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Practice the AP 10th Class Maths Bits with Answers Chapter 5 Quadratic Equations on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 5th Lesson Quadratic Equations with Answers

Question 1.
If x² – px + q = 0(p,q∈R and p ≠ 0, q ≠ 0) has distinct real roots, then write
the condition.
Answer: p² > 4q.

Question 2.
If one root of 2x² + kx – 6 is 2., then find k.
Answer:
k = – 1
Explanation:
2(2)² + k(2) – 6 = 0
⇒ 8 + 2k – 6 = O
⇒ 2k + 2 = 0 ⇒ k = -1

Question 3.
If the equation x² + 5x + k = 0 has real and distinct roots, then find the value of ‘k’.
Answer:
k > 6.25
Explanation:
Real and distinct roots so,
b² – 4ac > 0
⇒ 25 – 4 . 1. k > 0
⇒ 25 > 4k = k > \(\frac{25}{4}\) > 6.25

Question 4.
Frame a quadratic equation, whose roots are 2 + \(\sqrt{3}\) and 2 – \(\sqrt{3}\) ?
Answer:
x² – 4x + 1 = 0
Explanation:
x² – (2 + \(\sqrt{3}\) +2 – \(\sqrt{3}\))x + (2 + \(\sqrt{3}\)) (2 – \(\sqrt{3}\))
⇒ x² – 4x + 1 = 0

Question 5.
In a quadratic equation ax² + bx + c = 0, if b² – 4ac > 0, then write the nature of the roots.
Answer:
Roots are real and distinct.

Question 6.
Create the quadratic equation, whose zeroes are \(\sqrt{2}\) and – \(\sqrt{2}\) ?
Answer:
x² – 2 = 0.
Explanation:
\(x^{2}-(\sqrt{2}-\sqrt{2}) x+(\sqrt{2})(-\sqrt{2})=0\)
⇒ x² – 2 = 0

Question 7.
For which positive value of x the qua-dratic equation 4x\(\sqrt{3}\) -9 = 0 satisfies ?
Answer:
\(\frac{3}{2}\)

Question 8.
If the roots of x² + 6x + 5 = 0 are a and P, then find the value of sum of the roots.
Answer:
-6
Explanation:
α + β = \(\frac{-b}{a}\) = — 6

Question 9.
Write the discriminant of 6x² – 5x + 1 = 0.
Answer:
D = 1
Explanation:
D = b² – 4ac = 25 – 4 . 6 . 1
⇒ 25 – 24 = 1 > 0 D = 1

Question 10.
Write the quadratic polynomial having \(\frac { 1 }{ 3 }\) and \(\frac { 1 }{ 2 }\) as its zeroes.
Answer:
x² – \(\frac{5 x-1}{6}\) = 0 ⇒ 6x² – 5x + 1 = 0
Explanation:

Question 11.
If a number is 132 smaller than its square, then find the number.
Answer:
12
Explanation:
x + 132 = x²
⇒ x² – x – 132 = 0
By solve the equation, ∴ x = 12

Question 12.
Write the general form of a quadratic equation in variable ‘x’.
Answer:
ax² + bx + c = 0 (a ≠ 0).

Question 13.
Make the quadratic polynomial, whose zeroes are 2 and 3.
Answer:
x² – 5x + 6.

Question 14.
If α, β are the roots of x² – 10x + 9 = 0, thep find the value of | α – β |.
Answer:
8
Explanation:
x² – 9x – x + 9 = 0
⇒ x(x – 9) – 1 (x – 9) = 0
⇒ (x-9)(x- 1) = 0
x = 9 and 1, |α – β| = |9- 1| = 8

Question 15.
Write the discriminant of adjacent dia-gram indicates.

Answer:
b² – 4ac > 0.

Question 16.
If the roots of a quadratic equation px² + qx + r = 0 are imaginary, then write the condition of discriminant.
Answer:
q² < 4pr (or) q² – 4pr < 0.

Question 17.
Two angles are complementary. If the large angle is twice the measure of a smaller angle, then find the value of smaller angle.
Answer:
30°
Explanation:
x + y = 90°
⇒ x + 2x = 90°
⇒ 3x = 90° ⇒ x = 30°

Question 18.
Observe the following graphs.

Which as them are the graphs of qua-dratic polynomials ?
Answer:
(i) and (iv).

Question 19.
Write the possible number of roots to a quadratic equation.
Answer:
At a maximum of 2.

Question 20.
If 1 is a common root of ax² + ax + 2 = 0 and x² + x + 6 = 0, then find a-b.
Answer:
2

Question 21.
Find the product of roots of quadratic equation ax² + bx + c = 0.
Answer:
\(\frac{\text { c }}{\text { a }}\)

Question 22.
Write the number of diagonals in a polygon, having ‘n’ sides.
Answer:
\(\frac{n(n-3)}{2}\)

Question 23.
Find the discriminant of quadratic equation 2x² + x – 4 = 0.
Answer:
33

Question 24.
A quadratic equation ax² + bx + c = 0 has two distinct real roots, then write the condition.
Answer:
b² – 4ac >0.

Question 25.
Draw the shape of quadratic equation which having distinct roots ?
Answer:

Question 26.
The sum of a number and its reciprocal is \(\frac { 5 }{ 2 }\) then find the number.
Answer:
2 or \(\frac { 1 }{ 2 }\)
Explanation:
x + \(\frac{1}{x}=\frac{5}{2}\)
⇒ \(\frac{x^{2}+1}{x}=\frac{5}{2}\)
⇒ 2x² + 2 = 5x
⇒ 2x² – 5x + 2 = 0
⇒ 2x² – 4x r x + 2 = 0
⇒ 2x(x – 2) – 1 (x – 2) – 0 ⇒ (x – 2) (2x – 1) = 0 1
∴ x = 2 or 1/2.

Question 27.
Find the roots of the equation 4x² – 4\(\sqrt{3}\) x + 3 = 0.
Answer:
\(-\frac{\sqrt{3}}{2}\)

Question 28.
Find the positive root of \(\sqrt{3 x^{2}+6}=9\)
Answer:
5
Explanation:
3x² + 6 = 81
⇒ 3x² = 81 – 6 = 75
⇒ x² = \(\) = 25 ⇒ x = 5

Question 29.
Find the roots of the quadratic equation (7x – 1) (2x + 3) = 0.
Answer:
\(\frac{1}{7}, \frac{-3}{2}\)

Question 30.
If the sum of the squares of two con-secutive odd numbers is 74, then find the smaller number.
Answer:
5 (or)-7
Explanation:
(2x + 1)² + (2x + 3)² – 74
⇒ 4x² + 4x + 1 + 4x² + 12x + 9 — 74′
⇒ 8x² + 16x + 10 = 74
⇒ 8x² + 16x – 64 = 0
⇒ 8(x² + 2x – 8) = 0
⇒ x² + 4x – 2x – 8 = 0
⇒ x(x + 4) – 2 (x + 4) = 0
⇒ x = – 4, 2
∴ x = – 4, then smaller number
= 2 . (-4) + 1 = -8 + 1 = -7
∴ x = 2, then smaller number
= 2 . (2) + 1 = 4 + 1 = 5

Question 31.
Write the standard form of a cubic polynomial.
Answer:
ax³ + bx² + cx + d = 0; (a ≠ 0).

Question 32.
Write the discriminant of 5x²– 3x – 2 = 0.
Answer:
49

Question 33.
Create the quadratic equation whose roots are – 2 and – 3.
Answer:
x² + 5x + 6 = 0

Question 34.
Find the roots of the quadratic equation \(\frac{x^{2}-8}{x^{2}+20}=\frac{1}{2}\)
Answer:
±6
Explanation:
2x² – 16 = x² + 20
⇒ x² – 36 ⇒ x = ±6.

Question 35.
Find the roots of the equation 3x² – 2\(\sqrt{6}\) x + 2 = 0.
Answer:
\(\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}\)

Question 36.
Find the roots of the quadratic equa- tion \(\left(x-\frac{1}{3}\right)^{2}\) = 9.
Answer:
\(\frac { 10 }{ 3 }\) (or) \(\frac { -8 }{ 3 }\).
Explanation:

Question 37.
On solving x² + 5 = – 6x, find the value of ‘x’
Answer:
– 1 or – 5.

Question 38.
Simplified form of \(\frac{\mathbf{x}}{\mathbf{x}-\mathbf{y}}-\frac{\mathbf{y}}{\mathbf{x}+\mathbf{y}}\)
Answer:
\(\frac{x^{2}+y^{2}}{x^{2}-y^{2}}\)

Question 39.
Find the sum of roots of bx² + ax + c = 0.
Answer:
\(\frac{-a}{b}\)

Question 40.
Find the roots of 2x² – x + \(\frac { 1 }{ 8 }\) = 0.
Answer:
\(\frac{1}{4}, \frac{1}{4}\)

Question 41.
If x + \(\frac{1}{x}\) = 2, then find \(x^{2}+\frac{1}{x^{2}}\).
Answer:
2
Explanation:
x + \(\frac { 1 }{ x }\) = 2
⇒ \(x^{2}+\frac{1}{x^{2}}\) + 2 = 4 ⇒ x² + \(x^{2}+\frac{1}{x^{2}}\) = 2

Question 42.
If 3y² — 192, then find ‘y’.
Answer:
y = ± 8

Question 43.
How many diagonals has a pentagon?
Answer:
’9′

Question 44.
If α and β are the roots of the quadratic equation x² – 3x + 1 = 0, then find \(\left(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}\right)\)
Answer:
7

Question 45.
If \(\mathbf{a}^{\mathbf{x}^{2}-4 \mathbf{x}+3}\) = 1, then find x (a # 0).
Answer:
1 or 3.

Question 46.
Find discriminant of the quadratic equation x + \(\frac { 1 }{ x }\) = 3.
Answer:
5

Question 47.
Create the quadratic equation with 2 < x < 3.
Answer:
x² – 5x + 6 < 0.
Explanation:
x² – (2 + 3)x + 2 . 3 < 0
⇒ x² – 5x + 6 < 0

Question 48.
p(x) = x² + 2x + 1, then find p(x²).
Answer:
x⁴ + 2x² + 1

Question 49.
x² – 7x – 60 = 0, then find ‘x’.
Answer:
12 and -5.

Question 50.
\(\frac{1}{a+3}+\frac{1}{a-3}+\frac{6}{9-a^{2}}\) is equal to ?
Answer:
\(\frac{2}{a+3}\)

Question 51.
Find the roots of \(\sqrt{2} x^{2}+7 x+5 \sqrt{2}\) = 0.
Answer:
\(\frac{-5}{\sqrt{2}} \text { or }-\sqrt{2}\)

Question 52.
Find the roots of a quadratic equation \((\sqrt{2} x+3)(5 x-\sqrt{3})=0\)
Answer:
\(\frac{-3}{\sqrt{2}}, \frac{\sqrt{3}}{5}\)

Question 53.
4x² + ky – 2 = 0 has no real roots, then find ‘k’.
Answer:
k < – \(\sqrt{32}\)

Question 54.
The sum of a number and its reciprocal is \(\frac { 50 }{ 7 }\), then find the number.
Answer:
7 (or) \(\frac { 1 }{ 7 }\)
Explanation:
x + \(\frac{1}{x}=\frac{50}{7}\)
⇒ \(\frac{x^{2}+1}{x}=\frac{50}{7}\)
⇒ 7x² + 7 — 50x
⇒ 7x² – 50x + 7 — 0
⇒ 7x² – 49x – x + 7 = 0
⇒ 7x (x – 7) – 1 (x – 7) – 0
⇒ (x – 7) (7x – 1) – 0
=+ x = 7 (or) 1/7

Question 55.
Find the roots of the quadratic equation \(\frac{9}{x^{2}-27}=\frac{25}{x^{2}-11}\)
Answer:
±6;

Question 56.
Write the nature of the roots of a qua-dratic equation 4x² – 12x + 9 = 0.
Answer:
Real and equal.

Question 57.
3x² + (- k)x + 8 = 0 has no real roots, then find k’.
Answer:
k < 4\(\sqrt{6}\)
Explanation:
No real roots. So D < 0,
(-k)² – 4 . 3 . 8 < 0
⇒ k² – 96 < 0
⇒ k² < 96
⇒ k < \(\sqrt{96}\)
⇒ k < \(4 \sqrt{6}\)

Question 58.
Find the discriminant of 3x² – 2x = \(\frac{-1}{3}\).
Answer:
D = 0

Question 59.
Find the product of the roots of 1 =x².
Answer:
-1

Question 60.
x(x + 4) = 12, then find ‘x’.
Answer:
– 6 or 2.

Question 61.
Form a quadratic equation from, x³ – 4x² – x + 1 = (x – 2)³.
Answer:
2x² – 13x + 9 = 0.
Explanation:
x³ – 4x² – x + 1 = x³ – 3 . x² . 2 + 3 . x . 2² – 2³
⇒ x³ – 4x² – x + 1 = x³ – 6x² + 12x – 8
⇒ x³ – 4x² – x + 1 = x³ + 6x² – 12x + 8 = 0
⇒ 2x² – 13x + 9 = 0

Question 62.
Find the product of the roots of x² + 7x = 0.
Answer:
0

Question 63.
\(\frac{2 a^{2}+a-1}{a+1}+\frac{3 a^{2}+5 a+2}{3 a+2}+\frac{4-a^{2}}{a+2}\) is equal to ?
Answer:
2 (a +1)

Question 64.
1 and \(\frac { 3 }{ 2 }\) are the roots of which qua-dratic equation ?
Answer:
2x² – 5x + 3 = 0.

Question 65.
If b² < 4ac, then draw the shape of graph.
Answer:

Question 66.
\(\sqrt{\mathbf{k}+\mathbf{1}}\) = 3, then find ‘k’.
Answer:
k = 8

Question 67.
\(\sqrt{x}=\sqrt{2 x-1}\), then find ‘x’.
Answer:
x = 1

Question 68.
If \(\frac{1}{x-2}+\frac{2}{x-1}=\frac{6}{x}\) then find ‘x’
Answer:
3 or \(\frac { 4 }{ 3 }\)

Question 69.
Find the coefficient of ‘x’ in a pine qua-dratic equation.
Answer:
0

Question 70.
Write number of distinct line segments that can be formed out of n – points.
Answer:
\(\frac{n(n-1)}{2}\)

Question 71.
The product of two consecutive positive integers is 306, then find the largest number.
Answer:
18
Explanation:
x(x + 1) = 306 ⇒ x² + x – 306 = 0
by solving thik Q.E., x = 17
∴ Largest number = x + 1
= 17 + 1 = 18.

Question 72.
Write the nature of roots of 3x² + 13x – 2 = 0.
Answer:
Real and unequal.

Question 73.
If α and β are the roots of x² – 2x + 3 = 0, then find α² + β²
Answer:
α² + β² = – 2.

Question 74.
If (2x – 1) (2x + 3) = 0, then find ‘x’.
Answer:
\(\frac { 1 }{ 2 }\) or \(\frac { -3 }{ 2 }\)

Question 75.
Write the quadratic equation whose one root is 2 – \(\sqrt{3}\) .
Answer:
x² – 4x + 1 = 0

Question 76.
If b² – 4ac > 0, then write nature of the roots of the quadratic equation.
Answer:
Real and distinct.

Question 77.
Find product of the roots of ax² + bx + c = 0. c
Answer:
c/a

Question 78.
Write the nature of the roots of a qua-dratic equation 4x² + 5x + 1 = 0.
Answer:
Real and distinct.

Question 79.
Write the quadratic equation whose roots are – 1,6.
Answer:
x² – 5x – 6 = 0.

Question 80.
Create the quadratic equation whose roots are – 3 and – 4.
Answer:
x² + 7x 4- 12 = 0.

Question 81.
Find the roots of the quadratic equation (3x + 4)² – 49 = 0.
Answer:
1, \(\frac{-11}{3}\)

Question 82.
If x² – 2x + 1 = 0, then find x + \(\frac{1}{x}\).
Answer:
2

Question 83.
Write nature of the roots of 5x² – x + 1 = 0.
Answer:
Imaginary roots.

Question 84.
Write the nature of the roots of qua-dratic equation 3x² + x + 8 = 0.
Answer:
Imaginary roots.

Question 85.
Find product of the roots of the qua-dratic equation 3x² – 6x + 11 = 0.
Answer:
\(\frac{11}{3}\)

Question 86.
Form a quadratic equation whose roots are k and 1/k.
Answer:
x² – (\(\mathrm{k}+\frac{1}{\mathrm{k}}\))x + 1 = 0

Question 87.
If k² – 8kx + 16 = 0 has equal roots, then find the value of ‘k’.
Answer:
k = ± 1.
Explanation:
(-8k)² – 4(1) (16) = 0
⇒ 64k² = 64 ⇒ k² = 1 ⇒ k = ±1

Question 88.
If the roots of a quadratic equation ax² + bx + c = 0 are real and equal, then find ‘b²‘.
Answer:
4ac

Question 89.
3(x – 4)² – 5(x – 4) = 12, then find ‘x’.
Answer:
7 (or) 8/3.
Explanation:
3(x – 4)² – 5 (x – 4) = 12
3[x² + 16 – 8x] – 5x + 20 — 12
3x² + 48 – 24x – 5x + 20 – 12 — 0
⇒ 3x² – 29x + 56 = 0
⇒ 3x² – 21x – 8x + 56 — 0
⇒ 3x (x – 7) – 8 (x – 7) = 0
⇒ (x – 7) (3x – 8) = 0
⇒ x = 7 (or) x = \(\frac { 8 }{ 3 }\)

Question 90.
If a and pare the roots of x² + x + 1 = 0, then find α² + β².
Answer:
α² + β² = – 1.

Question 91.
\(\frac{1-\frac{1}{1+x}}{\frac{1}{1+x}}\) is equal to ?
Answer:
x

Question 92.
Find sum of the roots of a pure quadratic equation.
Answer:
0

Question 93.
\(\frac{\mathbf{x}}{\mathbf{a}-\mathbf{b}}=\frac{\mathbf{a}}{\mathbf{x}-\mathbf{b}}\) , then find ‘x’.
Answer:
b – a (or) – a

Question 94.
\(\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}\) x ≠ -4 x or 7 find x’.
Answer:
2 or 1

Question 95.
(1 – 5x) (9x +1) is equal to ?
Answer:
1 + 4x – 5x².

Question 96.
From the figure, find ’x’.

Answer:
± 10
Explanation:
By Pythagoras theorem,
x² = 6² + 8² = 64 + 36 = 100
x = \(\sqrt{100}\) = ± 10

Question 97.
Find the sum of the roots of the equation 3x² – 7x + 11 = 0.
Answer:
7/3

Question 98.
Find the roots of the quadratic equation \((\sqrt{5} x-3)(\sqrt{5} x-3)\) – 0.
Answer:
\(\frac{3}{\sqrt{5}}, \frac{3}{\sqrt{5}}\)

Question 99.
Write the nature of the roots of the quadratic equation \(\sqrt{3} x^{2}-2 x-\sqrt{3}\).
Answer:
Real and distinct.

Question 100.
If 5x² – kx + 11 = 0 has root x = 3, then find ’k’.
Answer:
k = \(\frac{56}{3}\)
Explanation:
5(3)² – k(3) + 11 = 0
⇒ 45 + 11 – 3k = 0
⇒ 56 – 3k = 0
⇒ 3k = 56 ⇒ k = \(\frac { 56 }{ 3 }\)

Question 101.
Find the value of ‘p’ for which 4x² – 2px + 7 = 0 has a real roots.
Answer:
p > 2\(\sqrt{7}\)

Question 102.
If one root of a quadratic equation is 7 – 7\(\sqrt{3}\) , then find the quadratic equation.
Answer:
x² – 14x + 46 = 0.

Question 103.
If b² – 4ac = 0, then write nature of the roots of the quadratic equation.
Answer:
Real and equal.

Question 104.
Find sum of the roots of ax² + bx + c = 0.
Answer:
\(-\frac{b}{a}\)

Question 105.
If the equation x² – kx + 1 = 0 has equal roots, then find the value of ‘k’.
Answer:
k = ± 2
Explanation:
b² – 4ac = (- k)² – 4 . 1 . 1 = 0
⇒ k² – 4 = 0
⇒ k² = 4 ⇒ k = \(\sqrt{4}\) = ± 2.

Question 106.
Find (he product of the roots of the qua-dratic equation \(\sqrt{2} \mathrm{x}^{2}-3 \mathrm{x}+5 \sqrt{2}\) = 0.
Answer:
5

Question 107.
Write the nature of roots of 3x² + 6x – 2 = 0.
Answer:
Real and distinct.

Question 108.
If the sum of the roots of the quadratic equation 3x² + (2k + 1)x – (k + 5) = 0 is equal to the product of the roots, then find the value of k.
Answer:
4
Explanation:
Sum of the roots = product of the roots
⇒ \(\frac{-(2 k+1)}{3}=\frac{-(k+5)}{3}\)
⇒ – 2k- 1 = -k – 5 ⇒ k = 4

Question 109.
Find the product of zeroes in the above equation.
Answer:
\(\frac{-11}{5}\)

Question 110.
Find the degree of any quadratic equation.
Answer:
2

Question 111.
In the quadratic equation
x² + x – 2 = 0, find the value of a + b + c.
Answer:
a + b + c = 0.

Question 112.
Find the value of \(\left(x+\frac{1}{x}\right)^{2}-\left(y+\frac{1}{y}\right)^{2}-\left(x y-\frac{1}{x y}\right) \cdot\left(\frac{x}{y}-\frac{y}{x}\right)\)
Answer:
0

Question 113.
Form a quadratic equation from x(2x + 3) = x² + 1.
Answer:
x² + 3x – 1 = 0.
Explanation:
2x² + 3x = x² + 1
⇒ x² + 3x – 1 = 0

Question 114.
(x – α) (x – β) = 0, then find the product.
Answer:
x² – (α + β)x + αβ = 0.

Question 115.
If α and β are die roots of x² – 5x + 6 = 0, then find the value of α – β.
Answer:
± 1.

Question 116.
For what values of m’ are the roots of the equation mx² + (m + 3)x + 4 = 0 are equal ?
Answer:
9 or 1.
Explanation:
(m + 3)² – 4 . m . 4 = 0
⇒ (m + 3)² – 16m = 0
⇒ m² + 9 + 6m- 16m = 0
⇒ m² – 10m + 9 = 0
⇒ m² – 9m – m + 9 = 0
⇒ m(m – 9) – 1 (m – 9) = 0
∴ m = 9 or 1

Question 117.
Find the roots of 2x² + x – 4 = 0.
Answer:
x = \(\frac{-1 \pm \sqrt{33}}{4}\)

Question 118.
If kx (x – 2) + 6 = 0 has equal roots, then find k’.
Answer:
k = 6.
Explanation:
kx² – 2kx + 6 = 0
⇒ (2k)² – 4 . k . 6 = 0
⇒ 4k² – 24k = 0
⇒ 4k (k – 6) = 0 ⇒ k = 6

Question 119.
If ‘2’ is a root of x² + 5x + r = 0, then find ‘r’.
Answer:
r = -14

Question 120.
(α + β)² – 2αβ is sequal to ……………
Answer:
α² + β²

Question 121.
Find the value of \(\sqrt{\mathbf{a}+\sqrt{\mathbf{a}+\sqrt{\mathbf{a + \ldots \ldots \infty}}}}\)
Answer:
\(\frac{1+\sqrt{1+4 a}}{2}\)

Question 122.
If the sum of the roots of kx² – 3x + 1 = 0 is \(\frac{-4}{3}\) then find ‘k’.
Answer:
\(\frac{-9}{4}\)
Explanation:
\(\frac{3}{\mathrm{k}}=\frac{-4}{3} \Rightarrow \frac{3 \times 3}{-4}=\mathrm{k} \Rightarrow \mathrm{k}=\frac{-9}{4}\)

Question 123.
\(\frac{n(n+1)}{2}\) = 55, then find ‘n’
Answer:
10
Explanation:
⇒ n² + n = 110 = 0
⇒ n² + n – 110 = 0
⇒ n = 10

Question 124.
If ‘α’ is β root of ax² + bx + c = 0, then find aα² + bα + c.
Answer:
0

Question 125.
If α and β are the roots of the quadratic equation 2x² + 3x – 7 = 0, then find \(\frac{\alpha^{2}+\beta^{2}}{\alpha \beta}\)
Answer:
\(\frac{-37}{14}\)

Question 126.
Find the sum of the roots of -7x + 3x² – 1 = 0.
Answer:
\(\frac{7}{3}\)

Question 127.
Find the roots of a quadratic equation \(\frac{\mathbf{x}}{\mathbf{p}}=\frac{\mathbf{p}}{\mathbf{x}}\)
Answer:
x = p
Explanation:
x² = p² ⇒ x = p

Question 128.
If (x – 3) (x + 3) = 16, then find the value of ‘x’.
Answer:
± 5.

