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Practice the AP 9th Class Maths Bits with Answers Chapter 3 The Elements of Geometry on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 9th Class Maths Bits 3rd Lesson The Elements of Geometry with Answers

Choose the correct answer :

Question 1.
Which of the following is an undefined term ?
A) Point
B) Right angle
C) Triangle
D) Circle
Answer:
A) Point

Question 2.
The word geometry is derived from
A) Greek
B) Latin
C) English
D) Sanskrit
Answer:
A) Greek

Question 3.
The Indian mathematician who used Pythagorean triplets
A) Boudhayan
B) Bhaskaracharya
C) Ramanujan
D) Aryabhatta
Answer:
A) Boudhayan

Question 4.
The author of ‘The Elements”
A) Pythagoras
B) Thales
C) Euclid
D) Plato
Answer:
C) Euclid

Question 5.
Statements which are self evident
A) Theorems
B) Conjectures
C) Axioms
D) Riders
Answer:
C) Axioms

Question 6.
Statements which are neither proved nor disproved are called
A) Axioms
B) Postulates
C) Conjectures
D) Theorems
Answer:
C) Conjectures

Question 7.
“Every even number greater than 4 can be written as sum of two primes” is a
A) Axiom
B) Conjecture
C) Postulate
D) Theorem
Answer:
B) Conjecture

Question 8.
“If equals are added to equals, the wholes are equal” is
A) Conjecture
B) Theorem
C) Statement
D) Axiom
Answer:
D) Axiom

Question 9.
The number of dimensions of a rect-angle is
A) 3
B) 2
C) 4
D) 1
Answer:
B) 2

Question 10.
The number of dimensions of a solid is
A) 2
B) 1
C) 0
D) 3
Answer:
D) 3

Question 11.
Pyramids are found in [ ]
A) China
B) Egypt
C) India
D) Japan
Answer:
B) Egypt

Question 12.
‘Sulba Sutras’ are in
A) Persia
B) Mathematics
C) Hindi
D) Vedic Sanskrit
Answer:
D) Vedic Sanskrit

Question 13.
Which of the following is a simple Pythagorean triple ?
A) 3, 4, 5
B) 5, 12, 16
C) 6, 10, 15
D) 8, 15, 20
Answer:
A) 3, 4, 5

Question 14.
Who wrote 13 books called The Ele-ments’ ?
A) Pythagoras
B) Alexander
C) Euclid
D) Boudhayana
Answer:
C) Euclid

Question 15.
Which of the following has no dimen-sions ?
A) Cube
B) Point
C) Cuboid
D) None of these
Answer:
B) Point

Question 16.
Which of the following is undefined ?
A) A plane surface
B) Line segment
C) Angle
D) Triangle
Answer:
A) A plane surface

Question 17.
The boundaries of a solid are called
A) curves
B) lines
C) surfaces
D) points
Answer:
C) surfaces

Question 18.
How many dimensions a solid has ?
A) 4
B) 3
C) 2
D) 1
Answer:
B) 3

Question 19.
Which of the following statements is false ?
A) All right angles are equal
B) Only one line can pass through a given point.
C) Circles with same radii are equal.
D) A line segment can be extended on either side to form a straight line.
Answer:
B) Only one line can pass through a given point.

Question 20.
The edges of surfaces are
A) lines
B) curves
C) circles
D) points
Answer:
A) lines

Question 21.
In the adjacent figure, according to Euclid’s 5^th postulate, the pair of angles having the sum less than 180° is

A) 2 and 4
B) 1 and 3
C) 3 and 4
D) 1 and 2
Answer:
B) 1 and 3

Question 22.
It is known that if a + b = 9, then a + b + z = 9 + z the Euclid’s axiom that il-lustrates this statement is
A) Second Axiom
B) Third Axiom
C) Fourth Axiom
D) First Axiom
Answer:
A) Second Axiom

Question 23.
Tushar’s age is the same as Karthik’s. Satwik’s age is also same as Karthik. State the Euclid’s axiom that illustrate the relative ages of Tushar and Satwik.
A) Second Axiom
B) Third Axiom
C) Fourth Axiom
D) First Axiom
Answer:
D) First Axiom

