 cm2
B) 30 cm2

Question 4.
Heron’s formula can be used to find the area of a triangle if you know the lengths of ________
A) All three sides
B) Two sides and an angle
C) Two angles and a side
D) One side and two angles
A) All three sides

Question 5.
In a triangle, if the semi perimeter is s and the lengths of the sides are a, b and c, then what is the value of s in terms of a, b and c ?
A) s = a + b
B) s = a + b + c
C) s = (a + b + c)/2
D) s = (a + b – c)/2
C) s = (a + b + c)/2

Question 6.
What is the maximum area of a triangle with sides of lengths 6 cm, 8 cm and 10 cm ?
A) 12 cm2
B) 24 cm2
C) 30 cm2
D) 48 cm2
B) 24 cm2

Question 7.
If a triangle has sides of lengths 7 cm, 8 cm and 9 cm and the altitude from the longest side is 6 cm, what is the area of the triangle using Heron’s formula ?
A) 20 cm2
B) 12√5 cm2
C) 30 cm2
D) 36 cm2
B) 12√5 cm2

Question 8.
Heron’s formula is named after
A) Euclid
B) Archimedes
C) Pythagoras
D) Hero of Alexandria
D) Hero of Alexandria

Question 9.
A triangle has sides of length 12, 13 and 14. What,is the area of the triangle using Heron’s formula ?
A) 72
B) 72.31
C) 90
D) 96
B) 72.31

Question 10.
The sides of a triangle are 25, 25 and 30. What is the area of the triangle using Heron’s formula?
A) 300
B) 192.3
C) 196.9
D) 200.2
A) 300

Question 11.
The sides of a triangle are 7, 9 and 13. What is the area of the triangle using Heron’s formula ?
A) 12.97
B) 15.25
C) 18.81
D) 29.95
D) 29.95

Question 12.
The sides of a triangle are in the ratio 3 : 8 : 9 and its perimeter, is 132. What is the area of the triangle using Heron’s formula?
A) 716
B) 716.4
C) 716.3
D) 716.5
B) 716.4

Question 13.
The sides of a right triangle are 5, 12, and 13. What is the area of the triangle using Heron’s formula?
A) 30
B) 24
C) 36
D) 40
A) 30

Question 14.
The perimeter of an isosceles triangle is 36 and its two equal sides are 13 each. What is the area of the triangle using Heron’s formula?
A) 84
B) 60
C) 96
D) 102
B) 60

Question 15.
The perimeter of a right triangle is 30 and its hypotenuse is 13. What is the area of the triangle using Heron’s formula ?
A) 30
B) 60
C) 84
D) 156
A) 30

Question 16.
What is Heron’s formula used for ?
A) Finding the area of a circle
B) Finding the area of a triangle
C) Finding the perimeter of a quadrilateral
D) Finding the volume of a sphere
B) Finding the area of a triangle

Question 17.
Which of the following is the correct formula for Heron’s formula?
A) A = bh
B) A = $$\frac{1}{2}$$bh
C) A = $$\sqrt{s(s-a)(s-b)(s-c)}$$
D) A = lb
C) A = $$\sqrt{s(s-a)(s-b)(s-c)}$$

Question 18.
What is the semi-perimeter of a triangle?
A) The sum of the lengths of the sides of a triangle
B) Half of the sum of the lengths of the sides of a triangle
C) Half of the difference between the lengths of the sides of a triangle
D) The product of the lengths of the sides of a triangle
B) Half of the sum of the lengths of the sides of a triangle

Question 19.
What is the maximum number of sides a triangle can have?
A) 2
B) 3
C) 4
D) 5
B) 3

Question 20.
If a triangle has sides of length 3, 4 and 5, what is its area using Heron’s formula ?
A) 6
B) 8
C) 10
D) 12
A) 6

Question 21.
What is the area of an equilateral triangle with sides of length 6 using Heron’s formula ?
A) 9.75
B) 15.59
C) 18
D) 20.78
A) 9.75

Question 22.
What is the area of an isosceles triangle with sides of length 5, 5 and 6 using Heron’s formula ?
A) 6
B) 7.8
C) 8.64
D) 9
C) 8.64

Question 23.
Which of the following is not required to use Heron’s formula ?
A) The length of all three sides of the triangle
B) The perimeter of the triangle
C) The semi-perimeter of the triangle
D) The height of the triangle
D) The height of the triangle

Question 24.
What is the area of a right triangle with sides of length 3 and 4 using Heron’s formula?
A) 3
B) 4
C) 5
D) 6
C) 5

Question 25.
Area of a triangle =
A) $$\frac{1}{2}$$ × Base × Height
B) Base × Height
C) $$\frac{1}{3}$$ × Base × Height
D) $$\frac{1}{4}$$ × Base × Height
A) $$\frac{1}{2}$$ × Base × Height

Question 26.
The area of ΔABC in which AB = BC = 4 cm and ∠B = 90° is
A) 16 cm2
B) 8 cm2
C) 4 cm2
D) 12cm2
B) 8 cm2

Question 27.
Area of a triangle having base 6 cm and altitude 8 cm is
A) 48 cm2
B) 24 cm2
C) 64 cm2
D) 36 cm2
B) 24 cm2

Question 28.
Area of a triangle is 60 cm2. Its base is 15 cm. Its altitude is
A) 30 cm
B) 4 cm
C) 8 cm
D) 10 cm
C) 8 cm

Question 29.
The area of a right triangle is 36 cm2 and its base is 9 cm. Find the length of the perpendicular
A) 8 cm
B) 4 cm
C) 16 cm
D) 32 cm
A) 8 cm

Question 30.
The side of an isosceles right angle triangle of hypotenuse 5√2 cm is
A) 10 cm
B) 8 cm
C) 5 cm
D) 3√2 cm
C) 5 cm

Question 31.
The base of a right triangle is 15 cm and its hypotenuse is 25 cm. Then its area is
A) 187.5 cm2
B) 375 cm2
C) 150 cm2
D) 300 cm2
C) 150 cm2

Question 32.
The side of an equilateral triangle is 6 cm. The area of the triangle is
A) 6√3 cm2
B) 9√3 cm2
C) 16√3 cm2
D) 3√3 cm2
B) 9√3 cm2

Question 33.
Side of an equilateral triangle is 4 cm. Its area is
A) 4√3 cm2
B) $$\frac{\sqrt{3}}{4}$$ cm2
C) √3 cm2
D) 2√3 cm2
A) 4√3 cm2

Question 34.
Find the length of the side of an equi-lateral triangle whose area is 9√3 m2
A) 1 cm
B) 2 cm
C) 3 cm
D) 6 cm
D) 6 cm

Question 35.
The area of an equilateral triangle is 16√3 m2. Its perimeter (in metres) is
A) 12
B) 48
C) 24
D) 306
C) 24

Question 36.
The perimeter of an equilateral triangle is 60 m. Its area is
A) 10√3m2
B) 100√3m2
C) 15√3m2
D) 20√3 m2
B) 100√3m2

Question 37.
If the length of median of an equilateral triangle be x cm, then its area is
A) x2
B) $$\frac{\sqrt{3}}{2}$$x2
C) $$\frac{\mathrm{x}^2}{\sqrt{3}}$$
D) $$\frac{x^2}{2}$$
C) $$\frac{\mathrm{x}^2}{\sqrt{3}}$$

Question 38.
The sides of a triangle are 7 cm, 24 cm and 25 cm. Its area is
A) 168 cm2
B) 84 cm2
C) 87.5 cm2
D) 300 cm2
B) 84 cm2

Question 39.
Area of an equilateral triangle of side ‘a’ is
A) $$\frac{\sqrt{3}}{4}$$a2
B) $$\frac{\mathrm{a} \sqrt{3}}{2}$$
C) √3a2
D) a√3
A) $$\frac{\sqrt{3}}{4}$$a2

Question 40.
Area of an equilateral triangle of side ‘a’ units can be calculated by using the formula
A) $$\sqrt{s^2(s-a)^2}$$
B) (s – a)$$\sqrt{s^2(s-a)}$$
C) $$\sqrt{\mathrm{s}(\mathrm{s}-\mathrm{a})^2}$$
D) (s – a)$$\sqrt{s(s-a)}$$
D) (s – a)$$\sqrt{s(s-a)}$$

Question 41.
Find the perimeter of the triangle whose sides are 17 cm, 33 cm and 20 cm
A) 70 cm
B) 50 cm
C) 53 cm
D) 37 cm
A) 70 cm

Question 42.
The semi-perimeter of a triangle having the length of its sides as 20 cm, 15 cm and 9 cm is
A) 44 cm
B) 21 cm
C) 22 cm
D) None
C) 22 cm

Question 43.
The perimeter of a triangular plot is 16 m. If the measures of its two sides are 5 m and 6 m, then find the third side
A) 2 m
B) 3 m
C) 5 m
D) 4 m
C) 5 m

Question 44.
Two sides of a triangle are 13 cm and 14 cm and its semi-perimeter is 18 cm. Then third side of the triangle is
A) 12 cm
B) 11 cm
C) 10 cm
D) 9 cm
D) 9 cm

Question 45.
The area of a triangle whose sides are 13 cm, 14 cm and 15 cm is
A) 42 cm2
B) 86 cm2
C) 84 cm2
D) 100 cm2
C) 84 cm2

Question 46.
Two equal rides of an isosceles triangle are 13 cm each and its perimeter is 36 cm. Find the area of the triangle.
A) 20 cm2
B) 30 cm2
C) 40 cm2
D) 60 cm2
D) 60 cm2

Question 47.
Find the area of an isosceles triangle whose equal sides are 6 cm each and the third side is 8 cm.
A) 8√5 cm2
B) 5√8 cm2
C) 3$$\sqrt{55}$$ cm2
D) 3√8 cm2
A) 8√5 cm2

Question 48.
The perimeter of a triangle is 36 cm and its sides are in the ratio a : b : c = 3 : 4 : 5, then a, b, c are respectively.
A) 9 cm, 15 cm, 12 cm
B) 15 cm, 12 cm, 9 cm
C) 12 cm, 9 cm, 15 cm
D) 9 cm, 12 cm, 16 cm
D) 9 cm, 12 cm, 16 cm

Question 49.
The sides of a triangular field are in the ratio 3 : 4 : 5. The perimeter of the f triangular field is 144 m. Find the longest side of the field.
A) 15 m
B) 30 m
C) 60 m
D) 90 m
C) 60 m

Question 50.
A) $$\frac{1}{2}$$ × a diagonal × sum of the perpendiculars on the diagonal
B) a diagonal × sum of the perpendiculars on the diagonal
C) $$\frac{1}{3}$$ × a diagonal × sum of the perpendiculars on the diagonal
D) $$\frac{1}{4}$$ × a diagonal × sum of the perpendiculars on the diagonal
A) $$\frac{1}{2}$$ × a diagonal × sum of the perpendiculars on the diagonal

Question 51.
Area of a trapezium =
A) $$\frac{1}{2}$$ × sum of parallel sides × distance between the parallel sides
B) sum of parallel sides × distance between the parallel sides
C) $$\frac{1}{3}$$ × sum of parallel sides × distance between the parallel sides
D) None of these
A) $$\frac{1}{2}$$ × sum of parallel sides × distance between the parallel sides

Question 52.
1 hectare =
A) 10 m2
B) 100 m2
C) 1000 m2
D) 10000 m2
D) 10000 m2

Question 53.
1 acre =
A) 10 m2
B) 100 m2
C) 1000 m2
D) 10000 m2
B) 100 m2

Question 54.
Find the area of a right angled triangle, if the radius of the semi-circle is 3 cm and altitude drawn to the hypotenuse is 2 cm

A) 4 cm2
B) 6 cm2
C) 8 cm2
D) 12 cm2
B) 6 cm2

Question 55.
The side of a square is 5 cm. Its perimeter is
A) 5 cm
B) 20 cm
C) 25 cm
D) 10 cm
B) 20 cm

Question 56.
Find the area of a quadrilateral whose one diagonal is 8 cm and the sum of perpendiculars from vertices is 10 cm
A) 20 cm2
B) 40 cm2
C) 80 cm2
D) 160 cm2
B) 40 cm2

Question 57.
The diagonals of a rhombus are 10 cm and 8 cm. Its area is
A) 80 cm2
B) 40 cm2
C) 9 cm2
D) 36 cm2
B) 40 cm2

Question 58.
The area of a rhombus is 96 cm2. If one of its diagonals is 16 cm, then the length of its sides is
A) 12 cm
B) 10 cm
C) 8 cm
D) 6 cm
B) 10 cm

Question 59.
The perimeter of a rhombus is 20 cm. If one of its diagonals is 6 cm, then its area is
A) 28 cm2
B) 36 cm2
C) 24 cm2
D) 20 cm2
C) 24 cm2

Question 60.
The parallel sides of a trapezium are 45.8 cm and 81.2 cm and the distance, between them is 22 cm. Find the area of the trapezium.
A) 1397 cm2
B) 1937 cm2
C) 3197 cm2
D) 139.7 cm2
A) 1397 cm2

Question 61.
A regular hexagon has a side 8 cm. Find its area.
A) 8√3 cm2
B) 96√3 cm2
C) 4√3 cm2
D) 12√3 cm2
B) 96√3 cm2

Question 62.
In the Heron’s formula which of the following is correct ?
A) S is perimeter of triangle
B) S is semi-perimenter of triangle
C) 2S is semi-perimeter of triangle S
D) $$\frac{\mathrm{S}}{2}$$ is perimeter of triangle
B) S is semi-perimenter of triangle

Assertion and Reason type questions :

Question 1.
Assertion : Heron’s formula can be used to find the area of any triangle.
Reason : Heron’s formula uses1 the , length of the sides of the triangle to calculate its area.
Assertion is true.
Reason is true and explains the assertion.
Explanation : Heron’s formula can be used to find the area of any triangle regardless of its shape or size. This is because the formula uses the length oi the suies of the triangle to calculate its area, which means that as long as you know the lengths of the sides, you can use the formula to find the area. Therefore, the assertion is true and the reason is also true and explains why Heron’s formula can be used to find the area of any triangle.

Question 2.
Assertion : Heron’s formula is named after the ancient Greek mathematician Euclid.
Reason : Euclid was the first person to discover the formula for finding the area of a triangle.
Assertion is false.
Reason is false.
Explanation : The assertion is false because Heron’s formula is actually named after the ancient Greek mathematician Heron of Alexandria, who discovered the formula. The reason is also false because while Euclid did write extensively on geometry, he did not discover Heron’s formula for finding the area of a triangle.

Question 3.
Assertion : Heron is formula can be used to find the area of a triangle even if only two sides and an angle opposite one of the sides are known.
Reason : Heron’s formula involves the use of the Pythagorean theorem to find the area of a triangle.
Assertion is false.
Reason is false.
Explanation ; The assertion is false because Heron’s formula requires the lengths of all three sides of a triangle to be known, not just two sides and an opposite angle. The reason is also false because while the Pythagorean theorem is used in some formulas for finding the area of a triangle, it is not used in Heron’s formula.

Question 4.
Assertion : Heron’s formula is more accurate for finding the area of a triangle than the formula A = 1/2bh.
Reason : Heron’s formula takes into account the lengths of all three sides of the triangle, while A = 1/2bh only takes into account the base and height.
Assertion is true.
Reason is true and explains the assertion.

Explanation : Heron’s formula is more accurate for finding the area of a triangle than the formula A = $$\frac{1}{2}$$ bh because it takes into account the lengths of all three sides of the triangle, which means that it can hp used to find the area of any triangle, regardless of its shape or size. In contrast, A = $$\frac{1}{2}$$ bh only takes into account the base and height of a triangle, which means that it can only be used to find the area of certain types of triangles, such as right triangles. Therefore, the assertion is true and the reason is , also true and explains why Heron’s formula is more accurate.

Question 5.
Assertion : Heron’s formula can be Used to find the area of a triangle with sides of length 0.
Reason : Heron’s formula involves dividing by zero in certain cases, which means that it can be used to find the area of any triangle, including those with sides of length 0.
Assertion is false.
Reason is false.
Explanation : The assertion is false because Heron’s formula cannot be used to find the area of a triangle with sides of length 0. This is because the formula involves taking the square root of a number and the square root of a negative number (which results when one or more sides are 0) is undefined. The reason is also false because Heron’s formula does not involve dividing by zero in any case, but rather involves taking the square root of a value that is always greater than or equal to zero.

Fill in the blanks :

1. Heron’s formula gives the area of a triangle in terms of the lengths of its ___________ .
sides

2. The semiperimeter of a triangle is defined as ___________ of the lengths of its sides.
half the sum

3. The area of a triangle with side lengths 5 cm, 12 cm and 13 cm is _________ cm2.
30

4. Heron’s formula can be used to find the area of a triangle when the lengths of ___________ are given.
all three sides

5. The formula for the semiperimeter of a triangle with sides a, b and c is s = ___________ .
(a + b + c) / 2

6. A triangle with side lengths 7 cm, 8 cm and 9 cm has an area of ___________ cm2.
26.83 (rounded to two decimal places)

7. If a triangle has sides of length 6 cm, 8 cm and 10 cm, then its area is __________ cm2.
24

8. Heron’s formula is named after _________ a Greek mathematician and engineer who lived in the first century AD.
Hero of Alexandria

9. The area of a triangle with sides of length 12 cm, 16 cm and 20 cm can be found using Heron’s formula to be _________ cm2.
96

10. The area of an equilateral triangle with side length 10 cm can be found using Heron’s formula to be __________ cm2.
43.30 (rounded to two decimal places)

11. The area of a triangle with sides of length 5, 7 and 8 can be found using Heron’s formula to be ___________ .
10√3 sq units

12. If one side of a right triangle are in 5 cm and the length of the hypotenuse is 13cm, then the area of the triangle can be found using Heron’s formula to be ___________ .
30

13. If the sides of a triangle are in the ratio 3:4:5 and the semiperimeter of the tri-angle is 30 cm .then its area is ____________ cm2.
150

14. If the length of two sides of a triangle are 15 cm and 8 cm and the angle between them is 90 degrees, then the area of the triangle can be found using Heron’s formula to be __________ .
60 cm2

15. The sides of a triangle are in the ratio 7 : 9 : 11 and the perimeter of the triangle is 54 cm. Then the area of the triangle can be found using Heron’s formula to be ___________ .
125.68 cm2

16. If the perimeter of an isosceles triangle is 26 cm and the length of the inequal sides is 10 cm, then the area of the triangle can be found using Heron’s for-mula to be ___________ .
31.22 cm2

17. If the area of a triangle is 12√3 cm2 and the semiperimeter is 12 cm and if two lengths of the sides are 6 and 10 cm, then the third side of the triangle will be ___________ .
9 cm

18. If the area of a triangle is 24 cm2 and two sides of the triangle are 6 cm and 8 cm then the third side of the triangle can have lengths ___________ .
10 cm

19. If the lengths of two sides of a triangle are 5 cm and 7 cm and the area of the triangle is 12 cm2, then the possible lengths of the third side of the triangle are ___________ .
11.17

## AP 9th Class Maths Bits 9th Lesson Circles

Multiple Choice Questions (MCQs) :

Question 1.
In a circle with center O, the length of a chord AB is 16 and its distance from the center O is 6. What is the radius of the circle ?
A) 6
B) 8
C) 10
D) 12
C) 10

Question 2.
In a circle, a chord of length 6 cm is at a distance of 4, cm from the center of the circle. What is the length of the radius of the circle ?
A) 2 cm
B) 3 cm
C) 4 cm
D) 5 cm
D) 5 cm

Question 3.
In a circle, if a chord is of length 8 cm and the radius of the circle is 5 cm, what is the distance between the chord and the center of the circle?
A) 3 cm
B) 4 cm
C) 5 cm
D) 6 cm
A) 3 cm

Question 4.
In a circle, if an angle at the center is 90 degrees, what is the measure of the subtended angle at the circumference?
A) 45 degrees
B) 90 degrees
C) 180 degrees
D) It cannot be determined from the given information.
A) 45 degrees

Question 5.
In a circle, if the angle at the center is 60 degrees, what is the measure of the subtended angle at the circumference?
A) 30 degrees
B) 60 degrees
C) 120 degrees
D) 180 degrees
A) 30 degrees

Question 6.
In a circle, the measure of an angle formed by a tangent and a chord is equal to
A) half the measure of the arc it subtends
B) the measure of the arc it subtends
C) twice the measure of the arc it subtends
D) the measure of the diameter of the circle
A) half the measure of the arc it subtends

Question 7.
In a circle, the measure of an angle formed by two chords that intersect inside the circle is equal to
A) half the measure of the sum of the arcs they subtend
B) half the measure of the difference of the arcs they subtend
C) the measure of the sum of the arcs they subtend
D) the measure of the difference of the arcs they subtend
B) half the measure of the difference of the arcs they subtend

Question 8.
In a circle, the measure of an angle formed by two chords that-intersect outside the circle is equal to
A) half the measure of the sum of the arcs they subtend
B) half the measure of the difference of the arcs they subtend
C) the measure of the sum of the arcs they subtend
D) the measure of the difference of the arcs they subtend
B) half the measure of the difference of the arcs they subtend

Question 9.
In a circle with center O, if angle AOB is 60 degrees, what is the measure of angle ACB, where C is point on circumference of circle?
A) 90 degrees
B) 60 degrees
C) 30 degrees
D) 120 degrees
C) 30 degrees

Question 10.
In a circle with center O, if angle AOB is 120 degrees, what is the measure of angle ACB, where C is point on circumference of circle?
A) 30 degrees
B) 60 degrees
C) 90 degrees
D) 120 degrees
B) 60 degrees

Question 11.
In a circle with center O, if angle APB is 45 degrees, where P is point on circumference of circle. Then what is the measure of angle AOB, where AB is a chord of the circle?
A) 45 degrees
B) 90 degrees
C) 135 degrees
D) 180 degrees
B) 90 degrees

Question 12.
In a circle with center O, if angle APB is 90 degrees, what is the measure of angle AOB, where AB is a chord of the circle?
A) 45 degrees
B) 90 degrees
C) 135 degrees
D) 180 degrees
D) 180 degrees

Question 13.
In a circle with center O, if angle AOB is 120 degrees, what is the measure of angle APB, where P is point on circumference of circle, and AB is a chord of the circle ?
A) 45 degrees
B) 90 degrees
C) 135 degrees
D) 60 degrees
B) 90 degrees

Question 14.
In a circle with center O, if angle APB is 66 degrees, what is the measure of angle AOB, where AB is a chord of the circle?
A) 60 degrees
B) 132 degrees
C) 150 degrees
D) 240 degrees
B) 132 degrees

Question 15.
In a circle with center O, if angle AOB is 90 degrees, what is the measure of angle ACB, where AC and BC are tangents to the circle at points A and B, respectively?
A) 30 degrees
B) 45 degrees
C) 60 degrees
D) 90 degrees
D) 90 degrees

Question 16.
In a circle with center O, if angle AOB is 60 degrees, what is the measure of angle ACB, where AC and BC are tangents to the circle at points A and B, respectively?
A) 120 degrees
B) 45 degrees
C) 60 degrees
D) 90 degrees
A) 120 degrees

Question 17.
In a cyclic quadrilateral, opposite angles sure are ______________ .
A) supplementary
B) complementary
C) congruent
D) equal
A) supplementary

Question 18.
If a quadrilateral is cyclic, then the sum of its opposite angles is ________ degrees.
A) 180
B) 360
C) 90
D) 270
B) 360

Question 19.
The opposite angles of a cyclic quadrilateral are _______________ .
A) equal and complementary
B) equal and supplementary
C) congruent and complementary
D) congruent and supplementary
D) congruent and supplementary

Question 20.
In a cyclic quadrilateral, the sum of two adjacent angles is ____________ degrees.
A) 90
B) 180
C) 270
D) 360
B) 180

Question 21.
If the opposite angles of a quadrilateral are congruent, then the quadrilateral is _____________ .
A) a parallelogram
B) a rectangle
C) a square
D) a rhombus
B) a rectangle

Question 22.
In a cyclic quadrilateral, the sum of any two opposite angles is _____________ degrees.
A) 180
B) 360
C) 90
D) 270
A) 180

Question 23.
A quadrilateral is cyclic if and only if its opposite angles are _________________ .
A) congruent
B) supplementary
C) complementary
D) equal
B) supplementary

Question 24.
In a cyclic quadrilateral, the measure of an angle subtended by an arc outside the quadrilateral is equal to the measure of the ____________ angle of the quadrilateral.
A) opposite
C) diagonal
D) complementary
A) opposite

Question 25.
What is the measure of the angle formed by a chord that is tangent to a circle at one of its endpoints?
A) 90 degrees
B) 180 degrees
C) 45 degrees
D) It depends on the length of the chord.
A) 90 degrees

Question 26.
In a circle with a diameter of 10 cm, what is the measure of the angle subtended by the diameter?
A) 90 degrees
B) 180 degrees
C) 45 degrees
D) 120 degrees
B) 180 degrees

Question 27.
In a circle with center O, what is the measure of the angle subtended by a semicircle at a point on circumference of circle?
A) 90 degrees
B) 180 degrees
C) 45 degrees
D) 120 degrees
A) 90 degrees

Question 28.
If a chord of a circle subtends an angle of 60 degrees at the center of the circle, what is the measure of the angle subtended by the chord at point on circumference of circle ?
A) 30 degrees
B) 60 degrees
C) 90 degree
D) 120 degrees
A) 30 degrees

Question 29.
If two chords of a circle intersect inside the circle, what is the relationship between the angles formed by the chord and their Intercepted arcs?
A) They are equal.
B) They are supplementary.
C) They are complementary.
D) They are congruent.
B) They are supplementary.