Question 129.
Write the roots of a quadratic equation ax² + bx + c = 0.
Answer:
x = \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

Question 130.
Find the sum of the roots of the quadratic equation 5x² + 4\(\sqrt{3}\)x – 11 = 0.
Answer:
\(\frac{-4 \sqrt{3}}{5}\)

Question 131.
If one root of x² – (p – 1)x + 10 = 0 is 5, then find ‘p’.
Answer:
7
Explanation:
5² – (p – 1) 5 + 10 = 0
⇒ 25 + 10 – 5p + 5 = 0
⇒ 35 = 5p ⇒ p = 7

Question 132.
If one root of x2 – x + k = 0 is square of other, then find ‘k’.
Answer:
k = cube of one root
Explanation:
α = x, β = x²
Product of roots = αβ = \(\frac{\mathrm{c}}{\mathrm{a}}\)
⇒ x.x² = k ⇒ k = x³
k is cube of the first root.

Question 133.
If α, β are the roots of x2 – px + q = 0, then find α³ + β³.
Answer:
p³ – 3pq

Question 134.
x² + (x + 2)² = 290, then find ‘x’.,
Answer:
11 or – 13

Question 135.
Find the value of \(\sqrt{\mathbf{a} \sqrt{\mathbf{a} \sqrt{\mathbf{a}} \ldots \ldots \infty}}\)
Answer:
a

Question 136.
If \(\frac{-7}{3}\) is a root of 6x² – 13x – 63 = 0, then find other root.
Answer:
\(\frac{9}{2}\)

Question 137.
If b²2 – 4ac < 0, then write nature of the roots of the quadratic equation.
Answer:
Imaginary roots.

Choose the correct answer satistying the following statements.

Question 138.
Statement (A) : The equation x² + 3x + 1 = (x – 2)² is a quadratic equation.
Statement (B) : Any equation of the form ax2 + bx + c = 0 where a ± 0, is called a quadratic equation.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
We have, x² + 3x + 1 = (x – 2)²
⇒ x² + 3x + 1 = x² – 4x + 4
⇒ 7x – 3 = 0, it is not of the form ax² + bx + c = 0
So, A is incorrect but B is correct.
Hence (iii) is the correct option.

Question 139.
Statement (A) : The roots of the qua-dratic equation x² + 2x + 2 = 0 are imaginary.
Statement (B) : If discriminant D = b² – 4ac < 0, then the roots of quadratic equation ax² + bx + c = 0 are imaginary.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
x² + 2x + 2 = 0
∴ Discriminant, D = b² – 4ac
= (2)² – 4 x 1 x 2
= 4 – 8 = -4 < 0
∴ Roots are imaginary.
So, both A and B are correct and B explains Answer: Hence, (i) is the correct option.

Question 140.
Statement (A) : The value of k = 2, if one root of the quadratic equation
6x² – x – k = 0 is \(\frac{2}{3}\)
Statement (B) : The quadratic equation ax² + bx + c = 0, a ≠ 0 has two roots.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
As one root is \(\frac{2}{3}\) ⇒ x = \(\frac{2}{3}\)

So, both A and B are correct but B does not explain Answer:
∴ Hence, (i) is the correct option.

Question 141.
Statement (A) : The equation 8x² + 3kx + 2 = 0 has equal roots, then the value of k is ± \(\frac{8}{3}\).
Statement (B) : The equation ax² + bx + c = 0 has equal roots if D = b² – 4ac = 0.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
8x² + 3kx + 2 = 0
∴ Discriminant, D = b² – 4ac
= (3k)² – 4 x 8 x 2
= 9k² – 64
For equal roots, D = 0
⇒ 9k² – 64 = 0
⇒ 9k² = 64
⇒ k² = \(\frac { 64 }{ 9 }\)
⇒ 9k² = ±\(\frac { 8 }{ 3 }\)
So, A and B both correct and B explains Answer: Hence, (i) is the correct option.

Question 142.
Statement (A) : The values of x are \(\frac{-a}{2}\), a for a quadratic equation 2x² + ax – a² = 0.
Statement (B) : For quadratic equation ax² + bx + c = 0.
x = \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
2x² + ax – a² =0

So, A is incorrect but B is correct. Hence, (iii) is the correct option.

Question 143.
Statement (A) : The equation (x – p) (x – r) + λ(x – q) (x – s) = 0, p < q < r < s, has non-real roots if λ > 0.
Statement (B) : The equation ax² + bx + c = 0, a, b,c ∈ R, has non-real roots if b² – 4ac < 0.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
Statement (A):
Let f(x) = (x – p) (x – r) + λ(x – q) (x- s)
f(p) = λ(p – q) (p – s)
f(q) = (q – p) (q – r)
f(s) = (s – p) (s – r)
f(r) = λ(r – q) (r – s)
If λ > 0, then f(p) > 0, f(q) < 0, f(r) < 0 and f(s) > 0.
⇒ f(x) = 0 has one real root between p and q and other real root between r and s.
Statement – B is obviously true. Option (iii) is correct.

Question 144.
Statement (A) : If roots of the equation x² – bx + c = 0 are two consecutive integers, then b² – 4c = 1.
Statement (B) : If a, b, c are odd integer, then the roots of the equation 4abc x² + (b² – 4ac)x – b = 0 are real and distinct.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
Statement (A) : Given equation , x² – bx + c = 0.
Let α, β be two roots such that |α – β| = 1. .
⇒ (α + β)² – 4αβ = 1.
⇒ b² – 4c = 1
Statement (B): Given equation
4abc x² + (b² – 4ac) x – b = 0
D = (b² – 4ac)² + 16 ab² c
D = (b² – 4ac)² > 0
Hence roots are real and unequal. Option (ii) is correct.

Question 145.
Statement (A) : If 1 ≤ a ≤ 2, then \(\sqrt{a+2 \sqrt{a-1}}+\sqrt{a-2 \sqrt{a-1}}=2\)
Statement (B) : If 1 ≤ a ≤ 2, then (a – 1) > 1.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
If 1 ≤ a ≤ 2 ⇒ 0 ≤ a- 1 ≤ 1

Statement – A is true.
Statement – B is false.
Option – (ii) is correct.

Question 146.
Statement (A): If one root is \(\sqrt{3}-\sqrt{2}\), then the equation of lowest degree with rational coefficients x⁴ – 10x² + 1 = 0.
Statement (B): For a polynomial equa-tion with rational coefficient irrational roots occurs in pairs.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
x = \(\sqrt{3}-\sqrt{2}\), x² = 5 – 2\(\sqrt{6}\)
(x² – 5)² = 24
x⁴ – 10x² + 25 = 24
x⁴ – 10x² + 1 = 0
For polynomial equation with rational coefficients irrational roots occurs in pairs.
Option (i) is correct.

Question 147.
Statement (A): Degree of the polynomial 5x² + 3x + 4 is 2.
Statement (B) : The degree of a poly-nomial of one variable is the highest value of the exponent of the variable.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Read the below passages and answer to the following questions.

Let us consider a quadratic equation x² + 3ax + 2a² = 0.
If the above equation has roots α,β and it is given that α² + β² = 5.

Question 148.
Find value of ‘a’.
Answer:
±1.
Explanation:
α + β = – 3a; αβ = 2a²
a² + p² = 5
⇒ (α + β)² – 2αβ = 5
⇒ (- 3a)² – 2(2a²) = 5
⇒ 9a² – 4a² = 5
⇒ 5a² = 5 ⇒ a = ± 1

Question 149.
Find value of ‘D’ for the above qua-dratic equation.
Answer:
D > 0.
Explanation:
D = (3a)² – 4(2a²)
= 9a² – 8a² = a² = 1 > 0

Question 150.
Find the product of roots.
Answer:
2
Explanation:
αβ = 2a² = 2(1) = 2

Let us consider a quadratic equation x² + λx + λ + 1.25 = O, where λ is a constant. The value of A such that the above quadratic equation has

Question 151.
Two distinct roots.
Answer:
λ > 5 or λ < – 1.
Explanation:
The equation has two distinct roots if b² – 4ac > 0.
∴ (λ – 5)(λ + 1) > 0
⇒ Either λ – 5 > 0 (or) λ + 1 > 0
⇒ λ > 5 (or) λ > -1
∴ λ > 5
⇒ λ – 5 <0 (or) λ + 1 < 0
⇒ λ < 5 (or) λ < – 1
∴ λ < -1 Hence the given equation has two dis-tinct roots for λ > 5 (or) λ < – 1

Question 152.
Two coincident roots.
Answer:
λ = 5 or λ = -1.
Explanation:
The equation has two coincident roots if b² – 4ac = 0
⇒ (λ – 5) (λ + 1) = 0
⇒ Either λ – 5 = 0 (or) λ = 5
⇒ λ + 1 = 0
⇒ λ = – 1
⇒ λ = 5 or – 1
Hence the given equation has coincident roots for λ = 5 or – 1.

The area of a rectangular plot is 528 m². The length of the plot is one more than twice its breadth.

Question 153.
Which mathematical concept is used to find area of above plot ?
Answer:
Quadratic equation.

Question 154.
Write the breadth and length of above given plot.
Answer:
Let breadth = x m, length = 2x + 1 m.

Question 155.
Write the equation of area of above given plot.
Answer:
Area = length x breadth
= x(2x + 1) – 2x² + x = 528 m².

The hypotenuse of a right triangle is 25 cm. We know that the difference in lengthof the other two sides is 5 cm.

Question 156.
Write the lengths of smaller and larger sides.
Answer:
Smaller side = x m
Larger side = (x + 5) cm.

Question 157.
Write the hypotenuse of the triangle.
Answer:
x² + (x + 5)² = (25)²
i.e., x² + 5x – 300 = 0

Question 158.
Which mathematical concept is used to find out the values of dimensions ?
Answer:
Quadratic equations.

Question 159.
Column -II give roots of quadratic equations given in column – I, match them correctly.

Answer:
A – (iv), B – (ii).

Question 160.
Column – II give roots of quadratic equations given in column -1, match them correctly.

Answer:
A – (i), B – (iii).

Question 161.
Write the correct matching.

Answer:
A – (ii), B – (iv).

Question 162.
Write die correct matching.

Answer:
A – (iii), B – (i).

Question 163.
Column – II give pair at two numbers for solution to problems given in column -I. Match them correctly.

Answer:
A – (iv), B – (ii).

Question 164.
Column – II give pair at two numbers for solution to problems given in column -I.
Match them correctly.

Answer:
A – (i), B – (iii).

Question 165.
D is the discriminait of the quadratic equation ax² + bx + e = 0.

Answer:
A – (ii), B – (i).

Question 166.
D Is the discriminant of the quadratic equation ax² + bx + c = O.

Answer:
A – (ii), B – (i).

Question 167.
Write a quadratic equation with roots 3 and 4.
Answer:
x² – 7x + 12 = 0

Question 168.
Draw the rough graph of the quadratic equation ax² + bx + c = 0, when b² – 4ac < 0.
Answer:

Practice the AP 10th Class Maths Bits with Answers Chapter 12 Applications of Trigonometry on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 12th Lesson Applications of Trigonometry with Answers

Question 1.
The ratio of the length of a rod and its shadow is 1 : √3 . Then find the angle of elevation of the sun.
Answer:
30°
Explanation:

tan θ = \(\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{1}{\sqrt{3}}\) = tan 30°
∴ θ = 30°

Question 2.
Find the angle ‘θ’ in the figure.

Answer:
30°
Explanation:
sin θ = \(\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{2}{4}=\frac{1}{2}\) = sin 30°
∴ θ = 30°

Question 3.
Find the angle made by the minuteshand in a clock during a period of 20 minutes.
Answer:
120°
Explanation:
Angle made by minutes hand in 1 minute is 6°.
Angle made by minutes hand in 20 minutes is = 20 × 6 = 120°

Question 4.
If the angle of elevation of Sun is 45°, then find the length of the shadow of a 12 m high tree.
Answer:
12 m
Explanation:
tan 45° = \(\frac{\text { height of tree }}{\text { shadow of tree }}\)
⇒ Shadow of tree = 12 m

Question 5.
In the given figure, find measurement of BC.

Answer:
7√3 cm
Explanation:
From figure, tan 30° = \(\frac{\mathrm{AB}}{\mathrm{BC}}\)
⇒ \(\frac{1}{\sqrt{3}}=\frac{7}{\mathrm{BC}}\) ⇒ BC = 7√3 cm

Question 6.
A boy observed 20 m away from the base of a 20 m high pole, find the angle of elevation of the top.
Answer:
45°
Explanation:
tan θ = \(\frac{20}{20}\) = 1 = tan 45° ⇒ θ = 45°

Question 7.
The length of shadow of a pole is equal to the length of the pole, then find the angle of the elevation of the Sun.
Answer:
45°

Question 8.
Ladder ‘x’ metres long is laid against a wall making an angle ‘θ’ with the ground. If we want to directly find the distance between the foot of ladder and foot of the wall, which trigonometrical ratio should be considered ?
Answer:
cos θ

Question 9.
Top of a building was observed at an angle of elevation ‘α’ from a point, which is at distance’d’ metres from the foot of the building. Which trigonometrical ratio should be considered for finding height of buildings?
Answer:
tan α

Question 10.
If the angle of elevation of sun in¬creases from 0° to 90°, then find the length of shadow of the tower.
Answer:
Decreases
Explanation:
Sine value decreases from 0° to 90°. So length of shadow of the tower also decreases.

Question 11.
A ladder touches a wall at a height of 5 m. Find the angle made by the ladder with the ground, if its length is 10 m.
Answer:
30°
Explanation:

sin θ = \(\frac{5}{10}=\frac{1}{2}\)= sin 30°
∴ θ = 30°

Question 12.
x = (sec θ + tan θ), y = (sec θ – tan θ), then find xy.
Answer:
1

Question 13.
From the top of a building with height 30( √3 +1 )m two cars make angles of depression of 45° and 30° due east. What is the distance between two cars?
Answer:
60 m

Question 14.
At a point 15 m away from the base of a 15 m high pole, find the angle of elevation of the top.
Answer:
45°

Question 15.
If cosec θ + cot θ = k, then find cos θ.
Answer:
\(\frac{\mathrm{k}^{2}-1}{\mathrm{k}^{2}+1}\) = cos θ

Question 16.
A pole 6 m high casts a shadow 2√3 m long on the ground, then find the sun’s elevation.
Answer:
60°
Explanation:

tan θ = \(\frac{6}{2 \sqrt{3}}\) = √3 = tan 60°
∴ θ = 60°

Question 17.
A tower is 50 m high. Its shadow is x m shorter when the sun’s altitude is 45°, then when it is 30°, then find x.
Answer:
100 cm

Question 18.
Suppose you are shooting an arrow from the top of a building at a height of 6m to a target on the ground at an angle of depression of 60°. What is the distance between you and the object?
Answer:
3√3 m

Question 19.
What change will be observed in the angle of elevation as.we move away from the object?
Answer:
Angle decreases.

Question 20.
In the given figure, the positions of the observer and the object are marked, find the angle of depression.

Answer:
60°

Question 21.
An object is placed above the observer’s horizontal, we call the angle between the line of sight and observer’s horizontal.
Answer:
Angle of elevation.

Question 22.
x = a sin θ, y = a cos θ, then find x² + y².
Answer:
a²

Question 23.
If AB = 4m and AC = 8 m, then find the angle of elevation of A as observed from C.
Answer:
30°
Explanation:

sin C = \(\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{4}{8}=\frac{1}{2}\) = sin 30°
∴ C = 30°

Question 24.
The given figure shows the observation of point ‘C’ from point A. Find the angle of depression from A.

Answer:
30°
Explanation:
tan C = θ = \(\frac{4}{4 \sqrt{3}}=\frac{1}{\sqrt{3}}\) = tan = 30°
∴ θ = 30°

Question 25.
Find the angle of elevation of tower at a point 40 m apart from it is cot^-1(\(\frac{3}{5}\)) . Obtain the height of the tower.
Answer:
\(\frac{200}{3}\) m
Explanation:

Question 26.
A ladder of 10 m length touches a wall at a height of 5 m. Find the angle made by it with the horizontal.
Answer:
30°

Question 27.
The ratio of length of a pole and its shadow is 1: √3 . Find the angle of elevation.
Answer:
30°

Question 28.
A wall of 8m long casts a shadow 5m long. At the same time, a tower casts a shadow 50 m long, then find the height of tower.
Answer:
80 m

Question 29.
In the below figure, find x.

Answer:
x = 10 m

Question 30.
An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°. After a flight of 10 seconds, its angle of elevation is observed to be 30° from the same point on the ground. Find the speed of the aeroplane.
Answer:
415.7 km/h

Question 31.
A building casts a shadow of length 50 √3m when the sun is 30° about the horizontal. Find the height of the building.
Answer:
50 m

Question 32.
A ladder 20 m long is placed against a vertical wall of height 10 m, then find the distance between the foot of the ladder and the wall.
Answer:
10√3 m

Question 33.
In the figure given below, if AB = 10 m and AC = 20 m, then find θ.

Answer:
30°

Question 34.
Find the length of the shadow of a tree is 8m long when the sun’s angle of elevation is 45°.
Answer:
8

Question 35.
Find the length of the string of a kite flying at 100m above the ground with the elevation of 60°.
Answer:
\(\frac{200}{\sqrt{3}}\)

Question 36.
In the figure given below, if AB = 10√3 m, then find CD. (take √3 = 1.732).

Answer:
7.32 m

Question 37.
From a bridge 25 m high, the angle of depression of a boat is 45°. Find the horizontal distance of the boat from the bridge.
Answer:
25 m

Question 38.
If the shadow of a tree is \(\frac{1}{\sqrt{3}}\) times the height of the tree, then find the angle of elevation of the sun.
Answer:
60°

Question 39.
A player sitting on the top of a tower of height 40m observes the angle of depression of a ball lying on the ground is 60°. Find the distance between the foot of the tower and ball.
Answer:
\(\frac{40}{\sqrt{3}}\)m

Question 40.
The length of the shadow of a tree is 7m high, when the sun’s elevation is
Answer:
45°
Explanation:
Length and shadow of a tree is same.
So sun’s elevation is 45°.

Question 41.
Write any one example of a Pythagorean triplet.
Answer:
5, 12, 13
Explanation:
5, 12, 13 (or) 3, 4, 5 (or) 7,24,25

Question 42.
When the angle of elevation of a light changes from 30° to 45°, the shadow of pole becomes 100 √3 m less. Find the height of the pole.
Answer:
100m

Question 43.
Find the elevation of the sun at the moment when the length of the shadow of a tower is just equal to its height.
Answer:
45°

Question 44.
The height of a tower is 10m. Find the length of its shadow when sun’s altitude is 45°.
Answer:
10 m

Question 45.
If the height and length of the shadow of a man are the same, then find the angle of elevation of the sun.
Answer:
45°

Question 46.
A boy observed the top of an electrical pole to be at angle of elevation of 60° when the observation point is 8m away from the foot of the pole, then find the height of the pole.
Answer:
8√3m

Question 47.
When the length of the shadow of a person is equal to his height, then find the elevation of source of light.
Answer:
45°

Question 48.
From the top of a building 50m from horizontal, the angle of depression made by a car is 30°. How far is the car from the building?
Answer:
50√3m

Question 49.
What change will be observed in the angle of elevation as we approach the foot of the tower ?
Answer:
Angle decreases.

Question 50.
The length of the shadow of a tower on the plane ground is √3 times the height of the tower. Find the angle of elevation of sun.
Answer:
30°

Question 51.
A pole of 12m high casts a shadow 4 √3 m on the ground, then find the sun’s angle of elevation.
Answer:
60°

Question 52.
Angle of elevation of the top of a build-ing from a point on the ground is 30°. Then find the angle of depression of this point from the top of the building.
Answer:
30°

Question 53.
In the figure given below, a man on the top of cliff observes a boat coming towards him. Then θ represents the angle of …………..

Answer:
Depression

Question 54.
The angle of elevation of a cloud from a point 200 m above the take is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud above the lake.
Answer:
400 m

Question 55.
In a rectangle, if the angle between a diagonal and a side is 30°, and the length of the diagonal is 6cm, find the area of the rectangle.
Answer:
9√3 cm²
Explanation:

In ΔABC, sin 30°= \(\frac{\mathrm{BC}}{\mathrm{AC}}\) ⇒ \(\frac{1}{2}=\frac{\mathrm{BC}}{6}\) ⇒ BC = 3 cm,
cos 30° = \(\frac{\mathrm{BC}}{\mathrm{AC}}\) ⇒ \(\frac{\sqrt{3}}{2}=\frac{\mathrm{AB}}{6}\)
⇒ AB = 3√3 cm.
∴ Area of rectangle = AB × BC
= 3√3 × 3 = 9√3 cm²

Question 56.
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance between the top of the tree and the ground is 10m. Find the height of the tree.
Answer:
10√3 m

Question 57.
The angle of depression of the top of a tower at a point 100m from the house is 45°, then find the height of the tower.
Answer:
36.6 m

Question 58.
In the given figure, find the value of angle θ.

Answer:
30°

Question 59.
If the angle of elevation of a tower from a distance of 100m from its foot is 60°, then find the height of the tower.
Answer:
100√3 m

Question 60.
In the figure given below, if AD = 7 √3 m, then find BC.

Answer:
28 m

Question 61.
If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then find the length of tangent.
Answer:
3√3 cm

Question 62.
An electric pole 20 m high stands up right on the ground with the help of steel wire to its top and affixed on the ground. If the steel wire makes 60° with the horizontal ground, find the length of steel wire.
Answer:
\(\frac{20}{\sqrt{3}}\)m

Question 63.
When the angle of elevation of a pole is 45°, what do you say about the length of the pole and its shadow.
Answer:
Both are equal.

Question 64.
If the ratio of height of a tower and the’length of its shadow on the ground is √3 : 1, then find the angle of elevation of the sun.
Answer:
60°

Question 65.
The ratio of the length of a rod and its shadow is 1 : √3, then find the angle of elevation of the sun.
Answer:
30°

Question 66.
In the figure given below, if AB = CD = 10√3m, then find BC.

Answer:
40 m
Explanation:
From ΔABM, tan 30° = \(\frac{\mathrm{AB}}{\mathrm{BM}}\)
\(\frac{1}{\sqrt{3}}=\frac{10 \sqrt{3}}{\mathrm{BM}}\)
⇒ BM = 30m

From Δ CDM, tan 60°= \(\frac{\mathrm{CD}}{\mathrm{MC}}\)
√3 = \(\frac{10 \sqrt{3}}{\mathrm{MC}}\) ⇒ MC = 10m
∴ BC = BM + MC
= 30 + 10
= 40 m

Question 67.
The angle of elevation of top of a tree is 30°. On moving 20m nearer, the angle of elevation is 60°. Find the height of the tree.
Answer:
10√3 m

Question 68.
Two posts are 15m and 25m high and the line joining their tops make an angle of 45° with the horizontal. Find the distance between the two posts.
Answer:
10m

Question 69.
If a pole 6m high casts a shadow, 2 √3 m long on the ground, then find the sun’s angle of elevation.
Answer:
60°

Question 70.
If the length of the shadow of a tower \(\frac{1}{\sqrt{3}}\) is times the height of the tower, then find the angle of elevation of the sun.
Answer:
60°

Question 71.
In the figure given below, the imaginary line through the object and eye of the observer is called

Answer:
Line of sight

Question 72.
A tower makes an angle of elevation equal to the angle of depression from the top of a cliff 25 m high. Find the height of the tower.
Answer:
50 m

Question 73.
If two towers of height X and Y subtend angles of 30° and 60° respectively at the centre of the line joining their feet, then find X : Y.
Answer:
1 : 3

Question 74.
An object is placed below the observer’s horizontal, then what is the angle between line of sight and observer’s horizontal?
Answer:
Angle of depression

Question 75.
The upper part of a tree is broken by wind and makes an angle of 30° with the ground and at a distance of 21 m from the foot of the tree. Find the total height of the tree.
Answer:
21√3 m

Question 76.
If the sun’s angle of elevation is 60°, then a pole of height 6 m, then find cast a shadow of length.
Answer:
2√3 m

Question 77.
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°. When he retires 40 in from the bank, he finds the angle to be 30°. Find the breadth of the river.
Answer:
20 m

❖ Choose the correct answer satisfying the following statements.
Question 78.
Statement (A) : If the below figure, if BC = 20m, then height AB is 11.56 m.