Question 24.
The number of books in Euclid’s The Elements
A) 13
B) 23
C) 31
D) 32
Answer:
A) 13

Question 25.
If the points P, Q, R and S are collinear, how many line segments are formed?
A) 3
B) 4
C) 6
D) 12
Answer:
C) 6

Question 26.
How many straight lines can be drawn through two points in a plane ?
A) 0
B) 1
C) 2
D) Many
Answer:
B) 1

Question 27.
According to Euclid’s postulate, which angles are equal to one another ?
A) All acute angles
B) All right angles
C) All obtuse angles
D) All reflex angles
Answer:
B) All right angles

Question 28.
If two points X and Y lie on the line segment AB such that AX = XY = YB. then AY = ?
A) \(\frac { 1 }{ 3 }\) AB
B) \(\frac { 1 }{ 2 }\) AB
C) \(\frac { 2 }{ 3 }\) AB
D) AB
Answer:
C) \(\frac { 2 }{ 3 }\) AB

Question 29.
No. of volumes in “Elements” written by Euclid’s is ……………
A) 23
B) 13
C) 103
D) 24
Answer:
B) 13

Practice the AP 9th Class Maths Bits with Answers Chapter 2 Polynomials and Factorisation on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 9th Class Maths Bits 2nd Lesson Polynomials and Factorisation with Answers

Choose the correct answer :

Question 1.
A polynomial with two terms is called
A) monomial
B) binomial
C) trinomial
D) multinomial
Answer:
B) binomial

Question 2.
For a polynomial p(x) if p(a) = 0 then a is called a…………….of the polynomial.
A) variable
B) zero
C) coefficient
D) term
Answer:
B) zero

Question 3.
The degree of the polynomial 7x²y⁴ + 3x²y² – 8
A) 4
B) 8
C) 6
D) 1
Answer:
C) 6

Question 4.
The degree of the polynomial (x² + 2x³) (x² – 5x²y² + y⁴) is
A) 3
B) 4
C) 2
D) 7
Answer:
D) 7

Question 5.
The coefficient of x² in (2x – 8) (7 – 3x) is
A) 2
B) 3
C) 6
D) – 6
Answer:
D) – 6

Question 6.
The zero of the polynomial 2x – 3 is
A) 3
B) \(\frac { 2 }{ 3 }\)
C) \(\frac { -3 }{ 2 }\)
D) \(\frac { 3 }{ 2 }\)
Answer:
D) \(\frac { 3 }{ 2 }\)

Question 7.
2x³ + 7x² – 8 is a
A) monomial
B) binomial
C) trinomial
D) zero
Answer:
C) trinomial

Question 8.
A quadratic polynomial in one vari¬able has………….terms.
A) 1
B) 2
C) 3
D) 4
Answer:
C) 3

Question 9.
Which of the following expressions is a polynomial ?
A) 3x^-2 + 7x + 1
B) 7x² + \(\frac{2}{\mathrm{x}}\) – 8
C) x² – x – 1
D) x² + \(\frac{x}{2}+\frac{1}{x}\)
Answer:
C) x² – x – 1

Question 10.
The value of p(x) = 7x² + 2x – 8 when x = 1
A) – 1
B) 1
C) 2
D) 17
Answer:
B) 1

Question 11.
The value of 5x² + 6x + 7 at x = -1
A) 18
B) – 6
C) 6
D) 0
Answer:
C) 6

Question 12.
Zeroes of the polynomial 9x² – 12x + 4 are
A) \(\frac{2}{3}, \frac{2}{3}\)
B) \(\frac{3}{2}, \frac{-3}{2}\)
C) \(\frac{-3}{2}, \frac{-3}{2}\)
D) (3,2)
Answer:
A) \(\frac{2}{3}, \frac{2}{3}\)

Question 13.
Zeros of the polynomial x² – 7x + 10 are
A) (2,-5)
B) (-2, 5)
C) (- 2,-5)
D) (2, 5)
Answer:
D) (2, 5)

Question 14.
The number of zeroes of the polynomial \(\frac{7 x}{3}=\) = -5
A) 1
B) 2
C) 3
D) many
Answer:
A) 1