Question 30.
In a circle, if two chord are equal in length, then
A) they are parallel
B) they are perpendicular
C) they are congruent
D) they intersect at the center of the circle
C) they are congruent

Question 31.
The distance from the center of a circle to a chord is called
B) the diameter
C) the circumference
D) the chord length

Question 32.
In a circle, if a chord is perpendicular to a radius, then
A) the chord bisects the radius
B) the chord is a diameter of the circle
C) the chord is tangent to the circle
D) the chord is a secant of the circle
B) the chord is a diameter of the circle

Question 33.
In a circle, if two chords are equidistant from the center, then
A) they are parallel
B) they are perpendicular
C) they are congruent
D) they intersect at the center of the circle
A) they are parallel

Question 34.
In a circle, an angle formed by two radii is called
A) a central angle
B) an inscribed angle
C) a tangent angle
D) a chord angle
A) a central angle

Question 35.
In a circle, the measure of a central angle is equal to
A) the measure of the arc it subtends
B) half the measure of the arc it subtends
C) twice the measure of the arc it subtends
D) the measure of the diameter of the circle
A) the measure of the arc it subtends

Question 36.
In a circle, the measure of an inscribed angle is equal to
A) half the measure of the arc it subtends
B) the measure of the arc it subtends
C) twice the measure of the arc it subtends
D) the measure of the diameter of the circle
A) half the measure of the arc it subtends

Question 37.
The path traced by the tip of the second’s hand is a
A) circle
B) square
C) rectangle
D) straight line
A) circle

Question 38.
The shape of the coin of ₹ 1 is
A) triangle
B) rhombus
C) circle
D) trapezium
C) circle

Question 39.
The wheels of a vehicle are in
A) rectangular
B) triangular
C) circular
D) trapezoidal shape
C) circular

Question 40.
The longest chord of a circle
B) Arc
C) Diameter
D) Segment
C) circular

Question 41.
The centre of a circle lies
A) outside the circle
B) inside the circle
C) on the circle
D) none of these
B) inside the circle

Question 42.
The minute hand of a clock is at 12 and the smaller hour’s hand is at 2. The angle between the hands of the clock is
A) 10°
B) 20°
C) 30°
D) 60°
D) 60°

Question 43.
The perpendicular from the centre of a circle bisects the
A) circle
B) circumference
C) chord
C) chord

Question 44.
Given a circle with centre ‘O’ and smallest chord AB is of length 6 cm and the longest chord CD of the circle is of length 10 cm, then the radius of the circle is
A) 15 cm
B) 6 cm
C) 5 cm
D) 3.5 cm
C) 5 cm

Question 45.
A chord of length 24 cm of a circle is at a distance of 5 cm from the centre. The radius of the circle is
A) 13 cm
B) 12 cm
C) 11 cm
D) 19 cm
A) 13 cm

Question 46.
The length of a chord of a circle is 16 cm and its distance from the centre is 6 cm. The measure of radius of the circle is
A) 6 cm
B) 8 cm
C) 10 cm
D) 12 cm
C) 10 cm

Question 47.
The length of the chord of a circle, of radius 13 cm, at a distance of 5 cm from the centre is
A) 12 cm
B) 18 cm
C) 20 cm
D) 24 cm
D) 24 cm

Question 48.
In the following figure, ‘O is the centre of the circle. OA = 10 cm and perpendicular OC on chord AB = 8 cm, then the length of the chord AB is

A) 8 cm
B) 10 cm
C) 12 cm
D) 16 cm
C) 12 cm

Question 49.
AD is a diameter of a circle and AB is a chord. If AD = 34 cm and AB = 30 cm, the distance of AB from the centre of the circle is
A) 17 cm
B) 15 cm
C) 4 cm
D) 8 cm
D) 8 cm

Question 50.
In figure if OA = 5 cm, AB = 8 cm and OD ⊥ AB, then CD is equal to

A) 3 cm
B) 2 cm
C) 4 cm
D) 5 cm
B) 2 cm

Question 51.
How many circles can pass through three given non-collinear points?
A) one and only one
B) two
C) three
D) infinitely many
B) two

Question 52.
To determine a unique circle, the number of points required is
A) 1
B) 2
C) 3 non – coflinear points
D) 3 collinear points
C) 3 non – coflinear points

Question 53.
How many points are sufficient to determine a line?
A) 1
B) 2
C) 3
D) none of these
B) 2

Question 54.
Three chords AB, CD and EF of a circle are respectively 3 cm, 3.5 cm and 3.8 cm away tram the centre. Then which of the following is correct?
A) AB > CD > EF
B) AB < CD < EF
C) AB = CD = EF
D) AB = CD < EF
A) AB > CD > EF

Question 55.
Equal chords of a circle are equidistant from
A) the centre
B) an extremity of a diameter
C) any point on the circumference
D) any point on the diameter
A) the centre

Question 56.
The measure of the angle of a semi circle is
A) 30°
B) 45°
C) 60°
D) 90°
D) 90°

Question 57.
The angle of a minor segment is
A) acute
B) right
C) obtuse
D) straight
C) obtuse

Question 58.
The angle of a major segment is
A) acute
B) right
C) obtuse
D) straight
A) acute

Question 59.
In the figure, AOB is a diameter of the semi circle, If ∠A s 60°, then ∠B is equal to

A)60°
B) 30°
C) 50°
D) 40°
B) 30°

Question 60.
In the following figure, O is the centre of the circle PAB and ΔOAB is equilateral. The measure of ∠APB is equal to

A) 60°
B) 45°
C) 40°
D) 30°
D) 30°

Question 61.
The length of a chord of a circle is equal to its radius. Find the measure of the angle subtended by that chord in major segment.
A) 30°
B) 60°
C) 45°
D) none of these
A) 30°

Question 62.
∠ADB = 90° and ∠ABC = 30°, then ∠ACB is

A) 30°
B) 45°
C) 90°
D) 60°
A) 30°

Question 63.
‘O’ is the centre of the circle, if ∠BOC – 140°, then x =

A) 70°
B) 40°
C) 35°
D) 20°
D) 20°

Question 64.
The value of ‘x’ in figure is :

A) 35°
B) 45°
C) 55°
D) 30°
A) 35°

Question 65.
In the following figure, there are two congruent circles whose centres are O and O’ and chord AB = chord CD. If ∠CO’D == 60°, then the measure of ∠APB is equal to

A) 45°
B) 40°
C) 30°
D) 15°
C) 30°

Question 66.
In the following figure, two circles C(O1, r) and C(O2, r) are congruent and arc $$\overparen{\mathbf{P Q}}$$ = arc $$\overparen{\mathbf{R S}}$$. If the measure of ∠RO2S = 62° and the measure of ∠PAQ = x°, then the value of x is

A) 28°
B) 31°
C) 38°
D) 62°
B) 31°

Question 67.
In the figure, ‘O’ is the centre of the circle. If ∠BAD = 48°, then ∠BCD =

A) 96°
B) 48°
C) 42°
D) 84°
B) 48°

Question 68.
In the figure, ‘O’ is the centre of the circle. The value of x is

A) 50°
B) 40°
C) 60°
D) 20°
A) 50°

Question 69.
In the given figure, AD || BC and ∠BCA = 40°. The measure of ∠DBC is equal to

A) 50°
B) 80°
C) 40°
D) 20°
C) 40°

Question 70.
In this figure AB = AC and ∠ABC = 50°. Then ∠BDC is equal to

A) 50°
B) 65°
C) 90°
D) 80°
D) 80°

Question 71.
The opposite angles of a cyclic quadrilateral
A) are complementary
B) are supplementary
C) are equal
D) form a linear pair
B) are supplementary

Question 72.
In the following figure, ABCD is a cyclic quadrilateral whose side AD is a diameter of the circle and the point O’ is the centre of the circle. If ∠OCD = 50°, then the measure of ∠ABC is

A) 100°
B) 120°
C) 110°
D) 130°
D) 130°

Question 73.
In the figure, the magnitude of angle ABC if angle AOC = 120° will be

A) 125°
B) 120°
C) 130°
D) 135°
B) 120°

Question 74.
In the below figure, ‘O’ is the centre of the circle and P, Q and R are points on , the circle such that ∠PQR = 100°, then ∠QPR equals :

A) 80°
B) 10°
C) 100°
D) 60°
B) 10°

Question 75.
In the following figure, ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle. If ∠ADC = 120°, then the value of ∠CAB is

A) 30°
B) 45°
C) 60°
D) 90°
A) 30°

Assertion and Reason type questions :

Question 1.
Assertion : If two circle in a circle are congruent, then the corresponding centred angles are also congruent.
Reason : The measure of an arc in a circle is equal to the measure of the central angle that subtends it.
Assertion is true.
Reason is true and explains the assertion.
Explanation : The assertion is true because if two arcs in a circle are congruent, then they have the same length. Since the measure of an arc is equal to the length of the s arc divided by the radius of the circle, and the radius of the circle is the same for both arcs, it follows that the measures of the two arcs are equal. The reason is also true and explains why the corresponding central angles are congruent.

Question 2.
Assertion : If two angles are in the same segment of a circle, then they are congruent.
Reason : The angles in the same segment of a circle are subtended by the same chord.
Assertion is true.
Reason is true and explains the assertion.
Explanation : The assertion is true because if two angles are in the same segment of a circle, then they are subtended by the same arc. Since the arc is part of a circle, it follows that the two angles are subtended by the same chord. Since the angles are inscribed in the same chord and subtend the same arc. they must be congruent. The reason is also true and explains why the angles in the same segment of a circle are congruent.

Question 3.
Assertion : In a cyclic quadrilateral, the opposite angles are supplementary.
Reason : The opposite angles in a cyclic quadrilateral are subtended by the same arc.
Assertion is true.
Reason is true and explains the assertion :
Explanation : The assertion is true because a cyclic quadrilateral is a quadrilateral that
can be inscribed in a circle. This means that the opposite angles of the quadrilateral are inscribed in the same arc of the circle, and the measure of an inscribed angle is half the measure of the subtended arc. Since the opposite angles of a cyclic quadrilateral are inscribed in the same arc, they must have equal measures, and since their sum is equal to 180 degrees (the measure of the entire circle), it follows that they are supplementary. The reason is also true and explains why the opposite angles in a cyclic quadrilateral are supplementary.

Question 4.
Assertion : If a quadrilateral has two pairs of congruent adjacent sides, then it must be a cyclic quadrilateral.
Reason : A cyclic quadrilateral is a quadrilateral that can the inscribed in a circle.
Assertion is false.
Reason is true.
Explanation : The assertion is false because a quadrilateral with two pairs of congruent adjacent sides is not necessarily a cyclic quadrilateral. For example, a rectangle has two pairs of congruent adjacent sides, but it is not a cyclic quadrilateral. The reason is true Because a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, and this definition does not depend on the lengths of its sides or angles.

Question 5.
Assertion : In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides.
Reason : The diagonals of a cyclic quadrilateral are perpendicular.
Assertion is true.
Reason is false.
Explanation : The assertion is true because in a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides. This property is known as Ptolemy’s theorem. The reason is false because the diagonals of a cyclic quadrilateral are not necessarily perpendicular. However, it is true that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent, which is known as the circumcentre of the quadrilateral.

Fill in the blanks :

1. Two chords in a circle are congruent if ‘ and only if they are equidistant from the _____________ of the circle.
Center

2. In a circle, the perpendicular bisector of a chord passes through the center of the circle if and only if the chord is a _______________ .
Diameter

3. The measure of an angle formed by a chord and a tangent at one of its end points is equal to half the measure of the _____________ .
Intercepted arc

4. Two chords intersect inside a circle. If the products of their segments are equal, then the chords are ______________ .
Equidistant

5. If two chords in a circle are parallel, then the segments they cut off on the circle are _______________ .
Congruent

6. The angle formed by two intersecting chords in a circle is equal to the average of the measures of the arcs ____________ and _______________ .
Intercepted by the chords section

7. If two chords in a circle are perpendicular, then the product of the segments of one chord is equal to the product of the segments of the ________________ .
Other chord

8. The measure of an angle formed by two intersecting chords in a circle is equal to half the sum of the measures of the arcs ______________ and _____________ .
Intercepted by the chord section

9. A quadrilateral is cyclic if and only if its opposite angles are ____________ .
Equal

10. In a cyclic quadrilateral, the sum of the measures of opposite angles is ______________ degrees.
180

11. If a quadrilateral is cyclic, then its opposite angles are ______________ .
Supplementary

12. A quadrilateral is cyclic if and only if its interior opposite angles are _______________ .
Supplementary

13. In a cyclic quadrilateral, the sum of any two adjacent angles is ______________ degrees.
180

14. The opposite angles of a cyclic quadrilateral are ______________ .
Congruent

15. In a cyclic quadrilateral, the sum of any two opposite angles is ____________ degrees.
180

16. A quadrilateral is cyclic if and only if a pair of opposite angles are ____________ .
Supplementary

17. The measure of an angle subtended by an arc outside the quadrilateral is equal to the measure of the ____________ angle of the quadrilateral.
Inscribed

18. In a cyclic quadrilateral, if one angle is right, then the opposite angle is ______________ .
Right

19. Angles in the same segment of a circle ______________ .
Equal

20. In a cyclic quadrilateral, the sum of pair of opposite angles is ______________ .
180°

## AP 9th Class Maths Bits 8th Lesson Quadrilaterals

Multiple Choice Questions (MCQs) :

Question 1.
What is the sum of interior angles of a parallelogram?
A) 360 degrees
B) 180 degrees
C) 270 degrees
D) 90 degrees
A) 360 degrees

Question 2.
What type of quadrilateral is a parallelogram?
A) Square
B) Trapezoid
C) Rhombus
D) Rectangle
B) Trapezoid

Question 3.
The diagonals of a parallelogram are _________ .
A) perpendicular
B) bisect each other
C) have equal length
D) all of the above
B) bisect each other

Question 4.
A parallelogram with all sides equal is called a ________ .
A) Square
B) Trapezium
C) Rhombus
D) Rectangle
C) Rhombus

Question 5.
In a parallelogram, the opposite angles are ________ .
A) Equal and parallel
B) Supplementary
C) Complementary
D) None of the above
A) Equal and parallel

Question 6.
In a parallelogram, if one of the angles measures 60 degrees, what is the measure of the opposite angle?
A) 60 degrees
B) 120 degrees
C) 90 degrees
D) 30 degrees
B) 120 degrees

Question 7.
The diagonals of a parallelogram bisect each other. This property is known as:
A) Opposite angles are congruent
B) Diagonals are perpendicular
C) Diagonals bisect each other
D) Opposite sides are parallel
C) Diagonals bisect each other

Question 8.
The opposite sides of a parallelogram are
A) Congruent
B) Perpendicular
C) Parallel
D) None of the above
C) Parallel

Question 9.
In a parallelogram, if one of the angles measures 110 degrees, what is the measure of the adjacent angle?
A) 70 degrees
B) 90 degrees
C) 80 degrees
D) 100 degrees
C) 80 degrees

Question 10.
In a rectangle, what is the measure of each angle?
A) 45 degrees
B) 60 degrees
C) 90 degrees
D) 120 degrees
C) 90 degrees

Question 11.
The diagonals of a rectangle are of equal length. This is because:
A) Opposite sides of a rectangle are parallel
B) Opposite sides of a rectangle are congruent
C) All angles of a rectangle are congruent
D) The diagonals of a rectangle bisect each other
D) The diagonals of a rectangle bisect each other

Question 12.
If the length of a rectangle is 16 cm and its width is 12 cm, what is the length of its diagonal?
A) 10 cm
B) 14 cm
C) 20 cm
D) 15 cm
C) 20 cm

Question 13.
Which of the following statements is true about a rectangle?
A) All sides are congruent
B) All angles are congruent
C) The Opposite sides are parallel and congruent
D) The diagonals are perpendicular bisectors of each other
D) The diagonals are perpendicular bisectors of each other

Question 14.
What is the are,a of a rectangle with a length of 6 cm and a width of 4 cm?
A) 12 cm2
B) 18 cm2
C) 20 cm2
D) 24 cm2
D) 24 cm2

Question 15.
In a rhombus, what is the measure of each angle?
A) 45 degrees
B) 60 degrees
C) 90 degrees
D) not always equal
D) not always equal

Question 16.
Which of the following statements is true about a rhombus?
A) All sides are congruent
B) The opposite sides are parallel and congruent
C) The diagonals are perpendicular bisectors of each other
D) All above
D) All above

Question 17.
The diagonals of a rhombus are perpendicular to each other. This is because:
A) Opposite sides of a rhombus are parallel
B) All sides of a rhombus are congruent
C) The opposite angles of a rhombus are congruent
D) The diagonals of a rhombus bisect each other
B) All sides of a rhombus are congruent

Question 18.
What is the area of a rhombus whose diagonals are x, x + 5?
{hint A = (d1.d2)/2}
A) x2
B) $$\frac{1}{2}$$(x2 + 5x)
C) $$\frac{1}{2}$$(2x2 – 5x)
D) $$\frac{1}{2}$$(2x2 + 5)
B) $$\frac{1}{2}$$(x2 + 5x)

Question 19.
In a square, what is the measure of each angle?
A) 45 degrees
B) 60 degrees
C) 90 degrees
D) 120 degrees
C) 90 degrees

Question 20.
If the perimeter of a square is 40 cm, what is the length of one of its sides?
A) 5 cm
B) 8 cm
C) 10 cm
D) 12 cm
D) 12 cm

Question 21.
How many sides does a quadrilateral have ?
A) 3
B) 5
C) 6
D) 4
D) 4

Question 22.
How many angles are there in a quadrilateral ?
A) 4
B) 2
C) 1
D) 3
A) 4

Question 23.
How many diagonals does a quadrilateral have ?
A) 1
B) 2
C) 3
D) 4
B) 2

Question 24.
How many pairs of opposite angles does a quadrilateral have ?
A) 4
B) 3
C) 2
D) 1
C) 2

Question 25.
The sum of all the angles of a quadrilateral ?
A) 360°
B) 180°
C) 540°
D) 720°
A) 360°

Question 26.
The three consecutive angles of a quadrilateral are 60°, 120° and 60°. The fourth angle of the quadrilateral is
A) 45°
B) 60°
C) 120°
D) none of these
C) 120°

Question 27.
Each angle of a rectangle is
A) 90°
B) 60°
C) 45°
D) 30°
A) 90°

Question 28.
The angle between the diagonals of a rhombus
A) 45°
B) 90°
C) 30°
D) 60°
B) 90°

Question 29.
If one angle of a parallelogram is 90°, then the parallelogram is called a
A) kite
B) rectangle
C) rhombus
D) square
B) rectangle

Question 30.
If in a quadrilateral, two pairs of adjacent sides are equal, then the quadrilateral is called a
A) kite
B) trapezium
C) rhombus
D) square
A) kite

Question 31.
Which of the following is false ?
A) A square is a rectangle
B) A square is a rhombus
C) A parallelogram is a trapezium
D) A kite is a parallelogram
D) A kite is a parallelogram

Question 32.
Which of the following is not true ?
A) A rectangle is not a square
B) A rhombus is not a square
C) A trapezium is a parallelogram
D) A kite is not a parallelogram
C) A trapezium is a parallelogram

Question 33.
Which of the following is false ?
A) If each pair of opposite sides of a quadrilateral is equal, then the quadrilateral is a parallelogrm.
B) If in a quadrilateral each pair of opposite angles is equal, then it is a parallelogrm.
C) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
D) If the sum of the consecutive interior angles of a quadrilateral is 90°, then the quadrilateral is a parallelogram.
D) If the sum of the consecutive interior angles of a quadrilateral is 90°, then the quadrilateral is a parallelogram.

Question 34.
A quadrilateral whose all the four sides and all the four angles are equal is called a
A) rectangle
B) rhombus
C) square
D) parallelogram
C) square

Question 35.
A quadrilateral, whose all the four sides are equal yet all the four angles are not equal, is called
A) square
B) rhombus
C) rectangle
D) parallelogram
B) rhombus

Question 36.
If the diagonals of a parallelogram are equal and mutually perpendicular, then it will be a
A) rectangle
B) rhombus
C) trapezium
D) square
D) square

Question 37.
ABCD is a rhombus such that ∠ABC = 40°, then ∠ADC is equal to
A) 40°
B) 45°
C) 50°
D) 20°
A) 40°

Question 38.
In a parallelogram ABCD, if ∠A = 70°, then the measure of ∠B is
A) 10°
B) 20°
C) 110°
D) 90°
C) 110°

Question 39.
Choose the correct statement :
A) Diagonals of a rectangle are perpendicular to each other.
B) Diagonals of a rhombus are perpendicular to each other.
C) A kite is parallelogram whose, adjacent sides are equal.
D) Diagonals of a trapezium are always equal.
B) Diagonals of a rhombus are perpendicular to each other.

Question 40.
If a pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a
A) parallelogram
B) rectangle
C) rhombus
D) square
A) parallelogram

Question 41.
The triangle formed by joining the midpoints of the sides of an equilateral triangle is
A) scalene
B) right
C) equilateral
D) isosceles right
C) equilateral

Question 42.
The triangle formed by joining the midpoints of the sides of a right angled triangle is a
A) scalene
B) isosceles
C) equilateral
D) right
D) right

Question 43.
The quadrilateral formed by joining the midpoints of the sides of a quadrilateral taken in order is a
A) kite
B) parallelogram
C) rectangle
D) square
B) parallelogram

Question 44.
The quadrilateral formed by joining the mid-points of the sides of a rectangle taken in order is a
A) rectangle
B) square
C) rhombus
D) kite
C) rhombus

Question 45.
The figure formed by joining the mid-points of sides of a Quadrilateral successively is _______
A) Parallelogram
B) Rhombus
C) Square
D) Rectangle
A) Parallelogram

Assertion and Reason type questions :

Question 1.
Assertion : A rhombus is a parallelogram.
Reason : A rhombus has opposite sides that are parallel and congruent.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 2.
Assertion : A square is a rhombus.
Reason : A square has four sides of equal length and opposite sides are parallel.
Both assertion and reason are true, but the reason is not the correct explanation of the assertion.