Statement (B) :
tan θ = \(\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{\text { perpendicular }}{\text { base }}\)
where θ is the ∠ACB.
(i) Both A and B are true
(ii) A istrue, B is false
(iii) A is false, B is true
(iv) Both A and B are false
Answer:
(i) Both A and B are true
Explanation:
Both A and B are correct, B is the correct explanation of the A.
tan30° = \(\frac{\mathrm{AB}}{\mathrm{BC}}\) ⇒ \(\frac{1}{\sqrt{3}}=\frac{A B}{20}\)
AB = \(\frac{1}{\sqrt{3}}\) × 20 = \(\frac{20}{1.73}\) = 11.56 m.
So, option (i) is correct.

Question 79.
Statement (A) : If the length of shadow of a vertical pole is equal to its height, then the angle of elevation of the sun is 45°.
Statement (B) : According to Pythagoras theorem, h² = l² + b², where h = hypotenuse, l = Length and b = base.
(i) Both A and B are true
(ii) A istrue, B is false
(iii) A is false, B is true
(iv) Both A and B are false
Answer:
(i) Both A and B are true
Explanation:
Both A and B are correct, but B is not the correct explanation of the A.
So, option (i) is correct.

❖ Read the below passages and answer to the following questions.
From the top of a tower, the angles of depresssion of two objects on the same side of the tower are found to be α and β where α > β.

Question 80.
If the distance between the objects is ‘P’ metres, then find the height ’h’ of the tower.
Answer:
\(\frac{P \tan \alpha \tan \beta}{\tan \alpha-\tan \beta}\)
Explanation:

Height of the tower (AB) = h m
Distance (CD) = P m
Let distance (BC) = x m
∠ACB = α and∠ADB = β
In right ΔABC,\(\frac{\mathrm{AB}}{\mathrm{BC}}\) = tan α
⇒ \(\frac{\mathrm{h}}{\mathrm{x}}\) = tan α ⇒ h = x tan α …………….. (i)
In right ΔABD,\(\frac{\mathrm{AB}}{\mathrm{BD}}\) = tan β
⇒ \(\frac{\mathrm{h}}{\mathrm{BC}+\mathrm{CD}}\) = tan β
⇒ h = (x + P) tan β …………..(ii)
From (i), we get x = \(\frac{\mathrm{h}}{\tan \alpha}\)
Hence, h = \(\frac{P \cdot \tan \alpha \cdot \tan \beta}{\tan \alpha-\tan \beta}\) proved.

Question 81.
Find the height of the tower if P = 150 m, α = 60° and β = 30°.
Answer:
130 m
Explanation:
Putting P = 150 m, α = 60° and β = 30°, we get
h = \(\frac{150 \times \tan 60^{\circ} \times \tan 30^{\circ}}{\tan 60^{\circ}-\tan 30^{\circ}}\) m
= 129.9m ≅ 130m

Question 82.
The distance of the extreme object from the top of the tower is
Answer:
260 m
A straight highway leads to the foot of a tower of height 50m. From the top of the tower, the angles of depression of two cars standing on the highway are 30° and 60°.

Explanation:
sin β = \(\frac{\mathrm{h}}{\mathrm{y}}\)
⇒ y = \(\frac{h}{\sin \beta}=\frac{130}{\sin 30^{\circ}}\) = 260m

Question 83.
Find the distance between the cars.
Answer:
57.7 m
Explanation:
From ΔABD, tan 60° = \(\frac{\mathrm{AB}}{\mathrm{BD}}\)
⇒ √3 = \(\frac{50}{\mathrm{BD}}\) ⇒ BD = \(\frac{50}{\sqrt{3}}\)
From ΔABC, tan 30° = \(\frac{\mathrm{AB}}{\mathrm{BC}}\)
⇒ \(\frac{1}{\sqrt{3}}=\frac{50}{\mathrm{BC}}\) ⇒ BC = 50√3
BC = BD + DC CD ⇒ BC – BD
= 50√3 = \(\frac{50}{\sqrt{3}}=\frac{150-50}{\sqrt{3}}=\frac{100}{\sqrt{3}}\) = 57.7 m

Question 84.
Find the distance between the second car from the tower.
Answer:
86.60 m
Explanation:
BC = 50 × 1.732 = 86.60 m

Question 85.
Which trignometrical concept was used to solve the given problem?
Answer:
Tangent and cosine.
The angle of elevation of a ladder against a wall is 60° and the foot of the ladder is 9.6 m from the wall.

Question 86.
Find the length of the ladder.
Answer:
19.2 m
Explanation:
From ΔOAD, cos 60° = \(\frac{\mathrm{OA}}{\mathrm{DA}}\)
⇒ \(\frac{1}{2}=\frac{9.6}{\mathrm{DA}}\) ⇒ DA = 9.6 × 2 = 19.2m

Question 87.
Which trigonometrical concept was used to solve the problem?
Answer:
cos θ
Observe the below figure and answer to the following questions.

Question 88.
In the above figure θ₁ is called
Answer:
Angle of elevation

Question 89.
In the above figure θ₂ is called
Answer:
Angle of depression

Question 90.
θ₁ and θ₂ are measured from where?
Answer:
Always horizontal line.

Question 91.
What is the relation between θ₁ and θ₂?
Answer:
Both θ₁ = θ₂
Write the correct matching options.

Question 92.
From a window, ‘h’m high above the ground, of a house in a street, the angles of elevation and depression of the top and bottom of another house on the opposite side of the street are a and P respectively, then match the column.

Answer:
A – (iv), B – (iii), C – (i), D – (ii)

Question 93.

Answer:
A – (ii), B – (iv).

Question 94.

Answer:
A – (i), B – (iii).
Explanation:
(A) cos θ = \(\frac{\mathrm{BC}}{\mathrm{AC}}=\frac{20}{40}=\frac{1}{2}\) = cos 60°
∴ θ = 60°

(B) In ΔABC, tan 45° = \(\frac{\mathrm{AB}}{\mathrm{BC}}\)
⇒ 1 = \(\frac{\mathrm{AB}}{2}\)
⇒ AB = 2
InΔABD, tan θ = \(\frac{\mathrm{AB}}{\mathrm{BD}}=\frac{2}{10}=\frac{1}{5}\)

Question 95.
A tower of height 100√3m casts a shadow of length 10073 m then what is the angle of elevation of the sun at that time?
(OR)
In the given figure, what is the value of angle θ?

Solution .
In ΔABC
tan θ = \(\frac{\mathrm{BC}}{\mathrm{AB}}\) ⇒ tan θ = \(\frac{100}{100 \sqrt{3}}=\frac{1}{\sqrt{3}}\)
θ = 30°

Question 96.
Name the ‘angle of depression’ from the figure given below in which
∠B = 90°.

Answer:
∠DAC

Practice the AP 10th Class Maths Bits with Answers Chapter 4 Pair of Linear Equations in Two Variables on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 4th Lesson Pair of Linear Equations in Two Variables with Answers

Question 1.
Pair of equations 4x + 6y = 7 and 2x + 3y = 8. How many solutions have ?
Answer:
No solution.

Question 2.
Find the point where the line 2x – 3y = 8 intersects X – axis ?
Answer:
(4,0)
Explanation:stitute y = 0 in 2x – 3y = 8
⇒ 2x – 3-0 = 8 ⇒ x = 4
∴ The point on X – axis = (4, 0).

Question 3.
Find the solution for the equations \(\sqrt{3} x+\sqrt{5 y}=0\) and \(\sqrt{7} \mathrm{x}+\sqrt{11} \mathrm{y}=0\)
Answer:
x = 0, y = 0.

Question 4.
Find the value of ‘x’ in the equation 3x- (x-4) = 3x + 1.
Answer:
3
Explanation:
3x – x + 4 = 3x + 1
⇒ 3x – 3x — x = 1 – 4
⇒ x = 3

Question 5.
The pair of equations a₁x + b₁y + C₁ = 0 and a₂x + b₂y + c₂ = 0 are consistent, then write the condition for that.
Answer:
\(\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}} \neq \frac{\mathrm{b}_{1}}{\mathrm{~b}_{2}}\) and \(\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{~b}_{2}}=\frac{\mathrm{c}_{1}}{\mathrm{c}_{2}}\)

Question 6.
The graph y = ax + b is a straight line, find the point x where it intersects x – axis ?
Answer:
(\(-\frac{b}{a}\), 0)

Question 7.
Find the value of ‘k’ for which the sys-tem of equations kx – y = 2 and
6x – 2y = 3 has no solution.
Answer:
3
Explanation:
\(\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{~b}_{2}}\) when pair of equations has no solution.
\(\frac{\mathrm{k}}{6}=\frac{1}{2} \Rightarrow \mathrm{k}=\frac{6}{2}\) = 3

Question 8.
Find the point of intersection of x + y = 6 and x – y = 4.
Answer:
(5,1)
Explanation:
x + y = 6 and
x-y = 4 ⇒ x = 4 + y
4 + y + y = 6 ⇒ 2y = 2 ⇒ y = 1
x = 6 – y = 6 – 1 ⇒ x = 5 .
∴ Point of intersection (x, y) = (5, 1).

Question 9.
If pair of equations 6x + 2y – 9 = 0 and kx + y – 7 = 0 has no solution, then find ‘k’.
Answer:
3
Explanation:
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \Rightarrow \frac{6}{k}=\frac{2}{1} \Rightarrow \frac{6}{2}\) = k ⇒ k = 3

Question 10.
If the pair of equations 2x+3y+k = 0, 6x + 9y + 3 = 0 having infinite solu-tions, find the value of ‘k’.
Answer:
1
Explanation:
Infinite solutions, so

Question 11.
A pair of linear equations in two variables are 2x – y = 4 and 4x – 2y = 6. The pair of equations are
Answer:
Inconsistent.
Explanation:
\(\frac{2}{4}=\frac{1}{2} \neq \frac{4}{6} \Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
∴ Pair of equations are inconsistent.

Question 12.
How many solutions to the pair of equations y = 0 and y = – 3 ?
Answer:
No solution.

Question 13.
If 7x – 8y = 9, then find ‘y’.
Answer:
\(\frac{7 x-9}{8}\)

Question 14.
Write the standard form of a linear equation.
Answer:
ax + by + c = 0

Question 15.
For which value of ‘k’ will the follow-ing pair of linear equations have no solution 3x + y = 1;
(2k- l)x + (k – l)y = 2k – 1 ?
Answer:
2
Explanation:
No, solution, so \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\)
⇒ \(\frac{3}{2 k-1}=\frac{1}{k-1}\)
⇒ 3(k – 1) = 1 (2k – 1)
⇒ 3k – 3 = 2k – 1 ⇒ 3k – 2k = 3 – 1 ⇒ k = 2

Question 16.
The lines 3x + 8y – 13 = 0 and – 6x – 16y + 23 = 0 are type of lines.
Answer:
Parallel
Explanation:
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{3}{-6}=\frac{8}{-16} \neq \frac{-13}{23}\)
∴ Parallel lines.

Question 17.
2x + 3y = 1, 3x – y = 7, then find (x, y).
Answer:
(2,-1)

Question 18.
Where the line x – y = 8 intersects X – axis ?
Answer:
(8,0)

Question 19.
x + \(\frac { 6 }{ y }\) = 6 and 3x – \(\frac { 8 }{ y }\) = 5, then find ‘y’.
Answer:
2

Question 20.
Write the nature of the graph of the line y = 5x.
Answer:
The graph of the line passes through the origin.
Explanation:
y – 5x ⇒ y = mx passes through the origin.

Question 21.
Pair of linear equations px + 3y – (p – 3) = 0, 12x + py – p = 0 has infinitely many solutions, then find p’.
Answer:
± 6.
Explanation:
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} \Rightarrow \frac{p}{12}=\frac{3}{p}=\frac{(p-3)}{p}\)
p² = 36 ⇒ p = \(\sqrt{36}\) = ± 6

Question 22.
In which quadrant (2, 0) belongs ?
Answer:
Q₁

Question 23.
x + y = 10, x – y = – 4, then find ‘x’.
Answer:
3
Explanation:
x – y = -4 ⇒ x = y – 4 = y – 4 + y= 10
⇒ 2y – 4 = 10 ⇒ 2y = 14 ⇒ y = 7,
x – 7 = -4 ⇒ x = 7 – 4 = 3
∴ x = 3

Question 24.
Find the solution of the equations \(\sqrt{2} x+\sqrt{3} y=0\) and \(\sqrt{3} x-\sqrt{8} y=0\).
Answer:
x = 0, y = 0 (or) (0, 0)

Question 25.
If 3x + 4y + 2 = 0and9x + 12y + k = 0 represent coincident lines, then find the value of ‘k’.
Answer:
x = 0, y = 0 (or) (0.0)
Explanation:

Question 26.
If ax + b = 0, then find ‘x’. b
Answer:
\(-\frac{b}{a}\)

Question 27.
If x + y = 7, x – y = 1, then find 2x.
Answer:
8
Explanation:
x – y = 1 ⇒ x = 1 + y and
x + y = 7 ⇒ 1 + y + y = 7
⇒ 1 + 2y = 7 ⇒ 2y = 6 ⇒ y = 3,
x = 1 + 3 = 4
∴ 2x = 2(4) = 8

Question 28.
If x – y = 0; 2x – y = 2, then find the value of ‘y’.
Answer:
2

Question 29.
How many solutions to the pair of lin-ear equations 3x + 4y + 5 = 0 and 12x + 16y +15 = 0 have ?
Answer:
No solution. They are parallel lines.

Question 30.
Find the solution to \(\frac{a^{2}}{x}-\frac{b^{2}}{y}=0\) ; \(\) = a + b, x ≠ 0, y ≠ 0
Answer:
(a², b²)

Question 31.
If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then find the value of ‘k’.
Answer:
k = \(\frac{15}{4}\)

Question 32.
The lines represented by 5x + 3y – 7 = 0 and 6y + 10x – 14 = 0 are type …………… of lines.
Answer:
Coincident
Explanation:
\(\frac{5}{10}=\frac{3}{6}=\frac{7}{14} \Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
So given pair of equations are coinci-dent lines.

Question 33.
Find slope of the line x = 2y.
Answer:
Slope (m) = \(\frac{1}{2}\)
Explanation:
x = 2y ⇒ y = \(\frac { 1 }{ 2 }\) x ⇒ m = \(\frac { 1 }{ 2 }\)

Question 34.
If x = 1 and y = \(\frac{1}{2}\), then find x – y.
Answer:
\(\frac{3}{2}\)
Explanation:
x – y = 1 – \(\frac { -1 }{ 2 }\) = 1 + \(\frac { 1 }{ 2 }\) = \(\frac { 3 }{ 2 }\)

Question 35.
Find the value of x if y = \(\frac{3}{4}\) x and 5x + 8y = 33.
Answer:
x = 3

Question 36.
Write the slope of X – axis.
Answer:
y = 0

Question 37.
Find the value of y in – 5x + 10y =100 at x = 0.
Answer:
y = 10

Question 38.
2u + 3v = 2 and 4u – 6v = 0, then find ‘v’.
Answer:
\(\frac{1}{3}\)

Question 39.
If – x + y = – 10, then write ‘x’ as sub-ject.
Answer:
x = y + 10
Explanation:
-x + y = -10 ⇒ x = y + 10

Question 40.
Write shape of the graph of 3x-y = -1.
Answer:
Straight line.

Question 41.
The pair of linear equations
px + 2y = 5 and 3x + y = 1 has unique solution, then find value of’p’.
Answer:
p ≠ 6.
Explanation:
\(\frac{p}{3} \neq \frac{2}{1}\) ⇒ p ≠ 6

Question 42.
The lines represented by 5x + 7y – 14 = 0 and 10x + 3y-8 = 0 are……………….lines.
Answer:
Consistent.

Question 43.
3x – 5y = – 1 and – y + x = – 1, then find (x, y).
Answer:
(-2,-1).

Question 44.
How many solutions to the pair of lin-ear equations – 3x + 4y = 7 and
\(\frac { 9 }{ 2 }\) x – 6y + \(\frac { 21 }{ 2 }\) = 0 ?
Answer:
Infinitely many solutions.
Explanation:

So infinitely, many solutions.

Question 45.
For which value of ‘p’ the lines repre-sented by 8x + 2py = 2 and 2x + 5y + 1 = 0 are parallel ?
Answer:
10
Explanation:
\(\frac{8}{2}=\frac{2 p}{5}\) ⇒ P = \(\frac{8 \times 5}{2 \times 2}\) ⇒ P = 10

Question 46.
If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\), then the lines are …………………. type of lilies.
Answer:
Coincident lines (or) dependent lines.

Question 47.
4m – 2n = 2 and 6m – 5n = 9, then find ‘n’.
Answer:
n = – 3.

Question 48.
If 99x + lOly = 499,101x+ 99y = 510, then find ‘x’.
Answer:
x = 3

Question 49.
Find the solution to x – y = 1 and 2x – 2y = 7.
Answer:
No solution (or) not possible.
Explanation:
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \Rightarrow \frac{1}{2}=\frac{1}{2}\),
so no solution.

Question 50.
141x + 93y = 189, 93x + 141y = 45, then find y’.
Answer:
y = -1

Question 51.
\(\frac{120}{x}+\frac{12}{x}\) = 11, then find ‘x’. x x
Answer:
x = 12.

Question 52.
Find the solution to 2x – 2y – 2 = 0, 4x – 4y – 5 = 0.
Answer:
No solution (or) not possible.

Question 53.
500x + 240y = 8, 130x + 240y = \(\frac{43}{10}\)
then find the value of ‘x’.
Answer:
\(\frac{1}{100}\)

Question 54.
Find the value of ‘y’ when \(\frac{x+y}{x y}\) = 2 and \(\frac{x-y}{x y}\) = 6.
Answer:
y = \(\frac { 1 }{ 4 }\)

Question 55.
Where the two lines 2x + y – 6 = 0 and 4x – 2y – 4 = 0 intersect, find that intersecting point.
Answer:
(2,2)

Question 56.
\(\frac{x+3}{2}-y=2, \frac{x-3}{2}+2 y=4 \frac{1}{2}\) then find ‘x’
Answer:
\(\frac { 14 }{ 3 }\)

Question 57.
How much the angle between any two parallel lines ?
Answer:
0°

Question 58.
Find the values of ’k’ for which the pair of linear equations 3x – 2y = 7, and 6x + ky + 11 = 0 has a unique solution.
Answer:
All numbers expect “’-“4r are the solution.

Question 59.
Find slope of the line y = x.
Answer:
m = 1

Question 60.
If x = 1, then find the value of’y’ satis- 5 3
fying the equation \(\frac{5}{x}+\frac{3}{y}\) = 6 ;
Answer:
y = 3.
Explanation:
\(\frac{5}{1}+\frac{3}{y}\) = 6 ⇒ \(\frac{3}{y}\) = 6 – 5 = 1 ⇒ y = 3

Question 61.
Write the point (- 3, – 8) is in the…………………quadrant.
Answer:
Q₃

Question 62.
Write the point (7, -5) is in the quadrant.
Answer:
Q₄

Question 63.
Find slope of the line ax + by + c = 0.
Answer:
m = \(\frac{-a}{b}\)

Question 64.
Write the nature of the line x = 2020.
Answer:
Slope not defined and it parallel to Y – axis.

Question 65.
Write the nature of the line x = 7.
Answer:
It is parallel to Y – axis.

Question 66.
Write nature of the graph of a pair of linear equations in two variables is represented by
Answer:
Straight lines.

Question 67.
Find the number of solutions to the pair of equations 6x – 7y + 8 = 0 and 12x – 14y +16 = 0.
Answer:
Infinitely many solutions.

Question 68.
If 2^x + 3^y = 17 and 2^{x + 2} – 3^{y + 1} = 5, then find y’.
Answer:
y = 2.
Explanation:
2^x + 3^y = 17
⇒ a + b = 17
⇒ 3a + 3b = 51
2^x . 2² – 3^y . 3¹ = 5
⇒ 4a – 3b = 5 ……………….. (ii)
Solving equations (1) & (2)
7a = 56 ⇒ a = 8,
b = 17-a = 17-8 = 9
3^y = b = 3² ⇒ y = 2

Question 69.
If \(\frac{2}{x}+\frac{3}{y}\) = 13 and \(\frac{5}{x}-\frac{4}{y}\) = -2
find the solution.
Answer:
( \(\frac{1}{2}, \frac{1}{3}\) )

Question 70.
If 5x + py + 8 = 0 and 10x + 15y + 12 = 0 has no solution, then find the value of
p’.
Answer:
P = 7\(\frac{1}{2}\)

Question 71.
The larger of two supplementary angles exceeds the smaller by 38°. Find them.
Answer:
71° and 109°.
Explanation:
x, 180 – x =⇒ y = 180 – x
x + 38 = 180 – x
⇒ 2x = 180-38 = 142
x = \(\) = 71°
⇒ y = 180-71 = 109°

Question 72.
Write the number of solutions to 4x + 6y – 7 = 0 and 8x + 5y – 8 = 0.
Answer:
Only one solution.

Question 73.
Find the number of solutions to the pair of equation llx – 7y = 6 and 4x + 9y = 8.
Answer:
Only one solution.

Question 74.
\(\frac{2}{x}+\frac{3}{y}\) = 2, \(\frac{12}{x}-\frac{9}{y}\) = 3, then find ’x’.
Answer:
x = 2.

Question 75.
Write the pair of equations 4x – 2y + 6 = 0 and 2x – y + 8 = 0 has ……………. solutions.
Answer:
No solution.

Question 76.
Sita has pencils and pens which are together 40 in number. If she has 5 less pencils and 5 more pens the number of pens become four times the number of pencils. Represent this situation in a linear equation form.
Answer:
x + y = 40

Question 77.
Which type of equations
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are …
Answer:
Pair of linear equations.

Question 78.
If ax + by = c and px + qy = r has unique solution, then write the condition.
Answer:
\(\frac{a}{b}=\frac{p}{q}\)

Question 79.
If the pair of equations kx + 14y + 8 = 0 and 3x + 7y + 6 = 0 has a unique so-lution, then find ’k’.
k ≠ 6

Question 80.
\(\frac{\mathbf{a x}}{\mathbf{b}}-\frac{\mathbf{b y}}{\mathbf{a}}\) = a + b, ax – by = 2ab, then find ‘x’.
Answer:
x = 3b

Question 81.
The pair of equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has a unique solution, then write the condition.
Answer:
\(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)

Question 82.
Write the type of pair of lines
3x – 2y + 6 = 0, 6x – 4y + 8 = 0 are represents
Answer:
Parallel lines.

Question 83.
The age of a daughter is one third the age of her father. If the present age of father is ‘x’ years, then write the age of the daughter after 18 years in linear form.
Answer:
y = \(\frac{\mathrm{x}}{3}\) + 18

Question 84.
How many solutions \(\frac{\mathbf{a}_{1}}{\mathbf{a}_{\mathbf{2}}}=\frac{\mathbf{b}_{1}}{\mathbf{b}_{\mathbf{2}}}=\frac{\mathbf{c}_{1}}{\mathbf{c}_{\mathbf{2}}}\) will have ?
Answer:
Infinitely many solutions.

Question 85.
\(\frac{x+1}{2}+\frac{y-1}{3}=9, \frac{x-1}{3}+\frac{y+1}{2}=8\) then find x’.
Answer:
x = 13.

Question 86.
If the pair of equations 2x + y = 7 and 6x – py – 21 = 0 has infinite number of solutions, then find p.
Answer:
p = – 3
Explanation:
Infinite solutions,
so \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \Rightarrow \frac{2}{6}=\frac{1}{-p}\)
⇒ \(\frac{1}{3}=\frac{-1}{\mathrm{p}}\)
⇒ p = – 3

Question 87.
The ratio of incomes of two persons is 11:7 and the ratio of their expendi-tures is 9 : 5. If each of them manages to save ₹400 per month, then find the monthly income of first person.
Answer:
₹ 2200
Explanation:
11x – 9y = 400 and 7x – 5y = 400
On solving these equations income of first person was ₹ 2200.

Question 88.
Where the two lines 2x – y = 1, x + 2y = 13 will intersect each other ?
Answer:
(3,5)

Question 89.
Find the lines x – y = 1; 2x + y = 8 where they intersects at each other ?
Answer:
(3,2)
Explanation:
x = y + 1, 2(y + 1) + y = 8
⇒ 2y + 2 + y = 8
⇒ 3y = 6 ⇒ y = 2
⇒ x – 2 = 1 ⇒ x = 3

Question 90.
Write a line parallel to the line x + 2y + 1 = 0.
Answer:
2x + 4y + 1 = 0

Question 91.
If the equations (2m – l)x + 3y-5=0, 3x + (n – 1 )y – 2 — 0 has infinite number of solutions, then find ‘n’.
Answer:
n = \(\frac{11}{5}\).