Question 15.
The number of zeroes of a quadratic equation
A) 1
B) 2
C) 3
D) many
Answer:
B) 2

Question 16.
If 2 is a zero of the polynomial 2x² – 3x – a then a =
A) 2
B) – 2
C) – 3
D) 4
Answer:
A) 2

Question 17.
The polynomial of degree ‘n’ has ………… zeroes.
A) 1
B) n + 1
C) n – 1
D) n
Answer:
D) n

Question 18.
If (x – 2) is a factor of 2x³ + 3x² – 8x + k then the value of k is
A) 12
B) – 12
C) 24
D) 44
Answer:
B) – 12

Question 19.
The remainder when x³ + 2x² + 3x + 4 is divided by (x – 1) is
A) 10
B) 6
C) 4
D) 2
Answer:
A) 10

Question 20.
The remainder when 2x³ + 3x² +4x + 5 is divided by (x – 2)
A) 31
B) 29
C) 36
D) 41
Answer:
D) 41

Question 21.
The remainder when 3x²+ 2x – 6 is divided by (2x – 1)
A) \(\frac{-37}{8}\)
B) \(\frac{-43}{8}\)
C) \(\frac{59}{3}\)
D) None
Answer:
A) \(\frac{-37}{8}\)

Question 22.
The value of p if px⁴ + 7x² – 18 is divisible by (x – 3)
A) \(\frac{9}{5}\)
B) \(\frac{-5}{9}\)
C) \(\frac{5}{9}\)
D) 1
Answer:
B) \(\frac{-5}{9}\)

Question 23.
The expansion of (5x – 1)² is
A) 25x² – 5x + 1
B) 25x² – 10x – 1
C) 25x² – 10x +1
D) 25x² + 10x + 1
Answer:
C) 25x² – 10x +1

Question 24.
Factors of 81x⁴ – 25 are
A) (9x² + 5)²
B) (9x² – 5)²
C) (9x² + 5) (9x² – 5)
D) None
Answer:
C) (9x² + 5) (9x² – 5)

Question 25.
103 x 97 =
A) 9991
B) 9791
C) 10197
D) 9997
Answer:
A) 9991

Question 26.
(P² + 2) (p² – 2) =
A) p² + 4
B) p⁴ – 4
C) p² – 4
D) p⁴ + 4
Answer:
B) p⁴ – 4

Question 27.
(2x + 3y + 4z)² =
A) 4x² + 9x² + 16z² + 12xy + 24yz + 16xz
B) 2x² + 34y² + 4z² + 6xy + 12yz + 8xz
C) 4x² + 9y² + 16z²
D) 4x² + 9y² + 16z² + xy + yz + zx
Answer:
A) 4x² + 9x² + 16z² + 12xy + 24yz + 16xz

Question 28.
102³ =
A) 10404
B) 1061208
C) 1061206
D) 1061202
Answer:
B) 1061208

Question 29.
x³ – y³ =
A) (x – y) (x² – xy – y²)
B) (x – y) (x² + xy + y²)
C) (x + y) (x² – xy + y²)
D) None
Answer:
C) (x + y) (x² – xy + y²)

Question 30.
(- 12)³ + (8)³ + (4)³ =
A) 1152
B) 1296
C) 1084
D) 1082
Answer:
A) 1152

Question 31.
Which of the following expressions is not a polynomial ?
A) \(\frac{1}{x+1}\)
B) 7
C) x² – 3
D) x² – 4x + 5
Answer:
A) \(\frac{1}{x+1}\)

Question 32.
The degree of the polynomial
3x²y³ + 4xy + 7 is
A) 2
B) 3
C) 5
D) 1
Answer:
C) 5

Question 33.
The coefficient of \(\sqrt{2}\) x² in x² + 5 x – 1 is
A) 5
B) – 1
C) 0
D) 72
Answer:
C) 0

Question 34.
An example for constant polynomial is
A) 0
B) – 12
C) 3x – 4
D) x² + 2x – 1
Answer:
B) – 12

Question 35.
The value of p(t) = 5t² – 6t + 7 at t = – 1 is
A) – 18
B) 16
C) 18
D) – 16
Answer:
C) 18