Question 3.
Assertion : The diagonals of a rhombus are perpendicular.
Reason : The opposite angles of a rhombus are congruent.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 4.
Assertion : A parallelogram with congruent diagonals is a rectangle.
Reason : A rectangle is a parallelogram with four right angles.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 5.
Assertion : The diagonals of a parallelogram bisect each other.
Reason : The opposite sides of a parallelogram are parallel.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Fill in the blanks :

1. A parallelogram is a quadrilateral with __________ pairs of parallel sides.
2

2. The opposite sides of a parallelogram are __________ .
Equal and parallel

3. In a parallelogram, the diagonals __________ each other.
Bisect

4. The area of a parallelogram is given by the formula __________ .
Base × Height

5. A parallelogram with perpendicular diagonals is a __________ .
Rhombus

6. The diagonals of a parallelogram bisect each other __________ .
internally

7. In a parallelogram, the opposite sides are __________ and __________ .
equal, parallel

8. The opposite angles of a parallelogram are __________ .
equal

9. In a parallelogram, the sum of the adjacent angles is __________ degrees.
180

10. The diagonals of a parallelogram __________ each other.
bisect

Match the following :

Match the geometric shape or theorem with its definition or property :

 (A) (B) 1. Rectangle A) A quadrilateral with two pairs of parallel sides and four right angles. 2. Parallelogram B) The diagonals of a rectangle are congruent and bisect each other. 3. Rhombus C) A quadrilateral with four right angles and all four sides congruent. 4. Square D) A quadrilateral with opposite sides parallel and congruent. 5. Opposite sides E) A line segment that joins the midpoint of two sides of a triangle. 6. Opposite angles F) A line segment that connects the midpoints of two sides of a triangle. 7. Diagonal G) A line segment that connects two non-adjacent vertices of a polygon. 8. Midpoint Theorem H) The midpoint of a line segment joining two sides of a triangle is parallel to the third side and half its length. 9. Rhombus Property I) The opposite sides of a parallelogram are congruent. 10. Rectangle Property J) A quadrilateral with four congruent angles and sides.

1 – A, 2 – D, 3 – C, 4 – J, 5 – F, 6 – E, 7 – G, 8 – H 9 – I 10 – B

 (A) (B) 1. Rectangle A) A quadrilateral with two pairs of parallel sides and four right angles. 2. Parallelogram D) A quadrilateral with opposite sides parallel and congruent. 3. Rhombus C) A quadrilateral with four right angles and all four sides congruent. 4. Square J) A quadrilateral with four congruent angles and sides. 5. Opposite sides F) A line segment that connects the midpoints of two sides of a triangle. 6. Opposite angles E) A line segment that joins the midpoint of two sides of a triangle. 7. Diagonal G) A line segment that connects two non-adjacent vertices of a polygon. 8. Midpoint Theorem H) The midpoint of a line segment joining two sides of a triangle is parallel to the third side and half its length. 9. Rhombus Property I) The opposite sides of a parallelogram are congruent. 10. Rectangle Property B) The diagonals of a rectangle are congruent and bisect each other.

Explanations:
1. A rectangle is a quadrilateral with two pairs of parallel sides and four right angles.
2. A parallelogram is a quadrilateral with opposite sides parallel and congruent.
3. A rhombus is a quadrilateral with four right angles and all four sides congruent.
4. A square is a quadrilateral with four congruent angles and sides.
5. The mid-segment is a line segment that connects the midpoints of two sides of a triangle.
6. The median is a line segment that joins the midpoint of two sides of a triangle.
7. A diagonal is a line segment that connects two non-adjacent vertices of a polygon.
8. The midpoint theorem states that the midpoint of a line segment joining two sides of a triangle is parallel to the third side and half its length.
9. The rhombus property states that the opposite sides of a rhombus are parallel and congruent.
10. The rectangle property states that the diagonal^ of a rectangle are congruent and bisect

## AP 9th Class Maths Bits 7th Lesson Triangles

Multiple Choice Questions (MCQs) :

Question 1.
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are :
A) Congruent
B) Similar
C) Neither (A) nor (B)
D) None
A) Congruent

Question 2.
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are :
A) Congruent
B) Similar
C) Neither
D) None
A) Congruent

Question 3.
The number of congruent parts in two congruent triangles is :
A) One
B) Two
C) Three
D) None
C) Three

Question 4.
In an isosceles triangle, the congruent sides are opposite to :
A) The smallest angle
B) The largest angle
C) The vertex angle
D) none
C) The vertex angle

Question 5.
If two angles of one triangle are congruent to two angles of another triangle, then the third angle in each triangle is :
A) Congruent
B) Supplementary
C) Complementary
D) None of above
B) Supplementary

Question 6.
If the sides of one triangle are proportional to the sides of another triangle, then the triangles are :
A) Congruent
B) Similar
C) Neither (A) nor (B)
D) None of above
B) Similar

Question 7.
If the corresponding sides of two triangles are proportional, then the triangles are :
A) Congruent
B) Similar
C) Neither (A) nor (B)
D) None of above
B) Similar

Question 8.
The sides opposite to the congruent angles of an isosceles triangle are :
A) Congruent
B) Parallel
C) Neither
D) None of above
A) Congruent

Question 9.
The measurement of any angle of an equilateral triangle is :
A) 30 degrees
B) 45 degrees
C) 60 degrees
D) none of above
C) 60 degrees

Question 10.
Two triangles are congruent if they have :
A) The same shape and size
B) The same shape but different size
C) The same size but different shape
D) None of above
A) The same shape and size

Question 11.
If two sides of a triangle are equal, then the angles opposite to them are :
A) Congruent
B) Complementary
C) Supplementary
D) None of above
A) Congruent

Question 12.
If AB = PQ, BC = QR and angle B = angle
Q are given, which condition will prove that ΔABC ≅ ΔPQR ?
A) SSS congruence condition
B) SAS congruence condition
C) ASA congruence condition
D) AAS congruence condition
B) SAS congruence condition

Question 13.
If angle B = 40, then what is the value of angle E that will prove by SAS congruency that ΔABC ≅ ΔDEF ?
A) 40°
B) 50°
C) 140°
D) 60°
A) 40°

Question 14.
If two angles and a side of one triangle, are equal to two angles and a side of another triangle, then the triangles are
A) Similar but not congruent
B) Congruent
C) Neither similar nor congruent
D) None of the above
A) Similar but not congruent

Question 15.
If in two triangles, the corresponding sides are proportional and their corresponding angles are equal, then the triangles are
A) Congruent
B) Similar
C) Neither similar nor congruent
D) None of the above
B) Similar

Question 16.
If angle Q, angle T are equal to 90 and PQ = ST and are given PR = SU, which condition will prove that ΔPQR ≅ ΔSTU ?
A) SSS congruence condition
B) SAS congruence condition
C) ASA congruence condition
D) RHS congruence condition
D) RHS congruence condition

Question 17.
If angle D = angle H , and also angle E = angle H, DE = GH, then from which condition will prove that ΔDEF ≅ ΔGHI?
A) SSS congruence condition
B) SAS congruence condition
C) ASA congruence condition
D) AAS congruence condition
C) ASA congruence condition

Question 18.
In the given figure, if AB = CD, BC = AD and ∠A = ∠C, then which condition will prove that ΔABC ≅ ΔCDA ?
A) SSS congruence condition
B) SAS congruence condition
C) ASA congruence condition
D) AAS congruence condition
B) SAS congruence condition

Question 19.
In the given figure, which of the following is not a criterion for congruence of triangles ?
A) SSS
B) SAS
C) AAA
D) RHS
C) AAA

Question 20.
In the given figure, if ∠1 = ∠3, then which of the following is true ?
A) ΔABC ≅ ΔCDA
B) ΔABD ≅ ΔCDA
C) ΔABD ≅ ΔACD
D) ΔABD ≅ ΔACB
B) ΔABD ≅ ΔCDA

Question 21.
In the given figure, which condition will prove that ΔABC ≅ ΔPQR ?
A) SSS congruence condition
B) SAS congruence condition
C) ASA congruence condition
D) AAS congruence condition
A) SSS congruence condition

Question 22.
If in two triangles, the corresponding sides are equal and their corresponding angles are also equal, then the triangles are
A) Congruent
B) Similar
C) Neither similar nor congruent
D) None
A) Congruent

Question 23.
Which of the following criteria is used to prove two triangles are congruent ?
A) ASA criterion
B) AAS Criterion
C) SSS criterion
D) All of the above
D) All of the above

Question 24.
If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are
A) Similar
B) Congruent
C) Neither similar nor congruent
D) Cannot be determined
B) Congruent

Question 25.
In a triangle ABC, if angle A is equal to angle C and side AB is equal to side BC, then which of the following is true ?
A) Angle B is greater than angle A
B) Angle B is greater than angle C
C) Angle B is equal to angle A
D) Angle B is equal to angle C
D) Angle B is equal to angle C

Question 26.
Two triangles are said to be congruent if
A) Their corresponding sides are proportioned
B) Their corresponding angles are equal
C) Both (A) and (B)
D) Neither (A) nor (B)
C) Both (A) and (B)

Question 27.
In triangle ABC, if angle A is equal to angle B and side AC is equal to side BC, then which of the following is trtie ?
A) Angle C is greater than angle A
B) Angle C is greater than angle B
C) Angle C is equal to angle A
D) Angle C is equal to angle B
D) Angle C is equal to angle B

Question 28.
In triangle ABC, if AB = AC and angle B = angle C, then which of the following is true ?
A) Triangle ABC is an isosceles triangle
B) Triangle ABC is a right triangle
C) Triangle ABC is an equilateral triangle
D) Triangle ABC is not a triangle
A) Triangle ABC is an isosceles triangle

Question 29.
Two triangles are said to be congruent if they have
A) The same area
B) The same perimeter
C) The same shape and size
D) The same height
C) The same shape and size

Question 30.
In triangle ABC, if AB = AC and angle A = 80 degrees, then which of the following is true ?
A) Angle B = Angle C = 50 degrees
B) Angle B = Angle C = 40 degrees
C) Angle B = Angle C = 60 degrees
D) Angle B = Angle C = 70 degrees
B) Angle B = Angle C = 40 degrees

Question 31.
In a triangle ABC, if AB = AC and angle A = 60 degrees, then which of the following is true ?
A) Angle B = Angle C = 60 degrees
B) Angle B = Angle C = 70 degrees
C) Angle B = Angle C = 50 degrees
D) Angle B = Angle C = 80 degrees
A) Angle B = Angle C = 60 degrees

Question 32.
If two triangles are congruent, then their corresponding angles are
A) Congruent
B) Supplementary
C) Complementary
D) None of the above
A) Congruent

Question 33.
In triangle ABC, if AB = AC and angle B = 50 degrees, then which of the following is true ?
A) Angle A = Angle C = 50 degrees
B) Angle A = Angle C = 60 degrees
C) Angle A = Angle C = 70 degrees
D) None of above
A) Angle A = Angle C = 50 degrees

Question 34.
In an isosceles triangle ABC, if AB = AC, then which angles are equal ?
A) ∠ABC and ∠ACB
B) ∠ABC and ∠BAC
C) ∠BAC and ∠ACB
D) None of the above
A) ∠ABC and ∠ACB

Question 35.
If ∠ABC = ∠ACB in a triangle ABC, then which of the following is true ?
A) AB = AC
B) AB = BC
C) AC = BC
D) None of the above
A) AB = AC

Question 36.
In an isosceles triangle ABC, AB = AC.
If ∠B = 60°, then what is the measure of ∠A?
A) 30°
B) 60°
C) 90°
D) 120°
B) 60°

Question 37.
In an isosceles triangle ABC, AB = AC.
If ∠A = 100°, then what is the measure of ∠B and ∠C?
A) ∠B = 40°, ∠C = 40°
B) ∠B = 90°, ∠C = 90°
C) ∠B = 80°, ∠C = 80°
D) ∠B = 100°, ∠C = 100°
A) ∠B = 40°, ∠C = 40°

Question 38.
In an isosceles triangle ABC, AB = AC.
If ∠B = 70°, then what is the measure of ∠C ?
A) 70°
B) 80°
C) 110°
D) 140°
A) 70°

Question 39.
If the angles opposite to equal sides of a triangle are equal, then the triangle is :
A) Acute-angled
B) Obtuse-angled
C) Right-angled
D) Isosceles
D) Isosceles

Question 40.
If a triangle is isosceles, then which angles are equal ?
A) Base angles
B) Opposite angles
C) Vertical angles
D) None of the above
A) Base angles

Question 41.
In a triangle ABC, if AB = AC, then which angles are equal ?
A) ∠ABC and ∠ACB
B) ∠ABC and ∠BAC
C) ∠BAC and ∠ACB
D) None of the above
A) ∠ABC and ∠ACB

Question 42.
If a triangle has two equal angles, then which sides are equal ?
A) Base sides
B) Opposite sides
D) None of the above
D) None of the above

Question 43.
In a triangle ABC, if AB = AC and ∠B = 50°, then what is the measure of ∠C ?
A) 50°
B) 80°
C) 100°
D) 130°
B) 80°

Question 44.
What does the mid-point theorem state ?
A) The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
B) In any triangle, the midpoint of a side is equidistant from the other two vertices.
C) The line joining the midpoints of two sides of a triangle is parallel to the third side.
D) The three medians of a triangle are concurrent at a point.
C) The line joining the midpoints of two sides of a triangle is parallel to the third side.

Question 45.
In triangle ABC, if D is the midpoint of AB and E is the midpoint of AC, then DE is ________ to BC.
A) perpendicular
B) parallel
C) bisecting v
D) none of the above
B) parallel

Question 46.
In a right-angled triangle, the mid-point of the hypotenuse is equidistant from the _________ .
A) acute angles
B) right angle
C) vertex opposite to the right angle
D) none of the above
B) right angle

Question 47.
In a triangle ABC, if D is the midpoint of BC and E is the midpoint of AC, then what is the length of DE if AB = 10 cm and AC = 12 cm ?
A) 6 cm
B) 5 cm
C) 7 cm
D) 8 cm
B) 5 cm

Question 48.
In a triangle ABC, if D is the midpoint of AB and E is the midpoint of AC, then what is the ratio of the area of triangle ADE to the area of triangle ABC ?
A) 1/2
B) 1/3
C) 1/4
D) 1/6
C) 1/4

Question 49.
What is the converse of the midpoint theorem ?
A) If a line is parallel to one side of a triangle, then it bisects the other two sides proportionally.
B) If a line bisects one side of a triangle proportionally, then it is parallel to the other two sides.
C) If a line is perpendicular to one side of a triangle, then it bisects the other two sides proportionally.
D) If a line bisects one angle of a triangle, then it is perpendicular to the other two sides.
B) If a line bisects one side of a triangle proportionally, then it is parallel to the other two sides.

Question 50.
In a triangle ABC, if the line joining the midpoints of AB and AC is parallel to BC, then the triangle is ___________.
A) equilateral
B) isosceles
C) acute-angled
D) none of the above
B) isosceles

Question 51.
In triangle ABC, if the line joining the midpoints of AB and AC is perpendicular to BC, then the triangle is __________ .
A) equilateral
B) isosceles
C) right-angled
D) none of the above
C) right-angled

Question 52.
In triangle ABC, if the line joining the midpoints of AB and AC is not parallel to BC, then the triangle is __________ .
A) acute-angled
B) obtuse-angled
C) equilateral
D) none of the above
B) obtuse-angled

Question 53.
In triangle ABC, if the line joining the midpoints of AB and AC is perpendicular to the angle bisector of angle A, then the triangle is _________ .
A) equilateral
B) isosceles
C) right-angled
D) none of the above
A) equilateral

Question 54.
A closed figure formed by three intersecting lines is called
A) circle
B) square
C) triangle
D) rhombus
C) triangle

Question 55.
“Tri” means
A) one
B) two
C) three
D) four
C) three

Question 56.
The symbol for congruence is
A) =
B) ~
C) 0
D) ≅
D) ≅

Question 57.
The symbol for correspondence is
A) →
B) ⇔
C) ↔
D) ≡
C) ↔

Question 58.
Two circles are congruent. If the radius of one circle is 3 cm, what is the radius of the other circle ?
A) 3 cm
B) 6 cm
C) 1.5 Cm
D) 1 cm
A) 3 cm

Question 59.
Two circles are congruent. If the radius of one circle is 1 cm, then the diameter of the other circle is
A) 1 cm
B) 2 cm
C) 4 cm
D) 0.5 cm
B) 2 cm

Question 60.
If the diameter of a circle is 2 cm, what is the diameter of circle congruent to it?
A) 1 cm
B) 2 cm
C) 4 cm
D) none of the these
B) 2 cm

Question 61.
If the side of a square is ‘a’ cm, then what is the side of a congruent square?
A) 1 cm
B) 2 cm
C) a cm
D) 2a cm
C) a cm

Question 62.
Δ ABC ≅ Δ PQR, then which of the following is true ?
A) A ↔ P
B) AB = QR
C) AC = PQ
D) AB = PQ
D) AB = PQ

Question 63.
Δ ABC ≅ Δ PQR. If AB = 5 cm, ∠B = 40° and ∠A = 80°, then which of the following is true ?
A) QP = 5 cm, ∠P = 60°
B) QP = 5 cm, ∠R = 60°
C) QR = 5 cm, ∠R = 60°
D) QR = 5 cm, ∠Q = 40°
B) QP = 5 cm, ∠R = 60°

Question 64.
Two triangles are congruent, if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle*. This rule is known as
A) SAS congruence rule
B) ASA congruence rule
C) SSS congruence rule
D) RHS congruence rule
A) SAS congruence rule

Question 65.
Two triangles are congruent, if any two pairs of angles and one pair of corresponding sides are equal. The rule is known as
A) SAS congruence rule
B) ASA congruence rule
C) AAS congruence rule
D) SSS congruence rule
C) AAS congruence rule

Question 66.
Among the following which is not criteria for congruence of two triangles ?
A) SAS
B) ASA
C) SSA
D) SSS
C) SSA

Question 67.
In ΔAOC and ΔXYZ, ∠A = ∠X, AO = XZ, AC = XY, then by which congruence rule Δ AOC ≅ ∠XZY ?
A) SAS
B) ASA
C) SSS
D) RHS
A) SAS

Question 68.
In ΔABC and ΔDEF, AB = DF and ∠A = ∠D, the two triangles will be congruent by SAS axiom if :
A) BC = EF
B) AC = DE
C) DC = DE
D) AC = EF
B) AC = DE

Question 69.
The measure of each angle of an equilateral triangle is
A) 30°
B) 45°
C) 60°
D) 90°
C) 60°

Question 70.
In ΔABC, BC = AB and ∠B = 80, then ∠A is equal to
A) 80°
B) 40°
C) 50°
D) 180°
C) 50°

Question 71.
In Δ PQR, PQ = PR and ∠Q = 65°, then ∠P is
A) 55°
B) 130°
C) 65°
D) 50°
D) 50°

Question 72.
In Δ ABC, ∠C = ∠A and BC = 6 cm and AC = 5 cm, then the length of AB is
A) 6 cm
B) 5 cm
C) 3 cm
D) 2.5 cm
A) 6 cm

Question 73.
In Δ ABC and Δ FDE, if AB = DE, BC = DE, AC = EF and ∠D = 55°, then ∠B =
A) 55°
B) 35°
C) 90°
D) 45°
A) 55°

Question 74.
In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are
A) isosceles but not congruent
B) isosceles and congruent
C) congruent but not isosceles
D) neither isosceles nor congruent
A) isosceles but not congruent

Question 75.
Which of the following is false ?
A) The mid-point of the hypotenuse of a right triangle is equidistant from its vertices.
B) Each angle of an equilateral trianlge is 60°.
C) The side opposite to the greater angle of a triangle is longer than the side opposite to the smaller angle.
D) The two altitudes corresponding to two equal sides of a triangle are not equal.
D) The two altitudes corresponding to two equal sides of a triangle are not equal.

Question 76.
In figure, if AB = AC and AP = AQ, then . by which congruence criterion ΔPBC ≅ ΔQCB ?

A) SSS
B) ASA
C) SAS
D) RHS
C) SAS

Question 77.
If the 3 altitudes of a triangle are equal, then triangle is
A) right angled triangle
B) isosceles traingle
C) acute angled triangle
D) equilateral triangle
D) equilateral triangle

Question 78.
If three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent
A) SAS congruence rule
B) ASA congruence rule
C) AAS congruence rule
D) SSS congruence rule
D) SSS congruence rule

Question 79.
In ΔABC is congruent to Δ DEF by SSS congruence rule, then
A) ∠C < ∠F
B) ∠B < ∠E
C) ∠A < ∠D
D) ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
D) ∠A = ∠D, ∠B = ∠E, ∠C = ∠F

Question 80.
If AB = QR, BG = PR and CA – PQ, then
A) Δ ABC ≅ Δ PQR
B) Δ CBA ≅ Δ PRQ
C) Δ BAC ≅ Δ RPQ
D) Δ PQR ≅ Δ BCA
B) Δ CBA ≅ Δ PRQ

Question 81.
If the lengths of the perpendiculars drawn from the middle point of a line to the other two sides are equal, then the triangle is
A) equilateral
B) isosceles
C) equiangular
D) scalene
B) isosceles

Question 82.
In figure, ABCD is a quadrilateral in which AB = BC and AD = DC. Measure of ∠BCD is

A) 150°
B) 30°
C) 105°
D) 72°
C) 105°

Question 83.
The sum of the three altitudes of a triangle is ________ the perimeter of the triangle.
A) greater than
B) equal to
C) half of
D) less than
D) less than

Question 84.
In Δ ABC, if ∠A = 35° and ∠B = 65°, then the longest side of the triangle is
A) AC
B) AB
C) BC
D) none of these
B) AB

Question 85.
In Δ ABC, if ∠B = ∠C = 45°, then the longest side is
A) AB
B) BC
C) CA
D) none of these
B) BC

Question 86.
Two sides of a triangle are of lengths 7 cm and 3.5 cm. The length of the third side of the triangle cannot be :
A) 3.6 cm
B) 4.1 cm
C) 3.4 cm
D) 3.8 cm
C) 3.4 cm

Question 87.
The length of the largest side of a triangle is 12 cm, then other two sides can be
A) 4.8 cm, 8.2 cm
B) 3.2 cm, 7.8 cm
C) 6.4 cm, 2.8 cm
D) 7.6 cm, 3.4 cm
A) 4.8 cm, 8.2 cm

Question 88.
In any triangle ABC, ∠A > ∠B and ∠B > ∠C, then the smallest side is
A) AB
B) BC
C) CA
D) none of these
A) AB

Question 89.
If Δ ABC is right angled at B, then
A) AB = AC
B) AC < AB
C) AB = BC
D) AC > AB
D) AC > AB

Question 90.
In Δ ABC, if AB > BC, then
A) ∠C < ∠A
B) ∠C = ∠A
C) ∠C > ∠A
D) ∠A = ∠B
C) ∠C > ∠A

Question 91.
In Δ PQR, if ∠R > ∠Q, then
A) QR > PR
B) PQ < PR
C) PQ < PR
D) QR < PR
B) PQ < PR

Question 92.
In Δ ABC if AB = B, then
A) ∠B > ∠C
B) ∠A = ∠C
C) ∠A = ∠B
D) ∠A < ∠C
B) ∠A = ∠C

Question 93.
If E is a point on side QR of a Δ PQR such that PE bisects ∠QPR, then
A) QE = ER
B) QP > QE
C) QE > QP
D) ER > RP
B) QP > QE

Question 94.
P is a point on side BC of Δ ABC such that AP bisects ∠BAC. Then
A) BP = CP
B) BA > BP
C) BP > BA
D) CP < CA
B) BA > BP

Question 95.
In Δ PQR, PE is the perpendicular bisector of ∠QPR, then
A) QE = PE
B) QP > QE
C) PQ = PR
D) PQ > PR
C) PQ = PR

Question 96.
∠X and ∠Y are exterior angles of Δ ABC at the points B and C respectively. Also ∠B > ∠C, then the relation between ∠X and ∠Y is
A) ∠X > ∠Y
B) ∠X < ∠Y
C) ∠X = ∠Y
D) ∠X ≥ ∠Y
B) ∠X < ∠Y

Question 97.
In Δ ABC, ∠B = 30°, ∠C = 80° and ∠A = 70°, then
A) AB > BC > AC
B) AB < BC > AC
C) AB > BC > AC
D) AB < BC < AC
C) AB > BC > AC

Question 98.
Which of the following options is/are appropriate criteria for congruence of triangles ?
i) S.S.S.
ii) A.A.A.
iii) A.S.A
iv) S.A.S

A) i, ii, iii
B) i, ii, iv
C) i, iii, iv
D) ii, iii, iv
C) i, iii, iv

Question 99.
ΔNVY = ΔAJU and both the triangles are scalene, then which of the following is true ?
A) NV = JU
B) VY = AJ
C) ∠VNY = ∠AUJ
D) ∠VYN = ∠JUA
C) ∠VNY = ∠AUJ

Assertion and Reason Type Questions :

Question 1.
Assertion : In a triangle, if the three sides are congruent to the three sides of another triangle, then the two triangles are congruent by SSS congruency.
Reason : The three sides of a triangle completely determine the triangle.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 2.
Assertion : In a right triangle, if the hypotenuse and one leg are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent by RHS congruency.
Reason : The hypotenuse and one leg of a right triangle completely determine the triangle.
Both assertion and reason are true and the reason is ‘the correct explanation of the assertion.