Question 92.
Find where the line 2x + y = 7 inter-sects X – axis ?
Answer:
(\(\frac{7}{2}\), 0)

Question 93.
If \(\frac{5}{x-1}+\frac{1}{y-2}=2, \frac{6}{x-1}+\frac{-3}{y-2}=1\) then find ‘x’.
Answer:
x = 4

Question 94.
For what value of It, 2x + 3y = 4 and (k + 2)x + 6y = 3k + 2 will have infi-nitely many solutions ?
Answer:
k = 2
Explanation:
Infinitely many solutions, so

k + 2 = 4
k = 4 – 2 = 2

Question 95.
A fraction becomes \(\frac { 9 }{ 11 }\) if ‘2’ is added to both numerator and denominator. If ‘3’ is added to both numerator and denominator it becomes \(\frac { 5 }{ 6 }\), then find
the fraction.
Answer:
\(\frac { 7 }{ 9 }\) = fraction.
Explanation:
\(\frac{x+2}{y+2}=\frac{9}{11}\)
⇒ 11x + 22 = 9y + 18
⇒ 11 x – 9y = -4
\(\frac{x+3}{y+3}=\frac{5}{6}\)
⇒ 6x + 18 = 5y + 15
⇒ 6x – 5y = – 3
On solving these equations x = 7 and y = 9
∴ Fraction = \(\frac{x}{y}=\frac{7}{9}\)

Question 96.
Find the value of It’ for which the sys-tem of equations kx + 3y = 1, 12x + ky = 2 has no solution.
Answer:
k = – 6

Question 97.
The age of a father 8 years ago was 5 times that of his son 8 years. Hence, his age will be 8 years more than twice the age of his son. Then find the present age of father.
Answer:
48 years.

Question 98.
In the above problem, find the age of son.
Answer:
16 years.

Question 99.
If ad ≠ be, then find the pair of linear equations ax + by = p and cx + dy =q has how many solutions ?
Answer:
2 solutions.

Choose the correct answer satisfying the following statements.

Question 100.
Statement (A): Pair of linear equations 9x + 3y 4- 12 = 0, 18x 4- 6y + 24 = 0 have infinitely many solutions.
Statement (B): Pair of linear equations a₁x + b₁y + c₁ = 0 , a₂x + b₂y + c₂ = 0 have infinitely many solutions, if
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
From the given equations, we have

So, both A and B are correct and B explains A. Hence, (i) is the correct option.

Question 101.
Statement (A) : For k = 6, the system of linear equations x + 2y + 3 = 0 and 3x + ky + 6 = 0 is inconsistent. Statement (B) : The system of linear equations a₁x + b₁y + c₁ = 0,
a₂x + b₂y + c₂ = 0 is inconsistent if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
For inconsistent solution we have bs
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
So, A is correct but B is incorrect.
Hence, (ii) is the correct option.

Question 102.
Statement (A): The value of q = ±2, if x = 3, y = 1 is the solution of the line 2x + y – q² – 3 = 0.
Statement (B): The solution of the line will satisfy the equation of the line.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
As x = 3, y = 1 is the solution of
2x + y – q² – 3 = 0
⇒ 2 x 3 + 1 – q² – 3 = 0
⇒ 4 – q² = 0 ⇒ q² = 4 ⇒ q = ±2
So, both A and B are correct and B explains A.
Hence, (i) is the correct option.

Question 103.
Statement (A): The value of k for which the system of equations kx – y = 2, 6x – 2y = 3 has a unique solution is 3.
Statement (B) : The system of linear equations a₁x + b₁y + c₁ = 0,
a₂x + b₂y + c₂ = 0 has a unique solution if \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
Given system of linear equations has a
unique solution, if \(\frac{\mathrm{k}}{6} \neq \frac{1}{2}\)
⇒ \(\frac{\mathrm{k}}{6} \neq \frac{1}{2}\) ⇒ k ≠ 3
So, A is incorrect and B is corrrect.
Hence, (iii) is the correct option.

Question 104.
Statement (A) : The lines 2x – 5y = 7 and 6x – 15y = 8 are parallel lines.
Statement (B) : The system of linear equations a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 have infinitely many
solutions if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\).
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel,
if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
So, both A and B are correct.
Hence, (i) is the correct option.

Question 105.
Statement (A) : The system of equations 2x + y + 3 = 0 and 2x 4- y – 3=0 has no solution.
Statement (B) : The system of equations a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 has a unique solution when \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 106.
Statement (A) : If the system of equations 2x + 3y = 7 and 2ax + (a + b)y = 28 has infinitely many solutions, then 2a – b = 0.
Statement (B) : The system of equations x – 5y = 3 and 2x- lOy = 5 has a unique solution.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)

Question 107.
Statement (A): If the pair of lines are coincident, then we say that pair of lin-ear equations is consistent and it has a unique solution.
Statement (B) : If the pair of lines are parallel, then the pair of linear equations has no solution and is called inconsistent pair of equations.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)

Question 108.
Statement (A) : 3x + 4y – 5 = 0 and 6x -I- ky + 9 = 0 represent parallel lines if k = 8.
Statement (B): a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 represent parallel lines if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
In statement (A), given lines represent parallel lines, if
\(\frac{3}{6}=\frac{4}{\mathrm{k}} \neq \frac{5}{9} \Rightarrow \mathrm{k}=\frac{6 \times 4}{3}=8\)
∴ Statement (A) is true.
∴ Statement (B) is also true.
∴ Since reason is the correct explana¬tion for statement (A).
Option (i) is true.

Question 109.
Statement (A) : x + y – 4 = 0 and 2x + ky – 3 = 0 has no solution if k = 2.
Statement (B): a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 are consistent if \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
For (A), given equation has no solution, if
\(\frac{1}{2}=\frac{1}{\mathrm{k}} \neq \frac{-4}{-3}, \text { i.e., } \frac{4}{3}\)
⇒ k = 2 [ \(\frac{1}{2} \neq \frac{4}{3}\) holds]
∴ (A) is true.
Since (B) does not give result of (A), so option (i) is true.

Question 110.
Statement (A): If the system of equa-tions 2x + 3y = 7 and 2ax + (a + b)y = 28 has infinitely many solutions, then 2a – b = 0.
Statement (B) : The system of equations 3x – 5y = 9 and 6x- lOy = 8 has a unique solution. –
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
For (A), given system of equations has infinitely many solutions, if
\(\frac{2}{2 a}=\frac{3}{a+b}=\frac{-7}{-28}, \text { i.e., }\)
⇒ \(\frac{1}{a}=\frac{3}{a+b}=\frac{1}{4}\)
⇒ 3a = a + b ⇒ 2a – b = 0
Also clearly a = 4 and
a + b = 12 ⇒ b = 8
∴ 2a – b = 8 – 8 = 0
∴ (A) is true.
But (B) is false ∵ \(\frac{3}{6}=\frac{-5}{-10}\)
[∵ 3(-10) = (-5) (6) = -30]
For unique solution, if
\(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)
∴ Option (ii) is true.

Question 111.
Statement (A): If kx – y – 2 = 0 and 6x – 2y – 3 = 0 are inconsistent, then k = 3.
Statement (B): a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 are inconsistent if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
Statement (A) is true.

⇒ k = 3.
Statement (B) is also true.
∴ Option (i) is true.

Question 112.
Statement (A): 3x – 4y = 7 and
6x – 8y = k have infinite number of solutions if k = 14.
Statement (B): a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 have a unique solution if \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 113.
Statement (A) : The linear equations x – 2y – 3 = 0 and 3x + 4y – 20 = 0 have exactly one solution.
Statement (B) : The linear-equations 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 have a unique solution.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)

Question 114.
Statement (A) : kx + 2y = 5 and 3x + y = 1 have a unique solution if
k = 6.
Statement (B) : x + 2y = 3 and 5x + ky + 7 = 0 have a unique solution k ≠ 1.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iv)

Read the below passages and answer to the following questions.

If we have two simultaneous equations ax + by = c and bx + ay — d, l (c + d c-d^
then x = \(\frac{1}{2}\left(\frac{\mathrm{c}+\mathrm{d}}{\mathrm{a}+\mathrm{b}}+\frac{\mathrm{c}-\mathrm{d}}{\mathrm{a}-\mathrm{b}}\right)\) and y = \(\frac{1}{2}\left(\frac{c+d}{a+b}-\frac{c-d}{a-b}\right)\)

Question 115.
Find the solution of 217x + 131y = 913 and 131x + 217y = 827.
Answer:
x = 3, y = 2
Explanation:
We have 217x + 131y = 913
131x + 217y = 827

Question 116.
Find the solution of 37x + 41y = 70 and 4lx + 37y = 86.
Answer:
x = 3, y = – 1
Explanation:
We have,
37x + 41y = 70
41x + 37y = 86

Question 117.
Find the solution of x + 2y = \(\frac { 3 }{ 2 }\) and 2x + y = \(\frac { 3 }{ 2 }\)
Answer:
x = \(\frac { 1 }{ 2 }\), y = \(\frac { 1 }{ 2 }\)
A system of linear equations is given as follows :
a₁x + b₁y + c₁ = 0 and
a₂x + b₂y + c₂ = 0
Explanation:
We have

Question 118.
Write the condition for two lines to have a unique solution.
Answer:
\(\frac{a_{1}}{a_{2}} \neq \frac{c_{1}}{c_{2}}\)

Question 119.
Write the condition for two lines to have infinitely many solutions.
Answer:
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

Question 120.
Write the condition both lines are par-allel only.
Answer:
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
6 pencils and 4 notebooks together cost ₹ 90 whereas. 8 pencils and 3 notebooks together cost ₹ 85.

Question 121.
Create an equation to first situation.
Answer:
6x + 4y = 90.

Question 122.
Create an equation to second situation.
Answer:
8x + 3y = 85

Question 123.
Which mathematical concept is used to find the cost of notebook and pencil ?
Answer:
Pair of linear equations.
A boat goes 30 km upstream and 44 km downstream in 10 hrs. In 13 hrs it can go 40 km upstream and 55 km downstream.

Question 124.
Prepare an equation to first condition.
Answer:
\(\frac{30}{x-y}+\frac{44}{x+y}=10\)

Question 125.
Prepare an equation to second condition.
Answer:
\(\frac{40}{x-y}+\frac{55}{x+y}=13\)

Write the correct option to match the column -I and column – II, which gave value of ‘x1 and ‘y’ for pair of equation given in column -I.

Question 126.

Answer:
A – (iv), B – (ii)

Question 127.

A – (iii), B – (i)

Question 128.

A – (ii), B – (iv)

Question 129.

A – (i), B – (iii)

Answer Questions 130 and 131 based on the data given below.
“The cost of 1 kg potatoes and 2kg to-matoes was ₹ 30 on a certain day. After two days the cost of 2 kg potatoes and 4 kg tomatoes was found to be ₹ 66”.

Question 130.
Write a pair of linear equations in two variables x and y from the datAnswer:
Solution:
x + 2y = 30, 2x + 4y = 66 (or) x + 2y = 33

Question 131.
Which system of linear equations in two variables does the data represent ?
Answer:
Parallel lines, inconsistent, no solution.

Question 132.
For what value of ‘k’ is the pair of linear equations x + 2y = 7 and3x – ky = 21 has infinitely many solutions ?
Answer:
Given equations are x + 2y = 7 and 3x – ky = 21 has infinitely many solutions, so

Question 133.
What is the value of ‘x’ in 4x – 7y = 9 ify = 3?
Answer:
Given , 4x – 7y = 9
If y = 3, then 4x – 7(3) = 9
=> 4x-21 = 9
=> 4x = 30
x = \(\frac{30}{4}=\frac{15}{2}\)

Question 134.
Lahari bought two pens and five pencils spending Rs. 30. Express this information as a linear equation in variables x and y.
Answer:
Let the cost of each pen be ₹ x and the cost of each pencil be ₹ y.
By problem, 2x + 5y = 30

Practice the AP 10th Class Maths Bits with Answers Chapter 13 Probability on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 13th Lesson Probability with Answers

Question 1.
Karishma and Reshma are playing chess. The probability of winning Karishma is 0.59. Then find probability of Reshma winning the match.
Answer:
0.41
Explanation:
Reshma winning match = 1 – 0.59
= 0.41

Question 2.
Vineeta said that probability of impos-sible events is 1. Dhanalakshmi said that probability of sure event is ‘0’ and Sireesha said that probability of any event lies in between 0 and 1. In the above with whom will you agree?
Answer:
Sireesha

Question 3.
A page is opened at a random from a book containing 90 pages. Then find the probability of a page number is a perfect square.
Answer:
\(\frac{1}{90}\)
Explanation:
\(\frac{\text { Required page }}{\text { Total pages }}\) = \(\frac{1}{90}\)

Question 4.
From the figure, find the probability of getting blue ball.

Answer:
\(\frac{3}{5}\)
Explanation:
\(\frac{\text { No. of blue balls }}{\text { Total balls }}\) = \(\frac{3}{5}\)

Question 5.
Find the probability of picking a red king card from a well shuffled deck of playing cards.
Answer:
\(\frac{1}{26}\)

Question 6.
Find the probability of getting a head when a coin is tossed once.
Answer:
\(\frac{1}{2}\)

Question 7.
Write any one value which cannot be the probability of an event?
Answer:
-1.5

Question 8.
If P(E) = 0.26, then find P (\(\overline{\mathbf{E}}\)).
Answer:
0.74

Question 9.
Find probability of getting 7, when a dice is rolled.
Answer:
0

Question 10.
Which of the following situations have equally likely events?
Answer:
Getting 1 or 2 or 3 or 4 or 5 or 6 when a dice is rolled and winning or loosing a game and Head or Tail, when a coin is tossed.

Question 11.
The probability of picking a letter from the set of english alphabets is \(\frac{5}{26}\). Find that alphabet can be.
Answer:
Vowel
Explanation:
In English letters 5 vowels are there.
So the probability is \(\frac{5}{26}\)

Question 12.
If a die is rolled, then find the probability of getting a prime number.
Answer:
\(\frac {1}{2}\)

Question 13.
Which of the following cannot be the probability of an event?
Answer:
1.\(\overline{3}\)

Question 14.
IfP(E) = 1, then p(\(\overline{\mathbf{E}}\)).
Answer:
0

Question 15.
When a die is rolled, find the probability of getting an odd prime number.
Answer:
\(\frac{1}{3}\)

Question 16.
Find the probability that the sum of two numbers appearing on the top of the dice is 13, when two dice are rolled at the same time.
Answer:
0

Question 17.
If P(E) = 0.05, then find P(\(\overline{\mathbf{E}}\)).
Answer:
0.95

Question 18.
Which of the following be the probability of an event?
– 1.5, 2.4, 0.7, 115%
Answer:
0.7

Question 19.
P(E) = 0.65, then find P(\(\overline{\mathbf{E}}\)).
Answer:
0.35

Question 20.
If P(E) = 0.82, then find P(\(\overline{\mathbf{E}}\))
Answer:
0.18

Question 21.
On random selection, find the probability of getting a composite number among the numbers from 51 to 100.
Answer:
\(\frac{4}{5}\)

Question 22.
Let E and \(\overline{\mathbf{E}}\) be the complementary events. If P(\(\overline{\mathbf{E}}\)) = 0.65, then find P(E).
Answer:
0.35
Explanation:
P(E) = 1 – P(E)
⇒ 0.65 = 1 – P(E)
⇒ P(E) = 1 – 0.65 = 0.35

Question 23.
At what value of ‘x’, \(\frac{5}{x}\) may possible, probability of an event?
Answer:
6

Question 24.
If P(E) is the probability of an event E, then write the relation.
Answer:
0 ≤ P(E) ≤ 1

Question 25.
The probability of getting right answer to a question is 0.68, then find the probability of getting a wrong answer.
Answer:
0.32 (or) 32%

Question 26.
In a single throw of two dice, find the probability of getting distinct numbers.
Answer:
\(\frac{5}{6}\)

Question 27.
Find the value of P(E) – 1 + P(\(\overline{\mathbf{E}}\)) .
Answer:
0

Question 28.
Getting a prime (or) composite number is example to event.
Answer:
Mutually exclusive event.

Question 29.
P(G) = \(\frac{4}{17}\) then find p(\(\overline{\mathbf{G}}\)).
Answer:
\(\frac{13}{17}\)

Question 30.
From a well shuffled pack of cards, a card is drawn at random, then find the probability of getting a red jack.
Answer:
\(\frac{1}{26}\)

Question 31.
Find the probability of raining in a day.
Answer:
\(\frac{1}{2}\)

Question 32.
P(E) = 0.45, then find P(\(\overline{\mathbf{E}}\)) .
Answer:
0.545

Question 33.
There are 50 cards numbered from 1 to 50. A card is drawn at random, then find the probability that the number on the card is divisible by 8.
Answer:
\(\frac{3}{25}\)
Explanation:
From 1 to 50 numbers 8 multiples are 8, 16, 24, 32, 40, 48
∴ Required probability = \(\frac{6}{50}=\frac{3}{25}\).

Question 34.
In a simultaneous toss to two coins, find the probability of at least one head.
Answer:
3/4

Question 35.
The set of all possible events is called
Answer:
Sample space

Question 36.
In a lucky dip of 30 tokens, Gopi purchased two tokens. Then find the probability of getting the first prize.
Answer:
17/26

Question 37.
Two fair dice are rolled and the face values are added. Find the probability of getting an odd number greater than 8.
Answer:
\(\frac{1}{6}\)

Question 38.
Two dice are thrown once together. What is the probability of getting a doublet?
Answer:
\(\frac{1}{6}\)

Question 39.
P(E) + P(\(\overline{\mathbf{E}}\)) is equals to
Answer:
1

Question 40.
When a coin is tossed, find the probability of getting a tail or head.
Answer:
\(\frac{1}{2}\)

Question 41.
P(Impossible event) is equal to
Answer:
0

Question 42.
P(Sure event) is equal to
Answer:
1

Question 43.
If one side is chosen at random from the sides of a right triangle, then find the probability that it is hypotenuse.
Answer:
1/3

Question 44.
In a box, there are 28 marbles of which ‘x’ are green and the rest are white. If the probability of getting a green marble is \(\frac{2}{7}\), then find the number of green marbles.
Answer:
8
Explanation:

∴ No. of green marbles = x = 8.

Question 45.
How many cards in a pack of playing cards?
Answer:
52

Question 46.
Getting a Tail or Head ……………..event.
Answer:
Equally likely

Question 47.
Find the probability of an event lies between …………… and …………..
Answer:
0,1

Question 48.
A card is pulled from a deck of 52 cards. Find the probability of obtaining a club.
Answer:
1/4

Question 49.
In a single throw of two dice, find the probability of getting even doublet.
Answer:
1/12

Question 50.
When a dice is rolled, find the probability of getting a composite number.
Answer:
1/3

Question 51.
When a coin is tossed, find the probability of getting a head.
Answer:
1/2

Question 52.
If a coin is tossed, then find the prob-ability that a head turns up.
Answer:
1/2

Question 53.
A box contains pencils and pens. The probability of picking out a pen at random is 0.65. Then find the probability of not picking a pen.
Answer:
0.35

Question 54.
In a single throw of two dice, find the probability of getting a total of 12.
Answer:
\(\frac{1}{36}\)

Question 55.
From a bag containing 6 red balls, 5 green balls and 3 blue balls, find the probability of getting a green ball at random.
Answer:
5/14

Question 56.
If a card is drawn from a pack, find the probability that it is a king.
Answer:
\(\frac{1}{13}\)

Question 57.
Getting a red card (or) black card is …………… event.
Answer:
Mutually exclusive.

Question 58.
When a dice is thrown, find the probability of getting neither a prime nor composite number.
Answer:
1/6

Question 59.
If E is an event whose probability is
\(\frac{2}{5}\), then find the probability of not E.
Answer:
3/5

Question 60.
Probability of switching on a bulb in a dark room is 0.35, then find the probability of not switching the bulb.
Answer:
0.65

Question 61.
Find the probability of a certain event.
Answer:
1

Question 62.
Find the probability that a leap year has 53 Sundays.
Answer:
2/7

Question 63.
Three different greeting cards and their corresponding covers are randomly-strewn about on a table. If Sita puts the greeting cards into the covers at random, find the probability of correctly matching of all the greeting cards and covers.
Answer:
\(\frac{1}{6}\)

Question 64.
In a simultaneous toss of two coins, find the probability of no tails.
Answer:
3/4

Question 65.
If a ball is drawn at random from a box containing 11 red balls, 6 white balls and 9 green balls, then find the probability that the ball is not green.
Answer:
\(\frac{17}{26}\)

Question 66.
From a deck of cards, a card is drawn at random, then find the probability of getting a black face card.
Answer:
\(\frac{3}{26}\)

Question 67.
If two dice are rolled at a time, then find the probability that the two faces show different numbers.
Answer:
5/16

Question 68.
Find the probability of a face card from red cards.
Answer:
3/13

Question 69.
A dice is tossed once, then find the probability of getting an even number or a multiple of 3.
Answer:
2/3

Question 70.
The event which can’t happen at all is known as …………….. event.
Answer:
Impossible

Question 71.
A baby is born, find the probability that it is a boy (or) girl.
Answer:
1/2

Question 72.
If an unbiased coin is tossed, find the probability of getting a tail.
Answer:
1/2

Question 73.
In a single throw of two dice, find the probability of getting a doublet.
Answer:
1/6

Question 74.
Find the probability of getting two tails when two coins are tossed.
Answer:
1/4

Question 75.
From a well shuffled pack of cards a card is drawn at random, then find the probability of getting a red coloured card.
Answer:
1/2

Question 76.
If a die is rolled, then find the probability of getting an even number.
Answer:
1/2

Question 77.
If two dice are rolled simultaneously, then find the sum with greatest possibility to happen.
Answer:
7

Question 78.
When two dice are rolled, find probability of getting odd doublet.
Answer:
1/12

Question 79.
Find the probability of an impossible event.
Answer:
0

Question 80.
Find the probability of getting a number less than 5 when a die is rolled.
Answer:
2/3

Question 81.
In a single throw of two dice, find the probability of getting a total of 11.
Answer:
1/18

Question 82.
Find the probability of drawing a black card from the black cards.
Answer:
1

Question 83.
If a card is drawn from a deck of 52 cards find the probability that it is a club card.
Answer:
1/4

Question 84.
Two dice are rolled, find the probability of getting 6 as the product.
Answer:
1/9

Question 85.
If the occurrence of one event prevents the occurrence of another event, then which type of element they are?
Answer:
Mutually exclusive.

Question 86.
In a single throw of two dice, find the probability of getting a total of 3 or 5.
Answer:
1/6

Question 87.
If two dice are thrown simultaneously, find the probability of showing the same numbers on their faces.
Answer:
1/6

Question 88.
Find the probability of drawing a black king from the deck.
Answer:
\(\frac{1}{26}\)

Question 89.
Find the event of getting a number less than or equal to 6.
Answer:
Sure event

Question 90.
If two events have same chances to happen, then they are called which type of events?
Answer:
Equally likely events.

Choose the correct answer satisfying the following statements.
Question 91.
Statement (A): The probability of winning a game is 0.4, then the probability of losing it is 0.6.
Statement (B) : P(E) + P(not E) = 1
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
i) Both A and B are true.

Question 92.
Statement (A) : When two coins are tossed simultaneously, then the probability of getting no tail is \(\frac{1}{4}\).
Statement (B): The probability of getting a head (i.e., no tail) in one toss of a coin is \(\frac{1}{2}\).
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
i) Both A and B are true.

Question 93.
Statement (A) : Card numbered as 1, 2, 3, ………….., 15 are put in a box and mixed thoroughly, one card is then drawn at random. The probability of drawing an even number is \(\frac{1}{2}\).
Statement (B) : For any event E, we have 0 ≤ P(E) ≤ 1.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
iii) A is false, B is true.

Question 94.
Statement (A) : In a simultaneously throw of a pair of dice. The probability of getting a double is \(\frac{1}{6}\) .
Statement (B): Probability of an event may be negative.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
ii) A is true, B is false.

Question 95.
Statement (A): The probability of getting a prime number. When a die is
thrown once is \(\frac{2}{3}\).
Statement (B) : Prime numbers on a die are 2, 3, 5.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
iii) A is false, B is true.

Question 96.
Statement (A) : If P(A) = 0.3 and P(A∪\(\bar{B}\))= 0.8, then P(B) is \(\frac{2}{7}\).
Statement (B): P(\(\overline{\mathbf{E}}\)) = 1 – P(E), where E is any event.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
i) Both A and B are true.