Question 36.
The number of zeroes of a polynomial of degree ‘n’ will have
A) n – 1
B) n + 1
C) 0
D) n
Answer:
D) n

Question 37.
The identity used in simplifying 101 x 99 is
A) (a + b) (a – b) = a² – b²
B) (a – b)² = a² – 2ab + b²
C) (a + b)² = a² + 2ab + b²
D) (x + a) (x + b) = x² + (a + b)x + ab
Answer:
A) (a + b) (a – b) = a² – b²

Question 38.
If x = 2, then the value of (x + 5) (x + 2)
A) 40
B) 28
C) 20
D) x² + 7x + 10
Answer:
B) 28

Question 39.
If p(x) = x² + 5x + 4, then p(- 4) =
A) 0
B) – 32
C) 40
D) – 20
Answer:
A) 0

Question 40.
Zero of the polynomial p(x) = 3x + 1 is
A) \(\frac{1}{3}\)
B) – 2
C) \(\frac{-1}{3}\)
D) 3
Answer:
C) \(\frac{-1}{3}\)

Question 41.
The first term in quotient when you divide 3x² + x – 1 by x+1 is
A) 1
B) – 2
C) 3x
D) 3
Answer:
C) 3x

Question 42.
(x – 3) (x + 4) =
A) x² + x – 12
B) x² – x – 12
C) x² + 7x + 12
D) x² – 7x + 12
Answer:
A) x² + x – 12

Question 43.
If a + b + c = 0, then a³ + b³ + c³ =
A) O
B) 3abc
C) 3(a + b + c)
D) 3(a+b)(b+c)(c+a)
Answer:
B) 3abc

Question 44.
x³ – 8 =
A) (x – 2) (x – 2) ( x – 2)
B) (x – 2) (x² + 2x + 8)
C) (x – 2) (x² + 2x + 4)
D) (x – 2) (x² – 2x – 4)
Answer:
C) (x – 2) (x² + 2x + 4)

Question 45.
If f(x) is divided by (x – a), then the remainder is
A) f(-a)
B) f(l)
C) f(0)
D) f(a)
Answer:
D) f(a)

Question 46.
The factor of 3x² + x – 2 is
A) x – 1
B) 3x – 2
C) 3x + 2
D) 2x – 3
Answer:
B) 3x – 2

Question 47.
(x + y)² + (x – y)² =
A) 2(x² + y²)
B) 4xy
C) x² + y²
D) 2xy
Answer:
A) 2(x² + y²)

Question 48.
9x² – 25 =
A) (3x + 25) (3x – 1)
B) (9x – 1) (x + 25)
C) (3x + 5) (3x – 5)
D) (9x + 1) (x – 25)
Answer:
C) (3x + 5) (3x – 5)

Question 49.
One of the factors of 3x³ – 12x is
A) x – 1
B) 4x
C) x + 2
D) x + 1
Answer:
C) x + 2

Question 50.
The degree of the polynomial 7 – 2x³ + 7x²y + xy³ is
A) 4
B) 3
C) 2
D) 1
Answer:
A) 4

Question 51.
The zero of the polynomial ax + b is
A) \(\frac{-b}{a}\)
B) \(\frac{a}{b}\)
C) – a
D) – b
Answer:
A) \(\frac{-b}{a}\)

Question 52.
If 2 is a zero of the polynomial x² – kx + 8, then k =
A) 1
B) 4
C) 6
D) 3
Answer:
C) 6

Question 53.
The coefficient of x² in 7x³ – 2x² + 3x + 1 is
A) 7
B) – 2
C) 3
D) 1
Answer:
B) – 2

Question 54.
The coefficient of x³ in (2x – 3y) (4x² + 6xy + 9y²) is
A) 8
B) – 27
C) 12
D) 9
Answer:
A) 8

Question 55.
101² =
A) 12001
B) 10001
C) 10201
D) 100201
Answer:
C) 10201

Question 56.
The value of p(x) = 4x³ + 3x² + 2x + 1 at x = 1 is
A) 9
B) 10
C) 6
D) 7
Answer:
B) 10

Question 57.
(1 + x) (1 + x) =
A) 2 + 2x
B) 1 + x²
C) 1 + x² + 2x
D) 1 + x + 2x²
Answer:
C) 1 + x² + 2x