Question 3.
Assertion : If two triangles are congruent by SSS congruency, then their corresponding angles are also congruent.
Reason : Corresponding parts of congruent triangles are congruent.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 4.
Assertion : If two triangles are congruent by RHS congruency, then their corresponding angles are also
congruent.
Reason : Corresponding parts of congruent triangles are congruent.
Both assertion and reason are not true. The RHS congruency applies only to right triangles and the right angle is already included in the congruent parts.

Question 5.
Assertion : If two triangles have the same perimeter, then they must be congruent by SSS congruency.
Reason : The perimeter of a triangle is the sum of the lengths of its sides and SSS congruency requires all three sides of two triangles to be congruent.
The assertion is not true, as two triangles can have the same perimeter but different side lengths. The reason is also not a correct explanation of the assertion.

Fill in the blanks :

1. Two triangles are congruent if all corresponding sides and ________ are equal.
angles

2. The congruence of two triangles can be proved by ________ congruence criteria.
different

3. In SAS congruence criterion, the two sides and the included ________ of one triangle are equal to the corresponding parts of the other triangle.
angle

4. In SSS congruence criterion, all three sides of one triangle are equal to the corresponding sides of the other triangle, then the two triangles are ________ .
congruent

5. If in two triangles, the corresponding angles ate equal and the corresponding sides are in the same ratio (or proportion), then the two triangles are ________ .
similar

6. In RHS congruence criterion, if the hypotenuse and one side of a rightangled triangle are respectively equal to the hypotenuse and one, side of another right-angled triangle, then the two triangles are ________ .
congruent

7. In ASA congruence criterion, two angles and the included side of one triangle are equal to the corresponding parts of the other triangle, then the two triangles are ________ .
congruent

8. In AAS congruence criterion, two angles and a non-included side of one triangle are equal to the corresponding parts of the other triangle, then the two triangles are ________ .
congruent

9. If the three sides of a triangle are respectively equal to the three sides of another triangle, then the two triangles are ________ .
congruent

10. The reflexive property of congruence states, that any segment or angle is congruent to ________ .
itself

11. Two triangles are congruent by SSS criterion if the ________ of their corresponding sides are equal.
lengths

12. In RHS congruency, two triangles are congruent if the hypotenuse and ________ of one triangle are equal to the hypotenuse and corresponding ________ of the other triangle.
one leg, angle

13. SSS congruence criterion holds true for ________ type of triangles?
Scalene, isosceles and equilateral triangles

14. In an isosceles triangle, if the two equal sides are congruent to two equal sides of another triangle, then the third side of both triangles are also ________ .
congruent

15. If two sides and the included angle of one triangle are equal to . the corresponding two sides and included angle of another triangle, then the two triangles are congruent by ________ criterion.
SAS

16. Two triangles are congruent by RHS criterion if the hypotenuse and ________ of one triangle are equal to the hypotenuse and corresponding ________ of the other triangle.
one leg, angle

17. In an isosceles triangle, the angle opposite to the equal sides is always ________ .
congruent

18. In RHS congruence, if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the other leg of both triangles are also ________
congruent

19. SSS congruence criterion is based on the principle of ________ .
equality

20. If two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are congruent by ________ criterion.
AAS

21. In a triangle ABC, if D is the midpoint of AB and E is the midpoint of AC, then DE is ________ to BC.
parallel

22. The mid-point theorem states that the line joining the midpoints of two sides of a triangle is ________ to the third side.
parallel

23. In a triangle ABC, if D is the midpoint of BC and E is the midpoint of AC, then AD is ________ to BE.
parallel

24. In a triangle ABC, if D is the midpoint of AB, then BD = ________ .

25. In a triangle ABC, if D is the midpoint of AB and E is the midpoint of AC, then the length of DE is ________ of the length of BC.
half

26. State Mid-point theorem.
“The line segment in a triangle joining the mid-point of any two sides of a triangle is said to be parallel to its third side and also half of the length of the third side.”

27. Write “ASA Congruence Rule”.
“Two triangles are congruent if two angles and the included side of the one triangle are equal to two angles and the included side of other triangle” (ASA).

Match the following :

Match the congruency law or triangle property with its definition:

 A B 1. Side-Side-Side (SSS) congruency A) Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle 2. Side-Angle-Side (SAS) congruency B) Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. 3. Angle-Side-Angle (ASA) congruency C) Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle 4. Hypotenuse-Leg (HL) congruency D) The sides opposite congruent angles in a triangle are congruent 5. Corresponding Parts of Congruent Triangles (CPCT) E) Two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and corresponding leg of the other triangle 6. Isosceles Triangle Property F) The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 7. Equilateral Triangle Property G) A triangle with at least two congruent sides is called an isosceles triangle 8. Scalene Triangle Property H) A triangle in which all sides are congruent is called an equilateral triangle 9. Angle-Angle-Side (AAS) congruency I) Two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle 10. Triangle Inequality Property J) If two triangles are congruent, then their corresponding parts (sides and angles) are congruent

1 – A, 2 – C, 3 – B, 4 – E, 5 – J, 6 – G, 7 – H, 8 – D, 9 – I, 10 – F

 A B 1. Side-Side-Side (SSS) congruency A) Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle 2. Side-Angle-Side (SAS) congruency C) Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle 3. Angle-Side-Angle (ASA) congruency B) Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. 4. Hypotenuse-Leg (HL) congruency E) Two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and corresponding leg of the other triangle 5. Corresponding Parts of Congruent Triangles (CPCT) J) If two triangles are congruent, then their corresponding parts (sides and angles) are congruent 6. Isosceles Triangle Property G) A triangle with at least two congruent sides is called an isosceles triangle 7. Equilateral Triangle Property H) A triangle in which all sides are congruent is called an equilateral triangle 8. Scalene Triangle Property D) The sides opposite congruent angles in a triangle are congruen 9. Angle-Angle-Side (AAS) congruency I) Two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle 10. Triangle Inequality Property F) The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Explanations :
1. SSS congruency states that if the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.
2. SAS congruency states that if two sides and the included angle of one triangle are, congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
3. ASA congruency states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
4. HL congruency states that if the hypotenuse and one leg of one right triangle are
congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent.
5. CPCT (Corresponding Parts, of Congruent Triangles) states that if two triangles are congruent, then their corresponding parts (sides and angles) are congruent.
6. The isosceles triangle property states that in an isosceles triangle, the two sides opposite the congruent angles are themselves congruent.
7. The equilateral triangle property states that in an equilateral triangle, all aides are congruent.
8. The scalene triangle property states that in a scalene triangle, none of the sides are
congruent.
9. AAS congruency states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.
10. The triangle inequality property states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

## AP 9th Class Maths Bits 6th Lesson Lines and Angles

Multiple Choice Questions (MCQs) :

Question 1.
Which of the following pairs of angles are complementary?
A) 60° and 30°
B) 75° and 105°
C) 45° and 135°
D) 120° and 30°
A) 60° and 30°

Question 2.
What is the measurement of angle x if x and y are complementary, where y = 30?
A) 90°
B) 60°
C) 45°
D) 30°
B) 60°

Question 3.
Which of the following pairs of angles are vertically opposite angles?
A) 40° and 40°
B) 120° and 60°
C) 180° and 100°
D) 30° and 60°
A) 40° and 40°

Question 4.
Which of the following statements is true about parallel lines?
A) They intersect at one point.
B) They have the same slope.
C) They are perpendicular to each other.
D) They form a right angle.
B) They have the same slope.

Question 5.
What Is the measurement of angle y if ” x ” and ” y ” are supplementary and x = 120?
A) 60°
B) 120°
C) 90°
D) 45°
A) 60°

Question 6.
Which of the following angles are supplementary?
A) 70° and 110°
B) 60° and 120°
C) 45° and 135°
D) all above
D) all above

Question 7.
Which of the following statements is true about a transversal?
A) It intersects two lines at three points.
B) It is perpendicular to both lines.
C) It forms a right angle with both lines.
D) None of bove
D) None of bove

Question 8.
Which of the following angles can be a pair of corresponding angles by a transversal cm parallel lines ?
A) 70° and 70°
B) 60° and 60°
C) 45° and 45°
D) all above
D) all above

Question 9.
Which of the following pairs of angles can be a pair of alternate interior angles by a transversal at a pair of straight lines ?
A) 70° and 110°
B) 63° and 63°
C) 76° and 67°
D) 81° and 09°
B) 63° and 63°

Question 10.
If two parallel lines are intersected by a transversal, then alternate interior angles are
A) Congruent
B) Supplementary
C) Equal
D) Complementary
A) Congruent

Question 11.
Given that, if AB | | CD and PQ is the transversal meets AB, CD at E and F respectively. And if the value of angle PEB = 30, then angle DFP =
A) 30
B) 60
C) 150
D) 120
A) 30

Question 12.
If two parallel lines are intersected by a transversal, then corresponding angles are :
A) Congruent
B) Supplementary
C) Equal
D) Complementary
A) Congruent

Question 13.
If two parallel lines are intersected by a transversal, then alternate exterior angles are :
A) Congruent
B) Supplementary
C) Equal
D) Complementary
A) Congruent

Question 14.
Sum of linear pair is ________
A) 90 degrees
B) 180 degrees
C) 270 degrees
D) 360 degrees
B) 180 degrees

Question 15.
If two parallel lines are intersected by a transversal, what is the sum of the measurements of the two interior angles on the same side of the transversal?
A) 90 degrees
B) 180 degrees
C) 270 degrees
D) 360 degrees
B) 180 degrees

Question 16.
What is the relationship between the measurement of alternate exterior angles formed when a transversal intersects two parallel lines?
A) They are supplementary.
B) They are complementary.
C) They are equal.
D) They are not related.
C) They are equal.

Question 17.
If angle ABD and angle CBE are vertical angles and angle ABD measurement 60°, what is the measurement of angle CBE?
A) 60°
B) 90°
C) 120°
D) 180°
A) 60°

Question 18.
What is the measurement of each angle in a regular hexagon?
A) 120°
B) 135°
C) 140°
D) 150°
A) 120°

Question 19.
If angle ABD and angle ACD are complementary angles and angle ABD means 40°, what is the measurement of angle ACD?
A) 40°
B) 50°
C) 90°
D) 140°
B) 50°

Question 20.
If angle ABC measurement 120 degrees and angle ABD measurement 60 degrees, what is the measurement of angle CBD?
A) 30 degrees
B) 60 degrees
C) 90 degrees
D) 120 degrees
B) 60 degrees

Question 21.
If angle ABD and angle DBC are complementary angles and angle ABD measurement 30 degrees, what is the measurement of angle DBC?
A) 30 degrees
B) 90 degrees
C) 60 degrees
D) 120 degrees
C) 60 degrees

Question 22.
Two angles whose sum is 90 degrees are called
A) Complementary angles
B) Supplementary angles
C) Vertical angles
D) Alternate angles
A) Complementary angles

Question 23.
Two angles whose sum is 180 degrees are called
A) Complementary angles
B) Supplementary angles
C) Vertical angles
D) Alternate angles
B) Supplementary angles

Question 24.
Two angles formed by intersecting lines and located opposite to each other are called
A) Alternate angles
B) Corresponding angles
C) Vertical angles
D) Supplementary angles
C) Vertical angles

Question 25.
Two angles on the same side of a transversal and inside the two parallel lines are called
A) Corresponding angles
B) Alternate angles
C) Interior angles
D) Exterior angles
C) Interior angles

Question 26.
Two angles on opposite sides of a transversal and outside the two parallel lines are called
A) Corresponding angles
B) Alternate angles
C) Interior angles
D) Exterior angles
D) Exterior angles

Question 27.
If a transversal intersects two parallel lines, then each pair of corresponding angles are
A) Congruent
B) Supplementary
C) Complementary
D) Vertical
A) Congruent

Question 28.
If a transversal intersects two parallel lines, then each pair of alternate interior angles are
A) Congruent
B) Supplementary
C) Complementary
D) Vertical
A) Congruent

Question 29.
If a transversal intersects two parallel lines, then each pair of alternate exterior angles are
A) Congruent
B) Supplementary
C) Complementary
D) Vertical
A) Congruent

Question 30.
If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal are
A) Congruent
B) Supplementary
C) Complementary
D) Vertical
B) Supplementary

Question 31.
If a transversal intersects two parallel lines, then each pair of exterior angles on the same side of the transversal are
A) Congruent
B) Supplementary
C) Complementary
D) Vertical
B) Supplementary

Question 32.
Two lines that intersect to form right angles are called
A) Perpendicular lines
B) Parallel lines
C) Skew lines
D) Intersecting lines
A) Perpendicular lines

Question 33.
Two lines that intersect to form acute angles are called
A) Perpendicular lines
B) Parallel lines
C) Skew lines
D) Intersecting lines
D) Intersecting lines

Question 34.
Two lines that intersect to form obtuse angles are called
A) Perpendicular lines
B) Parallel lines
C) Skew lines
D) Intersecting lines
D) Intersecting lines

Question 35.
Two lines that do not intersect and do not lie in the same plane are called
A) Perpendicular lines
B) Parallel lines
C) Skew lines
D) Intersecting lines
C) Skew lines

Question 36.
Two lines that do not intersect and lie in the same plane are called
A) Perpendicular lines
B) Parallel lines
C) Skew lines
D) Intersecting lines
B) Parallel lines

Question 37.
The minimum number of points required to draw a line is
A) 1
B) 2
C) 3
D) 4
B) 2

Question 38.
In how many points, two intersecting lines intersect ?
A) 4
B) 3
C) 2
D) 1
D) 1

Question 39.
How many types of angles are formed between the edges of plane surfaces ?
A) of different types
B) of only one type
C ) of only two types
D) of only three types
A) of different types

Question 40.
An acute angle
A) measures between 0° and 90°
B) is exactly equal to 90°
C) is greater than 90° but less than 180°
D) is equal to 180°
A) measures between 0° and 90°

Question 41.
A right angle
A) measures between 0° and 90°
B) is exactly equal to 90°
C) is greater than 90° but less than 180°
D) is equal to 180°
B) is exactly equal to 90°

Question 42.
An angle which is greater than 90° and less than 180° is called
A) a right angle
B) a straight angle
C) an acute angle
D) an obtuse angle
D) an obtuse angle

Question 43.
A straight angle
A) is greater than 90° but less than 180°
B) is exactly equal to 90°
C) measures between 0° and 90°
D) is exactly equal to 180°
D) is exactly equal to 180°

Question 44.
A reflex angle
A) is greater than 90° but less than 360°
B) is exactly equal to 180°
C) is exactly equal to 90°
D) is greater than 90° but less than 180°
A) is greater than 90° but less than 360°

Question 45.
The sum of two complementary angles is
A) 180°
B) 360°
C) 90°
D) none of these
C) 90°

Question 46.
The angles whose sum is 180° are called
A) supplementary angles
B) complementary angles
C) alternate angles
D) corresponding angles
A) supplementary angles

Question 47.
The complement of an angle m is
A) m
B) 90° + m
C) 90° – m
D) m × 90°
C) 90° – m

Question 48.
The angle supplementary to 60° is
A) 30°
B) 120°
C) 45°
D) 300°
B) 120°

Question 49.
Find the measure of the angle which is complement of itself.
A) 30°
B) 90°
C) 45°
D) 180°
C) 45°

Question 50.
The angle of supplementary to 90° – 9° is
A) 90° + 9°
B) 9°
C) 180° – 9°
D) 360° – 9°
A) 90° + 9°

Question 51.
In the following figure, the reflex angle AOB is equal to

A) 60°
B) 120°
C) 300°
D) 360°
C) 300°

Question 52.
If the measure of an angle is twice the measure of its supplementary angle, then the measure of the angle is
A) 60°
B) 90°
C) 120°
D) 130°
C) 120°

Question 53.
Two complementary angles are in the ratio 4 : 5, then angles are
A) 90°, 90°
B) 40°, 50°
C) 30°, 150°
D) 45°, 45°
B) 40°, 50°

Question 54.
In the given figure, XYZ is a straight line. If ∠XY + ∠ZYQ = 85°, then ∠PYQ is

A) 95°
B) 85°
C) 90°
D) 75°
A) 95°

Question 55.
We can draw two different lines in
A) only one way
B) two different ways
C) three different ways
D) none of these
B) two different ways

Question 56.
A line indicates
A) only one direction
B) two directions
C) no direction
D) none of these
B) two directions

Question 57.
The length of the common perpendiculars at different points on parallel lines is the same and is called
A) the distance between the parallel lines
B) the altitude
C) the median
D) none of these
A) the distance between the parallel lines

Question 58.
A pair of angles is called linear pair if the sum of two adjacent angles is
A) 90°
B) 180°
C) 230°
D) 360°
B) 180°

Question 59.
The value of x in figure is :

A) 80°
B) 20°
C) 25°
D) 40°
B) 20°

Question 60.
In the figure, the value of y is

A) 28°
B) 32°
C) 36°
D) 44°
A) 28°

Question 61.
In the given figure, AOB is a straight line, then ∠BOC is

A) 80°
B) 70°
C) 60°
D) 20°
A) 80°

Question 62.
In the given figure, the value of x which makes POQ a straight line is :

A) 35°
B) 30°
C) 25°
D) 40°
C) 25°

Question 63.
In the following figure, two straight lines AB and CD intersect each other at ‘O’ and angles formed at ‘O’ are marked. Here ∠x – ∠y has the value

A) 56°
B) 118°
C) 62°
D) 180°
A) 56°

Question 64.
In figure, the value of angle q is :

A) 60°
B) 90°
C) 50°
D) 40°
D) 40°

Question 65.
From the given figure, identify the incorrect statement, given that l || m and ‘t’ is the transversal :

A) ∠2 and ∠5 are supplementary
B) ∠2 and ∠8 are supplementary
C) ∠2 and ∠3 are supplementary
D) ∠2 and ∠1 are supplementary
B) ∠2 and ∠8 are supplementary

Question 66.
In the following figure, a transversal ‘c’ intersects two parallel lines ‘a’ and ‘b’ at points A and B. The angles formed at A and B have been marked. Tell which pair of angles need not be equal ?

A) ∠1, ∠2
B) ∠1, ∠3
C) ∠1, ∠5
D) ∠2, ∠8
A) ∠1, ∠2

Question 67.
If two parallel lines are cut by a transversal, then which of the following is not true ?
A) Corresponding angles are equal.
B) Alternate interior angles are equal.
C) Interior angles on the same side of the transversal are supplementary.
D) Interior angles on the same side of the transversal are complementary.
D) Interior angles on the same side of the transversal are complementary.

Question 68.
If two parallel lines are intersected by a transversal, then corresponding angles are :
A) equal
B) complementary
C) supplementary
D) sum of the two angles is 360°
A) equal

Question 69.
Given lines l1, l2 and l3 in figure are parallel. The value of x is

A) 40°
B) 140°
C) 50°
D) 80°
B) 140°

Question 70.
If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2 : 3, then the smaller of two angles is
A) 72°
B) 108°
C) 54°
D) 36°
A) 72°

Question 71.
In figure BC // DE, If ∠ABC = ∠CDE = 90° and ∠ACB = 30°, then the measure of ∠DCE is

A) 30°
B) 60°
C) 90°
D) 120°
B) 60°

Question 72.
In the given figure, PQ // RS and EF // QS. If ∠PQS = 60°, then the measure of ∠RFE is

A) 115°
B) 120°
C) 60°
D) 180°
B) 120°

Question 73.
In the figure, AB // CD, the value of x is

A) 35°
B) 40°
C) 60°
D) 75°
D) 75°

Question 74.
In figure, AB and CD are parallel to each other. The value of x is

A) 90°
B) 100°
C) 120°
D) 140°
B) 100°

Question 75.
Lines parallel to the same line are
A) perpendicular to each other
B) parallel to each other
C) opposite to each other
D) none of these
B) parallel to each other

Question 76.
If l, m, n are lines in the same plane such that ‘l’ intersects m and n // m, then l and n are
A) parallel
B) intersecting
C) always perpendicular
D) always intersecting at 60°
B) intersecting

Question 77.
The sum of the three angles of a triangle is
A) one right angle
B) two right angles
C) three right angles
D) four right angles
B) two right angles

Question 78.
Each angle of an equilateral triangle is
A) 50°
B) 90°
C) 80°
D) 60°
D) 60°

Question 79.
The measure of each angle of a regular octagon is
A) 120°
B) 60°
C) 135°
D) 108°
C) 135°

Question 80.
Which of the following are not the angles of a triangle ?
A) 45°, 45°, 90°
B) 60°, 30°, 90°
C) 40°, 50°, 100°
D) 60°, 60°, 60°
C) 40°, 50°, 100°

Question 81.
Which of the following can be the angles of a quadrilateral ?
A) 90°, 90°, 60°, 120°
B) 85°, 95°, 50°, 120°
C) 90°, 90°, 60°, 130°
D) 90°, 90°, 50°, 140°
A) 90°, 90°, 60°, 120°

Question 82.
In a regular polygon of ‘n’. sides the measure of each interior angle is
A) $$\frac{360^{\circ}}{n}$$
B) ($$\frac{2 n-4}{n}$$) right angle
C) n right angles
D) 2n right angles
B) ($$\frac{2 n-4}{n}$$) right angle

Question 83.
The exterior angle of a triangle is equal to the sum of two
A) exterior angles
B) interior angles
C) interior opposite angles
D) alternate angles
C) interior opposite angles

Question 84.
In a triangle ABC if ∠A = 53° and ∠C = 44°, then the value of ∠B is
A) 73°
B) 83°
C) 93°
D) 46°
B) 83°

Question 85.
In figure, measure of ∠ABC is

A) 60°
B) 70°
C) 80°
D) 50°
A) 60°

Question 86.
In figure, ∠PQR is

A) 40°
B) 50°
C) 30°
D) 105°
C) 30°

Question 87.
In figure, the value of x is

A) 120°
B) 130°
C) 110°
D) 100°
B) 130°

Question 88.
In the given figure, the measure of ∠ABC is

A) 80°
B) 20°
C) 100°
D) 60°
A) 80°

Question 89.
A, B, C are the three angles of a triangle. If A + B = 145° and B + C = 100°, then angles A, B, C are
A) 80°, 65°, 35°
B) 80°, 35°, 65°
C) 65°, 80°, 35°
D) 35°, 65°, 80°
A) 80°, 65°, 35°

Question 90.
An exterior angle of a triangle is 105° and its two interior opposite angles are equal. Each of these equal angles is
A) 37$$\frac{1}{2}$$°
B) 52$$\frac{1}{2}$$°
C) 72$$\frac{1}{2}$$°
D) 75°
B) 52$$\frac{1}{2}$$°

Question 91.
In figure l1 // l2, the value of x is

A) 80°
B) 100°
C) 110°
D) 70°
A) 80°

Question 92.
In figure, if PQ // RS, then the measure of m is

A) 110°
B) 100°
C) 90°
D) 133°
A) 110°

Question 93.
The ratio of the measures of the three angles of a triangle is 2 : 3 : 4. The measure of the largest angle is
A) 80°
B) 60°
C) 40°
D) 180°
A) 80°

Question 94.
The ratio of the four angles of a quadrilateral is 1 : 2 : 3 : 4. The measure of its smallest angle is
A) 120°
B) 36°
C) 18°
D) 10°
B) 36°

Question 95.
One interior angle of a hexagon is 165° and each of the remaining interior angles is of x°. Find the measure of each of the remaining angles.
A) 111°
B) 109°
C) 107°
D) 115°
A) 111°

Question 96.
In the figure, the measure of (a + b + c + d + e + f + g + h + i + j) is

A) 900°
B) 720°
C) 540°
D) 360°
C) 540°

Question 97.
In ΔABC, the bisectors of ∠ABC and ∠BCA intersect each other at O. The measure of ∠BOC is
A) 90° + ∠A
B) 90° + $$\frac{\angle \mathrm{A}}{2}$$
C) 180° – ∠A
D) 90° – $$\frac{\angle \mathrm{A}}{2}$$
B) 90° + $$\frac{\angle \mathrm{A}}{2}$$

Question 98.
Which of the following are complementary angles?
A) 90°, 110°
B) 30°, 60°
C) 130°, 90°
D) 35°, 45°
B) 30°, 60°

Question 99.
Supplementary angle of 80° is
A) 10°
B) 20°
C) 60°
D) 100°
D) 100°

Question 100.
Assertion (A) : The complementary angle of 60° is 30°.
Reason (R) : The complementary angle of x° is 90° – x°.
A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B) Both Assertion and Reason are correct and Reason is not the correct explanation for Assertion.
C) Assertion is true but reason is false.
D) Assertion is false but the reason is true.
A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

Assertion and Reason Type Questions :

Question 1.
Assertion : Two lines intersecting at a point cannot be parallel to a third line.
Reason : If two lines are parallel, they will never intersect and hence, cannot form a point of intersection with any other line.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 2.
Assertion : All right angles are congruent.
Reason : The measure of all right angles is 90 degrees.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 3.
Assertion : An angle bisector of an angle of a triangle divides the opposite side into two equal parts.
Reason : The angle bisector of an angle of a triangle divides the angle into two equal parts.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 4.
Assertion : If two angles are complementary, then their sum is 90 degrees.
Reason : Complementary angles are two angles whose sum ¡s 180 degrees.
Both assertion and reason are true, but the reason is not the correct explanation of the assertion.