Question 97.
Statement (A): If P(A) = 0.25, P(B) = 0.50 and P(A∩B) = 0.14, then the probability that either A or B occurs is 0.39.
Statement (B) : \(\overline{\mathrm{A} \cup \mathrm{B}}=\overline{\mathrm{A}} \cup \overline{\mathrm{B}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
iii) A is false, B is true.

Question 98.
Statement (A) : If A and B are two independent events and it is given that
P(A) = \(\frac{2}{5}\), P(B) = \(\frac{3}{5}\) , then P(A∩B) = \(\frac{6}{25}\).
Statement (B) : P(A∩B) =P(A).P(B), where A and B are two independent events.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
i) Both A and B are true.

Question 99.
Statement (A) : If a box contains, 5 white, 2 red and 4 black marbles, then the probability of not drawing a white marble from the box is \(\frac{5}{11}\).
Statement (B): p(\(\overline{\mathbf{E}}\)) = 1 – P(E), where E is any event.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
iii) A is false, B is true.

Question 100.
Statement (A) : In rolling a dice, the probability of getting number 8 is zero.
Statement (B) : Its an impossible event.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
i) Both A and B are true.

Read the below passages and answer to the following questions.
A die has two faces each with number ‘1’, three faces each with number ‘2’ arid one face with number ‘3’. Die is rolled once.
Question 101.
The probability of obtaining the number 2 is
Answer:
\(\frac{1}{2}\)

Question 102.
The probability of getting the number 1 or 3 is
Answer:
\(\frac{1}{2}\)

Question 103.
The probability of getting the number 3 is
Answer:
\(\frac{5}{6}\)
A student hds a dice whose six faces show the letters A, B, C, D, E and E The dice is thrown once.

Question 104.
What is the probability of getting ‘A’?
Answer:
\(\frac{1}{6}\)

Question 105.
What is the probability of getting ‘D’?
Answer:
\(\frac{1}{6}\)

Question 106.
How many times the experiment was done?
Answer:
Once only
A box contains 3 blue and 4 red balls.

Question 107.
Find the probability of taken blue ball out.
Answer:
\(\frac{3}{7}\)

Question 108.
Find the probability of taken red ball out.
Answer:
\(\frac{4}{7}\)
A page is opened at random from a book containing 100 pages.

Question 109.
Find the probability that the page number is a perfect square.
Answer:
\(\frac{10}{100}\) = 0.1

Question 110.
Find the probability that the page number is a even number.
Answer:
\(\frac{50}{100}=\frac{1}{2}\)

Question 111.
Find the probability that the page number is a odd number.
Answer:
\(\frac{50}{100}=\frac{1}{2}\)
Observe the given table and answer to the following questions.

Question 112.
Find the probability of selecting ‘B’ blood group student.
Answer:
\(\frac{13}{40}\)

Question 113.
Find the probability of selecting ‘A’ blood group student.
Answer:
\(\frac{10}{40}\)

Question 114.
Find the probability of selecting ‘AB’ blood group student.
Answer:
\(\frac{12}{40}\)

Question 115.
Find the probability of selecting ‘O’ blood group student.
Answer:
\(\frac{5}{40}=\frac{1}{8}\)
A die is thrown once

Question 116.
Find the probability of getting an even number.
Answer:
1/2

Question 117.
Find the probability of getting an odd prime number.
Answer:
\(\frac{1}{2}\)
Write the correct matching options.

Question 118.

Answer:
A – (iv), B – (v), C – (i)

Question 119.

A – (iii), B – (ii)

Question 120.
Written description Probability

Answer:
A – (ii), B – (i)

Question 121.

Answer:
A – (ii), B – (iv)

Question 122.

Answer:
A – (ii), B – (i)

Question 123.

Answer:
A – (iv), B – (iii)

Question 124.

Answer:
A – (vi), B – (v)

Question 125.
A card is drawn from a well-shuffled deck of 52 cards randomly. What is the probability of getting a card, which is neither an ace nor a king card?
Answer:
\(\frac{11}{13}\)

Practice the AP 10th Class Maths Bits with Answers Chapter 3 Polynomials on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 3rd Lesson Polynomials with Answers

Question 1.
Write the sum of zeroes of
bx² + ax + c.
Answer:
\(\frac{-a}{b}\)

Question 2.
Find product of the zeroes of
p(x) = (x-2).(x + 3)
Answer:
= -6
Explanation:
p(x) = x² + x – 6,
Product of zeroes = \(\frac{\mathrm{c}}{\mathrm{a}}\) = – 6.

Question 3.
5x – 3 represents which type of poly¬nomial ?
Answer:
Linear.

Question 4.
Find the value of ‘x’ which satisfies 2(x – 1) – (1 – x) = 2x + 3 ?
Answer:
6
Explanation:
2x – 2 – 1 + x = 2x + 3
⇒ 3x – 2x = 3 + 3 = 6
⇒ x = 6.

Question 5.
Write the degree of the polynomial
\(\sqrt{2}\)x² – 3x + 1.
Answer:
2

Question 6.
Write order of the polynomial 5x⁷ – 6x⁵ + 7x – 6.
Answer:
7

Question 7.
Find the product of zeroes of 2x² – 3x + 6.
Answer:
3
Explanation:
Product of zeroes = \(\frac{c}{a}=\frac{6}{2}\) = 3.

Question 8.
Find sum of zeroes of a polynomial x³ – 2x² + 3x – 4.
Answer:
2
Explanation:
Sum of zeroes = α + β + γ
= \(\frac{-b}{a}=\frac{-(-2)}{1}\) = 2

Question 9.
Find a quadratic polynomial, the sum of whose zeroes is zero and one zero is 4, is
Answer:
x² – 16 = 0
Explanation:
Sum of zeroes = \(\frac{-b}{a}\) = 0,
α + β = 0, β = 4
⇒ α = -4 x2 – (α + β)x + αβ = 0
⇒ x² – 0(x) -16 = 0
⇒ x² – 16 = 0.

Question 10.
If p(x) = x² – ax – 3 and p(2) = – 3, then find Answer:
Answer:
2
Explanation:
p(x) = (2)² – 2a – 3 = – 3
⇒ 1 — 2a = — 3
⇒ 4 – 2a = 0
⇒ 4 = 2a ⇒ a = 2

Question 11.
Write the zero value of polynomial px + q.
Answer:
\(\frac{-\mathrm{q}}{\mathrm{p}}\)

Question 12.
In a division, if divisor is x + 1, quo¬tient is x and remainder is 4, then find dividend.
Answer:
x (x + 1) + 4 = x² + x + 4
Explanation:
Dividend = Divisor x Quotient
+ Remainder
= (x + 1) x + 4 = x² + x + 4.

Question 13.
Find the zero value of linear polynomial ax – b.
Answer:
\(\frac{\mathrm{b}}{\mathrm{a}}\)

Question 14.
Find the sum of the zeroes of the poly-nomial x² + 5x + 6.
Answer:
– 5

Question 15.
4y² – 5y + 1 is an example for this type of polynomial.
Answer:
Quadratic polynomial.

Question 16.
Write the degree of the polynomial 5x⁷ – 6x⁵ + 7x – 4.
Answer:
7

Question 17.
How much the sum of zeroes of the polynomial 2x² – 8x + 6 ?
Answer:
4

Question 18.
f α,β are the zeroes of x² + x +1, then find \(\frac{1}{\alpha}+\frac{1}{\beta}\)
Answer:
-1
Explanation:
α + β = -1, αβ = 1
⇒ \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}=\frac{-1}{1}\) = -1

Question 19.
Write the number of zeroes of the poly¬nomial in the given graph.

Answer:
3

Question 20.
Write product of zeroes of the cubic polynomial 3x³ – 5x² – 11x – 3.
Answer:
1
Explanation:
Product of zeroes = αβγ
= \(\frac{-d}{a}=\frac{-(-3)}{3}=\frac{3}{3}\) = 1

Question 21.
The following is the graph of a poly¬nomial. Find the zeroes of the poly¬nomial from the given graph.

Answer:
– 2,1

Question 22.
Find the value of p(x) = 4x² + 3x + 1 at x = -1.
Answer:
2

Question 23.
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d and (a ≠ 0), then find αβγ

Question 24.
4x + 6y = 18 doesn’t pass through ori-gin, then its graph indicates
Answer:
A straight line

Question 25.
If ‘3’ is one of the zeroes of
p(x) = x² + kx – 9, then find the value of k.
Answer:
0
Explanation:
p(3) = 3² + 3k – 9 = 0
⇒ 3k = 0 ⇒ k = 0

Question 26.
When p(x) = x² – 8x + k leaves a re-mainder when it is divided by (x – 1), then find k.
Answer:
k > 7
Explanation:
p(1) = 1 – 8 + k = 0
⇒ k > 7

Question 27.
If α, β, γ are zeroes of x³ + 3x² – x + 2, then find the value of αβγ.
Answer:
-2

Question 28.
Make a quadratic polynomial having 2, 3 as zeroes.
Answer:
x² – 5x + 6 = 0
Explanation:
x² – (2 + 3)x + 2 . 3 = 0
⇒ x² – 5x + 6 = 0.

Question 29.
Write a quadratic polynomial, whose zeroes are \(\sqrt{2}\) and –\(\sqrt{2}\).
Answer:
x² – 2 = 0
Explanation:
x² – \((\sqrt{2}-\sqrt{2}) x+\sqrt{2}(-\sqrt{2})\) = 0
⇒ x² – 0(x) -2 = 0
⇒ x² – 2 = 0.

Question 30.
Find the coefficient of x7 in the poly¬nomial 7x¹⁷ – 17x¹¹ + 27x⁵ – 7.
Answer:
0

Question 31.
If α, β are the zeroes of the polyno¬mial x² – x – 6, then find α²β²
Answer:
36
Explanation:
αβ = \(\frac{\mathrm{c}}{\mathrm{a}}\) = – 6 ⇒ (αβ)² = (-6)² = 36

Question 32.
If ‘4’ is one of the zeroes of p(x) = x² + kx – 8, then the value of k.
Answer:
-2

Question 33.
If the polynomial p(x) = x³ – x² + 3x + k is divided by (x – 1), the remainder obtained is 3, then find the value of k.
Answer:
0
Explanation:
p(1) = 1-1 + 3 + k = 3 ⇒ k = 0

Question 34.
If one zero of the polynomial f(x) = 5x² + 13x + k is reciprocal of the other, then find the value of k.
Answer:
5
Explanation:
α = x, β = 1/x
⇒ α . β = x . 1/x = k/5 ⇒ k = 5

Question 35.
Find the number of zeroes of the po!y: nomial, whose graph is given below.
Answer:

Question 36.
Number of zeroes that can be identi¬fied by the given figure.
Answer:

Question 37.
Observe the given rectangular figure, then write its area in polynomial func-tion.

Answer:
x² – 7x – 30 = 0
Explanation:
(10 – x)(x + 3) = 10x + 30 – x² – 3x = 0
⇒ x² – 7x – 30 = 0

Question 38.
f(x) = x + 3, then find zero of f(x).
Answer:
-3

Question 39.
One zero of the polynomial 2x² + 3x + k is 1/2, then find k.
Answer:
-2
Explanation:
im 19
α + β + γ = \(\frac{-b}{a}\) = -3
\(\sqrt{5}-\sqrt{5}+\gamma\) = -3
γ = -3

Question 40.
Find factors of x² + x(a + b) + ab.
Answer:
x + a and x + b

Question 41.
f(x) = 4x² + 4x – 3, then find f(\(\frac{-3}{2}\))
Answer:
0

Question 42.
If \(\sqrt{5}\) and \(-\sqrt{5}\) are two zeroes of the polynomial x³ + 3x² – 5x – 15, then find its third zero.
Answer:
-3
Explanation:

Question 43.
If \(\sqrt{3}\) and – \(\sqrt{3}\) are the zeroes of a polynomial p(x), then find p(x).
Answer:
x² – 3

Question 44.
If ‘m’ and ‘n’ are zeroes of the polyno-mial 3x² + 11x – 4, then find the value \(\frac{\mathbf{m}}{\mathbf{n}}+\frac{\mathbf{n}}{\mathbf{m}}\)
Answer:
\(\frac{-145}{12}\)
Explanation:
3x² + 11x – 4 = 3x² + 12x – x – 4 = 0
⇒(x + 4)(3x – 1)= 0
⇒x = -4, \(\frac { 1 }{ 2 }\)

Question 45.
In the product (x + 4) (x + 2). Write the constant term.
Answer:
8

Question 46.
Find the polynomial whose zeroes are – 5 and 4.
Answer:
x² + x – 20

Question 47.
Flnd product of zeroes of 3x² = 1
Answer:
\(-\frac{1}{3}\)

Question 48.
Find the sum of the zeroes of x² + 7x + 10.
Answer:
-7

Question 49.
If one root of the polynomial
fix) = 5x² + 13x + k is reciprocal of the other, then find ‘k’.
Answer:
5

Question 50.
f(x) = 3x – 2. then find zero of f(x).
Answer:
\(\frac{2}{3}\)

Question 51.
Write the zeroes from the below graph.

Answer:
-2, 0 and 2

Question 52.
Make a quadratic polynomial whose zeroes are 5 and -2.
Answer:
x² – 3x – 10.

Question 53.
If α, β are (1w zero ai polynomial
f(x) = x² – p(x + 1) – c then find (α + 1)(β + 1).
Answer:
1 – c
Explanation:
α + β = \(\frac{-b}{a}=\frac{+p}{1}\) = +p
αβ = \(\frac{c}{a}=\frac{-p-c}{1}\) = -(p+c)
(α + 1) (β + 1) = αβ + α + β + 1
= -p – c + p + 1
= 1 – c

Question 54.
Write number of constant polynomial x² + 7x – 7.
Answer:
1

Question 55.
Write the quadratic polynomial, whose sum and product of zeroes are 1 and – 2 respectively.
Answer:
x² – x – 2

Question 56.
Find the product of the zeroes of x³ + 4x² + x – 6.
Answer:
6

Question 57.
p(x) = \(\frac{x+1}{1-x}\) , then find p(0).
Answer:
1

Question 58.
Write the maximum number of zeroes that a polynomial of degree 3 can have.
Answer:
Three

Question 59.
If x + 2 is a factor of x² + ax + 2 b and a 4 b = 4, then find Answer:
Answer:
3
Explanation:
(- 2)² – 2a + 2b = 0
⇒ 4 – 2a + 2b = 0
⇒ -a + b = -2

⇒ b = 1,a = 4- b = 4 – 1 = 3.

Question 60.
The graph of ax + b represents which type of polynomial ?
Answer:
linear polynomial.

Question 61.
If ‘1’ is the zero of the quadratic poly- nomlal x² + kx – 5, then find the value ofk.
Answer:
4
Explanation:
1 + k – 5 = 0 ⇒ k = 4.

Question 62.
Find a quadratic polynomial, the sum of whose zeroes is W and product of zero is 3.
Answer:
x² – 9

Question 63.
If y = p(x) is represented by the given pqih, then find die number of zeroes.

Answer:
4

Question 64.
If α + β = 0, αβ = \(\sqrt{3}\), then write the quadratic polynomial.
Answer:
x² + \(\sqrt{3}\)

Question 65.
Find the degree of the polynomial ax⁴ + bx³ + cx² + dx + e.
Answer:
4.

Question 66.
If α, β, γ are the zeroes of the polyno- mid f(x) = x³ – px² + qx – r, then find
\(\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}\)
Answer:
\(\frac{\mathbf{p}}{\mathrm{r}}\)
Explanation:
\(\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}=\frac{\alpha+\beta+\gamma}{\alpha \beta \gamma}=\frac{\frac{p}{1}}{\frac{r}{1}}=\frac{p}{r}\)

Question 67.
If α and β are the zeroes of the poly¬nomial p(x) = 2x² + 5x + k satisfying the relation α² + β² + αβ = \(\frac{21}{4}\), then find k.
Answer:
2
Explanation:
α² + β² + αβ

Question 68.
If p(t) = t³ , then find p(- 2).
Answer:
– 9

Question 69.
If the polynomial f(x) = ax³ – bx – a is divisible % the polynomial g(x) = x² + bx + c, then find ah.
Answer:
1

Question 70.
Write degree of a linear polynomial.
Answer:
1

Question 71.
If the sum off the zeroes off die polyno¬mial fix) = 2x³ – 3kx² + 4x – 5 is 6, then find k.
Answer:
4.

Question 72.
If p and q are the zeroes of the poly¬nomial t² – 4t + 3, then find \(\frac{1}{p}+\frac{1}{9}-2 p q+\frac{14}{3}\)
Answer:
0
Explanation:
t² – 3t – t + 3
= t(t – 3) – 1(t-3)
= (t – 3) (t – 1)
t = 3,1
p = 3, q = 1

Question 73.
If the inudnct of zeroes rf9x2+3x + p is 7, then find p”.
Answer:
63

Question 74.
If α, β, γ are the zeroes of the polyno-mial f(x) = ax³ + hx² + cx + d, then find α² + β² + γ²
Answer:
\(\frac{b^{2}-2 a c}{a^{2}}\)

Question 75.
p(x) = x² + 5x + 6, then find zeroes of p(x).
Answer:
-2,-3.

Question 76.
If one of the zeroes of the quadratic polynomial ax² + bx + c is ‘0’, then find the other zero.
Answer:
-b

Question 77.
α = a – b, β = a 4 b, then make the quadratic polynomial.
Answer:
x² – 2ax + a² – b²

Question 78.
Find the quadratic polynomial whose zeroes are \(4+\sqrt{5}\) and \(4-\sqrt{5}\)
Answer:
x² – 8x + 11

Question 79.
Find die remainder when
3x³ + x² + 2x + 5 is divided by x² + 2x + 1.
Answer:
9x + 10

Question 80.
What must be subtracted or added to p(x) = 8x⁴ + 14x³ – 2x² + 8x – 12, so that 4x² + 3x – 2 is a factor off p(x) ?
Answer:
15x – 14

Question 81.
Prepare a quadratic polynomial whose zeroes are \(\sqrt{15}\) and \(-\sqrt{15}\).
Answer:
x² – 15

Question 82.
Divide(x³ – 6x² + 11x – 12) by (x² – x + 2)r then find quotient.
Answer:
x – 5

Question 83.
If -1 is a zero of the polynomial f(x) = x² – 7x – 8, then find the other zero.
Answer:
8

Question 84.
If a > 0, dim draw the shape off ax² + bx + c = 0.
Answer:

Question 85.
If 2x + 3 is a factor of 2x³ – x – b + 9x², then find the value erf b.
Answer:
-15

Question 86.
If the order erf ax⁵ + 3x⁴ + 4x³ + 3x² + 2x + 1 is 4, then find Answer:
Answer:
0
Explanation:
If a = 0, then given equation order becomes 4.

Question 87.
Find the zeroes erf the polynomial p(x) = 4x² – 12x + 9.
Answer:
\(\frac{3}{2}, \frac{3}{2}\)

Question 88.
If α, β, γ are roots erf a cubic polynomial ax³ + bx² + cx + d, then find αβ + βγ + γα
Answer:
\(\frac{\mathbf{c}}{\mathbf{a}}\)

Question 89.
If one of the zeroes of the quadratic polynomial f(x) = 14x² – 42k²x – 9 is negative of the other, then find k.
Answer:
0
Explanation:
Let α = x, β = – x
α + β = \(-\frac{b}{a}\)
x – x = \(\frac{-\left(-42 \mathrm{k}^{2}\right)}{14}\)
⇒ + 3k² = 0
⇒ k² = \(\frac{0}{3}\) = 0 ⇒ k = 0

Question 90.
p(x) = 4x² + 3x – 1, then find P(\(\frac{1}{4}\))
Answer:
0

Question 91.
Find the sum of zeroes of the polynomial 3x² – 8x + 1 = 0.
Answer:
\(\frac{8}{3}\)
Explanation:
3x² – 8x + 1 = 0
Sum of zeroes = \(-\frac{b}{a}\)
= \(\frac{-(-8)}{3}=\frac{8}{3}\)

Question 92.
If the product of zeroes of the polynomial f(x) = ax³ – 6x² + 11x – 6 is 4, then find Answer:
Answer:
\(\frac{3}{2}\)
Explanation:
αβγ = \(\frac{-\mathrm{d}}{\mathrm{a}}\) = 4
\(\frac{-(-6)}{a}\) = 4 ⇒ 4a = 6 ⇒ a = \(\frac{6}{4}=\frac{3}{2}\)

Question 93.
Name a polynomial of degree ‘3’.
Answer:
Cubic polynomial.

Question 94.
Is the below graph represents a polynomial ?

Answer:
No, it is not a polynomial.

Question 95.
Find the quotient when x⁴ + x³ + x² – 2x – 3 is divided by x² – 2.
Answer:
x² + x + 3

Question 96.
If α, β, γ are roots of a cubic polyno-mial ax³ + bx² + cx + d, then find α + β + γ
Answer:
\(\frac{-b}{a}\)

Question 97.
If one zero of the quadratic polynomial 2x² + kx – 15 is 3, then find the other zero.
Answer:
\(\frac{-5}{2}\)

Question 98.
If α, β, γ are the zeroes of the polynomial f(x) = ax³ + bx² + cx + d, then
find \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\)
Answer:
\(-\frac{c}{d}\)

Question 99.
The graph

represents, which type of polynomial ?
Answer:
Cubic polynomial.

Question 100.
Write the product and sum of the zeroes of the quadratic polynomial ax² + bx + c respectively.
Answer:
\(\frac{c}{a}, \frac{-b}{a}\)

Question 101.
Write the number of zeroes of the poly¬nomial in the given graph.

Answer:
0

Question 102.
If f(x) = ax² + bx + c has no real zeroes and a + b + c < 0, then write the condition.
Answer:
c < 0

Question 103.
Which type of polynomial ax² + bx + c?
Answer:
Quadratic polynomial.

Question 104.
If one zero of the polynomial
f(x) = (k² + 4)x2 + 13x + 4k is recip¬rocal of the other, then find k.
Answer:
2
Explanation:
\(x \cdot \frac{1}{x}=\frac{4 k}{k^{2}+4}\)
⇒ k² + 4 = 4k
=» k² – 4k + 4 = 0
⇒ (k – 2)² = 0
⇒ k = 2

Question 105.
Write the number of zeroes of the poly-nomial function p(x), whose graph is given below.

Answer:
3

Question 106.
If two zeroes of x³ + x² – 5x – 5 are \(\sqrt{5}\) and \(-\sqrt{5}\) then find its third zero.
Answer:
-1

Question 107.
If one zero of the quadratic polynomial x² – 5x – 6 is 6, then find the other zero.
Answer:
– 1

Question 108.
Find the degree of the polynomial \(a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\ldots .+a_{n} x^{n}\)
Answer:
n

Question 109.
If α and β are the zeros of the polyno-mial f(x) = x² + px + q, then find a
polynomial having \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) as its a p
zeroes.
Answer:
qx² + px + 1

Question 110.
If both the zeroes of a quadratic poly-nomial ax² + bx + c are equal and opposite in sign, then find ‘b’.
Answer:
0

Question 111.
Write the number of zeroes of the poly-nomial in the given graph.

Answer:
1

Question 112.
Find the sum and product of the ze¬roes of polynomial x² – 3 respectively.
Answer:
0,-3

Question 113.
If a < 0, then draw the shape of ax² + bx + c = 0.
Answer:

Question 114.
From the graph write the number of zeroes of the polynomial.

Answer:
2

Question 115.
If α and β are zeroes of the polynomial p(x) = x² – 5x + 6, then find the value of α + β – 3αβ.
Answer:
– 13
Explanation:
α + β = \(\frac{-b}{a}=\frac{-(-5)}{1}\) = 5
αβ = \(\frac{\mathrm{c}}{\mathrm{a}}\) = 6
= α + β – 3αβ = 5 – 3(6)
= 5- 18 = – 13.

Question 116.
What should be subtracted from the polynomial x² – 16x + 30 so that 15 is the zero of the resulting polynomial ?
Answer:
15

Question 117.
Find the number of zeroes lying be-tween – 2 and 2 of the polynomial f(x) whose graph is given below.

Answer:
2

Question 118.
If the zeroes of a quadratic polynomial are equal in magnitude but opposite in sign, then write the relation be¬tween zeroes.
Answer:
Sum of its zeroes is 0.

Question 119.
Find where the graph of the polyno¬mial f(x) = 2x – 5 is a straight line which intersects the x – axis.
Answer:
[\(\frac{5}{2}\),0]

Question 120.
If α,β are the zeroes of the polyno¬mial f(x) = ax² + bx + c, then find \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}=\)
Answer:
\(\frac{b^{2}-2 a c}{c^{2}}\)

Choose the correct answer satis¬fying the following statements.