Question 58.
3y² + 2y – 5
A) (y – 1)(3y + 5)
B) (y + 1) (3y – 5)
C) (y – 1)(3y-5)
D) (y + 1) (3y + 5)
Answer:
A) (y – 1)(3y + 5)

Question 59.
(t-1)(t-1) =
A) t² + 1
B) t² – 2t + 1
C) t² + 2t + 1
D) t² – 1
Answer:
B) t² – 2t + 1

Question 60.
One of the factors of a³ – 3a²b + 3ab² – b³ is
A) – 3ab
B) 3ab
C) a + b
D) a – b
Answer:
D) a – b

Question 61.
If 2x³ – 2x² – 2x – 5 is divided by x + 1, then the remainder is
A) 0
B) -7
C) 6
D) -6
Answer:
B) -7

Question 62.
The factor of a – b – a² + b² is
A) a – b
B) a² + b²
C) 2ab
D) 4ab
Answer:
A) a – b

Question 63.
If a + b = 8, ab = 6 then the value of a³ + b³
A) 656
B) 576
C) 368
D) 638
Answer:
C) 368

Question 64.
The degree of Polynomial 3x²y³ + 4xy³ + 7 is….
A) 3
B) 5
C) 4
D) 7
Answer:
B) 5

Question 65.
The zero of the polynomial px + q is…
A) \(\frac { p }{ q }\)
B) \(\frac { q }{ p }\)
C) \(-\frac { p }{ q }\)
D) \(-\frac { q }{ p }\)
Answer:
D) \(-\frac { q }{ p }\)

Question 66.
Which of the following figure represent the algebrical identity (x – y)² = x² – 2xy + y² ?

Answer:
(B)

7-11 Questions:
I. (a + b)² = a² + 2ab + b²
ii. (a – b)² = a² – 2ab + b²
III. (a + b) (a – b) = a² – b²
V. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
VI. (a + b)³ = a³ + b³ + 3ab (a + b)
VII. (a – b)³ = a³ – b³ – 3ab (a – b)
Without actual multiplication which algebraic identity can be used to find the following:

Question 67.
99 x 99
A) II
B) III
C) I and II
D) I
Answer:
D) I

Question 68.
103 x 97
A) II
B) III
C) IV
D) I
Answer:
B) III

Question 69.
Expand (2a – 3b + 4c)²
A) 4a² – 9b² + 16c² – 12ab – 24bc + 16ac
B) 4a² + 9b² + 16c² – 12ab – 24bc + 16ca
C) 4a² – 9b² – 16c² – 12ab – 24bc + 16ca
D) 4a² + 9b² + 16c² – 12ab + 24bc + 16ac
Answer:
B) 4a² + 9b² + 16c² – 12ab – 24bc + 16ca

Question 70.
1 – 64a³ – 12a + 48a² is obtained by using which of the following identity:
A) IV
B) V
C) VII
D) III
Answer:
C) VII

Question 71.
x² – 5x + 6
A) (x – 3) (x – 2)
B) (x + 3) (x + 2)
C) (x – 3) (x + 2)
D) (x + 3) (x-2)
Answer:
A) (x – 3) (x – 2)

Question 72.
Assertion (A) : 6x⁵ + 5x⁴ + 3x² + \(\frac{4}{\mathrm{x}}\) + 5 is a polynomial of degree 5.
Reason (R) : Exponent of ‘x’ is negative integer, it is multinomial.
A) Both A and R are true but ‘R’ is not correct explanation of A.
B) A is correct, R is incorrect.
C) A is incorrect, R is correct.
D) Both A and R are true and ‘R’ is the correct explanation of A.
Answer:
B) A is correct, R is incorrect.

Question 73.
Match the following group A to B. Choose correct mapping of the following
Group – A — Group – B
i) (a + b)² – (a – b)² — A) 2(a² + b²)
ii) (a + b)² + (a – b)² — B) 2ab
iii) (a + b)² – (a² + b²) — C) 4ab
Choose the correct answer :
A) i – B, ii – C, iii – A
B) i – C, ii – A, iii – B
C) i – C, ii – B, iii – A
D) i – A, ii – B iii – C
Answer:
B) i – C, ii – A, iii – B

Question 74.
The degree of the polynomial (x³ + 7) (3 – x²) is
A) 3
B) 5
C) 6
D) 2
Answer:
B) 5

Question 75.
If length, breadth and height of the cuboid are (x – 1), (x – 10) and (x – 12) units, then volume of the cuboid in cubic units.