Question 5.
AssertIon : In a triangle, the sum of the measures of the three angles is 180 degrees.
Reason : Euclid’s fifth postulate states that if a straight line intersects two other straight lines and the sum of the interior angles on one side ¡s less than 180 degrees, then the two straight lines will eventually meet on that side.
The assertion is true, but the reason is not related to the assertion.

Fill in the blanks :

1. ________ angles are formed when parallel lines are intersected by a transversal.
Corresponding

2. A ________ is a pair of angles whose sum is 180 degrees.
Linear pair

3. The sum of ________ angles is always 180 degrees.
Supplementary

4. ________ angles are opposite angles formed by two intersecting lines.
Vertical

5. If two parallel lines are intersected by a transversal, then alternate ________ are congruent.
Exterior

6. If two parallel lines are intersected by a transversal, then alternate _________ are congruent.
Interior

7. A ___________ is a line that intersects two or more lines at distinct points.
Transversal

8. If two lines are intersected by a transversal and the alternate interior angles are congruent then the lines are ___________ .
Parallel

9. If two lines are parallel, then the alternate ___________ angles are congruent.
Exterior

10. If two lines are parallel, then the ___________ angles are congruent.
Corresponding

11. ___________ angles are formed when parallel lines are intersected by a transversal.
Corresponding

12. A ___________ is a pair of angles whose sum is 180 degrees.
Linear pair

13. The sum of ___________ angles is always 180 degrees.
Supplementary

14. ___________ angles are opposite angles formed by two intersecting lines.
Vertical

15. If two parallel lines are intersected by a transversal, then alternate ___________ are congruent.
Exterior

16. If two parallel lines are intersected by a transversal, then alternate ___________ are congruent.
Interior

17. A ___________ is a line that intersects two or more lines at distinct points.
Transversal

18. If two lines are intersected by a transversal and the alternate interior angles are congruent, then the lines are ___________ .
Parallel

19. If two lines are parallel, then the alternate ___________ angles are congruent.
Exterior

20. If two lines are parallel, then the ___________ angles are congruent.
Corresponding

Match the following :

Match the type of angle pair or line pair with its definition:

 A B 1. Supplementary angles A) Two lines that never intersect 2. Complementary angles B) Two angles that add upto 180 degrees 3. Vertical angles C) Two angles that are on the same side of the transversal and in corresponding positions. 4. Adjacent angles D) Two angles that add upto 90 degrees 5. Parallel lines E) Two angles with the same vertex and a common side, but no common interior points 6. Perpendicular lines F) Two angles that are opposite each other and have the same measure 7. Transversal G) Two lines that intersect at a right angle 8. Corresponding angles H) A line that intersects two or more other lines at different points 9. Alternate interior angles I) Two angles that are on opposite sides of the transversal and inside the two intersected lines 10. Alternate exterior angles J) Two angles that are on opposite sides of the transversal and outside the two intersected lines

1 – B, 2 – D, 3 – F, 4 – E, 5 – A, 6 – G, 7 – H, 8 – C, 9 – I, 10 – J

 A B 1. Supplementary angles B) Two angles that add upto 180 degrees 2. Complementary angles D) Two angles that add upto 90 degrees 3. Vertical angles F) Two angles that are opposite each other and have the same measure 4. Adjacent angles E) Two angles with the same vertex and a common side, but no common interior points 5. Parallel lines A) Two lines that never intersect 6. Perpendicular lines G) Two lines that intersect at a right angle 7. Transversal H) A line that intersects two or more other lines at different points 8. Corresponding angles C) Two angles that are on the same side of the transversal and in corresponding positions. 9. Alternate interior angles I) Two angles that are on opposite sides of the transversal and inside the two intersected lines 10. Alternate exterior angles J) Two angles that are on opposite sides of the transversal and outside the two intersected lines

Explanations
1. Supplementary angles are two angles that add upto 180 degrees.
2. Complementary angles are two angles that add upto 90 degrees.
3. Vertical angles are two angles that are opposite each other and have the same measure.
4. Adjacent angles are two angles with the same vertex and a common side, but no common interior points.
5. Parallel lines are two lines that never intersect.
6. Perpendicular lines are two lines that intersect at a right angle.
7. A transversal is a line that intersects two or more other lines at different points.
8. Corresponding angles are two angles that are on the same side of the transversal and in corresponding positions.
9. Alternate interior angles are two angles that are on opposite sides of the transversal and inside the two intersected lines.
10. Alternate exterior angles are two angles that are on opposite sides of the transversal and outside the two intersected lines.

## AP 8th Class Social Bit Bank Pdf Download in Telugu & English Medium

AP 8th Class History Bit Bank

• 1st Lesson How, When and Where Bits
• 2nd Lesson From Trade to Territory The Company Establishes Power Bits
• 3rd Lesson Ruling the Countryside Bits
• 4th Lesson Tribals, Dikus and the Vision of a Golden Age Bits
• 5th Lesson When People Rebel 1857 and After Bits
• 6th Lesson Weavers, Iron Smelters and Factory Owners Bits
• 7th Lesson Civilising the “Native”, Educating the Nation Bits
• 8th Lesson Women, Caste and Reform Bits
• 9th Lesson The Making of the National Movement: 1870s-1947 Bits
• 10th Lesson India After Independence Bits

AP 8th Class Geography Bit Bank

• 1st Lesson Resources
• 2nd Lesson Land, Soil, Water, Natural Vegetation and Wildlife Resources Bits
• 3rd Lesson Mineral and Power Resources Bits
• 4th Lesson Agriculture Bits
• 5th Lesson Industries Bits
• 6th Lesson Human Resources Bits

AP 8th Class Politics Bit Bank

• 1st Lesson The Indian Constitution Bits
• 2nd Lesson Understanding Secularism Bits
• 3rd Lesson Why do we need a Parliament Bits
• 4th Lesson Understanding Laws Bits
• 5th Lesson Judiciary Bits
• 6th Lesson Understanding Our Criminal Justice System Bits
• 7th Lesson Understanding Marginalisation Bits
• 8th Lesson Confronting Marginalisation Bits
• 9th Lesson Public Facilities Bits
• 10th Lesson Law and Social Justice Bits

## AP 9th Class Maths Bits 5th Lesson Introduction to Euclid’s Geometry

Multiple Choice Questions (MCQs) :

Question 1.
Which of the following is not an axiom of Euclid?
A) A straight line segment can be extended indefinitely in a straight line.
B) A circle can be drawn with any given center and radius.
C) Two lines intersect at exactly one point.
D) All right angles are equal to each other.
B) A circle can be drawn with any given center and radius.

Question 2.
Euclid’s second axiom is known as the _________
A) Common Notion
B) Postulate
C) Theorem
D) Definition
B) Postulate

Question 3.
Euclid’s fifth axiom, the parallel postulate is :
A) Two straight lines cannot intersect at more than one point.
B) A straight line can be drawn between any two points.
C) Given any straight line and a point not on that line, there exists exactly one straight line passing through that point and never intersecting the given line.
D) If two straight lines intersect a third straight line in such a way that the sum of the inner angles on one side is less than two right angles, then the two straight lines will eventually intersect each other on that side if they are extended far enough.
C) Given any straight line and a point not on that line, there exists exactly one straight line passing through that point and never intersecting the given line.

Question 4.
Which of the following statements is not an example of Euclid’s axiom?
A) Given any two distinct points, there exists more than 1 straight line passing through them.
B) If two lines are parallel, they will never intersect.
C) If two angles are congruent, then their measurements are equal.
D) The sum of the angles in a triangle is 180 degrees.
A) Given any two distinct points, there exists more than 1 straight line passing through them.

Question 5.
Which of the following statements is not a consequence of Euclid’s axioms?
A) The sum of the angles in a triangle is 180 degrees.
B) The Pythagorean theorem holds for right triangles.
C) Every prime number is either 1 or a product of distinct primes.
D) Two circles can intersect at most two points.
C) Every prime number is either 1 or a product of distinct primes.

Question 6.
Euclid’s fourth postulate is ________
A) all right angles are equal to one another
B) Axiom of congruence
C) Axiom of symmetry
D) Axiom of transitivity
A) all right angles are equal to one another

Question 7.
Euclid’s first axiom is known as the
A) Axiom of parallels
B) Axiom of congruence
C) Axiom of symmetry
D) Axiom of common notions
D) Axiom of common notions

Question 8.
Which of the following is not a basic concept in Euclid’s geometry?
A) Point
B) Line
C) Angle
D) Curve
D) Curve

Question 9.
Euclid’s axioms are the foundation of
A) Analytic geometry
B) Differential geometry
C) Topology
D) Euclidean geometry
D) Euclidean geometry

Question 10.
Which of the following statements is true according to Euclid’s axioms?
A) A straight line can be drawn between any two points.
B) A point can be extended indefinitely in any direction.
C) The sum of the interior angles of a polygon is always greater than 360 degrees.
D) A circle can be drawn with any point as centre.
C) The sum of the interior angles of a polygon is always greater than 360 degrees.

Question 11.
Which postulate states that a straight line can be drawn from any point to any other point?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 4
A) Postulate 1

Question 12.
Which postulate states that a finite straight line can be extended indefinitely in a straight line?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 4
B) Postulate 2

Question 13.
Which postulate states that acircle can be drawn with any point as its center and any radius?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 4
C) Postulate 3

Question 14.
Which postulate states that all right angles are equal to one another?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 4
D) Postulate 4

Question 15.
Which postulate states that if a straight line intersects two other straight lines and the sum of the interior angles on one side is less than two right angles, then the two lines will eventually intersect on that side ?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 5
D) Postulate 5

Question 16.
Which postulate deals with the equality of angles?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 4
D) Postulate 4

Question 17.
Which postulate deals with the parallel lines?’
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 5
D) Postulate 5

Question 18.
Which postulate states that if a straight line falls on two straight lines in such a way that the sum of the interior angles on the same side is less than two right angles, then the two lines will eventually meet on that side?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 5
D) Postulate 5

Question 19.
Which postulate is known as the postulate of superposition?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 4
D) Postulate 4

Question 20.
Which postulate states that if two straight lines Intersect a third straight line in such a way that the sum of the interior angles on one side is less than two right angles, then the two lines will eventually meet on that side?
A) Postulate 1
B) Postulate 2
C) Postulate 3
D) Postulate 5
D) Postulate 5

Question 21.
In Indus valley civilisation (about 300 B.C the bricks used for construction work were having dimensions in the ratio
A) 1 : 3 : 4
B) 4 : 2 : 1
C) 4 : 4 : 1
D) 4 : 3 : 2
B) 4 : 2 : 1

Question 22.
Euclid belonged to the country
A) Babylonia
B) Egypt
C) Greece
D) India
C) Greece

Question 23.
Pythagoras was a student of
A) Thales
B) Euclid
C) Both (A) and (B)
D) Archimedes
A) Thales

Question 24.
Number of dimension(s) a surface has
A) 0
B) 1
C) 2
D) 3
C) 2

Question 25.
The number of lines that can pass through a given point is
A) Two
B) None
C) Only one
D) Infinitely many
D) Infinitely many

Question 26.
The number of line segments determined by three collinear points is
A) Two
B) Three
C) Only one
D) Four
B) Three

Question 27.
“Lines are parallel if they do not intersect” is stated in the form of
A) an axiom
B) a definition
C) a postulate
D) a proof
B) a definition

Question 28.
A proof is required for
A) postulate
B) axiom
C) theorem
D) definition
C) theorem

Question 29.
The things which coincide with one another are
A) equal to one another
B) unequal
C) double of same thing
D) triple of same thing
A) equal to one another

Question 30.
Euclid stated that things which are equal to the same thing are equal to one another in the form of
A) an axiom
B) a definition
C) a postulate
D) a proof
A) an axiom

Question 31.
The things which are double of same thing are
A) equal
B) halves of the same thing
C) unequal
D) double of the same thing
A) equal

Question 32.
Which of the following statements is incorrect ?
A) A line segment has definite length.
B) Three lines are concurrent if and only if they have a common point.
C) Two lines drawn in a plane always intersect at a piont.
D) One and only one line can be drawn passing through a given point and parallel to a given line.
C) Two lines drawn in a plane always intersect at a piont.

Question 33.
Select the wrong statement.
A) Only one line can pass through a single point.
B) Only one line can pass through two distinct points.
C) A terminated line can be produced indefinitely on both the sides.
D) If two circles are equal, then their radii are equal.
A) Only one line can pass through a single point.

Question 34.
If the point ‘P’ lies in between M and N and C is mid-point of MP, then :
A) MC + PN = MN
B) MP + CP = MN
C) MC + CN = MN
D) CP + CN = MN
C) MC + CN = MN

Question 35.
“Two intersecting lines cannot be parallel to the same line” is stated in the form of
A) an axiom
B) a definition
C) a postulate
D) a proof
C) a postulate

Question 36.
John Playfair was a
A) French mathematician
B) Scottish mathematician
C) Indian mathematician
D) Egyptian mathematician
B) Scottish mathematician

Question 37.
“There exists a pair of straight lines that are everywhere equidistant from one another” is a direct consequence of Euclid’s
A) First postulate
B) Second postulate
C) Third postulate
D) Fifth postulate
D) Fifth postulate

Question 38.
Assertion (A) : If AB = PQ and PQ = XY, then AB = XY.
Reason (R) : According to Euclid’s 1st axiom – “Things Which are equal to the same thing are also equal to one another”.
A) Both A and R are correct and R is the correct explanation of A.
B) Both A and R are correct and R is not the correct explanation of A.
C) A is true but the R is false.
D) A is false but the R is true.
A) Both A and R are correct and R is the correct explanation of A.

Assertion and Reason Type Questions :

Question 1.
Assertion : Euclid’s second axiom is known as the axiom of “equals”.
Reason : Euclid’s first axiom is based on the idea that things which are equal to the same thing are equal to each other.
Both assertion and reason are true and reason is not the correct explanation for the assertion.

Question 2.
Assertion : Euclid’s fifth postulate is known as the parallel postulate.
Reason : Euclid’s fifth postulate states that if a line intersects two other lines and the interior angles on one side of the intersecting line add upto less than 180 degrees, then the two lines will eventually intersect if extended far enough.
Both assertion and reason are true, but reason is not the correct explanation for the assertion.

Question 3.
Assertion : Euclid’s first postulate states that a straight line can be drawn between any two points.
Reason : Euclid’s first postulate is based on the idea that two points in space define a unique line.
Both assertiori and reason are true and reason is the correct explanation for the assertion.

Question 4.
Assertion : Euclid’s first postulate is known as the transitive property.
Reason : Euclid’s third postulate is based on the idea that if A is equal to B and B is equal to C, then A is equal to C.
Both assertion and reason are wrong.

Question 5.
Assertion : The side-angle-side congruence postulate.
Reason : If two triangles have two sides and the included angle of one triangle congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Both assertion and reason are true and reason is the correct explanation for the assertion.

Question 6.
Assertion : Two lines intersecting at a point cannot be parallel to a third line.
Reason : If two lines are parallel, they will never intersect and hence, cannot form a point of intersection with any other line.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 7.
Assertion : All right angles are congruent.
Reason : The measurements of all right angles is 90 degrees.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 8.
Assertion : An angle bisector of an angle of a triangle divides the opposite side into two equal parts.
Reason : The angle bisector of an angle of a triangle divides the ahgle into two equal parts.
Both assertion and reason are true and the reason is the correct explanation of the assertion.

Question 9.
Assertion : If two angles are comple-mentary, then their sum is 90 degrees.
Reason : Complementary angles are two angles whose sum is 180 degrees.
Both assertion and reason are true but the reason is not the correct explanation of the assertion.

Question 10.
Assertion : In a triangle, the sum of the measurements of the three angles is 180 degrees.
Reason : Euclid’s fifth postulate states that if a straight line intersects two other straight lines and the sum of the interior angles on one side is less than 180 degrees, then the two straight lines will eventually meet on that side.
The assertion is true but the reason is not related to the assertion.

Fill in the blanks :

1. From The axiom ” if equals are added to equals, then __________ ”.
wholes are equal

2. From the axiom , that “which states that given any __________ points, there exists a unique __________ passing through them.”
two points and a line

3. From the axioms we can conclude “__________ distinct lines cannot have more than one point in common.”
two

4. Euclid’s axioms form, the foundation of __________ geometry.
Euclidean

5. Euclid’s __________ is “A straight line may be drawn from anyone point to any other point.”
first postulate

6. The first postulate of Euclid states that a straight line can be drawn between any __________ points.
Two

7. The second postulate of Euclid states that any straight line segment can be extended indefinitely in a __________ direction.
Straight

8. The third postulate of Euclid states that a circle can be drawn with any __________ and any radius.
Centre

9. The fourth postulate of Euclid states that all right angles are __________ .
Equal to one another

10. The fifth postulate of Euclid is also known as the __________ postulate.
Parallel

11. The parallel postulate states that if a straight line intersects two other straight lines and the interior angles on the same side of the transversal add up to less than two right angles, then the two straight lines, if extended indefinitely, will _________ intersect on that side of the transversal.
Eventually

12. According to the parallel postulate, if a straight line intersects two other straight lines and the interior angles on the same side of the transversal add upto exactly two right angles, then the two straight lines are __________ .
Parallel

13. Euclid’s postulates were first introduced in his book _________ .
Elements

14. The first axioms of Euclid are also known as the __________ .
Common notions

15. Euclid’s fifth postulate was the subject of much debate for centuries and eventually led to the development of __________ geometry.
Non-Euclidean

16. Euclid’s first axiom is “Two points determine a __________ .”
line

17. The third axiom is known as the ” __________ postulate,” which states that given a line and a point not on the line, there is exactly one line passing through The point that is parallel to the given line.
parallel

18. The fourth axiom is the ” __________ and __________ postulate,” which states that given any two points, there exists a unique line passing through them.
point and line

19. The fifth axiom is the ” __________ postulate,” which states that if a line intersects two other lines and the sum of the interior angles on one side is less than two right angles, then the two other lines will eventually intersect each other on that side.
parallel

20. How many lines can pass through a single point? __________
Infinite

21. The word ‘geometry’ comes from the Greek words ‘geo’ and ‘metrein’. Here ‘geo means ________ .
Earth

22. Self evident statements which doesn’t require any proof are Called __________ .
axioms

## AP 9th Class Maths Bits 4th Lesson Linear Equations in Two Variables

Multiple Choice Questions (MCQs) :

Question 1.
What is the maximum number of solutions for a system of linear equations in two variables?
A) 0
B) 1
C) 2
D) infinit
C) 2

Question 2.
Which of the following statements is true?
A) A system of linear equations with no solution has only one equation.
B) A system of linear equations with infinite solutions has no unique solution.
C) A system of linear equations with one unique solution has one equation.
D) A system of linear equations with no solution has two equations.
B) A system of linear equations with infinite solutions has no unique solution.

Questions 3.
Which of the following is true about a system of linear equations in two variables with infinite solutions?
A) The two lines intersect at one point.
B) The two lines are parallel.
C) The two lines are coincident.
D) None of the above.
C) The two lines are coincident.

Question 4.
How many solutions does the following system of linear equations 3x + 5y = 7, 6x + 10y = 14 , have?
A) 0
B) 1
C) 2
D) infinite
A) 0

Question 5.
Which of the following is true about a system of linear equations in two variables with no solution?
A) The two lines intersect at one point.
B) The two lines are parallel.
C) The two lines are coincident.
D) None of the above.
B) The two lines are parallel.

Question 6.
How many solutions does the following system of linear equations
x + 2y = 5, 2x – 4y = 10 have?
A) 0
B) 1
C) 2
D) infinite
B) 1

Question 7.
Co-ordinates of origin is ________
A) (x, y)
B) (x, 2y)
C) (0, 0)
D) (4, 0)
C) (0, 0)

Question 8.
What is the solution of the following system of linear equations?
3x + 2y = 7, 2x + 3y = 8
A) (1, 2)
B) (2, 1)
C) (-1, 3)
D) (3, -1)
A) (1, 2)

Question 9.
How many solutions does the following system of linear equations x + y = 6, x – y = 2 have?
A) 0
B) 1
C) 2
D) infinite
B) 1

Question 10.
What is the solution of the following system of linear equations, x + 2y = 4 ; 2x + 4y = 8 have?
A) (2, 1)
B) (1, 2)
C) (1, 3)
D) none
D) none

Question 11.
Which of the following is not a solution to the equation 2x – 3y = 12?
A) (0, -4)
B) (6, 0)
C) (3, -2)
D) (-6, -10)
D) (-6, -10)

Question 12.
Which of the following is not a solution to the equation x + 5y = 15?
A) (0, 3)
B) (5, 2)
C) (10, 1)
D) (-15, 6)
D) (-15, 6)

Question 13.
Which of the following is not a solution to the equation 4x – y = 7?
A) (2, 1)
B) (-1, -11)
C) (3, -5)
D) (4, 9)
C) (3, -5)

Question 14.
Which of the following is not a solution to the equation 2x + 3y = 6?
A) (0, 2)
B) (3, 0)
C) (1, 2)
D) (-3, 4)
C) (1, 2)

Question 15.
Which of the following is not a solution to the equation 5x – 2y = 8?
A) (4, 6)
B) (6, -7)
C) (-2, -9)
D) (0, -4)
B) (6, -7)

Question 16.
If x = 2 and y = -1 in the equation 2x + 4y = k, what is the value of k?
A) 0
B) 2
C) 4
D) 6
A) 0

Question 17.
Given that 3x – 2y = 7, find the value of y when x = -1.
A) 5
B) -5
C) 1
D) -1
B) -5

Question 18.
If 4x – 3y = 5 and 2x + 5y = 9, what is the value of x + y?
A) 3
B) 2
C) 13
D) 4
A) 3

Question 19.
For the equation 5x – 3y = -2, what is the value of y when x = 2?
A) 3
B) 4
C) 5
D) 6
B) 4

Question 20.
If 3x + 2y = 8 and the point (x, -2) lies on the line, then, what is the value of x?
A) -2
B) -1
C) 4
D) 2
C) 4

Question 21.
The value of y at x = -10 in the equation x + 5y = 0 is
A) 2
B) 25
C) 10
D) 0
A) 2

Question 22.
Equation of a line which is parallel to x-axis and 8 units distance below the x-axis is
A) x = 8
B) x + 8 = y
C) y = 8
D) y = – 8
D) y = – 8

Question 23.
x = 30 and y = 20 is a solution of the equation 4px – 3y = 180, then the value of p is
A) 0
B) 12
C) 2
D) 3
C) 2

Question 24.
Which of the following is the equation of a line parallel to x-axis?
(where p is distance from x -axis)
A) y = p
B) x + y = z
C) y = x
D) x = a
A) y = p

Question 25.
Any point on the line y = 3x is of the form
A) (b, 3b)
B) (3a, a)
C) (c/3, 3c)
D) (a, -a)
A) (b, 3b)

Question 26.
Any point of the form (2a, -a) always lie on the graph of the equation
A) x = -a
B) y = a
C) y = x
D) x + 2y = 0
D) x + 2y = 0

Question 27.
The graph of the equation 2x + 3y = 6 cuts the y-axis at the point
A) (0, 3)
B) (3, 0)
C) (2, 0)
D) (0, 2)
D) (0, 2)