Question 121.
Statement (A): (2 – \(\sqrt{3}\))is °ne zero of the quadratic polynomial, then other zero will be (2 + \(\sqrt{3}\)).
Statement (B) : Irrational zeroes (roots) always occurs in pairs.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
As irrational roots / zeroes always oc¬curs in pairs therefore, when one, zero
is 2 – √3 , then other will be 2 + √3
So, both A and B are correct and B explains A.
Hence, (i) is the correct option.

Question 122.
Statement (A) : If both zeroes of the quadratic polynomial x² – 2kx + 2 are equal in magnitude but opposite in sign, then value of k is \(\frac { 1 }{ 2 }\).
Statement (B) : Sum of zeroes of a quadratic polynomial ax² + bx + c is \(\frac{-b}{a}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
As the polynomial is x² – 2kx + 2 and
its zeros are equal but opposition sign.
∴ Sum of zeroes = 0 = \(\frac{-(-2 \mathrm{k})}{1}\) = 0
⇒ 2k = 0 ⇒ k = 0
So, A is incorrect but B is correct.
Hence, (iii) is the correct option.

Question 123.
Statement (A): p(x) = 14x³ – 2x² + 8x⁴ + 7x – 8 is a polynomial of degree 3. Statement (B) : The highest power of x in any polynomial p(x) is the degree of the polynomial.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
The highest power of x in the polyno¬mial p(x) = 14x³ – 2x² + 8x⁴ + 7x – 8 is 4.
∴ Degree of p(x) is 4.
So, A is incorrect but B is correct.
Hence, (iii) is the correct option.

Question 124.

Statement (A) : The graph y = f(x) is shown in figure, for the polynomial f(x). The number of zeroes of f(x) is 4.
Statement (B): The number of zero of polynomial f(x) is the number of point of which f(x) cuts or touches the axes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
As the number of zero of polynomial f(x) is the number of {mints at which f(x) cuts (intersects) the x-axis and number of zero in the given figure is 4. So A is correct but B is incorrect Hence, (ii) is the correct option.

Question 125.
Statement (A) : The sum and product of the zeroes of a quadratic polynomial
are – \(\frac { 1 }{ 4 }\) and \(\frac { 1 }{ 4 }\) respectively. Then the
quadratic polynomial is 4x² + x + 1.
Statement (B): The quadratic polyno-mial whose sum and product of zeroes are given is x² – (sum of zeroes)x + product of zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false,
Answer:
(i)
Explanation:
Sum of zeroes = –\(\frac { 1 }{ 4 }\) and
Product of zeroes = \(\frac { 1 }{ 4 }\)
∴ Quadratic polynomial be

∴ Quadratic polynomial be 4x² + x + 4.
So, both A and B are correct and B explains A. Hence, (i) is the correct option.

Question 126.
Statement (A) : If α, β are zeroes of x² – 3x + p and 2α + 3β = 15, then p – 54.
Statement (B) : If α, β are the zeroes of the polynomial ax² + bx + c, then
α + β = \(\frac{-b}{a}\) and αβ = \(\frac{c}{a}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)

Question 127.
Statement (A) : A quadratic polyno¬mial having 4 and – 2 as zeroes is 2x² – 4x – 16.
Statement (B): The quadratic polyno-mial having a and (3 as zeroes is given by p(x) = x² – (α + β)x + αβ
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 128.
Statement (A) : The polynomial p(x) = x³ + x has one real zero.
Statement (B) : A polynomial of nth degree has at most n zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)

Question 129.
Statement (A): If α, β, γ are the zeroes of x³ – 2x² – qx – r and α + β = 0, then 2q= r.
Statement (B) : If α, β, γ are the ze¬roes of ax³ + bx² + cx + d, then
α + β + γ = \(\frac{-b}{a}\), αβ, βγ, γα = \(\frac{c}{a}\), αβγ = \(\frac{-d}{a}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
Clearly, (B) is true, [standard result] cc + p + y = -(-2) = 2 ⇒ 0 + y = 2 ‘ Y – 2 ‘
α + β + γ = – (- 2) = 2=
o + γ = 2
γ = 2
αβγ = -(-r) = r
∴ αβ(2) = r
⇒ αβ = \(\frac{r}{2}\)
⇒ αβ + βγ + γα = -q
⇒\(\frac{r}{2}\) + γ(α+β) = -q
⇒\(\frac{r}{2}\) + 2(0) = -q
⇒ r = 2q
∴ (A) is false
Hence, (iii) is the correct option.

Question 130.
Statement (A) : If one zero of polyno¬mial p(x) = (k² + 4)x² + 13x + 4k is reciprocal of other, then k = 2.
Statement (B) : If x – a is a factor of p(x), then p(α) = 0 i.e., a is a zero of p(x).
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
(B) is true
Let α, 1/α be the zeroes of p(x), then
\(\alpha \cdot \frac{1}{\alpha}=\frac{4 \mathbf{k}}{\mathbf{k}^{2}+4} \Rightarrow \mathbf{1}=\frac{4 \mathbf{k}}{\mathbf{k}^{2}+4}\)
∴ k² -4k + 4 = 0 ⇒ (k- 2)² = 0
∴ k = 2
∴ (A) is true. So, (i) is correct option.

Question 131.
Statement (A) : The polynomial x⁴ + 4x² + 5 has four zeroes. Statement (B) : If p(x) is divided by (x – k), then the remainder = p(k).
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
(B) is true by remainder theorem. Again, x⁴ + 4x² + 5
= (x² + 2)² + 1 > 0 for all x.
∴ Given polynomial has no zeroes.
∴ (A) is not true.
Hence, (iii) is the correct option.

Question 132.
Statement (A) : x³ + x has only one real zero.
Statement (B) : A polynomial of nth degree must have ‘n’ real zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
(B) is false [v a polynomial of n<sup<th degree has at most x zeroes]
Again, x³ + x = x (x² + 1) which has only one real zero (x = 0)
[∵ x² + 1 ≠ 0 for all x ∈ R]
(A) is true.
Hence, (ii) is the correct option.

Question 133.
Statement (A) : If 2, 3 are the zeroes of a quadratic polynomial, then poly¬nomial is x² – 5x + 6.
Statement (B) : If a, P are the zeroes of a monic quadratic polynomial, then polynomial is x² – (a + p)x + ap.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 134.
Statement (A): Degree of a zero poly-nomial is not defined.
Statement (B) : Degree of a non-zero constant polynomial is ‘0’.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)

Question 135.
Statement (A) : Zeroes of f(x) = x³ – 4x – 5 are 5, – 1.
Statement (B): The polynomial whose zeroes are \(2+\sqrt{3}, 2-\sqrt{3}\) is x² – 4x + 7.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 136.
Statement (A) : x² + 4x + 5 has two zeroes.
Statement (B) : A quadratic polyno¬mial can have at the most two zeroes.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Read the below passages and an¬swer to the following questions.

If α, β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, then
α + β = \(\frac{-b}{a}\), αβ = \(\frac{c}{a}\)

If α, β are the zeroes of the quadratic polynomial f(x) = x² – px + q, then find \(\frac{1}{\alpha}+\frac{1}{\beta}\)
Answer:
\(\frac{p}{q}\)
Explanation:
α + β = p, αβ = q
∴ \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}=\frac{p}{q}\)

Question 138.
If α, β are the zeroes of the quadratic polynomial fix) = x² + x – 2, then find \(\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}\)
Answer:
\(\frac{9}{4}\)
Explanation:
α + β = -1, αβ = -2

Question 139.
If α, β are the zeroes of the quadratic polynomial fix) = x² – 5x + 4, then
find \(\frac{1}{\alpha}+\frac{1}{\beta}-2 \alpha \beta\)
Answer:
–\(\frac{27}{4}\)

If α, β, γ are the zeroes of ax³ + bx² + cx + d, then \(\Sigma \alpha=\frac{-b}{a}\), \(\Sigma \alpha \beta=\frac{\mathbf{c}}{\mathbf{a}}, \Sigma \alpha \beta \gamma=\frac{-\mathbf{d}}{\mathbf{a}}\)
Explanation:
α + β = 5, αβ = 4

Question 140.
If α, β, γ are the zeroes of x³ – 5x² – 2x + 24 and ap = 12, then
find αβ = 12, then find ‘γ’
Answer:
-2

Question 141.
If a – b, a, a + b are the roots of x³ – 3x² + x + 1, then find a + b².
Answer:
3

Question 142.
If two zeroes of the polynomial x³ – 5x² – 16x + 80 are equal in magni¬tude but opposite in sign, then find ze¬roes.
Answer:
4,-4, 5.

Manow says that the order of the polynomial (x² – 5) (x³ + 1) is 6.

Question 143.
Do you agree with Manow ?
Answer:
No.

Question 144.
Which mathematical concept is used to judge Manow ?
Answer:
Polynomial.

Question 145.
How much the actual order of given problem ?
Answer:
Degree is 5.

The length of a rectangle is ’5’ more than its breadth.

Question 146.
Express the information in the form of polynomial.
Answer:
(x + 5 + x) = 2x + 5.

Question 147.
Find the perimeter of the rectangle given above.
Answer:
(4x + 10)m

Question 148.
To solve this given problem which mathematical concept was used by you?
Answer:
Polynomial.

Question 149.
Write the correct matching option.

Answer:
A – (ii), B – (iv).

Question 150.
Write the correct matching option.

A – (iii), B – (i).

Question 151.
Write the correct matching option.

Answer:
A – (iv), B – (iii).

Question 152.
Write the correct matching option.

Answer:
A – (ii), B – (i).

Question 153.
Write the correct matching option.

Answer:
A – (iii), B – (i).

Question 154.
Write the correct matching option.

Answer:
A – (iii), B – (v).

Question 155.
Write the correct matching option.

Answer:
A – (iv), B – (i).

Question 156.
If α, β, γ are the zeroes of the polyno-mial px³ + qx² + rx + s then, which of the following matching is correct ?

a) A(i), B(ii), C(iii)
b) A(ii), B(iii), C(i)
c) A(iii), B(i), C(ii)
d) A(ii), B(i), C(iii)
Answer:
(b)

Question 157.
What is the zero of the polynomial 3x – 2 ?
Solution:
f(x) = 3x – 2; f(x) = 0
3x – 2 = 0 ⇒ 3x = 2
⇒ x = 2/3

Question 158.
Write the polynomial in variable ‘x’ whose zero is \(\frac{-k}{a}\).
Solution:
x – \(\frac{-k}{a}\) = 0 ⇒ x + \(\frac{k}{a}\) = 0
⇒ ax + k = 0
∴ ax + k = 0 is a polynomial with degree T in variable ‘x’ whose zero is \(\frac{-k}{a}\).

Practice the AP 10th Class Maths Bits with Answers Chapter 14 Statistics on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 14th Lesson Statistics with Answers

Question 1.
If mean of 8, 6, 4, x, 3, 6 and ‘0’ is ‘4’, then find the value of x.
Answer:
1
Explanation:
\(\frac{6+8+4+x+3+6+0}{7}\) = 4
⇒ 27 + x = 28 ⇒ x = 1

Question 2.
Where the extreme values of some data influences high?
Answer:
In AM

Question 3.
In a data ‘n’ scores are given and if ‘n’ is odd, then find median.
Answer:
(\(\frac{n+1}{2}\))^th event

Question 4.
Find the class mark of 10 – 25.
Answer:
17.5

Question 5.
Mode of the data 5, 3, 4, – 2, 3, 2, 2, 1, p is 3, then the value of ‘p’.
Answer:
3

Question 6.
Find the class interval of the class 11 – 20.
Answer:
10

Question 7.
Mean of 1,2, x, 3 is ‘0’, then the value of ‘x’.
Answer:
-6

Question 8.
If the sum of 15 observations is 420, then find their mean.
Answer:
28
Explanation:
Mean = \(\frac{420}{15}\) = 28

Question 9.
Find the mid value of the class 10 – 19.
Answer:
14.5

Question 10.
Find the median of 2, 3, 4, 5, 6, 7.
Answer:
4.5
Explanation:
Median = \(\frac{4+5}{2}\) = 4.5

Question 11.
To elect the leader of your class from 3 contestants, which central measures is to be considered ?
Answer:
Mode

Question 12.
From the given graph of ogives, find median.

Answer:
40

Question 13.
Find median of the scores
1, 3, 5, 7, 9, ……………. 99.
Answer:
50

Question 14.
Find the mode of the values sin 0°, cos 0°, sin 90° and tan 45°.
Answer:
1
Explanation:
Mode in the values at 0, 1, 1, 1 is 1.

Question 15.
Find the mean of first four odd prime number.
Answer:
6.5

Question 16.
Find A.M. of x – 5, x, x + 5.
Answer:
x

Question 17.
Mode of 3, 4, 5 and x is 5, then find x.
Answer:
5
Explanation:
x = 5

Question 18.
Find the mode of the data
5, 6, 9, 10, 6, 11, 4, 6, 10, 4.
Answer:
6

Question 19.
The letter that represents \(\frac{\mathbf{x}_{\mathbf{i}}-\mathbf{a}}{\mathbf{h}}\)
which is used in measuring mean is …………….
Answer:
u_i

Question 20.
In “more than ogive curve” we consider in drawing …………………..
Answer:
More than cumulative frequency, lower limits.

Question 21.
Observe the following tables.


For finding Arithmetic Mean by Direct method, the suggested frequency distribution table is ……………….
Answer:
only (1) is true.

Question 22.
If \(\overline{\mathbf{x}}\), is the mean of x₁ x₂, x₃, ……………… x_n
(n times), then find \(\sum_{i=1}^{n}\)(x₁ – \(\overline{\mathbf{x}}\))
Answer:
0

Question 23.
Mode can be calculated by = l + (\(\frac{\mathbf{f}_{1}-\mathbf{f}_{0}}{2 \mathbf{f}_{1}-\mathbf{f}_{0}-\mathbf{f}_{2}}\)) × h
here f₁ represents ……………..
Answer:
Frequency of the modal class.

Question 24.
Find the x – coordinate of the point of intersection of the two ogives of grouped data.
Answer:
Median of the data.

Question 25.
3, 2, 4, 3, 5, 2, x, 6. If the mode of this data is 3, then find ‘x’.
Answer:
3

Question 26.
For the terms, x + 1, x + 2, x – 1, x + 3 and x – 2 (x G N), if the median of the data is 12, then find x.
Answer:
11

Question 27.
Which one of the following is NOT a measure of central tendency?
Mean, Median, Mode, Range
Answer:
Range

Question 28.
Find the most stable measure of central tendency.
Answer:
Mean

Question 29.
Find mode of 2004, 2005, 2006, ……………
2019.
Answer:
No mode

Question 30.
Mode = 24.5, Mean = 29.75, then find median.
Answer:
28

Question 31.
Unimodal data may have ……………… modes.
Answer:
1

Question 32.
Who is known as father of statistics ?
Answer:
Fisher

Question 33.
Find AM of \(\frac{1}{3}\),\(\frac{7}{12}\),\(\frac{3}{4}\),\(\frac{1}{2}\),\(\frac{5}{6}\)
Answer:
\(\frac{3}{5}\)

Question 34.
Find the mean of first 5 odd multiples of 5.
Answer:
25
Explanation:
\(\overline{\mathbf{x}}\) = \(\frac{5+15+25+35+45}{5}\) = \(\frac{125}{5}\) = 25

Question 35.
Find the mean of 6,-4,\(\frac{2}{3}\),\(\frac{5}{4}\),\(\frac{7}{6}\)
Answer:
\(\frac{11}{20}\)

Question 36.
In the figure, find the value of median of the data using the graph of less than ogive and more than ogive.

Answer:
20

Question 37.
Find mode of the following distribution.

Answer:
52

Question 38.
Find the measure of central tendency which take into account all data terms.
Answer:
Mean

Question 39.
6, 3, 5, 6, 7, 5, 8, 7, 6, 2k + 1, 9, 7, 13. If the mode of this data is 7, then find ‘k’
Answer:
3
Explanation:
Mode = 7, so 2k + 1 = 7 ⇒ k = 3

Question 40.
For a given data with 50 observations ‘the less than ogive’ and the more than ogive intersect at (15.5, 20). Find the median of the data.
Answer:
15.5

Question 41.
Find median of first 8 prime numbers.
Answer:
9

Question 42.
In a data mean = 72.5 and median = 73.9, then find mode.
Answer:
76.7
Explanation:
Mode = 3 Median – 2 Mean
= 3 × 73.9 – 2 × 72.5
= 221.7 – 145 = 76.7

Question 43.
How much the sum of all deviations taken from AM?
Answer:
0

Question 44.
Find the modal class in the following frequency distribution.

Class	Frequency
0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60	3 9 15 30 18 5

Answer:
30 – 40

Question 45.
Mean – mode is equal to
Answer:
3 (Mean – Median)

Question 46.
In an arranged series of an even num¬ber 2n terms write the median.
Answer:
\(\frac{1}{2}\)(n^th and (n + 1)^th term)

Question 47.
In a data maximum value = x, minimum value = y, then find range.
Answer:
x – y

Consider the following frequency distribution.

Monthly income	Number of families
More than or equal to 10000 More than or equal to 13000 More than or equal to 16000 More than or equal to 19000 More than or equal to 22000	100 85 69 50 33

Question 48.
Find the number of families having income range from ₹ 16000 to ₹ 19000.
Answer:
19

Question 49.
C.I of 1 – 10 is …………….
Answer:
10

Question 50.
In calculating mode Δ₁ is equal to
Answer:
f – f₁

Question 51.
|| represents ………………
Answer:
7

Question 52.
Find mode of any 3 consecutive numbers.
Answer:
No mode.

Question 53.
Find the mean of the following data.

Answer:
8.1

Question 54.
Find AM of first n odd numbers.
Answer:
n

Question 55.
The class mark of 10 – 25 is ……………..
Answer:
17.5

Question 56.
Mean of 5,7,9,x is 9, then find ‘x’.
Answer:
15

Question 57.
Find AM of a – 2, a, a + 2.
Answer:
a

Question 58.
Cumulative frequency curves are called as ………………. curves.
Answer:
Ogive

Question 59.
Find the class marks of a class interval.
Answer:
\(\frac{\text { Upper boundary + lower boundary }}{2}\)

Question 60.
Write the abscissa of the point of intersection of the ‘less than’ type and ‘more than type’ cumulative frequency curves of a grouped data.
Answer:
Mode

Question 61.
Find AM of 1², 2², 3³, 4², …………….. , 20².
Answer:

Question 62.
Write mode of first ‘n’ natural numbers.
Answer:
No mode.

Question 63.
Find mid value of the class 10 – 20.
Answer:
15

Question 64.
Mid values are used to calculate ……………
Answer:
Mean

Question 65.
Find mode of 1, 2, 3, ……………. 10, 10.
Answer:
10

Question 66.
Find the mean of the following data.

Answer:
15.1

Question 67.
Find the median of the data 5, 3, 10, 7, 2, 9, 11,2, 6.
Answer:
6

Question 68.
Histograiri’consists of ………………..
Answer:
Rectangles

Question 69.
Write empirical relation among mean, median and mode.
Answer:
Mode = 3 median – 2 mean

Question 70.
Write the class marks of a class x – y.
Answer:
\(\frac{x+y}{2}\)

Question 71.
Find mean of 1, 2, 3, …………… n.
Answer:
\(\frac{n+1}{2}\)

Question 72.
Find AM of 23, 24, 24, 22, 10.
Answer:
22.6

Question 73.
Find the modal class for the following distribution.

Answer:
30 – 40
Explanation:
Maximum frequency is 57,
i.e., 30 – 40 class.

Question 74.
For a given data with 60 observations, ‘the less than’ ogive and ‘the more than ogive’ intersect at (66.5, 30). Find the median of the data.
Answer:
66.5

Question 75.
Data having two modes is called ……………….
Answer:
Bimodal

Question 76.
pie diagram consists of …………….
Answer:
Sectors

Question 77.
Write the Range of first 5 natural num-bers.
Answer:
4

Question 78.
If the mean of 10, 12, 18, 13, P and 17 is 15, then find ‘P’.
Answer:
20
Explanation:
\(\frac{10+12+18+13+\mathrm{P}+17}{6}\) = 15
⇒ 70 + P = 90 ⇒ P = 20

Question 79.
Find range of 1,2,3, …………….. 10.
Answer:
9

Question 80.
For a distribution with odd numbers (n) of observations, find the median is ………………. observation.
Answer:
\(\frac{n+1}{2}\)^th

Question 81.
If each observation of a data is increased by ‘a’, then mean is increases by ……………….
Answer:
a

Question 82.
Find mode of 5, 6, 9,10,6,12,3,6,11, 10, 4, 6, 7.
Answer:
6

Question 83.
……………….. is effected by extreme values.
Answer:
Mean

Question 84.
1-8, 9-16, 17-24, ………………… then find C.I.
Answer:
8
Explanation:
9 – 1 = 8

Question 85.
Mode is the value of variate which occurs …………….. number of times.
Answer:
Maximum

Question 86.
Find the mean of first five prime numbers.
Answer:
5.6

Question 87.
Find mean of -8, -4 and 4, 8.
Answer:
0

Question 88.
Mean of n-observations x₁, x₂, …………….. x_n repeated , f₁,f₂, f₃, ……………. , f_n times is
Answer:

Question 89.
Find mean of 7, 6, 5, 0, 7, 8, 9.
Answer:
6

Question 90.
Representing the data with the help of pictures is called
Answer:
pictograph

Question 91.
If the mean of the data 2, a + 1, a, a – 2 is 4, then find a.
Answer:
5
Explanation:
\(\overline{\mathbf{x}}\) = \(\frac{2+a+1+a+a-2}{4}\) = 4
⇒ 3a + 1 = 16 ⇒ a = 5

Question 92.
£f(x) = 200, n=20, then find \(\overline{\mathbf{x}}\).
Answer:
10
\(\overline{\mathbf{x}}\) = \(\frac{\Sigma \mathrm{fx}}{\mathrm{n}}=\frac{200}{20}\) = 10

Question 93.
Median = 52.5, Mean = 54, then find Mode.
Answer:
49.5

Question 94.
In the formula of mode in the grouped ……………… data l represents
Answer:
Lower limit of the class with highest frequency.

Question 95.
Cumulative frequency is used to calculate in ……………….
Answer:
Median

Question 96.
Find the lower limit of the modal class of the following data.

Answer:
10

Question 97.
Construction of cumulative frequency table is useful in determining the ………………
Answer:
Median

Question 98.
\(\frac{\text { Sum of observations }}{\text { Number of observations }}\) is equal to
Answer:
Mean

Question 99.
If mode of a distribution is 8 and its mean is 8, then find median.
Answer:
8

Question 100.
Find mode of 0, 1, 2, 3, 3, 3, 7.
Answer:
3

Question 101.
For a distribution with even number (n) of observation, the median is ……………….
Answer:

Question 102.
If the mean of x₁ x₂, …………….. x_n is \(\overline{\mathbf{x}}\), then find the mean of \frac{\mathbf{x}_{1}}{\mathbf{a}}, \(\frac{\mathbf{x}_{2}}{\mathbf{a}}\) ……………….. \(\frac{\mathbf{x}_{n}}{\mathbf{a}}\)
Answer:
\(\frac{\overline{\mathrm{x}}}{\mathrm{a}}\)

Question 103.
Find AM of 3 and 4.
Answer:
3.5

Question 104.
For a given data with 120 observations, the “less than ogive” and the “more than ogive” intersect at (42.5,60). Find the median of the data.
Answer:
42.5

Question 105.
Find the AM of 10 consecutive num¬bers starting with n + 1.
Answer:
x + 5.5

Question 106.
………………. is based on all observations.
Answer:
Mean

Question 107.
Mode of a continuous grouped distribution is ………………….
Answer:
l + \(\frac{\mathrm{f}_{1}-\mathrm{f}_{0}}{\left(\mathrm{f}_{1}-\mathrm{f}_{0}\right)+\left(\mathrm{f}_{1}-\mathrm{f}_{2}\right)}\) × h

Question 108.
Find mode of 20, 3, 7, 13, 3, 4, 6, 7, 19, 15, 7, 18, 3.
Answer:
3,7

Question 109.
Find the sum of lower limit of median class and upper limit of modal class is ………………

Answer:
90

Question 110.
The information collected is called ……………..
Answer:
data

Question 111.
Find AM of 1, 2, x, 3 is 0, then find ‘x’.
Answer:
-6

Question 112.
For a symmetrical distribution, which is correct?
A) Mean < Mode < Median B) Mean > Mode > Median
C) Mode = \(\frac{\text { Mean }+\text { Median }}{2}\)
D) Mean = Median = Mode
Answer:
D) Mean = Median = Mode

Question 113.
\(\overline{\mathbf{x}}\) = 2p + q, M = p + 2q, then find Z.
Answer:
4q – p

Question 114.
Find the median class of the following distribution.