A) x³ + 23x² – 142x + 120
B) x³ – 23x² + 142x – 120
C) x³ – 23x² – 142x + 120
D) x³ + 23x² – 142x – 120
Answer:
B) x³ – 23x² + 142x – 120

Question 76.
The ratio of molecular weight of hy-drogen and oxygen in water is 1 : 8. (The relation between hydrogen and oxygen in linear equations in two vari-ables is). If the quantities of hydrogen and oxygen are ‘x’ and ‘y’
A) y = 8x
B) x + y = 8
C) xy = 8
D) x = 8y
Answer:
A) y = 8x

Question 77.
Which Geometrical figure related to Algebraic Identity
(x – y)² = x² + y² – 2xy ?

Answer:
(D)

Question 78.
If ‘2’ is the zero of the polynomial P(x) = 2x² – 3x + 7a, then the value of a is
A) \(\frac{-2}{7}\)
B) \(\frac{7}{2}\)
C) \(\frac{-7}{2}\)
D) \(\frac{2}{7}\)

Question 79.
If a + b + c’ = 6, then the value of (2 – a)³ + (2 – b)³ + (2 – c)³ – 3 (2 – a) (2 – b) (2 – c) is
A) -1
B) 1
C) 2
D) 0
Answer:
D) 0

Question 80.
The remainder when x¹⁰¹ + 101 is divided by x + 1 is
A) 1
B) 100
C) 101
D) 0
Answer:
A) 1

Question 81.
If p(x) = x + 3, then p(-x) + p(x) is equal to
A) 0
B) 3
C) 2x
D) 6
Answer:
A) 0

Question 82.
If 3 is zero of the polynomial x² + 2x – a, then “a” is
A) -3
B) -15
C) 15
D) 3
Answer:
C) 15

Question 83.
Statement A : Cubic polynomial has at most three zeroes.
Statement B : Degree of the zero poly-nomial is zero.
A) Both A and B are true
B) Both A and B are false
C) A is true and B is false
D) A is false and B is true
Answer:
A) Both A and B are true

Question 84.
The degree of the polynomial (y² + 7) (-y⁵ + 3) is
A) 5
B) 7
C) 10
D) -10
Answer:
B) 7

Question 85.
If the area of a square field is x² + y² + z² + 2xy – 2yz – 2zx sq. units, then its perimeter is
A) (x + y + z)²
B) 4 (x + y + z)
C) 4 (x + y – z)
D) (x + y – z)²
Answer:
C) 4 (x + y – z)

Question 86.
If x + 1 is a factor of polynomial 2x² + Kx then the value of K is
A) -4
B) -2
C) 2
D) +4
Answer:
C) 2

Question 87.
x + 1 is a factor of xⁿ + 1 only if
A) ‘n’ is an odd integer
B) ‘n’ is an even integer
C) ‘n’ is a negative integer
D) ‘n’ is a positive integer
Answer:
A) ‘n’ is an odd integer

Question 88.
If x² + 1 has no zeroes then ‘x’ is a
A) Real number
B) Natural number
C) Not Real
D) Integer
Answer:
C) Not Real

Question 89.
The volume of a cube is
x³ – 6x² + 12x – 8 then its edge in units
A) x – 4
B) x – 2
C) x + 2
D) x + 4
Answer:
B) x – 2

Question 90.
If x = 2 then the value of (x + 5) (x + 2) = ……………..
A) 28
B) 40
C) 20
D) x² + 7x + 10
Answer:
A) 28

Question 91.
The identity used in simplifying
103 x 97
A) (a + b)² = a² + 2ab + b²
B) (a – b)² = a² – 2ab + b²
C) (a – b) (a + b) = a² – b²
D) (x + a) (x + b) = x² + (a + b) x + ab
Answer:
C) (a – b) (a + b) = a² – b²