Question 28.
Which of the following points lies on the equation x – 2y – 6?
A) (2, 4)
B) (0, 3)
C) (-4, 1)
D) (4,-1)
D) (4,-1)

Question 29.
How many linear equations in x and y can be satisfied by x = 5 and y = 4 ?
A) only one
B) two
C) infinitely many
D) three
C) infinitely many

Question 30.
Solution of linear equation 0x + 3y – 9 = 0 is
A) (3, -3)
B) (3, -9)
C) (0, 3)
D) (-3, 0)
C) (0, 3)

Question 31.
If (3, 2) is the solution 4x + ky = 8, then k equals of the equation
A) 2
B) 4
C) 3
D) -2
D) -2

Question 32.
Cost of book (x) exceeds twice the cost of pen (y) by Rs. 10. This statement can be expressed as linear equation.
A) x = 2y + 10
B) 2x – y = 10
C) 2x + y – 10 = 0
D) x – 2y – 10 = 0
D) x – 2y – 10 = 0

Question 33.
√2y + √3 = 0 is
A) a linear equation in one variable
B) not a linear equation in one variable
C) a linear equation in two variables
D) none of these
A) a linear equation in one variable

Question 34.
The equation x + √2 = 0 has
A) no solution
B) infinitely many solutions
C) only one solution
D) only two solutions
C) only one solution

Question 35.
A linear equation in two variables has infinitely many solutions which can be represented on
A) a number line
B) a circle
C) a square
D) the cartesian plane
D) the cartesian plane

Question 36.
The general form of a linear equation in two variables is :
A) ax + by + c = 0, where a, b, c are real numbers and a, b ≠ 0
B) ax + b = 0, where a, b are real numbers and a ≠ b
C) ax2 + bx + c = 0, where a, b, c are reed numbers and a, b ≠ 0
D) None of these
A) ax + by + c = 0, where a, b, c are real numbers and a, b ≠ 0

Question 37.
The condition that the equation ax + by + c = 0 represents a linear equation in two variables is :
A) a ≠ 0, b = 0
B) b ≠ 0, a = 0
C) a = 0, b = 0
D) a ≠ 0, b ≠ 0
D) a ≠ 0, b ≠ 0

Question 38.
Write a, b, c for the equation 2x = 5
A) 2, 0, -5
B) 0, 2, -5
C) 0, 0,-5
D) 2, 0, 5
A) 2, 0, -5

Question 39.
The equation x = 7 in two variables can be written as :
A) 1 . x + 1 . y = 7
B) 1 . x + 0 . y = 7
C) 0 . x + 1 . y = 7
D) 0 . x + 0 . y = 7
B) 1 . x + 0 . y = 7

Question 40.
Which of the following is a linear equation in one variable ?
A) 2x + 3y = 0
B) x2 = 5x + 3
C) 5x = y2 + 3
D) x + 5 = 6
D) x + 5 = 6

Question 41.
A linear equation in two variables has
A) a unique solution
B) no solution
C) two solutions
D) infinitely many solutions
D) infinitely many solutions

Question 42.
The equation 3x + 4y = 12 has
A) a unique solution
B) no solution
C) two solutions
D) infinitely many solutions
D) infinitely many solutions

Question 43.
The sum of the ages of Amala and Meenu is 48. Write a linear equation in two variables to represent the statement.
A) x + y = 48
B) x – y = 48
C) 2x + y = 48
D) x + 2y = 48
A) x + y = 48

Question 44.
The salary of Dr. Harikishan is three times the salary of Manish Goyal. Write a lineal4 equation in two variables to represent the statement.
A) x = 3y
B) x + 3y = 0
C) x = 3y + 3
D) x = y + 3
A) x = 3y

Question 45.
The opposite angles of a parallelogram are equal. Write a linear equation in two variables to represent the statement
A) x + y = 0
B) x = y
C) x = 2y
D) x + y + 1= 0
B) x = y

Question 46.
The line y = x passes through
A) (0, 0)
B) (0, 1)
C)(1, 0)
D)(1, -1)
A) (0, 0)

Question 47.
Equation of a line passing through origin is
A) x + y = 1
B) x = 2y – 4
C) x + y = 0
D) y = x – 1
C) x + y = 0

Question 48.
The line y = mx + c
A) passes through origin
B) does not pass through origin
C) is parallel to x – axis
D) is parallel to y – axis
B) does not pass through origin

Question 49.
The pair satisfying 2x + y = 6 is
A) (1, 2)
B) (2, 1)
C) (2, 2)
D) (1, 1)
C) (2, 2)

Question 50.
Which of the following is a solution of the equation x – y = -1?
A) (0, 1)
B) (1, 0)
C) (1, 1)
D) (2, 1)
A) (0, 1)

Question 51.
The linear equation 5x – 3y = 2 has a solution
A) (1, 2)
B) (1, 1)
C) (2, 1)
D)(1, -1)
B) (1, 1)

Question 52.
x = 2 and y = – 1 in the solution of the equation
A) x – y = 3
B) 2x + y = – 3
C) x – 2y = 0
D) x + y = 3
A) x – y = 3

Question 53.
Solution of linear equation 2x + 0y + 9 = 0 is
A) ($$\frac{-9}{2}$$, m)
B) (n, $$\frac{-9}{2}$$)
C) (0, $$\frac{-9}{2}$$,)
D) ($$\frac{-9}{2}$$, 0)
A) ($$\frac{-9}{2}$$, m)

Question 54.
If the point (4, 1) lies on the line x – ky = 2, then the value of k is
A) 1
B) -1
C) 2
D) 2
D) 2

Question 55.
If the units and ten’s digits of a two digit number are y and x, then the number is
A) 10x + y
B) 10y + x
C) x + y
D) xy
B) 10y + x

Question 56.
If x = 1, then the value of y from the equation $$\frac{4}{x}$$ + $$\frac{3}{y}$$ = 5 is
A) 1
B) $$\frac{1}{3}$$
C) 3
D) -3
C) 3

Question 57.
If y = 2x – 3 and y = 5, then the value of x is
A) 1
B) 2
C) 3
D) 4
D) 4

Question 58.
The equation of x – axis is
A) x = 0
B) y = 0
C) x = 1
D) y = 1
B) y = 0

Question 59.
The equation of y – axis is
A) x = 0
B) y = o
C) x = 1
D) y = 1
A) x = 0

Question 60.
How many lines may pass through ‘O ’ ?
A) 1
B) 2
C) 4
D) Infinitely many
D) Infinitely many

Question 61.
The maximum number of points that lie on the graph of a linear equation in two variables is
A) two
B) infinite
C) three
D) none of these
B) infinite

Question 62.
The linear equation 2x + 5y = 7 has
A) a unique solution
B) no solution
C) two solutions
D) infinitely many solutions
D) infinitely many solutions

Question 63.
The equation whose graph is
im1
A) x + y = 6
B) x + y = 3
C) x + y + 3 = 0
D) x + y = 0
B) x + y = 3

Question 64.
In the figure, the graph of the equation is drawn. Choose the correct equa¬tion for which the graph has been drawn.
im2
A) y = – x
B) y = x
C) y = 2x
D) y = 3x
D) y = 3x

Question 65.
To which linear equation does the graph represent ?
im3
A) 3x – 7y = 10
B) y – 2x = 3
C) 8y – 6x = 4
D) 5x + $$\frac{35}{2}$$y = 25
C) 8y – 6x = 4

Question 66.
The graph y = mx is a
A) straight line parallel to x – axis
B) straight line parallel to y – axis
C) line that passes through the origin
D) line that coincides with the x – axis
C) line that passes through the origin

Question 67.
Any point on the line y = 3x is of the form
A) (a, 3a)
B) (3a, a)
C) (a, $$\frac{a}{3}$$)
D) ($$\frac{a}{3}$$, a)
A) (a, 3a)

Question 68.
The equation of the line, whose graph passes through the origin is
A) 2x + 3y = 1
B) 2x + 3y = 0
C) 2x + 3y = 6
D) none of these
B) 2x + 3y = 0

Question 69.
The graph of the linear equation y – x = 0 passes through the point
A) (0, $$\frac{3}{2}$$)
B) ($$\frac{3}{2}$$, $$\frac{-3}{2}$$)
C) ($$\frac{-1}{2}$$, $$\frac{1}{2}$$)
D) ($$\frac{1}{2}$$, $$\frac{1}{2}$$)
D) ($$\frac{1}{2}$$, $$\frac{1}{2}$$)

Question 70.
Any point on the line y = x is of the form:
A) (a, a)
B) (0, a)
C) (a, 0)
D) (a, -a)
A) (a, a)

Question 71.
Straight line passing through the points (-1, 1), (0, 0) and (1, -1) has equation
A) y = x
B) x + y = 0
C) y = 2x
D) 2 + 3y = 7x
B) x + y = 0

Question 72.
The point of intersection of the lines represented by the equations 3x = 2y + 1 and 2x = 3y – 1 is
A) (2, 3)
B) (3, 2)
C) (1, 1)
D) (0,0)
C) (1, 1)

Question 73.
The point of the form ($$\frac{\mathbf{p}}{2}$$, p) always lies on the graph of the equation
A) 2x = y
B) x = 2y
C) x = y + 2
D) x + 2 = y
A) 2x = y

Question 74.
Where does the line – 2x + y – 7 = 0 intersect y – axis ?
A) at (0, 7)
B) at (7, 0)
C) at (7, 7)
D) at (-7,-7)
A) at (0, 7)

Question 75.
Graph of a linear equation ax + by + c = 0, a ≠ 0, b ≠ 0 cuts x – axis and y – axis respectively at the points
A) ($$\frac{-\mathrm{c}}{\mathrm{a}}$$, 0), (0, $$\frac{-\mathrm{c}}{\mathrm{a}}$$)
B) (0, $$\frac{-\mathrm{c}}{\mathrm{a}}$$), ($$\frac{-\mathrm{c}}{\mathrm{a}}$$, 0)
C) (-c, 0), (0, -c)
D) (x, 0), (y, 0)
A) ($$\frac{-\mathrm{c}}{\mathrm{a}}$$, 0), (0, $$\frac{-\mathrm{c}}{\mathrm{a}}$$)

Question 76.
Solve the equation 3x = 20 – x
A) 1
B) 2
C) 5
D) 4
C) 5

Question 77.
Express y in terms of x in the equation 5y – 3x – 10 = 0
A) y = $$\frac{3 x+10}{5}$$
B) y = $$\frac{3 x-10}{5}$$
C) y = $$\frac{10 x+3}{5}$$
D) y = $$\frac{-(3 x+10)}{5}$$
A) y = $$\frac{3 x+10}{5}$$

Question 78.
Solution to the equation 2y + 6= $$\frac{2}{3}$$(3y + 9) + x :
A) lies on the x – axis
D) on the y – axis
D) on the y – axis

Question 79.
The value of y at x = -1 in the equation 5y = 2 is
A) $$\frac{5}{2}$$
B) $$\frac{2}{5}$$
C) 10
D) 0
B) $$\frac{2}{5}$$

Question 80.
The graph of x = a is a straight line parallel to
A) x – axis
B) y – axis
C) line x = y
D) line x + y = 0
B) y – axis

Question 81.
The graph of y = a is a straight line parallel to
A) x – axis
B) y – axis
C) line y = x
D) line x + y = 0
A) x – axis

Question 82.
Which of the following is the equation of a line parallel to y – axis ?
A) y = 0
B) x + y = z
C) y = x
D) x = a
D) x = a

Question 83.
The equation of a line parallel to y – axis is
A) x = 1
B) x + y = 0
C) y = 0
D) y = 1
A) x = 1

Question 84.
x = 4 is a line
A) parallel to y = – 4
B) parallel to x – axis
C) passing through origin
D) parallel to x = – 4
D) parallel to x = – 4

Question 85.
Equation of a line which is at 5 units distance above the x – axis is
A) x = 5
B) x + 5 = y
C) y = 5
D) x – y = 0
C) y = 5

Question 86.
The number of solutions of linear equation 0.2x + 100 = 0 is _______ .
A) 0
B) 1
C) 2
D) Infinite
B) 1

Fill in the blanks :

1. A linear equation in two variables can be written in the form __________ .
ax + by + c = 0, (a ≠ 0)

2. The values of x and y that satisfy a linear equation are called its __________ .
solutions

3. The graphical representation of a linear equation is a __________ .
line

4. If two linear equations have the same graph, they are called __________ .
consistent

5. A linear equation in two variables has __________ or infinitely many solutions.
one solution

6. The number of solutions for a system of two linear equations in two variables can be __________ .
one, none, or infinitely many

7. If the lines represented by two linear equations are parallel, then the system has __________ solutions.
no

8. If the lines represented by two linear equations coincide, then the system has __________ solutions.
infinitely many

9. If the lines represented by two linear equations intersect at one point, then the system has _________ solution.
one

10. The solution to a system of two linear equations can be found by solving for __________ .
x and y

11. A system of linear equations can be solved by __________ method.
elimination

12. A system of linear equations can be solved by __________ method.
substitution

13. The point of intersection of two lines is the solution of the __________ system.
linear equation

14. If a system of linear equations has no solution, it is called __________ .
inconsistent

15. The solution to a system of linear equations can be represented as a __________ .
point in the coordinate plane

16. “Express x + 1 = 2y in the form of ax + by + c = 0 __________.
x + 1 = 2y
x – 2y + 1 = 0
It is the form
ax + by + c = 0

17. Express the following statement as linear equation in two variables. “The cost of a note book is 3 more than twice the cost of pen __________ “.
Let cost of pen ₹ x
Cost of note book ₹ y
∴ y = 2x + 3

## AP 9th Class Maths Bits 3rd Lesson Co-Ordinate Geometry

Multiple Choice Questions (MCQs) :

Question 1.
The coordinate system used to plot points on a plane is known as the:
A) Cartesian coordinate system
B) polar coordinate system
C) spherical coordinate system
D) cylindrical coordinate system
A) Cartesian coordinate system

Question 2.
The horizontal axis on a Cartesian coordinate system is known as the:
A) ordinate
B) abscissa
C) latitude
D) longitude
B) abscissa

Question 3.
The vertical axis on a Cartesian coordinate system is known as the:
A) ordinate
B) abscissa
C) latitude
D) longitude
A) ordinate

Question 4.
In. the point (3, 4), which number represents the abscissa?
A) 3
B) 4
C) Both 3 and 4
D) Neither 3 nor 4
A) 3

Question 5.
In the point (3, 4), which number represents the ordinate?
A) 3
B) 4
C) Both 3 and 4
D) Neither 3 nor 4
B) 4

Question 6.
What is the distance between the points (-2, 3) lies in which quadrant ?
A) 1
B) 2
C) 3
D) 4
B) 2

Question 7.
Which of the following is an example of a point on a Cartesian coordinate system?
A) (x, y, z)
B) (x, y)
C) (r, θ)
D) (p, Φ, θ)
B) (x, y)

Question 8.
The co-ordinates of origin are
A) (0, 9)
B) (0, 1)
C) (0, 0)
D) none of above
C) (0, 0)

Question 9.
The x-coordinate of all points on the y-axis on a Cartesian coordinate system is:
A) y = 0
B) x = 0
C) y = 1
D) x = 1
B) x = 0

Question 10.
The y-coordinate of all points on the x-axis on a Cartesian coordinate system is:
A) y = 0
B) x = 0
C) y = 1
D) x = 1
A) y = 0

Question 11.
Abscissa of a point is positive in

Question 12.
The points (-5, 2) and (2, -5) lie in the
B) II and III quadrants respectively.
C) II and IV quadrants respectively.
D) I and IV quadrants respectively
C) II and IV quadrants respectively.

Question 13.
If (x + 2, 4) = (5, y – 2), then coordinates (x, y) are
A) (7, 12)
B) (6, 3)
C) (3, 6)
D) (2, 1)
C) (3, 6)

Question 14.
Mirror image of the point (9, -8) in y – axis is
A) (-9, -8)
B) (9, 8)
C) (-9, 8)
D) (-8, 9)
A) (-9, -8)

Question 15.
The coordinates of the point which lies on y-axis at a distance of 4 units in negative direction of y-axis is
A) (5, 4)
B) (4, 0)
C) (0, -4)
D) (-4, 0)
C) (0, -4)

Question 16.
If the points A(2, 0), B(-6, 0) and C(3, a – 3) lie on the x-axis, then the value of a is
A) 0
B) 2
C) 3
D) -6
C) 3

Question 17.
Which of the following points lies on the negative side of x-axis?
A) (-4, 0)
B) (3, 2)
C) (0, -4)
D) (5, -7)
A) (-4, 0)

Question 18.
The Coordinates of point Q are

A) (3, 3.5)
B) (3.5, 3)
C) (-3, 3.5)
D) (-3, -3.5)
D) (-3, -3.5)

Question 19.
The point M lies in the IV quadrant. The coordinates of point M are
A) (a, b)
B) (-a, b)
C) (a, -b)
D) (-a, -b)
C) (a, -b)

Question 20.
Write the name of the quadrant in which the point (-3, -5) lies.

Question 21.
The number of parts the coordinates axes divide the plane is
A) Two parts
B) Four parts
C) Six parts
D) Eight parts
B) Four parts

Question 22.
Point (0, 4) lies
B) on x-axis
C) on y-axis
B) on x-axis

Question 23.
The mirror image of the point (-3, -4) in x-axis is
A) (-4, -3)
B) (3, -4)
C) (3, 4)
D) (-3, 4)
D) (-3, 4)

Question 24.
In which quadrant does the point (-1, 2) lie?

Question 25.
Abscissa of a point is negative in

Question 26.
Abscissa of all the points on y-axis is
A) 1
B) any number
C) 0
D) -1
C) 0

Question 27.
Which is the example of geometrical line?
A) Blackboard
B) Sheet of paper
C) Meeting place of two walls
D) Tips of sharp pencil
C) Meeting place of two walls

Question 28.
Rene Descartes was a
A) French Mathematician
B) Indian Mathematician
C) Russian Mathematician
D) British Mathematician
A) French Mathematician

Question 29.
Rene Descartes belonged to
A) 15th century
B) 16th century
C) 17th century
D) 18th century
C) 17th century

Question 30.
Which of the following is an example of a geometrical line ?
A) Blackboard
B) Sheet of paper
C) Metting place of two walls
D) Tip of the sharp pencil
C) Metting place of two walls

Question 31.
If x and y, both are positive, then the point (x, y) lines in

Question 32.
If x is positive and y is negative, then the point (x, y) lies in

Question 33.
The line of intersection of II and III quadrants is
A) x – axis
B) y – axis
C) vertical axis
D) none of these
A) x – axis

Question 34.
The line of intersection of III and IV quadrants is
A) y – axis
B) x – axis
C) horizontal axis
D) none of these

Question 35.
The line of intersection of IV and I quadrants is
A) x – axis
B) y – axis
C) vertical axis
D) none of these
A) x – axis

Question 36.
Where do the II and IV quadrants meet ?
A) at ‘0’
B) in y – axis
C) in x – axis
D) do not intersect
A) at ‘0’

Question 37.
Where do the four quadrants meet ?
A) at ‘0’
B) in x – axis
C) in y – axis
D) do not meet
A) at ‘0’

Question 38.
A) +ve
B) -ve
C) 0
D) none of these
A) +ve

Question 39.
A) -ve
B) +ve
C) 0
D) none of these
A) -ve

Question 40.
The points (- 5, 2) and (2, – 5) lie in the
B) II and III quadrants respectively
C) II and IV quadrants respectively
D) IV and III quadrants respectively
C) II and IV quadrants respectively

Question 41.
Abscissa of a point is positive in

Question 42.
In which quadrant does the poipt (- 1, 2) lie ?
A) I
B) II
C) III
D) IV
B) II

Question 43.
In which quadrant does the point (-1,-1) lie?
A) I
B) II
C) III
D) IV
C) III

Question 44.
In which quadrant does the point (1,-2) lie?
A) I
B) II
C) III
D) IV
D) IV

Question 45.
Point (0, 4) lies :
B) on x – axis
C) on y – axis
C) on y – axis

Question 46.
Abscissa of all the points on y – axis is:
A) 1
B) any number
C) 0
D) -1
C) 0

Question 47.
Which of the following points lies on the negative side of x – axis ?
A) (-4, 0)
B) (3, 2)
C) (0, -4)
D)(5, -7)
A) (-4, 0)

Question 48.
The coordinates of point Q are

A) (3, 3.5)
B) (3.5, 3)
C) (-3, 3.5)
D) (-3, -3.5)
D) (-3, -3.5)

Question 49.
The coordinates of the point which lies on y – axis at a distance of 4 units in negative direction of y – axis is
A) (0, 4)
B) (4, 0)
C) (0, -4)
D) (-4, 0)
C) (0, -4)

Question 50.
If a point is on negative side of y-axis at a distance of 3 units from origin then, the coordiantes of the point are
A) (0, 3)
B) (0, -3)
C) (3 ,0)
D) (-3, 0)
B) (0, -3)

Question 51.
The point which lies on y – axis at a distance of 6 units in the negative direction of y – axis is
A) (0, 6)
B) (6, 0)
C) (0, -6)
D) (-6,0)
C) (0, -6)

Question 52.
If the point A(2, 0), B(-6, 0) and C(3, a – 3) lies on the x – axis, then the value of ‘a’ is
A) 0
B) 2
C) 3
D) -6
C) 3

Question 53.
If (x + 2, 4) = (5, y – 2), then the coordinates (x, y) are
A) (7, 12)
B) (6, 3)
C) (3, 6)
D) (2, 1)
C) (3, 6)

Question 54.
Mirror image of the point (-1, 2) in y – axis is
A) (1, 2)
B) (1, -2)
C) (2, 1)
D)(2, -1)
A) (1, 2)

Question 55.
Mirror image of the po1nt (3, 9) In x – axis is
A) (-3, 9)
B) (9, 3)
C) (3, 9)
D) (3, -9)
D) (3, -9)

Question 56.
Write the coordinates of p.

A) (2, 2)
B) (-1, -2)
C) (1, -2)
D) (-1, 2)
A) (2, 2)

Question 57.
Write the coordinates of p.

A) (1, -3)
B) (1, 3)
C) (-1, -3)
D) (-1, -3)
A) (1, -3)

Question 58.
The distance of the point (1, 0) from ‘0’ is
A) 0
B) 1
C) 2
D) none of these
B) 1

Question 59.
The distance of the point (-1, 0) from ‘0’ is
A) 0
B) 1
C) -1
D) none of these
B) 1

Question 60.
The distance of the point (0, -1) from ‘0’ is
A) 1
B) 0
C) -1
D) none of these
A) 1

Question 61.
By plotting the points 0(0, 0), A(1, 0), B(1, 1), C(0, 1) and joining OA, AB, BC and CO, the figure we obtain is
A) square
B) rectangle
C) trapezium
D) rhombus
A) square

Question 62.
Match the following.

 i) (3, -3) a) Q1 ii) (-2, -2) b) Q2 iii) (1, 1 ) c) Q3 iv) (-5, 5) d) Q4

A) i – d, ii – c, iii – b, iv – a
B) i – b, ii – a, iii – d, iv – c
C) i- d, ii – c, iii – a, iv – b
D) i – d, ii – b, iii – c, iv – a
C) i – d, ii – c, iii – a, iv – b

 i) (3, -3) d) Q4 ii) (-2, -2) c) Q3 iii) (1, 1 ) a) Q1 iv) (-5, 5) b) Q2

Assertion and Reason Type Questions :

Question 1.
Assertion : The x-axis of a Cartesian plane is also known as the horizontal axis.
Reason : The x-axis represents the values of the horizontal coordinate or abscissa.
Both assertion and reason are correct and, the reason is the correct explanation for
the assertion.

Question 2.
Assertion : The origin of a Cartesian plane is the point where the x-axis and the y-axis intersect.
Reason : The coordinates of the origin is (0, 0).
Both assertion arid reason are correct and the reason is the correct explanation for
the assertion.

Question 3.
Assertion : The y-axis of a Cartesian plane is also known as the vertical axis.
Reason : The y-axis represents the values of the vertical coordinate or ordinate.
Both assertion and reason are correct and the reason is the correct explanation for the assertion.

Question 4.
Assertion : The distance between two points on the Cartesian plane can be calculated using the formula {(x2 – x1)2 + (y2 – y1)2}.
Reason : The Pythagorean theorem is used to calculate the lengths of sides of right triangle.
Both assertion and reason are correct and the reason is not the correct explanation for the assertion.