Answer:
30 – 40

Question 115.
A data has 13 observations arranged in descending order which observation represents the median of data?
Answer:
7^th

Question 116.
For a continuous grouped frequency distribution, the median is given by
Answer:
l + (\(\frac{\frac{\mathrm{n}}{2}-\mathrm{cf}}{\mathrm{f}}\)) × h

Question 117.
A data arrange in descending order has 25 observations. Which observation represents the median?
Answer:
13^th

Question 118
……………… of all bars is same in bar graph.
Answer:
Width

Question 119.
Find mean of a + 1, a + 3, a + 4 and a + 8.
Answer:
a + 4
Explanation:
Mean = \(\frac{a+1+a+3+a+4+a+8}{4}\)
= \(\frac{4 a+16}{4}\) = a + 4

Question 120.
If assumed mean is ‘a’, then write the mean.
Answer:
a + \(\frac{\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}}{\Sigma \mathrm{f}_{\mathrm{i}}}\)

Question 121.
Find the upper limit of median class of the following distribution.

Answer:
18

❖ Choose the correct answer satisfying the following statements.
Question 122.
Statement (A) : The arithmetic mean of the following given frequency distribution table is 13.81.

Statement (B) : \(\overline{\mathbf{x}}\) = \(\frac{\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\Sigma \mathrm{f}_{\mathrm{i}}}\)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i) Both A and B are true.
Explanation:
Both A and B are true, B is the correct explanation of the A.
Hence, (i) is the correct option.

Question 123.
Statement (A) : If the number of runs scored by 11 players of a cricket team of India are 5, 19, 42, 11, 50, 30, 21, 0, 52, 36, 27, then median is 30.
Statement (B) : Median = \(\frac{(n+1)^{t h}}{2}\)
value, if n is odd.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
iii) A is false, B is true.
Explanation:
Arranging the terms in ascending order, 0, 5, 11, 19, 21, 27, 30, 36, 42, 50, 52
Median value = (\(\frac{11+1}{2}\))^th = 6^th value = 27
∴ Option (iii) is true.

Question 124.
Statement (A) : If the value of mode and mean is 60 and 66 respectively, then the value of median is 64.
Statement (B) :
Median = \(\frac{1}{2}\)(mode + 2 mean)
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
ii) A is true, B is false.
Explanation:
Median = \(\frac { 1 }{ 3 }\) (Mode + 2 Mean)
= \(\frac { 1 }{ 3 }\) (60 + 2 × 66) = 64
∴ Option (ii) is correct.

Read the below passages and answer to the following questions.
The following table gives the weekly wages of workers in a factory.

Question 125.
Find the mean.
Answer:
69
Explanation:
Mean = \(\frac{\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\Sigma \mathrm{f}_{\mathrm{i}}}=\frac{5520}{80}\) = 69

Question 126.
Find the modal class.
Answer:
55 – 60
Explanation:
Modal Class : We know that class of maximum frequency is called the modal class, i.e., 55 – 60 is the modal class.

Question 127.
Find the number of workers getting weekly wages, below ₹ 80.
Answer:
60
The marks of 20 students in a test were as follows : 5, 6, 8, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 18, 19, 20.
Explanation:
Number of workers getting weekly wages below ₹ 80 according to table = 60 workers.

Question 128.
Calculate the mean.
Answer:
13
Explanation:
Arrange in ascending order

Question 129.
Calculate the median.
Answer:
13.5
Explanation:
Here, n = 20 which is an even number

= 13.5

Question 130.
Calculate the mode.
Answer:
15
A professor keeps data on students tabulated by performance and sex of the student. The data is kept on a computer disk.

Explanation:
In the data, 15 occurs the maximum times i.e., 3 times.
∴ Mode = 15

Question 131.
How many students are both female and excellent?
Answer:
0
Explanation:
There is no female excellent student in the class.

Question 132.
What proportion of good students are male?
Answer:
0.73
Explanation:
Proportion of good male students
= \(\frac{22}{30}\) = 0.73

Question 133.
What proportion of female students are good?
Answer:
0.26
Explanation:
Proportion of good female students
\(\frac{8}{30}\) = 0.26

Question 134.
Write the short form of the expansion in symbolic form.
1 + 2 + 3 + 4 + ……………………. + n
Answer:
Σn

Practice the AP 10th Class Maths Bits with Answers Chapter 2 Sets on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 2nd Lesson Sets with Answers

Question 1.
Which type of set of human being that reside on moon is ……………….
Answer:
null set

Question 2.
Write the number of subsets of the null set Φ.
Answer:
1

Question 3.
Ifn(A) = 8, n(B) = 3, n(A ∩ B) = 2, then find n(A ∪ B).
Answer:
9
Explanation:
n(A∪B) = n (A) + n (B) – n (A ∩ B)
= 8 + 3-2 = 9

Question 4.
The number of subsets of a set is 16, then find the set has ………… elements.
Answer:
4
Explanation:
2ⁿ = 16 = 2⁴
⇒ no. of elements in the set = 4

Question 5.
Write the number of subsets of the set A = {l, 2,3, 4}.
Answer:
16
Explanation:
n (A) = 4, no. of subsets = 2ⁿ = 2⁴ = 16

Question 6.
If A⊂ B, n(A) = l2and n(13) = 20, then find the value of n (B – A).
Answer:
8
Explanation:
A ⊂ B, son (B – A) = 20- 12 .= 8

Question 7.
Roster form of (x: x is a prime number and a divisor of 6).
Answer:
{2,3}

Question 8.
Write an example for finite set in your own.
Answer:
{x/x∈N and x² = 9}

Question 9.
If A⊂B,n(A) = 4 and n(B) = 6,then find n(A∪ B).
Answer:
6
Explanation:
A ⊂ B, so n (A ∪ B) = n (B) = 6

Question 10.
If A⊂B, then A∩B is
Answer:
A
Explanation:
A⊂B, so A∩B = A

Question 11.
If the union of two sets is one of the set itself, write the relation between the two sets.
Answer:
One set is a subset of other set.

Question 12.
The following venn diagram indicates

Answer:
A⊂B

Question 13.
If A From the venn diagram, find A ∪ B.

Answer:
{5, , 7, 8}

Question 14.
If A = {x : x is a letter in the word EX¬AMINATION}, then write its roster form.
Answer:
A = {e, x, m, i, n, a, t, o}

Question 15.
If A = {x : x is a letter in the word HEADMASTER}; then write its ros-ter form.
Answer:
A — {h, e, a, d, m, s, t, r}

Question 16.
If n (A) = 12 and n (A ∩ B) = 5, then find n (A – B).
Answer:
7
Explanation:
n (A – B) = n(A) – n(A∩B) = 12 – 5 = 7

Question 17.
The following venn diagram indicates

Answer:
A, B are disjoint sets.

Question 18.
The shaded region in the given figure shows.

Answer:
μ – B = B’

Question 19.
Write the relation between sets in the following venn diagram.

Answer:
A ∩ B = Φ

Question 20.
If A ={1,2, 3}, B = (3,4, 5), then find A Δ B.
Answer:
A Δ B = {1,2, 4, 5}
Explanation:
A Δ B = (A ∪ B) – (A ∩ B)
= {1, 2, 3, 4, 5}- {3} = {1,2, 4, 5}

Question 21.
(A’)’is equal to …………….
Answer:
A

Question 22.
An object of a set is called ……………..
Answer:
Element

Question 23.
2 is ………………. of set of natural numbers.
Answer:
An element

Question 24.
Number of elements in a singleton set is …………………..
Answer:
1

Question 25.
If A, B are disjoint sets such that n (A) = 4 and n (A ∪ B) = 7, then find n(B).
Answer:
3
Explanation:
n(A∪B) = n (A) + n (B) – n (A ∩ B)
⇒ 7 = 4 + n (B) – 0
⇒ n(B) = 7 – 4 = 3

Question 26.
‘O’ is to set of whole numbers.
Answer:
belong

Question 27.
n (A) = 4, then write n(p(A)).
Answer:
16
Explanation:
n(P(A)) = 2ⁿ = 2⁴ = 16

Question 28.
If A = {1, 2, 3} and B = {1, 2, 3, 4}, then we say A is a …………….. of B.
Answer:
Subset

Question 29.
Φ is equal to A.
Answer:
μ

Question 30.
A set is a ……………… of objects.
Answer:
Well defined collection.

Question 31.
{2, 4,6, 8, 10} is an example of which type of set ?
Answer:
Finite

Question 32.
If A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, then find A – B.
Answer:
{1,3}

Question 33.
A = {2, 4, 6, 8, 10}, then write its rule form.
Answer:
A = {x / x is an even number, x ≤ 10}

Question 34.
If B = {1, 7, 2, 0, 6}, then find n(B).
Answer:
5

Question 35.
A – (A -B) is equal to ……………..
Answer:
A ∩ B

Question 36.
The objects in the set are called ……………….. of the set.
Answer:
Elements

Question 37.
Let A, B are two sets such that n (A) = 5, n(B) = 7, then write the maximum number of elements in A ∪ B.
Answer:
12

Question 38.
Empty set is denoted by ………………..
Answer:
Φ

Question 39.
Write A Δ B.
Answer:
(A – B) ∪

(B – A) (or) (A∪B)-(A ∩ B)

Question 40.
A = {1, 2, 3}, B = {3, 4, 5}, then find A ∩ B.
Answer:
{3}

Question 41.
– 3 is of the set of whole numbers.
Answer:
not an element

Question 42.
If n(A ∪ B) = 8, n(A) = 6, n(B) = 4, then find n(A ∩ B).
Answer:
2

Question 43.
The number of elements in a set is called the…………….of the set.
Answer:
Cardinal number

Question 44.
A ∪ Φ is equal to …………………
Answer:
A

Question 45.
{x / x ≠ x} is which type of set ?
Answer:
Empty

Question 46.
B = {x/x ∈ N and x < 1000} is a ……………type of set.
Answer:
Finite

Question 47.
Write the symbol used for belongs to’.
Answer:
∈

Question 48.
n (Φ) is equal to ………………..
Answer:
0

Question 49.
Write (2, 6, 10} ∩ (8, 9, 11, 12, 13}.
Answer:
Φ

Question 50.
{x / x is a student of your school} is in which form ?
Answer:
Set Builder

Question 51.
Every set is ……………. of itself.
Answer:
Subset

Question 52.
A = {1, 2,7, 10}, then use symbol be-tween 7 and A.
Answer:
∈

Question 53.
If A = {1, 2, 3, 4}, then find the cardi-nality of set A.
Answer:
4

Question 54.
A≠B means, set A and B do not contains same elements. This statement is true (or) false.
Answer:
True

Question 55.
A = {1, 2, 3}, B = {12, 0, 5}, then find A-B.
Answer:
A

Question 56.
A = {x / x + 4 = 4}, then write Roster form of A.
Answer:
{0}

Question 57.
Which type of set has no elements in it ?
Answer:
Null set

Question 58.
If A ∪ B = A ∪ C and A ∩ B = A ∩ C, then write the relation between these sets.
Answer:
B = C

Question 59.
A set with only one element is known as ……………. set.
Answer:
singleton

Question 60.
The set of all real numbers is, which type of set ?
Answer:
Infinite set

Question 61.
Roster form of B = \(\left\{\frac{x}{x}+3=6\right\}\), B = ?
Answer:
{3}

Question 62.
‘μ’ is equal to
Answer:
Φ

Question 63.
If A ⊂ B and A ≠ B, then A’ is called the ………………. of B.
Answer:
Proper subset

Question 64.
{x / x is a natural number} is which type of set ?
Answer:
Infinite

Question 65.
Write the number of elements in the empty set.
Answer:
0

Question 66.
The null set is sometimes denoted as .
Answer:
{} = Φ

Question 67.
If in two sets A and B, every element of A is in B and every element of B is in A, then write it as
Answer:
A = B

Question 68.
Another name to Roster form is ……………….
Answer:
List

Question 69.
A’ – B’ is equal to
Answer:
B-A

Question 70.
If every element of A is also an element of B, then write this symboliically.
Answer:
A⊂B

Question 71.
A ⊂ B, then find A – B.
Answer:
Φ

Question 72.
“0 does not belong to the set of natural numbers”. Write the statement sym¬bolically.
Answer:
0 ∉ N

Question 73.
A = {1,2, 4}, B = {3, 5, 6}, then write the relation.
Answer:
A ∩ B = Φ

Question 74.
If A ⊂ B, then find A ∪ B.
Answer:
B

Question 75.
If B = {1,7, 2, 0,6}, then find n(B).
Answer:
5

Question 76.
Write Roster form of the set of natu¬ral numbers less than 6.
Answer:
(1, 2, 3, 4, 5}

Question 77.
If A ⊂ B, then find A-B.
Answer:
Φ

Question 78.
A ∪ Φ is equal to ……………
Answer:
A

Question 79.
A ∪ B = B ∪ A is called ……………. law.
Answer:
Commutative

Question 80.
If A = {1, 2, 2, 1, 3, 4, 3, 4}, then find n(A).
Answer:
4

Question 81.
Write cardinal number of null set.
Answer:
0

Question 82.
K = {x/x is a prime number less than 13}. Write list form of K.
Answer:
K= {2, 3, 5, 7, 11}

Question 83.
W – {0} is equal to …………………
Answer:
N

Question 84.
In the rule form, the slant bar stands for
Answer:
such that

Question 85.
A = {a, b, c}, B = {c, a, b}, then write the relation between A and B.
Answer:
A – B

Question 86.
Write the set formed from the letters of the word “SCHOOL “.
Answer:
{S, C, H, O, L}

Question 87.
A = {1, 2, 7}, B = {2, 1}, then write the relation between A and B.
Answer:
B⊂A

Question 88.
If A ⊂ B, B ⊂ C, then write the relation between A and C.
Answer:
A ⊂ C

Question 89.
If A ⊂ B, then find A∪(B – A).
Answer:
B

Question 90.
Write the set builder form of D = \(\left\{1, \frac{1}{2}, \frac{1}{3} ; \frac{1}{4}, \frac{1}{5}, \frac{1}{6}\right\}\)
Answer:
D = {x / x ∈ 1/n ,n ∈ N,n < 7}

Question 91.
In set builder form, the letter “X” denotes any………… that belongs to the set.
Answer:
Arbitrary element.

Question 92.
Write the Roster form of the set of multiples of 5 which lie between 25 and 50 is
Answer:
{30, 35, 40, 45}

Question 93.
Write the name of German mathemati¬cian who developed the theory of sets.
Answer:
George Cantor.

Question 94.
N∩W is equal to ………………..
Answer:
N

Question 95.
A = Φ, B = Φ, then find A∩B.
Answer:
Φ

Question 96.
Write the identity element under union of sets.
Answer:
Φ

Question 97.
A ∩ B = Φ, then find B ∩ A’.
Answer:
B

Question 98.
A = {all primes less than 20}
B = {all whole numbers less than 10}, then find A∩B.
Answer:
{2,3, 5, 7}

Question 99.
μ’ = Φ is called …………….. law.
Answer:
Complementary

Question 100.
A ∪ A = A is called……………law.
Answer:
Idempotent.

Question 101.
If A and B are disjoint sets, then write n (A ∪ B).
Answer:
n (A) + n (B)
Explanation:
n (A ∪ B) = n (A) + n (B)

Question 102.
If A = Φ, B = Φ, then find A∪B.
Answer:
Φ

Question 103.
n (A) = 3, then write the number of proper subsets of A.
Answer:
7
Explanation:
Proper subsets are 2ⁿ – 1 = 2³ – 1
= 8 – 1
= 7

Question 104.
A ∪ B = A ∩ B, then write the relation between A and B.
Answer:
A = B

Question 105.
n (A ∪ B) = 51, n (A) = 20, n (A ∩ B) = 13, then find n (B).
Answer:
44
Explanation:
n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
⇒ 51 = 20 + n(B)- 13
⇒ 31 + 13 = n(B) = 44

Question 106.
A’ = B, then find A ∪ B.
Answer:
μ
Explanation:
A’= B ⇒ μ – A = B
⇒ A u B = μ

Question 107.
n (A) = 10, n (B) = 4, n (A ∩ B) = 2, then find n (A ∪ B).
Answer:
12

Question 108.
μ ∪ Φ is equal to
Answer:
μ

Question 109.
If A ∩ B = Φ then find n (A ∩ B).
Answer:
n (A) + n (B)

Question 110.
Write the identity element under intersection of sets.
Answer:
μ

Question 111.
A∪B = B, then write the relation between A and B.
Answer:
A⊂B

Question 112.
The given venn diagram represents.

Answer:
AΔB

Question 113.
Φ Δ Φ is equal to
Answer:
Φ

Question 114.
(A ∪ B)’ is equal to
Answer:
A’ ∩ B’
Explanation:
(A ∪ B)’ = A’∩B’

Question 115.
Draw the venn diagram of A – B.
Answer:

Question 116.
If the number of proper subsets of a given set is 31, then how many ele-ments the set contains ?
Answer:
5
Explanation:
2ⁿ – 1 = 31 ⇒ 2ⁿ = 32 = 2⁵
∴ no. of elements are 5.

Question 117.
Write the intersection of set of ratio-nal numbers and set of irrational num¬bers.
Answer:
Real numbers

Question 118.

This venn diagram represents
Answer:
A∩B

Question 119.
From the venn diagram, write the set A∪B.

Answer:
A ∪ B = {1, 2, 4, 5, 6, 7, 10}
Explanation:
A ∪ B = {1,2, 4, 5, 6, 7, 10}

If A = {x : x is a natural number}
B = {x : x is an even natural number}
C = {x : x is an odd natural number) and
D = {x : x is a prime number}

Question 120.
Find A∩B.
Answer:
A∩B = (1, 2, 3, 4, } ∩ (2, 4, 6, 8…………… }
= {2,4,6, 8, ….} = B{∵B⊂A}

Question 121.
Find A ∩C.
Answer:
A ∩ C = {1, 2, 3,4, …} ∩ {1, 3, 5, 7,…} = (1,3, 5, 7,…} = C(∵C⊂A}

Question 122.
Find A ∩D.
Answer:
A∩D = {1,2, 3, 4, …} ∩ {2, 3,5,7,…} = {2, 3, 5, 7,…} = D{∵ D⊂A)

By observing the below diagram and answer the following questions :

Question 123.
Find A∪B.
Answer:
A∪B = {2, 3, 4, 5, 6} ∪ {7, 8, 9, 10} = {2,3,4,5,6,7,8,9,10}

Question 124.
Find A∩B.
Answer:
A∩B = {2, 3, 4, 5, 6} ∩ {7, 8, 9, 10}
= { } = Φ

Question 125.
Find A Δ B.
Answer:
A Δ B = (A ∪ B) – (A ∩ B) = A ∪ B . { ∵ A ∩ B = Φ}

By observing the below diagram and answer the following questions.

Question 126.
What do you observes in P and Q ?
Answer:
There are no common elements

Question 127.
Name the type of sets P and Q.
Answer:
P and Q are disjoint sets.

Question 128.
Write the relation between P and Q.
Answer:
P ∩ Q = Φ

Question 129.
Define disjoint sets.
Answer:
There is no common elements in any two sets such type of sets are called disjoint sets.

By observing the below information and answer the following questions.

D = The set of all letters in the word TRIGONOMETRY

Question 130.
Write the Roster form of set ‘D’.
Answer:
D = {T, R, I, G, O, N, M, E, Y}

Question 131.
Write the cardinal number of set D.
Answer:
n (D) = 9
By observing the below diagram and answer the following questions.

Question 132.
Find n(A).
Answer:
n (A) = 2

Question 133.
Find n(B).
Answer:
n (B) = 1

Question 134.
Find n(A ∩ B).
Answer:
n(A ∩ B) = Φ

Question 135.
Find n(A ∪ B).
Answer:
n(A ∪ B) = 3

Question 136.
Write the relation between n (A), n (B), n (A ∩ B) and n (A ∪ B).
Answer:
n (A) + n (B) = n(A∪B) + n(A∩B)
1 + 2 = 3 + 0 = 3

Write the correct matching options.

Question 140.
Roster form
A) {a, e, i, o, u} []
B) {2, 5, 10, 17} []

Choose the correct answer satistying the following statements.

Question 137.
Statement (A): If A = {1,2,3, 4, 5,6}, B = {7,8,9, 10, 11} and C = {6,8, 10, 12,14}, then A and B are disjoints sets.
Statement (B) : Two sets A and B are said to be disjoint, if A ∩ B = Φ
i) Both A and B are true
ii) A’ is true, ‘B’ is false
iii) A is false,’B’is true
iv) Both A and B are false
Answer:
i)

Question 138.
Statement (A) : The set of all rect-angles in contained in the set of all squares.
Statement (B) : The sets P = {a} and B = {{a}} are equal.
i) Both A and B are true
ii) A is true, ‘B’ is false
iii) A’ is false, ‘B’ is true
iv) Both A and 6 are false
Answer:
ii)

Question 139.
Statement (A) : For any two sets A and B, we have A – B = {x : x ∉ A and x∈B}
Statement (B) : For any two sets A and B, we have A-B = {x:x∈A and x∉B) andB-A = {x:x ∈ B and x ∉ A}
i) Both A and B are true
ii) A’ is true, ‘B’ is false
iii) A’ is false, ‘B’ is true
iv) Both A and B are false
Answer:
iii)

Write the correct matching options.

Question 140.
Roster form

Answer:
A – (i), B – (iv).

Question 141.

Answer:
A – (iii), B – (ii).

Question 142.

Answer:
A – (iv), B – (iii).

Question 143.

Answer:
A – (i), B – (ii).

Question 144.

Answer:
A – (ii), B – (iii).

Question 145.

Answer:
A – (i), B – (iv).

Question 146.
If A = {1,2,3} and Φ = { }, find A∩Φ
Answer:
Φ

Question 147.
Find n(A ∪ B) from the figure

Answer:
5

Question 148.
How many subsets does a set of three distinct elements have ?
Answer:
8 sub-sets

Question 149.
If A = {1,2,3} andB = {2,4,6}. What is n(A ∪ B) ?
Solution:
A = {1, 2, 3}, B = {2, 4, 6}
A∪B = {1, 2, 3}∪{2, 4,6} = {1,2,3,4,61
n(A ∪ B) = 5

Practice the AP 10th Class Maths Bits with Answers Chapter 1 Real Numbers on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 1st Lesson Real Numbers with Answers

Question 1.
Find the rational number in between \(\frac { 1 }{ 2 }\) and √1
Answer:
\(\frac { 3 }{ 4 }\)

Question 2.
Write the name set of rational and ir-rational numbers.
Answer:
Real numbers.

Question 3.
Write the logarithmic form of 3⁵ = 243.
Answer:
log₃243 = 5

Question 4.
Write the symbol of “implies”.
Answer:
⇒

Question 5.
Write the prime factorisation of 729.
Answer:
3⁶

Question 6.
If ‘x’ and ‘y’ are two prime numbers, then find their HCF.
Answer:
1
Explanation:
HCF of any two prime numbers is always 1.

Question 7.
Find the value of log₁₀ 0.01.
Answer:
-2
Explanation:
log₁₀0.01 = log₁₀ \(\frac{1}{10^{2}}\)
= log₁₀10^-2 = – 2

Question 8.
Find the number of odd numbers in between ‘0’ and 100.
Answer:
50

Question 9.
Write the exponential form of log₄8 = x.
Answer:
4^x = 8.
Explanation:
Exponential form of log₄8 = x is 4^x = 8

Question 10.
How much the value of \(\frac{36}{2^{3} \times 5^{3}}\) in decimal form ?
Answer:
0.036.

Question 11.
LCM of two numbers is 108 and their HCF is 9 and one of them is 54, then find the second one.
Answer:
18
Explanation:
LCM x HCF = one number x second number
108 x 9 = 54 x second one , 108 x 9
⇒ Second one = \(\frac{108 \times 9}{54}\) = 18.

Question 12.
\(\frac { 3 }{ 8 }\) is example for decimal.
Answer:
Terminating decimal.

Question 13.
If \(\mathbf{a} \sqrt{\mathbf{c}}=\sqrt{\mathbf{a c}}\) , then find the value of ‘a’, (a, c are positive integers),
Answer:
a = 1

Question 14.
Find the value of 9 – \(0 . \overline{9}\).
Answer:
8
Explanation:
9 – \(\frac{9}{9}\) = 9 – 1 = 8.