Question 92.
Which of the following is NOT a polynomial ?
A) 5
B) \(\sqrt{3}\)x² + 5y
C) \(\sqrt{x}\)
D) 3xyz
Answer:
C) \(\sqrt{x}\)

Question 93.
Which of the given polynomials has \(-\frac{2}{3}\) as its ‘zero’ ?
A) 2x + 3
B) 3x + 2
C) 2x – 3
D) 3x – 2
Answer:
B) 3x + 2

Question 94.
What is the value of ‘k’ if x – 3 is a factor of x³ – 2x² + k ?
A) 45
B) – 45
C) 9
D) – 9
Answer:
C) 9

Question 95.
The remainder when the polynomial f(x) in x is divided by 3x + 2 is ……………..
A) f(2)
B) f(-2)
C) f(\(\frac{2}{3}\))
D) f(\(-\frac{2}{3}\))
Answer:
D) f(\(-\frac{2}{3}\))

Question 96.
x² + y² + z² + 2xy – 2yz – 2zx =
A) (x + y + z)²
B) (x – y + z)²
C) (x + y – z)²
D) (x – y – z)²
Answer:
C) (x + y – z)²

Question 97.
Which of the given polynomials is of degree 3 ?
A) 3
B) 5xy
C) xyz
D) 3x² + 2x + 1
Answer:
C) xyz

Practice the AP 9th Class Maths Bits with Answers Chapter 13 Geometrical Constructions on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 9th Class Maths Bits 13th Lesson Geometrical Constructions with Answers

Choose the correct answer :

Question 1.
If in the figure, \(\overline{\mathbf{X Y}}\) is the perpendicular bisector to AB then AX =

A) AY
B) BY
C) BX
D) XY
Answer:
C) BX

Question 2.
In the above figure, ΔAXB is
A) equilateral
B) isosceles
C) scalene
D) right triangle
Answer:
B) isosceles

Question 3.
In the above figure, ΔAXO ≅
A) ΔAYO
B) ΔBYO
C) ΔAXB
D) ΔBXO
Answer:
D) ΔBXO

Question 4.
In the figure, \(\overrightarrow{\mathrm{BF}}\) is the bisector of ∠ABC then

A) ∠ABF = ∠CBF
B) ∠ABF = ∠ABC
C) ∠ABF + ∠CBF = 90°
D) ∠ABF + ∠CBF = 180°
Answer:
A) ∠ABF = ∠CBF

Question 5.
From the figure, ΔABD ΔCBD by

A) S.S.S
B) A.S.A
C) S.A.S
D) R.H.S
Answer:
A) S.S.S

Question 6.
∠DAC =

A) ∠B + ∠A
B) ∠A + ∠C
C) ∠B + ∠C
D) ∠A + ∠B + ∠C
Answer:
C) ∠B + ∠C

Question 7.
Angles in same circle segment are
A) Complementary
B) Supplementary
C) Equal
D) Unequal
Answer:
C) Equal

Question 8.
Angle in the semi-circle is
A) right angle
B) 180°
C) acute angle
D) obtuse angle
Answer:
A) right angle

Question 9.
The process of drawing a geometrical figure by using a compass and straight
edge is called
A) geometrical proof
B) hypothesis
C) geometrical construction
D) none
Answer:
C) geometrical construction

Question 10.
If in ΔABC \(\overrightarrow{\mathrm{BX}}\) and \(\overrightarrow{\mathrm{CY}}\) are bisectors to the base angles then ∠BXC =
A) 90° + \(\frac{1}{2}\)∠A
B) 90° – \(\frac{1}{2}\)∠A
C) 180°- \(\frac{1}{2}\)∠A
D) None
Answer:
A) 90° + \(\frac{1}{2}\)∠A

Question 11.
No. of circles drawn from a given point is
A) 1
B) 2
C) 3
D) Infinite
Answer:
A) 1

Question 12.
The circumcentre‘O’shown in the figure is formed by the concurrence of

A) Perpendicular bisectors
B) Angular bisectors
C) Medians
D) Altitudes
Answer:
A) Perpendicular bisectors