Question 5.
Question Assertion : The direction of the x-axis on the Cartesian plane is from left to right.
Reason : The direction of the x-axis is determined by the positive direction of the horizontal coordinate.
Both assertion and reason are correct and the reason is the correct explanation for the assertion.

Fill in the blanks :

1. The development of coordinate geometry is often attributed to the French mathematician _________ .
Rene Descartes

2. The Cartesian coordinate system is named after Rene Descartes’ Latinized name, _________ .
Cartesian

3. The development of coordinate geometry revolutionized mathematics by allowing geometric problems to be solved using _________ .
algebraic equations

4. The concept of using coordinates to represent geometric figures was first introduced by the Greek mathematician _________ .
Apollonius of Perga

5. The study of algebraic geometry, which combines algebra and geometry using coordinate systems, was further developed in the 19th century by mathematicians such as _________ .
Carl Friedrich Gauss and Arthur Cayley

6. The _________ axis is the horizontal line on the Cartesian plane.
X-axis

7. The _________ axis is the vertical line on the Cartesian plane.
Y-axis

8. The point where the X-axis and Y-axis intersect is called the _________ .
origin

9. A point on the Cartesian plane is represented as a pair of numbers in the form of _________ .
(x, y)

10. The Cartesian plane is divided into four regions called _________ .

11. We can plot the point (2, 3) on the Cartesian plane in _________ quadrant.
Q1

12. Which quadrant does the point located at (-4, 5) on the Cartesian plane ?
Q2

13. Drawn line connecting the points (-3, 2) and (1, 6) goes through the _________ , _________ quadrants on the Cartesian plane.
Q1 and Q2

14. Which quadrant does the point (-2, -7) lie on the Cartesian plane?
Q3

15. Plot the points (0, 4), (2, 1) and (-3, 2) on the Cartesian plane and connect them, we get _________ .
a triangle

16. What are the names of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane ? _________
Horizontal line – x-axis, Vertical line – y-axis

17. The sum of the abscissa and ordinate of a point of intersection of coordinate axes is _________ .
0 [That point is origin (0, 0)]

18. Draw a rough diagram of Cartesian plane.

19. The position of (2, 3) and (3, 2) represent same on the graph paper. (True or False) _________ .
False

## AP 9th Class Maths Bits 2nd Lesson Polynomials

Multiple Choice Questions (MCQs) :

Question 1.
What is the degree of the polynomial f(x) = 2x3 + 5x2 – 3x + 4?
A) 1
B) 2
C) 3
D) 4
C) 3

Question 2.
Which of the following is a 3rd degree polynomial?
A) 2x2 + 3x – 1
B) 4x3 – 2x
C) x4 + x3 – x2 + x
D) 5x2 – 7x + 3
B) 4x3 – 2x

Question 3.
What are the roots of the polynomial f(x) = x3 – 6x2 + 11x – 6?
A) 1, 2, 3
B) -1, -2, -3
C) 1, 2, -3
D) -1, 2, 3
A) 1, 2, 3

Question 4.
Which of the following is a possible factorization of the polynomial f(x) = x3 + 2x2 – 5x – 6?
A) (x + 3) (x – 2) (x + 1)
B) (x + 3) (x – 2) (x – 1)
C) (x – 3) (x + 2) (x – 1)
D) (x – 3) (x – 2) (x + 1)
B) (x + 3) (x – 2) (x – 1)

Question 5.
If the leading coefficient of a 3rd degree polynomial is negative, what can you
say about the end behavior of the graph?
A) The graph rises to the left and falls to the right.
B) The graph falls to the left and rises to the right.
C) The graph rises to both the left and right.
D) The graph falls to both the left and right.
B) The graph falls to the left and rises to the right.

Question 6.
Which of the following is the identity for the sum of two cubes?
A) a3 + 3a2b + 3ab2 + b3 = (a + b)2
B) a3 – 3a2b + 3ab2 – b3 = (a – b)3
C) a2 + 2ab + b2 = (a + b)2
D) a2 – 2ab + b2 = (a – b)2
B) a3 – 3a2b + 3ab2 – b3 = (a – b)3

Question 7.
Which of the following is the identity for the difference of two squares?
A) a2 – b2 = (a + b)(a – b)
B) a3 – b3 = (a – b)(a2 + ab + b2)
C) a3 + b3 = (a + b)(a2 – ab + b2)
D) a2 + b2 = (a – b)2 + 2ab
A) a2 – b2 = (a + b)(a – b)

Question 8.
Which of the following is the identity for the square of a binomial?
A) (a + b)2 = a2 + 2ab + b2
B) (a – b)2 = a2 – 2ab + b2
C) (a + b)2 = a2 – 2ab + b2
D) (a – b)2 = a2 + 2ab + b2
A) (a + b)2 = a2 + 2ab + b2

Question 9.
Which of the following is the identity for the product of two binomials?
A) (a + b)(a – b) = a2 – b2
B) (a + b)2 = a2 + 2ab + b2
C) (a – b)2 = a2 – 2ab + b2
D) (a + b)(c + d) = ac + ad + bc + bd
D) (a + b)(c + d) = ac + ad + bc + bd

Question 10.
Which of the following is the identity for the cube of a binomial?
A) (a + b)3 = a3 – 3a2b + 3ab2 – b3
B) (a – b)3 = a3 – b3
C) (a + b)3= a3 + 3a2b + 3ab2 + b3
D) (a – b)2 = a2 – 2ab + b2
C) (a + b)3= a3 + 3a2b + 3ab2 + b3

Question 11.
What identity can be used to simplify the expression (x + 3) (x – 3) without multiplying?
A) Difference of two squares
B) Slim of two cubes
C) Cube of a binomial
D) Product of two binomials
A) Difference of two squares

Question 12.
What identity can be used to simplify the expression (a + b)2 without multi-plying?
A) Difference of two squares
B) Sum of two cubes
C) Cube of a binomial
D) Square of a binomial
D) Square of a binomial

Question 13.
What identity can be used to simplify the expression (5y – 3z)2 without multiplying?
A) Difference of two squares
B) Sum of two cubes
C) Cube of a binomial
D) Square of a binomial
D) Square of a binomial

Question 14.
What identity can be used to simplify the expression (2a – 5b)(2a + 5b) without multiplying?
A) Difference of two squares
B) Sum of two cubes
C) Cube of a binomial
D) Product of two binomials
A) Difference of two squares

Question 15.
What identity can be used to simplify the expression (4x + 3)3 without multiplying?
A) Difference of two squares
B) Sum of two cubes
C) Cube of a binomial
D) Square of a binomial
C) Cube of a binomial

Question 16.
What is the degree of the zero polynomial?
A) 1
B) 0
C) -1
D) Undefined
B) 0

Question 17.
Which of the following is a zero of polynomial (x2 – 25)?
A) 5
B) -5
C) 0x
D) 0x2
A) 5 & B) -5

Question 18.
What is the degree of the polynomial x3 + 2x – 5?
A) 1
B) 2
C) 3
D) 4
C) 3

Question 19.
What is the degree of the polynomial 7x4 – 2x3 + 5x2 – 8x + 3?
A) 1
B) 2
C) 3
D) 4
D) 4

Question 20.
Which of the following is not a polynomial?
A) 3x3 – 2x + 1
B) 2x-1 – √2
C) x2 + 5x – 6
D) 4
B) 2x-1 – √2

Question 21.
What is the value of the polynomial 2x3 – 3x2 + 4x – 5 when x = 2?
A) 7
B) -7
C) 3
D) 21
A) 7

Question 22.
Which of the following is not a zero of the polynomial x3 – 2x2 – x + 2?
A) 1
B) -1
C) 2
D) -2
D) -2

Question 23.
What is the value of the polynomial 3x2 – 2x + 1 when x = -1?
A) -4
B) 16
C) 0
D) 6
D) 6

Question 24.
If (x – 3) is a factor of the polynomial x3 – 5x2 + px – 15, what is the value of P?
A) 11
B) -11
C) 15
D) -15
A) 11

Question 25.
What are the zeroes of the polynomial x2 + 5x + 6?
A) -2 and -3
B) 2 and 3
C) -2 and -4
D) 2 and 4
A) -2 and -3

Question 26.
Which of the following statements is true about the Factor Theorem?
A) If a polynomial is of degree 1, then it has no factors.
B) If a polynomial has a factor (x – a), then (a, 0) is a point on the graph of the polynomial.
C) A polynomial can have only one factor.
D) A polynomial with degree n can have at most n -1 factors.
B) If a polynomial has a factor (x – a), then (a, 0) is a point on the graph of the polynomial.

Question 27.
Which of the following polynomials is not a factor of x3 – 6x2 + 11x – 6?
A) (x – 1)(x – 2)
B) (x – 2)(x – 3)
C) (x – 1)(x – 3)
D) (x – 2)(x – 4)
D) (x – 2)(x – 4)

Question 28.
What is the possible value of k in the factorization of the polynomial x2 – 7x + k into two linear factors?
A) 1
B) 2
C) 3
D) 12
D) 12

Question 29.
Which of the following is the correct factorization of the polynomial x3 – 3x2 – 4x + 12?
A) (x – 2)(x + 1)(x – 3)
B) (x + 2)(x – 2)(x – 3)
C) (x + 2)(x – 3)(x + 3)
D) (x + 2)(x + 2)(x + 3)
B) (x + 2)(x – 2)(x – 3)

Question 30.
Which of the following is not a factor of the polynomial x4 – 81?
A) x + 3
B) x – 3
C) x2 + 9
D) x – 9
D) x – 9

Question 31.
Which of the following is the identity for the sum of two cubes?
A) (a – b)(a2 + 2ab + b2)
B) (a + b)(a2 -ab + b2)
C) (a – b)(a2 – ab + b2)
D) (a + b)(a2 + 2ab + b2)
B) (a + b)(a2 -ab + b2)

Question 32.
Which of the following is the identity for the product of two binomials with a common term?
A) (a + b)2 = a2 + 2ab + b2
B) (a + b)(a – b) = a2 – b2
C) (a + b)(a + d) = a2 + ad + ba + bd
D) (a + b)3 = a3 + 3a2b + 3ab2 + b3
C) (a + b)(a + d) = a2 + ad + ba + bd

Question 33.
Which of the following is the identity for the product of the sum and difference of two terms?
A) (a + b)(a – b) = a2 – b2
B) (a + b)2 = a2 + 2ab + b2
C) (a – b)2 = a2 – 2ab + b2
D) (a + b)(a + b) = a2 + 2ab + b2
A) (a + b)(a – b) = a2 – b2

Question 34.
What is a zero of a polynomial?
A) A value of x that makes the polynomial equal to zero.
B) A value of y that makes the polynomial equal to zero.
C) A value of x that makes the polynomial equal to one.
D) A value of y that makes the polynomial equal to one.
A) A value of x that makes the polynomial equal to zero.

Question 35.
What is the degree of a polynomial?
A) The coefficient of the highest degree term in the polynomial.
B) The sum of the exponents of the terms in the polynomial.
C) The product of the coefficients of the terms in the polynomial.
D) The number of terms in the polynomial.
B) The sum of the exponents of the terms in the polynomial.

Question 36.
Which of the following is a factor of the polynomial x2 – 4x + 3?
A) (x – 1)
B) (x + 1)
C) (x – 3)
D) (x + 3)
A) (x – 1) & C) (x – 3)

Question 37.
What is the quadratic formula used for?
A) Finding the zeroes of a quadratic polynomial.
B) Finding the degree of a quadratic polynomial.
A) Finding the zeroes of a quadratic polynomial.

Question 38.
What is the sum of the zeroes of the polynomial x2 – 3x + 2?
A) -2
B) -1
C) 0
D) 1
B) -1

Question 39.
Which of the following is an algebraic identity?
A) (a + b)2 = a2 + b2
B) (a + b)(a – b) = a2 – b2
C) (a + b)2 = a2 – b2
D) (a + b)(a + b) = a2 – b2
B) (a + b)(a – b) = a2 – b2

Question 40.
What is the identity (a + b)2 equal to?
A) a2 + b2
B) a2 – b2
C) a2 + 2ab + b2
D) a2 – 2ab + b2
C) a2 + 2ab + b2

Question 41.
What is the identity (a – b)2 equal to?
A) a2 + b2
B) a2 – b2
C) a2 + 2ab + b2
D) a2 – 2ab + b2
D) a2 – 2ab + b2

Question 42.
What is the identity (a + b)(a – b) equal to?
A) a2 + b2
B) a2 – b2
C) a2 + 2ab + b2
D) a2 – 2ab + b2
B) a2 – b2

Question 43.
What is the identity (a + b)3 equal to?
A) a3 + b3
B) a3 + 3a2b + 3ab2 + b3
C) a3 – b3
D) a3 – 3a2b + 3ab2 – b3
B) a3 + 3a2b + 3ab2 + b3

Question 44.
Which of the following is an algebraic identity ?
A) (x + y)2 = x2 – 2xy + y2
B) (x + y)2 = x2 + 2xy – y2
C) (x + y)2 = x2 + 2xy + y2
D) (x + y)2 = -x2 + 2xy + y2
C) (x + y)2 = x2 + 2xy + y2

Question 45.
The compact form of (x + y) (x – y) is
A) x2 + y2
B) x2 – 2xy + y2
C) x2 + 2xy + y2
D) x2 – y2
D) x2 – y2

Question 46.
Which of the following is a polynomial?
A) x2 + x + $$\frac{3}{x^2}$$
B) √x + 5
C) x3/4 – 7x + 4
D) $$\frac{3}{2}$$x3 – $$\frac{4}{3}$$x2 + 2x – 1
D) $$\frac{3}{2}$$x3 – $$\frac{4}{3}$$x2 + 2x – 1

Question 47.
Which of the following Is a polynomial in one variable?
A) 3 – x2 + x
B) √3x + 4
C) x3 + y3 + 7
D) x + $$\frac{1}{x}$$
A) 3 – x2 + x

Question 48.
The number ‘0’ is called a
A) zero polynomial
B) binomial
C) trinomial
D) linear polynomial
A) zero polynomial

Question 49.
Select the correct statement from the following
A) Degree of a zero polynomial is zero
B) Degree of a zero polynomial is not defined
C) Degree of a constant polynomal is not defined
D) Zero of the zero polynomial is not defined
B) Degree of a zero polynomial is not defined

Question 50.
A cubic polynomial has number of zeroes :
A) 2
B) 1
C) 3
D) atleast three
C) 3

Question 51.
Which of the following is a binomial in my?
A) y2 + 2
B) y + $$\frac{1}{y}$$ + 2
C) √y + √2y
D) y√y + 1
A) y2 + 2

Question 52.
Which of the following is a trianomial in x ?
A) x3 + 1
B) x3 + x2 + x
C) x√x – √x + 1
D) x3 + 2x
B) x3 + x2 + x

Question 53.
Which of the following is cubic polynomial ?
A) x3 + 3x2 – 4x + 3
B) x2 + 4x – 7
C) 3x2 + 4
D) 3(x2 + x + 1)
A) x3 + 3x2 – 4x + 3

Question 54.
(1 + 3x)3 is an example of
A) Monomial
B) Binomial
C) Trinomial
D) None of these
D) None of these

Question 55.
The degree of the polynomial p(x) = 3 is
A) 3
B) 1
C) 0
D) 2
C) 0

Question 56.
A cubic polynomial is a polynomial with degree
A) 1
B) 3
C) 0
D) 2
B) 3

Question 57.
If p = 17, the degree of the polynomial p(x) = (p – x)3 + 14 is
A) 17
B) 14
C) 0
D) 3
D) 3

Question 58.
The maximum number of terms in a polynomial of degree 10 is
A) 9
B) 10
C) 11
D) 1
C) 11

Question 59.
In the polynomial 1 – $$\sqrt{11}$$x, the coefficient of x is
A) 1
B) 11
C) –$$\sqrt{11}$$
D) $$\sqrt{11}$$
C) –$$\sqrt{11}$$

Question 60.
The coefficient of x2 in the polynomial 7 + 4x – x2 + x3 is
A) -1
B) 1
C) 7
D) 4
A) -1

Question 61.
The coefficient of x in the expansion of (x + 2)3 is
A) 1
B) 6
C) 8
D) 12
D) 12

Question 62.
The coefficient of x2 in (3x + x3) (x + $$\frac{1}{x}$$) is
A) 3
B) 1
C) 4
D) 2
C) 4

Question 63.
Zero of the zero polynomial is
A) 0
B) 1
C) any real number
D) not defined
C) any real number

Question 64.
The maximum number of zeroes of the polynomial p(y) = my a is
A) a + 1
B) m
C) m + 1
D) a
D) a

Question 65.
Zero of the polynomial p(x) = cx + d is
A) -d
B) -c
C) $$\frac{d}{c}$$
D) –$$\frac{d}{c}$$
D) –$$\frac{d}{c}$$

Question 66.
Which of the following-polynomials has – 3 as a zero ?
A) x – 3
B) x2 – 9
C) x2 – 3x
D) x2 + 3
B) x2 – 9

Question 67.
The value of polynomial 6a2 + 7a – 3 when a = 1 is
A) 10
B) 4
C) -13
D) -4
C) -13

Question 68.
If p(x) = 7 – 3x + 2x2, then value of p(-2)is
A) 12
B) 31
C) 21
D) 22
C) 21

Question 69.
If p(x) = x3 – x2 + x + 1, then value of P(1) + P(-1)is
A) $$\frac{1}{4}$$
B) 4
C) 0
D) -2
C) 0

Question 70.
If p(x) = 3x – 7, then p(x) + p(-x) is
A) 7
B) 6x
C) 0
D) -14
D) -14

Question 71.
If (2t + 1) is the factor of the polynomial p(t) = 4t3 + 4t2 – t – 1, then the p(-$$\frac{1}{2}$$) is
A) $$\frac{-1}{2}$$
B) $$\frac{1}{2}$$
C) –$$\frac{1}{2}$$
D) 0
D) 0

Question 72.
The zeroes of the polynomial x2 + 2x + 3 are
A) real
B) not real
C) irrational
D) rational
B) not real

Question 73.
A zero of 2x3 – 7x2 – 16x + 5 is
A) 5
B) 4
C) 1
D) -1
A) 5

Question 74.
If a polynomial f(x) is divided by x – a, then remainder is
A) f(0)
B) f(a)
C) f(-a)
D) f(a) – f(0)
B) f(a)

Question 75.
If the polynomial p(x) is divided by (x + 3), then the remainder will be
A) p(- 2)
B) p(2)
C) p(- 3)
D) p(3)
C) p(- 3)

Question 76.
The remainder obtained when the polynomial p(x) is divided by (b – ax) is
A) p($$\frac{-b}{a}$$)
B) P($$\frac{a}{b}$$)
C) p($$\frac{b}{a}$$)
D) p($$\frac{-a}{b}$$)
C) p($$\frac{b}{a}$$)

Question 77.
On dividing 5y3 – 2y2 – 7y + 1 by ‘y’, the remainder we get is
A) -1
B) 1
C) 0
D) 2
B) 1

Question 78.
If x31 + 51 is divided by (x + 1), the remainder is
A) 0
B) 1
C) 49
D) 50
D) 50

Assertion and Reason Type Questions:

Question 1.
Assertion: The degree of a polynomial with only one term is the exponent of the variable in that term.
Reason : A polynomial with only one term is called a monomial and the degree of a monomial is the exponent of the variable in that monomial.
Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion.

Question 2.
Assertion : A zero polynomial has no degree.
Reason : A zero polynomial is a polynomial in which all the coefficients are zero.
Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion.

Question 3.
Assertion : The degree of a polynomial is the highest power of the variable in that polynomial.
Reason : The degree of a polynomial is determined by the term with the highest power of the variable.
Both Assertion and Reason are true, but the Reason is not the correct explanation of the Assertion. The degree of a polynomial is determined by the term with the highest power of the variable, but it is not necessarily the highest power of the variable in the polynomial. For example, the polynomial 2x3 – 5x + 1 has a degree of 3, even though the highest power of the variable is 3.

Question 4.
Assertion : If a polynomial has a factor (x- a), then, ‘a’ is a zero of the polynomial.
Reason : If (x – a) is a factor of a polynomial, then the polynomial can be written as (x – a) times another polynomial.
Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion.

Question 5.
Assertion : A polynomial of degree ‘n’ can have at most n zeros.
Reason : The number of zeroes of a polynomial of degree n is equal to the degree of the polynomial.
The Assertion is true, but the Reason is false. A polynomial of degree n can have at most n distinct zeros, but it can have repeated zeros, which means the total number of zeros can be greater than n.

Question 6.
Assertion : (a + b)2 = a2 + 2ab + b2
Reason : This is the expansion of the algebraic identity (a + b)2. It can be proven using the distributive law of multiplication over addition.
Both assertion and reason are true and the reason is a correct explanation of the assertion.

Question 7.
Assertion : (a – b)2 = a2 – 2ab + b2
Reason : This is the expansion of the algebraic identity (a – b)2. It can be proven using the distributive law of multiplication over subtraction.
Both assertion and reason are true and the reason is a correct explanation of the assertion.

Question 8.
Assertion: (a + b)3 = a3 + 3a2b + 3ab2 + b3.
Reason : This is the expansion of the algebraic identity (a + b)3. It can be proven using the binomial theorem.
Both assertion and reason are true and the reason is a correct explanation of the assertion.

Question 9.
Assertion : a2 – b2 = (a + b)(a – b)
Reason : This is the factorization of the algebraic identity a2 – b2. It can be proven ‘ using the difference of squares formula.
Both assertion and reason are true and the reason is a correct explanation of the assertion.