Question 15.
Write rational number that equals to \(2 . \overline{6}\)
Answer:
\(\frac { 8 }{ 3 }\)

Question 16.
Write the value of log₂₅ 5.
Answer:
\(\frac { 1 }{ 2 }\)
Explanation:
log₂₅5 = log₅5¹ = \(\frac{1}{2}\)log₅5 = \(\frac{1}{2}\)

Question 17.
The fundamental theorem of arithmetic is applicable to, which least number ?
Answer:
2

Question 18.
Find the last digit of 6⁵⁰.
Answer:
6

Question 19.
Which of the following is a terminat¬ing decimal ?
A) \(\frac { 10 }{ 81 }\)
B) \(\frac { 41 }{ 75 }\)
C) \(\frac { 8 }{ 125 }\)
D) \(\frac { 3 }{ 14 }\)
Answer:
C)

Question 20.
Find the value of log2 32.
Answer:
5

Question 21.
Which of the following is not irrational ?
A) √2
B) √3
C) √4
D) √5
Answer:
(C)

Question 22.
Find the value of log₁₀ 0.001.
Answer:
-3

Question 23.
Find the number of prime factors of 36.
Answer:
2 and 3
Explanation:
36 = 2² x 3²
∴ Two prime numbers i.e., 2 and 3.

Question 24.
Write the exponential form of
Iog₁₀ = -3.
Answer:
10^-3 = 0.001

Question 25.
Define an irrational number.
Answer:
Which cannot be written in the form
of p/q where p, q ∈ Z, q ≠ 0.

Question 26.
Find the LCM of 24 and 36.
Answer:
72

Question 27.
Find the logarithmic form of a^b = c.
Answer:
log_ac = b.

Question 28.
If 3 log (x + 3) = log 27, then find the value of x.
Answer:
0
Explanation:
3 log (x + 3) = log 27
⇒ log (x + 3)3 = log 33
⇒ x + 3 = 3 ⇒ x = 0
log₃729 = x ⇒ 3^x = 729 = 3⁶ ⇒ x = 6

Question 29.
If P₁ and P₂ are two odd prime num-bers, such that P₁ > P₂, then the value of \(\mathbf{P}_{1}^{2}-\mathbf{P}_{\mathbf{2}}^{2}\) results number
Answer:
An even number.

Question 30.
Find the value of \(\log _{10} 2+\log _{10} 5\)
Answer:
1

Question 31.
If log₃ 729 = x, then find the value of x.
Answer:
6
Explanation:
log₃ 729 = x ⇒ 3^x = 729 = 3⁶ ⇒ x = 6

Question 32.
Write the number of digits in the fractional part of the decimal form of \(\frac{7}{40}\).
Answer:
3
Explanation:
\(\frac{7}{40}=\frac{7}{2^{3} \times 5^{1}}\)
In Denominator 2ⁿ x 5^m is equal to 3.

Question 33.
Write the prime factorization of 144.
Answer:
2⁴ x 3²

Question 34.
Find the number of prime factors of 72.
Answer:
2

Question 35.
log₃ x² = 2, then find the value of x.
Answer:
3
Explanation:
log₃x² = 2 ⇒ 3² = x² ⇒ x = 3

Question 36.
Find the value of \(9 \sqrt{2} \times \sqrt{2}\)
Answer:
18

Question 37.
Find the value of log_0.1 0.01.
Answer:
2
Explanation:
log _0.1 0.01 = log_10^-1 10^-2
= \(\frac{-2}{-1}\) log₁₀ 10 = 2

Question 38.
0.3030030003 ………………. is an ………………. number.
Answer:
Irrational.

Question 39.
\(\frac{27}{82}\) is a …………. decimal.
Answer:
Non-terminating

Question 40.
If log 2 = 0.30103, then find log 32.
Answer:
1.50515
Explanation:
log 32 = log 2⁵ = 5 log 2
= 5 x 0.30103 .
= 1.50515

Question 41.
Expand log 15.
Answer:
log5 + log3

Question 42.
Find the value of log₁₀ 10.
Answer:
1

Question 43.
Calculate the value of log8 128.
Answer:
\(\frac{7}{3}\)

Question 44.
Find the value of \(5 \sqrt{5}+6 \sqrt{5}-2 \sqrt{5}[/latex[
Answer:
9√5

Question 45.
743.2111111 … is a number.
Answer:
Rational

Question 46.
Find the value of log₅ 125.
Answer:
3

Question 47.
Expand log₁₀ [latex]\frac{125}{16}\)
Answer:
3 log 5 – 4 log 2
Explanation:
log₁₀ \(\frac{125}{16}\) = log₁₀ 125 – log₁₀16
= log₁₀5³ – log₁₀2⁴
= 3 log 5 – 4 log 2

Question 48.
Find the L.C.M of the numbers 2⁷ x 3⁴ x 7 and 2³ x 3⁴ x 11.
Answer:
2⁷ x 3⁴ x 7 x 11

Question 49.
If log_a a^x2 – 5x + 8 = 2, then find x.
Answer:
2 or 3.
Explanation:
\(\log _{a} a^{x^{2}-5 x+8}=\log _{a} a^{2}\)
{2 was write down as 2-log_a a}
x² – 5x + 8 = 2
x² – 5x + 6 = 0
by solving equation x = 2 or 3

Question 50.
Find the value of log_a \(\frac{1}{a}\).
Answer:
– 1

Question 51.
Find the value of log₁ 1.
Answer:
Not defined.

Question 52.
If log₁₀ 0.00001 = x, then find x.
Answer:
-5

Question 53.
Find the value of log_b a • log_a b.
Answer:
1.

Question 54.
16 x 64 = 4^k, then find the value of k.
Answer:
5
Explanation:
16 x 64 = 4^k
⇒ 4² x 4³ = 4^k
⇒ 4⁵ = 4^k
⇒ k = 5

I’m

Question 55.
Write exponential form of log₄64 = 3.
Answer:
4³ = 64

Question 56.
Calculate the value of \(\log _{9} \sqrt{3 \sqrt{3 \sqrt{3}}}\)
Answer:
\(\frac{7}{16}\)

Question 57.
If ‘m’ and ‘n’ are co-primes, then find H.C.F of m² and n².
Answer:
1

Question 58.
\(\sqrt{5}+\sqrt{7}\) is number.
Answer:
An irrational.

Question 59.
Find the H.C.F. of the numbers
3⁷ x 5³ x 2⁴ and 3² x 7⁴ x 2⁸.
Answer:
2⁴ x 3²

Question 60.
\(\frac{13}{125}\) is a ……………… decimaL
Answer:
Terminating

Question 61.
Write the decimal expansion of 0.225 in its rational form.
Answer:
\(\frac{9}{40}\)

Question 62.
How many prime factors are there in the prime factorization of 240.
Answer:
3

Question 63.
14.381 may certain the denominator when expressed in p/q form.
Answer:
2³ x 5³

Question 64.
By which numbers 7 x 11 x 17 +34 is divisible, write them.
Answer:
17 and 79
Explanation:
Given number = 7 x 11 x 17 + 34
= 17 (7 x 11 + 2)
= 17 x 79
Given number has 17 and 79 are factors.

Question 65.
Write log\(\frac{x^{2} y^{3} z^{4}}{w^{5}}\) in the expanded form.
Answer:
2 log x + 3 log y + 4 log z – 5 log w
Explanation:
log x²y³z⁴ – log w⁵ = log x² + log y³ + log z⁴ – log w⁵
= 2log x + 3log y + 4log z – 51og w

Question 66.
Write the logarithmic form of 12² = 144.
Answer:
log₁₂ 144 = 2

Question 67.
Expand log 81 x 25.
Answer:
4log 3 + 2 log 5

Question 68.
What is the L.C.M. of greatest two digit number and the greatest three digit number.
Answer:
9 x 11 x 111

Question 69.
Write logarithmic form of 19² = 361.
Answer:
log₁₉361 = 2

Question 70.
3 X 5 x 7 x 11 + 35 is number.
Answer:
Composite

Question 71.
Write the decimal expansion of \(\frac{101}{99}\).
Answer:
\(1 . \overline{02}\)

Question 72.
If P₁, p₂, p₃, …………… p_n are co-primes, then
their LCM is
Answer:
P₁p₂ …………… p_n

Question 73.
In the above problem find HCF.
Answer:
1

Question 74.
n² – 1 is divisible by 8, if ‘n’ is number.
Answer:
An odd number.

Question 75.
If x and y are any two co-primes, then find their L.C.M.
Answer:
x.y

Question 76.
Write \(\frac{70}{71}\) is which type of decimal ?
Answer:
Non-terminating, repeating.

Question 77.
0.12 112 1112 11112………………is…………… type of number.
Answer:
Irrational

Question 78.
Write \(\frac{123}{125}\) is which type of decimal ?
Answer:
Terminating.

Question 79.
Write the product of L.C.M. and H.C.F. of the least prime and least composite number.
Answer:
8

Question 80.
\(\sqrt{2}-2\) is…………………number.
An irrational.

Question 81.
Find the number of prime factors of 1024.
Answer:
Only one number, i.e., ‘2’. (i.e., 2¹⁰)

Question 82.
Write the LCM of 208 and 209.
Answer:
208 x 209 (Product of even and odd number is its product)

Question 83.
Write the expansion of \(\frac{87}{625}\) terminates after how many places ?
Answer:
4 places.

Question 84.
The decimal expansion of \(\frac{87}{625}\) terminates after how many places ?
Answer:
4 places.

Question 85.
What is the H.C.F. of ‘n’ and ‘n + 1’, where ‘n’ is a natural number ?
Answer:
1

Question 86.
What is the prime factorisation of 20677.
Answer:
23 x 29 x 31

Question 87.
Find the HCF of 1001 and 1002.
Answer:
1

Question 88.
p, q are co-primes and q = 2ⁿ . 5^m, where m > n, then write the decimal expansion of p/q terminates after how many places ?
Answer:
‘m’ places.

Question 89.
Write the decimal fprm of \(\frac{80}{81}\) and write repeats after how many places ?
Answer:
81 = 3⁴, so not possible.

Question 90.
If a rational number p/q has a termi¬nating decimal, then write the prime factorisation of ‘q’ is of the form.
Answer:
q = 2^m . 5ⁿ

Question 91.
Write the decimal expansion of \(\frac{7}{16}\) without actual division.
Answer:
0.4375

Question 92.
In the expansion of \(\frac{123}{125}\) terminates after how many places ?
Answer:
3 places.

Question 93.
What is the L.C.M of least prime and the least composite number ?
Answer:
Least composite

Question 94.
Write the decimal expansion of \(\frac{27}{14}\)
Answer:
\(1.9 \overline{285714}\)

Question 95.
Which type of number was \(5.6789 \overline{1}\) ?
Answer:
Rational number

Question 96.
After how many places the decimal expansion of \(\frac{23}{125}\) terminates ?
Answer:
3 places.

Question 97.
Write the type of decimal expansion of \(\frac{9}{17}\)
Answer:
Non-terminating & repeating.

Question 98.
Write the period of the decimal expansion of \(\frac{19}{21}\)
Answer:
904761

Question 99.
After how many digits will the deci-mal expansion of \(\frac{11}{32}\) terminates ?
Write it.
Answer:
5 places.

Question 100.
If \(\sqrt{2}\) = 1.414, then find \(3 \sqrt{2}\).
Answer:
4.242

Question 101.
Find the value of \(\frac{3}{8}\)
Answer:
0.375

Question 102.
Find the value of log 64 – log 4.
Answer:
16

Question 103.
Find the value of 128 ÷ 32.
Answer:
4

Question 104.
Find the value of 104.
Answer:
10000

Question 105.
Find the value of \(\sqrt{\mathbf{5}}\) .
Answer:
2.236

Question 106.
Find the value of log₂₇ 9.
Answer:
\(\frac{2}{3}\)

Question 107.
Find the value of | – 203 |.
Answer:
203

Question 108.
Complete the rule a(b + c).
Answer:
ab + ac

Question 109.
Find the value of log₃ \(\frac{1}{9}\).
Answer:
-2

Question 110.
How much the LCM of 12, 15 and 21.
Answer:
420

Question 111.
Find the value of log_a 1, a > 0.
Answer:
0

Question 112.
a + (-a) = 0 = (- a) + a is called ……………… property.
Answer:
Inverse

Question 113.
Find the value of \(\sqrt{\mathbf{a}} \times \sqrt{\mathbf{b}}\)
Answer:
\(\sqrt{a b}\)

Question 114.
Find the value of 5⁵.
Answer:
3125

Question 115.
Find the value of \(\frac{13}{4}\) .
Answer:
3.25

Question 116.
Find the value of \(\sqrt{12544}\)
Answer:
112

Question 117.
Find the value of log₆1.
Answer:
0

Question 118.
Find the value of log₁₀10000.
Answer:
4

Question 119.
How much the HCF of 12 and 18.
Answer:
6

Question 120.
Which number has no multiplicative inverse ?
Answer:
0

Question 121.
How much the LCM of 306 and 657.
Answer:
22338

Question 122.
Find the value of log_x \(\frac{\mathbf{a}}{\mathbf{b}}\).
Answer:
log_xa – log_xb

Question 123.
Find the value of log₃₂ \(\frac{1}{4}\)

Question 124.
Find the value of \(\sqrt{2025}\)
Answer:
\(\frac{-2}{5}\)

Question 125.
Find the value of \(2 \sqrt{3}+7 \sqrt{3}+\sqrt{3}\)
Answer:
\(10 \sqrt{3}\)

Question 126.
Find the value of 2² x 5 x 7.
Answer:
140

Question 127.
Find the value of log₁₀ 100.
Answer:
2

Question 128.
6ⁿ cannot end with this number. What is that number ? (When ‘n’ is a posi¬tive number).
Answer:
0

Question 129.
If 2^x = y and log₂ y = 3 then find (x – y)².
Answer:
25

Question 130.
Find the value of log₃ \(\frac{1}{27}\).
Answer:
-3

Question 131.
Expanded form of log₁₀1000.
Answer:
3 log 2 + 3 log 5

Question 132.
Find the value of log₂512.
Answer:
9

Question 133.
Write \(\frac{3}{2}\) (log x) – (log y) as single form.
Answer:
\(\log \sqrt{\frac{x^{3}}{y^{2}}}\)

Question 134.
Find HCF of 1 and 143.
Answer:
1

Question 135.
Which of the following is a correct one ?
A) N⊂Z⊂W
B) N⊂W⊂Z
C) R⊂N⊂W
D) All the above
Answer:
(B)

Question 136.
Find the value of log₂16.
Answer:
4

Question 137.
Find the value of \(\log _{7} \sqrt{49}\)
Answer:
1

Question 138.
Find the value of log₂1024.
Answer:
10

Question 139.
Find the value of log₁₈324.
Answer:
2

Question 140.
Express logarithmic form of a^x = b.
Answer:
loga^b = x .

Question 141.
Find the value of \((\sqrt{7}+\sqrt{5}) \cdot(\sqrt{7}-\sqrt{5})\)
Answer:
2

Choose the correct answer satisfying the following statements.

Question 142.
Statement (A) : 6n ends with the digit zero, where ‘n’ is natural number. Statement (B): Any number ends with digit zero, if its prime factor is of the form 2^m x 5ⁿ, where m, n are natural numbers.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
6ⁿ = (2 x 3)ⁿ = 2ⁿ x 3ⁿ
Its prime factors do not contain 5n i.e., of the form 2^m x 5ⁿ, where m, n are natural numbers. Here (A) is incorrect but (B) is correct.
Hence, (iii) is the correct option.

Question 143.
Statement (A) : \(\sqrt{a}\) is an irrational number, where ‘a’ is a prime number.
Statement (B) : Square root of any prime number is an irrational number.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
As we know that square root of every prime number is an irrational number. So, both A and B are correct and B explains A. Hence (i) is the correct option.

Question 144.
Statement (A) : For any two positive integers a and b,
HCF (a, b) x LCM (a, b) – a x b.
Statement (B) : The HCF of two num-bers is 5 and their product is 150. Then their LCM is 40.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
We have,
LCM (a, b) x HCF (a, b) = a xb LCM x 5 – 150 150
∴ LCM = \(\frac{150}{5}\) = 30
=> LCM = 30, i.e., (B) is incorrect and (A) is correct.
Hence, (ii) is the correct option.

Question 145.
Statement (A) : When a positive inte-ger ’a’ is divided by 3, the values of re-mainder can be 0, 1 (or) 2.
Statement (B) : According to Euclid’s Division Lemma a = bq + r, where 0 ≤ r < b and ‘r’ is an integer.
i) Both A and B are true.
ii) A is true, 3 is false
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
Given positive integers A and B, there exists unique integers q and r satisfy¬ing a = bq + r where 0 ≤ r < b.
This is known as Euclid’s Division Algorithm. So, both A and B are cor¬rect and B explains A.
Hence, (i) is the correct option.

Question 146.
Statement (A): A number N when di¬vided by 15 gives the remainder 2. Then the remainder is same when N is divided by 5.
Statement (B) : \(\sqrt{3}\) is an irrational number.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)

Question 147.
Statement (A): \(\frac{41}{1250}\) is a terminating decimal.
Statement (B) : The rational number p/q is a terminating decimal if q = 2^m x 5ⁿ, where m, n are non-nega¬tive integers.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 148.
Statement (A) : \(\sqrt{3}\) is an irrational number.
Statement (B) : The square root of a prime number is an irrational.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
Clearly, both A and B are correct but B does not explain A.
Hence, (i) is correct option.

Question 149.
Statement (A) : \(\frac{27}{250}\) is a terminating decimal.
Statement (B) : The rational number p/q is a terminating decimal, if q = (2^m x 5ⁿ) for some whole number m and n.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 150.
Statement (A): \(\frac{13}{3125}\) is a terminating decimal fraction.
Statement (B): If q = 2ⁿ . 5^m where n, m are non-negative integers, then p/q is terminating decimal fraction.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
(B) is correct.
Since the factors of the denominator 3125 is of the form 2° x 5⁵.
∴ \(\frac{13}{3125}\) is a terminatmg decimal.
∴ Since (A) follows from (B).
∴ Hence, (i) is the correct option.

Question 151.
Statement (A) : Denominator of 34.12345 is of the form 2^m x 5ⁿ, where m, n are non-negative integers.
Statement (B) : 34.12345 is a termi-nating decimal fraction.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)
Explanation:
(B) is clearly true.
Again 34.12345 = \(\frac{3412345}{100000}\)
= \(\frac{682469}{20000}=\frac{682469}{2^{5} \times 5^{4}}\)
Its denominator is of the form 2^m x 5ⁿ
[m = 5, n = 4 are non-negative integers.]
∴ (A) is true.
Since (B) gives (A).
Hence, (i) is the correct option.

Question 152.
Statement (A): The H.C.F. of two num-bers is 16 and their product is 3072. Then their L.C.M. = 162.
Statement (B): If a, b are two positive integers, then H.C.F x L.C.M = a x b.
i) Both A and B are true.
ii) A is true, B is false.-
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(iii)
Explanation:
Here (B) is true (standard result)
(A) is false.
∴ \(\frac{3072}{16}\) = 192 ≠ 162
Hence, (iii) is the correct option.

Question 153.
Statement (A) : 2 is a rational num¬ber.
Statement (B): The square roots of all positive integers are irrationals.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)
Explanation:
Here (B) is not true.
∴ \(\sqrt{4} \neq 2\) which is not an irrational
number.
Clearly, (A) is true.
∴ (ii) is the correct option.
Question 154.
Statement (A) : If L.C.M. {p, q} = 30 and H.C.F. {p, q} = 5, then pq = 150.
Statement (B): L.C.M. of a, b x H.C.F. of a, b = a • b.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Question 155.
Statement (A) : n² – n is divisible by 2 for every positive integer.
Statement (B): \(\sqrt{2}\) is a rational num¬ber.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(ii)

Question 156.
Statement (A): n² + n is divisible by 2 for every positive integer n.
Statement (B): If x and y are odd posi-tive integers, from x² + y² is divisible by 4.
i) Both A and B are true.
ii) A is true, B is false.
iii) A is false, B is true.
iv) Both A and B are false.
Answer:
(i)

Read the below passages and answer to the following questions.
If p is prime, then \(\sqrt{\mathbf{p}}\) is irrational and if a, b are two odd prime num-bers, then a² – b² is composite.

Question 157.
Is \(\sqrt{7}\) is a rational number ?
Answer:
No, it is an irrational number.

Question 158.
The results of 119² – 111² is a ………..
number.
Answer:
Composite

L.C.M. of several fractions
= \(\frac{\text { LCM of their numerators }}{\text { HCF of their denominators }}\)
H.C.F. of several fractions = \(\frac{\text { HCF of their numerators }}{\text { LCM of their denominators }}\)

Question 159.
Find the LCM of the fractions \(\frac{5}{16}, \frac{15}{24}\) and \(\frac{25}{8}\)
Answer:
\(\frac{75}{8}\).
Explanation:
L.C.M. of \(\frac{5}{16}, \frac{15}{24}\) and \(\frac{25}{8}\)
= \(\frac{\text { L.C.M. of numerators }}{\text { H.C.F. of denominators }}\)
L.C.M’. of 5, 15 and 25 is 75.
H.C.F. of 16, 24 and 8 is 8.
The H.C.F. of the given fractions = \(\frac{75}{8}\)

Question 160.
Find the HCF of \(\frac{2}{5}, \frac{6}{25}\) and \(\frac{8}{35}\).
Answer:
\(\frac{2}{175}\)
Explanation:

Question 161.
Find the HCF of the fractions \(\frac{8}{21}, \frac{12}{35}\) and \(\frac{32}{7}\)
Answer:
\(\frac{4}{105}\)
[H⁺] ion concentration in a soap used by Sohan is 9.2 x 10^-22.
Explanation:
H.C.F. of given fraction is
\(\frac{\text { H.C.F. of } 8,12,32}{\text { L.C.M. of } 21,35,7}\)
= \(\frac{4}{105}\)

Question 162.
Which mathematical concept is used to find pH of a soap ?
Answer:
Logarithms.

Question 163.
How much the pH of soap used by Sohan ?
Answer:
pH = 21.04.

Question 164.
Write the correct matching options :

Answer:
A – (iii), B – (iv)

Question 165.
Write the correct matching options :

Answer:
A – (i), B – (ii)

Question 166.
Write the correct matching options :

Answer:
A – (iii), B – (i)

Question 167.
Write the correct matching options :

Answer:
A – (ii), B – (v), C – (iii)

Question 168.
Write the correct matching options:

A – (i), B – (iv)

Question 169.
Write the correct matching options :

Answer:
A – (ii), B – (iii)

Question 170.
Write the correct matching options :

Answer:
A – (ii), B – (v)

Question 171.
Write the correct matching options:

Answer:
A – (iii), B – (iv)

Question 172.
Write the correct matching options:

Answer:
A — (i), B – (ii), C — (v)

Question 173.
What is the value of \(\log _{\frac{2}{3}}\left(\frac{27}{8}\right)\)
Answer:
-3

Question 174.
Write the decimal form of the rational number \(\frac{7}{2^{2} \times 5}\)
AP Model Paper
Answer:
0.35

Question 175.
What is the value of \(\log _{\sqrt[3]{5}} \sqrt{5}\) ?
Solution:

Question 176.
Which statement do you agree with ? P: The product of two irrational num-bers is always a rational number.
Q : The product of a rational and an irrational number is always an irra-tional number,
i) Only P ii) Only Q iii) Both P and Q
Answer:
(ii)

Question 177.
Express 3 log₂2 = x in exponential form.
Solution:
3 log₂2 = x
log₂2³ = x ⇒ log₂8 = x ⇒ 2^x = 8

Andhra Pradesh SCERT AP State Board Syllabus 9th Class Maths Important Bits with Answers in English and Telugu Medium are part of AP Board 9th Class Textbook Solutions.

Students can also read AP Board 9th Class Maths Solutions for board exams.

AP State Syllabus 9th Class Maths Important Bits with Answers in English and Telugu

• Chapter 1 Real Numbers Bits
• Chapter 2 Polynomials and Factorisation Bits
• Chapter 3 The Elements of Geometry Bits
• Chapter 4 Lines and Angles Bits
• Chapter 5 Co-Ordinate Geometry Bits
• Chapter 6 Linear Equation in Two Variables Bits
• Chapter 7 Triangles Bits
• Chapter 8 Quadrilaterals Bits
• Chapter 9 Statistics Bits
• Chapter 10 Surface Areas and Volumes Bits
• Chapter 11 Areas Bits
• Chapter 12 Circles Bits
• Chapter 13 Geometrical Constructions Bits
• Chapter 14 Probability Bits
• Chapter 15 Proofs in Mathematics Bits