Fill in the blanks :

1. The factor theorem states that if a polynomial P(x) has afactor (x – a), then, P(a) = _________ .
0

2. To factorize a polynomial completely, we must first find all of its _________ .
factors

3. A polynomial of degree n can be factored into _________ linear factors.
n

4. If a polynomial has a factor (x + a), then f(-a) = _________ .
0

5. The process of breaking down a polynomial into its factors is called _________ .
Factorization

6. If p(x) = x3 – 1 then the value of p(-1) = _________ .
p(x) = x3 – 1
p(x) = (-1)3 – 1 = – 1 – 1 = -2

7. $$\frac{2024^2-2023^2}{2024+2023}$$ = _________ .
a2 – b2 = (a – b) (a + b)
= $$\frac{(2024)^2-(2023)^2}{2024+2023}$$ = $$\frac{(2024+2023)(2024-2023)}{(2024+2023)}$$ = 1

8. Value of the polynomial p(x) = 5 + 2x at x = 1 is _________ .
p(1) = 5 + 2(-1) = 5 – 2 = 3
∴ p(-1) = 3

## AP 9th Class Maths Bits 1st Lesson Number System

Multiple Choice Questions (MCQs) :

Question 1.
What is the smallest prime number ?
A) 1
B) 2
C) 3
D) 4
B) 2

Question 2.
Which of the following number is an odd number ?
A) 20
B) 25
C) 32
D) 48
B) 25

Question 3.
Which of the following is not a natural number ?
A) 0
B) 1
C) 2
D) 3
A) 0

Question 4.
What is the HCF of 12 and 18 ?
A) 2
B) 4
C) 6
D) 12
C) 6

Question 5.
Which of the following is a composite number ?
A) 7
B) 9
C) 13
D) 17
B) 9

Question 6.
What is the LCM of 4 and 6 ?
A) 12
B) 16
C) 20
D) 24
A) 12

Question 7.
What is the prime factorization of 36 ?
A) 22 × 32
B) 23 × 3
C) 22 × 33
D) 24 × 3
A) 22 × 32

Question 8.
Which of the following is a perfect square ?
A) 28
B) 36
C) 43
D) 51
B) 36

Question 9.
What is the next prime number after 23 ?
A) 25
B) 27
C) 29
D) 31
C) 29

Question 10.
Which of the following is not a rational number ?
A) 0.5
B) 1.25
C) 2/3
D) √2
D) √2

Question 11.
What is the decimal expansion of 5/8?
A) 0.625
B) 0.75
C) 0.5
D) 0.6250
A) 0.625

Question 12.
Which of the following is an irrational number?
A) $$\frac{2}{3}$$
B) 0.5
C) √7
D) 0.75
C) √7

Question 13.
What is the value of (-2)3 ?
A) -8
B) -6
C) 6
D) 8
A) -8

Question 14.
What is the absolute value of -6?
A) 6
B) -6
C) 0
D) 1
A) 6

Question 15.
What is the value of 5! (5 factorial)?
A) 15
B) 120
C) 720
D) 1440
B) 120

Question 16.
Which of the following is a prime factor of 100?
A) 2 & 5
B) 5
C) 10
D) 20
A) 2 & 5

Question 17.
What is the sum of the first 10 natural numbers?
A) 45
B) 50
C) 55
D) 60
C) 55

Question 18.
Which of the following is a perfect cube?
A) 27
B) 32
C) 36
D) 49
A) 27

Question 19.
On the number line, which direction is to the right of zero?
A) Positive
B) Negative
C) Both positive and negative
D) Neither positive nor negative
A) Positive

Question 20
What is the distance between -3 and 5 on the number line?
A) 2
B) 5
C) 8
D) 10
C) 8

Question 21.
Which of the following points is closest to zero on the number line?
A) -2
B) -5
C) 3
D) 6
A) -2

Question 22.
Which of the following numbers is represented by the point located at -16/4 on the number line?
A) -4
B) 4
C) -5
D) 5
A) -4

Question 23.
What is the midpoint between -6 and 4 on the number line?
A) -1
B) 0
C) 1
D) 5
A) -1

Question 24.
Which of the following is a rational number?
A) √4
B) √3
C) $$\sqrt{14}$$
D) √5
A) √4

Question 25.
What is the decimal representation of the rational number 3/5?
A) 0.66
B) 0.3
C) 0.6
D) 0.8
C) 0.6

Question 26.
Which of the following is a rational number?
A) $$\frac{2}{\sqrt{3}}$$
B) –$$\frac{4}{\sqrt{3}}$$
C) $$\sqrt{1.2}$$
D) 0.32
D) 0.32

Question 27.
What is the reciprocal of 7/8?
A) 18/17
B) 8/7
C) 1/56
D) 56/1
B) 8/7

Question 28.
What is the product of 2/3 and -5/8?
A) -5/12
B) -25/12
C) 10/24
D) -30/24
A) -5/12

Question 29.
What is the sum of -3/4 and 1/6?
A) -5/12
B) -1/12
C) 1/12
D) -7/12
D) -7/12

Question 30.
What is the value of (-$$\frac{1}{2}$$)2 + ($$\frac{1}{2}$$)2?
A) -7/6
B) -1/6
C) 25/36
D) 7/6
C) 25/36

Question 31.
Which of the following is a fraction is odd one?
A) 14/6
B) 5/8
C) 6/9
D) 8/12
A) 14/6

Question 32.
What is the simplest form of the mixed number 3 1/2?
A) 1/2
B) 3/2
C) 5/2
D) 7/2
D) 7/2

Question 33.
Which of the following is a rational number lies between -1 and 0?
A) $$\sqrt{.0196}$$
B) -√3
C) -√4
D) -√5
B) -√3

Question 34.
What is the product of a non-zero rational number and an irrational number?
A) Always irrational
B) Always rational
C) Sometimes irrational, sometimes rational
D) None of the above
A) Always irrational

Question 35.
What is the sum of a rational number and an irrational number?
A) Always irrational
B) Always rational
C) Sometimes irrational, sometimes rational
D) None of the above
A) Always irrational

Question 36.
What is the decimal expansion of the rational number 1/7?
A) 0.142857142857…
B) 0.166666666666…
C) 0.125
D) 0.111111111111…
A) 0.142857142857…

Question 37.
Which of the following fractions is not equivalent to 2/3?
A) 4/6
B) 5/8
C) 8/12
D) 10/15
B) 5/8

Question 38.
What is the value of (2/3) ÷ (4/5)?
A) 5/6
B) 3/5
C) 8/15
D) 10/12
A) 5/6

Question 39.
Which of the following is not a rational number?
A) 0.125
B) -0.25
C) 0.75
D) √6
D) √6

Question 40.
What is the simplest form of the fraction 24/36?
A) 2/3
B) 3/4
C) 4/5
D) 5/6
A) 2/3

Question 41.
What is the reciprocal of 0.6?
A) 5/3
B) 3/4
C) 4/5
D) 2/7
A) 5/3

Question 42.
$$(64)^{\frac{1}{2}}+(27)^{\frac{1}{3}}$$ =
A) 11
B) 5
C) 17
D) Done
A) 11

Question 43.
(√a +√b)2 = a + b + ……..
A) 2ab
B) 2$$\sqrt{\mathrm{ab}}$$
C) -2ab
D) -√2 ab
B) 2$$\sqrt{\mathrm{ab}}$$

Question 44.
Which of the following numbers is an irrational number?
A) 2
B) 3
C) 4
D) √5
D) √5

Question 45.
What is the decimal expansion of the irrational number √2 ?
A) 1.41421356…
B) 2.71828182…
C) 3.14159265…
D) 4.66920160…
A) 1.41421356…

Question 46.
What is the value of √8 ÷ √2 ?
A) 2
B) √2
C) √8
D) 4
A) 2

Question 47.
Which of the following numbers is not an irrational number?
A) √3
B) √5
C) √7
D) √9
D) √9

Question 48.
What is the decimal expansion of the irrational number √3 ?
A) 1.73205081…
B) 2.23606798…
C) 3.14159265…
D) 4.47213600…
A) 1.73205081…

Question 49.
What is the value of √2 × √8 ?
A) 4
B) 8
C) 16
D) 64
A) 4

Question 50.
Which of the following numbers is irrational?
A) 0.5
B) 0.75
C) 1.5
D) √2 /2
D) √2 /2

Question 51.
What is the value of $$\sqrt{(4 / 9)}$$ ?
A) 2/3
B) 4/3
C) 2/√3
D) √3/3
A) 2/3

Question 52.
What is the simplest form of number (√2 + √3)(√2 – √3)?
A) √5/2
B) 2√5/5
C) 5/2√5
D) -1
D) -1

Question 53.
What is the product, of the irrational numbers √2 and √3 ?
A) √5
B) √6
C) √8
D) $$\sqrt{12}$$
B) √6

Question 54.
Which of the following fractions is not in the form of p/q?
A) 3/4
B) 1/3
C) 2/5
D) 0.122
D) 0.122

Question 55.
What is the simplified form of the fraction 24/36?
A) 2/3
B) 3/4
C) 4/5
D) 5/6
A) 2/3

Question 56.
Which of the following fractions is equivalent to 5/6?
A) 10/12
B) 12/15
C) 1/18
D) 18/21
A) 10/12

Question 57.
What is the decimal equivalent of the fraction 3/8?
A) 0.25
B) 0.375
C) 0.5
D) 0.625
B) 0.375

Question 58.
What is the simplest form of the fraction 10/15?
A) 2/3
B) 22/3
C) 3/4
D) 4/5
A) 2/3

Question 59.
Which of the following decimals is a terminating decimal?
A) 0.3333
B) 0.4040
C) 0.5000
D) 0.6666…
C) 0.5000

Question 60.
What is the decimal equivalent of the fraction 1/2?
A) 0.25
B) 0.5
C) 0.75
D) 0.9
B) 0.5

Question 61.
Which of the following decimals is a non-terminating decimal?
A) 0.125
B) 0.5
C) 0.3333
D) 0.8
C) 0.3333

Question 62.
What is the decimal equivalent of the fraction 2/5?
A) 0.4
B) 0.5
C) 0.6
D) 0.8
A) 0.4

Question 63.
Which of the following fractions has a terminating decimal expansion?
A) 1/7
B) 1/3
C) 1/5
D) 1/9
C) 1/5

Question 64.
What is the rationalized form of the fraction 1/√3?
A) (√3)/2
B) (√3)/3
C) 3/√3
D) 2/√3
B) (√3)/3

Question 65.
What is the rationalized form of the fraction 2/√5?
A) $$\frac{(\sqrt{5})}{2}$$
B) $$\frac{2 \sqrt{5}}{3}$$
C) $$\frac{5}{2 \sqrt{5}}$$
D) $$\frac{2 \sqrt{5}}{5}$$
D) $$\frac{2 \sqrt{5}}{5}$$

Question 66.
What is the rationalized form of the fraction $$\frac{(\sqrt{2}+\sqrt{3})}{\sqrt{2}}$$?
A) $$\frac{(2+\sqrt{6})}{2}$$
B) $$\frac{(\sqrt{2}-\sqrt{3})}{2}$$
C) $$\frac{(\sqrt{3}-\sqrt{2})}{2}$$
D) $$\frac{(\sqrt{6}+\sqrt{3})}{2}$$
A) $$\frac{(2+\sqrt{6})}{2}$$

Question 67.
What is the rationalized form of the fraction $$\frac{1}{(2+\sqrt{3})}$$?
A) (2 – √3)
B)(√3 – 2)/5
C)(√3 – 2)/√3
D)(√3 – 2)/4
A) (2 – √3)

Question 68.
What is the rationalized form of fraction (3 + 2√2)/(1 – √2)?
A) (5√2)/3
B) -7-5√2
C) 5 + 2√2
D) 5 – √2
B) -7-5√2

Question 69.
What is the value of 24 × 23?
A) 64
B) 128
C) 256
D) 512
B) 128

Question 70.
What is the value of (52)3?
A) 25
B) 125
C) 15625
D) 3125
C) 15625

Question 71.
What is the value of 40?
A) 0
B) 1
C) 2
D) 4
B) 1

Question 72.
What is the value of 3-2?
A) 1/3
B) 1/9
C) 9
D) 6
B) 1/9

Question 73.
What is the value of (23)-2?
A) 1/8
B) 1/64
C) 8
D) 64
B) 1/64

Question 74.
What is the value of 52 × 5-3?
A) 1/25
B) 1/5
C) 5
D) 25
B) 1/5

Question 75.
What is the value of (2-1)-3?
A) 1/2
B) 2/3
C) 23
D) 2-3
C) 23

Question 76.
What is the value of 4-1/2?
A) 1/2
B) 1/4
C) 2
D) 4
A) 1/2

Question 77.
What is the value of (32 × 3-3)-1?
A) 1/9
B) 3
C) 27
D) 81
B) 3

Question 78.
What is the value of (a2b3)4?.
A) a8b12
B) a16b12
C) a8b16
D) a16b24
A) a8b12

Question 79.
What is the value of (2a3b2)(3ab3)?
A) 6a4b5
B) 6a3b2
C) 5a4b5
D) 5a3b2
A) 6a4b5

Question 80.
What is the value of (a2)-1 (a3)-2?
A) a-8
B) a-3
C) a5
D) a3
A) a-8

Question 81.
What is the value of (x2y3)/(x5y2)?
A) x-3y
B) x-3y-1
C) x3y
D) x3y-1
A) x-3y

Question 82.
What is the value of (2x2y3z4)-2 (3xyz3)3?
A) 54x5y7z17
B) 54x4y4z6
C) $$\frac{27}{4}$$(x-1y-3z)
D) 54x4y5z9
C) $$\frac{27}{4}$$(x-1y-3z)

Question 83.
What is the value of (5a2b3c4)-3 (25abc)2?
A) 5a-4b-7c-10
B) $$\frac{1}{25}$$ a5b7c13
C) $$\frac{1}{125}$$ a7b10c7
D) $$\frac{1}{25}$$ a6b7c1
A) 5a-4b-7c-10

Question 84.
Which of the following is a rational number ?
A) 1 + √3
B ) π
C) 2√3
D) 0
D) 0

Question 85.
Every rational number is
A) a natural number
B) an integer
C) a real number
D) a whole number
C) a real number

Question 86.
A rational number lying between – 3 and 3 is
A) 0
B) -4.3
C) -3.4
D) 1.1011oo11ooo1 ………
A) 0

Question 87.
A rational number lying between √2 and √3 is
A) $$\frac{\sqrt{2}+\sqrt{3}}{2}$$
B) √6
C) 1.6
D) 1.9
C) 1.6

Question 88.
The rational number between $$\frac{-1}{5}$$ and $$\frac{-2}{5}$$ is
A) 0
B) $$\frac{-1}{4}$$
C) $$\frac{-3}{10}$$
D) $$\frac{-7}{25}$$
C) $$\frac{-3}{10}$$

Question 89.
Two rational numbers between $$\frac{2}{3}$$ and $$\frac{5}{3}$$ are
A) $$\frac{1}{6}$$ and $$\frac{2}{6}$$
B) $$\frac{1}{2}$$ and $$\frac{2}{7}$$
C) $$\frac{5}{6}$$ and $$\frac{7}{6}$$
D) $$\frac{2}{3}$$ and $$\frac{4}{3}$$
C) $$\frac{5}{6}$$ and $$\frac{7}{6}$$

Question 90.
If √x is irrational number, then x is
A) rational
B) irrational
C) 0
D) real
D) real

Question 91.
The decimal form of $$\frac{56}{1000}$$ is
A) 0.56
B) 0.056
C) 0.0056
D) 5.6
B) 0.056

Question 92.
A terminating decimal is
A) natural number
B) a whole number
C) a rational number
D) an integer
C) a rational number

Question 93.
A number is an irrational if and only if its decimal representation is
A) non-terminating
B) non-terminating and repeating
C) non-terminating and non-repeating
D) terminating
C) non-terminating and non-repeating

Question 94.
Which of the following is a rational number ?
A) √5
B) π
C) 0.101001000100001 ……………
D) 0.853853853 …………..
D) 0.853853853 …………..

Question 95.
Which one of the following is an irrational number ?
A) 0.14
B) $$0 . \overline{1416}$$
C) $$0 . \overline{1461}$$
D) 0.4014001400014….
D) 0.4014001400014….

Question 96.
Which of the following is mi irrational number ?
A) 0.15
B) $$0.15 \overline{16}$$
C) $$0 . \overline{1516}$$
D) 0.501500150001…….
D) 0.501500150001…….

Question 97.
Which of the following numbers is an irrational number ?
A) $$\sqrt{23}$$
B) $$\sqrt{225}$$
C) 0.3796
D) $$7 . \overline{478}$$
A) $$\sqrt{23}$$

Question 98.
Which of the following is an irrational number ?
A) $$3 . \overline{3}$$
B) 3.763
C) $$3 . \overline{763}$$
D) 3.101100110001 ………..
D) 3.101100110001 ………..

Question 99.
The decimal expansion of √2 is
A) finite decimal
B) 1.4121
C) non-terminating recurring
D) non-terminating non-recurring
D) non-terminating non-recurring

Question 100.
π is
A) a rational number
B) an integer
C) an irrational number
D) a whole number
C) an irrational number

Question 101.
An irrational number between $$\frac{5}{7}$$ and $$\frac{7}{9}$$ is
A) 0.75
B) √6
C) 0.750750075000….
D) 0.7512
C) 0.750750075000….

Question 102.
The value of $$2 . \overline{9}$$ in the form $$\frac{\mathbf{p}}{\mathbf{q}}$$, where p and q are integers and q ≠ 0 is
A) $$\frac{2999}{1000}$$
B) $$\frac{19}{10}$$
C) 3
D) $$\frac{26}{9}$$
C) 3

Question 103.
The sum of $$0 . \overline{3}$$ and $$0 . \overline{2}$$ is
A) $$\frac{5}{99}$$
B) $$\frac{5}{9}$$
C) $$\frac{5}{10}$$
D) $$\frac{5}{100}$$
B) $$\frac{5}{9}$$

Question 104.
The process of visualisation of representation of numbers on the number line through a magnifying glass is called
A) successive magnification
B) approximation
C) imagination
D) none of these
A) successive magnification

Question 105.
$$5.3 \overline{7}$$ lies most accurately
A) between 5.37 and 5.38
B) between 5-3 and 5.4
C) between 5.377 and 3.378
D) between 5.3777 and 5.3778
D) between 5.3777 and 5.3778

Question 106.
(√a + √b) (√a – √b) is
A) a + b
B) a – b
C) 2√a
D) 2√b
B) a – b

Question 107.
(a + √b) (a – √b) is equal to
A) b2 – a2
B) a2 – b2
C) a2 – b
D) b2 – a
C) a2 – b

Question 108.
Which of the following numbers is an irrational number ?
A) $$\sqrt{16}$$ – 4
B) (3 – √3) (3 + √3)
C) √5 + 3
D) –$$\sqrt{25}$$
C) √5 + 3

Question 109.
(- 2 – √3) (- 2 + √3) when simplified is
A) positive and irrational
B) positive and rational
C) negative and irrational
D) negative and rational
B) positive and rational

Question 110.
(5 + √8) + (3 – √2) – (√2 – 6) when simplified is
A) positive and irrational
B) negative and irrational
C) positive and rational
D) negative and rational
C) positive and rational

Question 111.
If x = $$\frac{\sqrt{7}}{5}$$ and $$\frac{5}{x}$$ = P√7 , then the value of P is
A) $$\frac{5}{\sqrt{7}}$$
B) $$\frac{25}{7}$$
C) $$\frac{7}{25}$$
D) $$\frac{\sqrt{7}}{5}$$
B) $$\frac{25}{7}$$

Question 112.
(√2 + $$\frac{1}{\sqrt{7}}$$)2 is equal to
A) 4√2
B) $$\frac{9}{2}$$
C) $$\frac{4}{\sqrt{12}}$$
D) 9
B) $$\frac{9}{2}$$

Question 113.
$$\sqrt{12}$$ × √8 is equal to
A) 2√6
B) 3√6
C) 4√6
D) 6√6
C) 4√6

Question 114.
Value of $$\frac{1}{\sqrt{18}-\sqrt{32}}$$ is equal to
A) √2
B) -√2
C) $$\frac{1}{\sqrt{2}}$$
D) $$\frac{-1}{\sqrt{2}}$$
D) $$\frac{-1}{\sqrt{2}}$$

Question 115.
If √3 = 1.732 and √2 = 1.414, the value of $$\frac{1}{\sqrt{3}-\sqrt{2}}$$ is
A) 0.318
B) 3.146
C) $$\frac{1}{3.146}$$
D) $$\sqrt{1.732}$$ – $$\sqrt{1.414}$$
B) 3.146

Question 116.
Rationalisation of the denominator of $$\frac{1}{\sqrt{5}+\sqrt{2}}$$ gives
A) $$\frac{1}{\sqrt{10}}$$
B) √5 + √2
C) √5 – √2
D) $$\frac{\sqrt{5}-\sqrt{2}}{3}$$
D) $$\frac{\sqrt{5}-\sqrt{2}}{3}$$

Question 117.
If b > 0 and b2 = a, then √a is equal to
A) -b
B) b
C) √b
D) b2
B) b

Question 118.
If a = 2 and b = 3, then the value of ba is
A) 4
B) 9
C) 2
D) 3
B) 9

Question 119.
The value of $$\frac{2^0 \times 7^0}{5^0}$$ is
A) 1
B) 0
C) $$\frac{9}{5}$$
D) $$\frac{1}{5}$$
A) 1

Question 120.
The value of $$\frac{2^0+7^0}{5^0}$$ is
A) 2
B) 0
C) $$\frac{9}{5}$$
D) $$\frac{1}{5}$$
A) 2

Question 121.
If xa/b = 1, then the value of ‘a’ is
A) 0
B) 1
C) -1
D) None of these
A) 0

Question 122.
When 15$$\sqrt{15}$$ is divided by 3√3, the quotient is
A) 5√3
B) 3√5
C) 5√5
D) 3√3
C) 5√5

Question 123.
The value of $$\sqrt[3]{216}-\sqrt[3]{125}$$ is
A) 1
B) 0
C) 2
D) -1
A) 1

Question 124.
Simplified value of (25)1/3 × (5)1/3 is
A) 25
B) 3
C) 1
D) 5
D) 5

Question 125.
(0.001)1/3 is equal to
A) 0.1
B) 0.001
C) 0.01
D) 0.0001
A) 0.1

Question 126.
The value of $$\sqrt[4]{\sqrt[3]{2^2}}$$ is equal to
A) 2-1/6
B) 2-6
C) 21/6
D) 26
C) 21/6

Question 127.
If 8x = $$\frac{64}{2^x}$$, then the value of x is
A) 3
B) 1
C) $$\frac{1}{2}$$
D) $$\frac{3}{2}$$
D) $$\frac{3}{2}$$

Question 128.
Value of $$\left[\left[(81)^{-1 / 2}\right]^{-1 / 4}\right]^2$$ is
A) 3
B) $$\frac{1}{3}$$
C) 9
D) $$\frac{1}{9}$$
A) 3

Question 129.
Which of the following is not correct ?
A) 0.25 is terminating decimal
B) 1.3333… is non terminating recurring decimal
C) 1.121231234 . is non terminating re-curring decimal .
D) 1.234… is non terminating and non-recurring decimal
C) 1.121231234 . is non terminating re-curring decimal .

Question 130.
Classify the following numbers as rational and irrational.
a) $$\sqrt{32}$$
b) 7.4777777…
Solution:
a) $$\sqrt{32}$$ = $$\sqrt{16 \times 2}$$ = 4√2 Irrational number.
b) 7.477…, Rational and Non-terminat-ing recurring decimal.

Question 131.
Simplest rationalising factor of √8 is ………….
A) √8
B) √2
C) 2√2
D) 2√3
B) √2

Assertion and Reason Type Questions :

Question 1.
Assertion : Every integer is a rational number.
Reason: A rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero.
True, because every integer can be expressed as a fraction in the form p/1, where p is an integer.

Question 2.
Assertion : The sum of two rational numbers is always a rational number.
Reason : The sum of two rational numbers can be expressed as a fraction in the form p/q, where p and q are integers and q is not equal to zero.
True, because the sum of two fractions can be expressed as a fraction in the form (p1q2 + p2q1) / (q1q2), where p1, p2, q1 and q2 are integers and q1 and q2 are not equal to zero.

Question 3.
Assertion : The product of two rational numbers is always a rational number.
Reason : The product of two rational numbers can be expressed as a fraction in the form p/q, where p and q are integers and q is not equal to zero.
True, because the product of two fractions can be expressed as a fraction in the form (p1p2) / (q1q2), where p1, p2, q1 and q2 are integers and q1 and q2 are not equal to zero.

Question 4.
Assertion : The reciprocal of a non-zero rational number is also a rationed number.
Reason : The reciprocal of a non-zero rational number can be expressed in the form q/p, where p and q are integers and q is not equal to zero.
True, because the reciprocal of a fraction in the form p/q is q/p, which can also be expressed as a fraction with integers in the numerator and denominator.

Question 5.
Assertion: The quotient of two rational’ numbers is always a rational number, provided the denominator of the second rational number is non-zero.
Reason : The quotient of two rational numbers can be expressed in the form p/q, where p and q are integers and q is not equal to zero.
True, because the quotient of two fractions can be expressed as a fraction in the form (p1q2) / (q1p2), where p1, p2, q1 and q2 are integers and q1 and q2 are not equal to zero.

Question 6.
Assertion : The square root of 2 is an irrational number.
Reason : The square root of 2 cannot be expressed as a fraction of two integers.
True, because the square root of 2 cannot be expressed as a fraction of two integers and therefore cannot be written in the form p/q where p and q are integers and q is not equal to zero.

Question 7.
Assertion : The sum of an irrational number and a rational number is always an irrational number.
Reason : The sum of a rational number and an irrational number cannot be expressed in the form p/q, where p and q are integers and q is not equal to zero.
True, because the sum of a rational number and an irrational number is never a rational
number, as the sum cannot be expressed in the form p/q, where p and q are integers and q is not equal to zero.

Question 8.
Assertion : The product of an irrational number and a non-zero rational number is always an irrational number.
Reason : The product of an irrational number and a non-zero rational number cannot be expressed in the form p/q, where p and q are integers and q is not equal to zero.
True, because the product of an irrational number and a non-zero rational number cannot be expressed as a fraction in the form p/q where p and q are integers and q is not equal to zero.

Question 9.
Assertion : The square of an irrational number is always irrational.
Reason : If a number is rational, then its square root is either rational or irrational.
True, because if the square of a number is rational, then the number itself is rational, which is a contradiction to the assumption that the number is irrational.

Question 10.
Assertion : The difference of two irrational numbers is not necessarily an irrational number.
Reason: The difference of two irrational numbers can be expressed in the form p/q, where p and q are integers and q is not equal to zero.
False, because the difference of two irrational numbers can be irrational. For example, the difference between √2 and $$\frac{\sqrt{2}}{2}$$ is $$\frac{(\sqrt{2})}{2}$$, which is irrational.