Practice the AP 9th Class Maths Bits with Answers Chapter 14 Probability on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 9th Class Maths Bits 14th Lesson Probability with Answers

Choose the correct answer:

Question 1.
Getting a ‘head’ when an unbiased coin is tossed is …………………
A) less likely
B) equally likely
C) more likely
D) certain
Answer:
B) equally likely

Question 2.
When a die is thrown, getting a number more than 6 is
A) less likely
B) equally likely
C) more likely
D) impossible
Answer:
D) impossible

Question 3.
The number of possible outcomes when a die is thrown is
A) 1
B) 4
C) 6
D) 4
Answer:
C) 6

Question 4.
When two dice are thrown, the total outcomes are
A) 12
B) 6
C) 1
D) 36
Answer:
D) 36

Question 5.
When a coin is tossed, the total possible outcomes are
A) 1
B) 2
C) 3
D) 4
Answer:
B) 2

Question 6.
The experiment in which all possible outcomes are known but exact outcome can’t be predicted is called
A) Mathematical experiment
B) Trial
C) Even
D) Outcome
Answer:
B) Trial

Question 7.
The each outcome of a random experiment is called
A) event
B) trial
C) chance
D) none
Answer:
A) event

Question 8.
The probability of getting a head when a coin is tossed is
A) 1
B) \(\frac{1}{4}\)
C) \(\frac{1}{2}\)
D) \(\frac{1}{8}\)
Answer:
C) \(\frac{1}{2}\)

Question 9.
If two identical coins are tossed, the probability of getting two heads is
A) \(\frac{1}{2}\)
B) \(\frac{1}{4}\)
C) \(\frac{1}{6}\)
D) \(\frac{1}{8}\)
Answer:
B) \(\frac{1}{4}\)

Question 10.
The probability of getting atmost two heads when three coins are tossed is
A) \(\frac{1}{8}\)
B) 8
C) \(\frac{7}{8}\)
D) \(\frac{3}{8}\)
Answer:
C) \(\frac{7}{8}\)

Question 11.
If two coins are tossed, the chance of getting np heads is
A) \(\frac{3}{4}\)
B) \(\frac{1}{4}\)
C) \(\frac{1}{2}\)
D) None
Answer:
B) \(\frac{1}{4}\)

Question 12,
Sum of the probabilities of getting a head and a tail when an unbiased coin is tossed is
A) \(\frac{1}{2}\)
B) \(\frac{1}{4}\)
C) 1
D) 0
Answer:
C) 1

Question 13.
Sum of probabilities of getting an even number and an odd number when a die is rolled is
A) \(\frac{1}{2}\)
B) \(\frac{1}{6}\)
C) \(\frac{1}{3}\)
D) 1
Answer:
D) 1

Question 14.
The sum of the probabilities of all outcomes of a Random experiment is always
A) 0
B) – 1
C) 1
D) \(\frac{1}{2}\)
Answer:
C) 1

Question 15.
The probability of an event which is certain
A) 0
B) 1
C) -1
D) Can’tsay
Answer:
B) 1

Question 16.
The probability of an event which is impossible
A) 0
B) -1
C) 1
D) 2
Answer:
A) 0

Question 17.
The probability of an event always lie between
A) 1 and 2
B) -1 and 1
C) -1 and 0
D) 0 and 1
Answer:
D) 0 and 1

Question 18.
The probability of drawing a prime number from a pack of cards numbered from 1 to 10
A) \(\frac{2}{10}\)
B) \(\frac{1}{10}\)
C) \(\frac{5}{10}\)
D) \(\frac{2}{5}\)
Answer:
D) \(\frac{2}{5}\)

Question 19.
The probability of “the last day of a month is Sunday”
A) \(\frac{2}{7}\)
B) \(\frac{1}{7}\)
C) \(\frac{3}{7}\)
D) \(\frac{4}{7}\)
Answer:
B) \(\frac{1}{7}\)

Question 20.
When a die is thrown, equally likely events are getting
A) even number and odd number
B) prime and composite
C) multiple of 3 and multiple of 2
D) numbers less than 3 and numbers greater than 3
Answer:
A) even number and odd number

Question 21.
Which of the following cannot be the probability of an event?
A) 0.9
B) – 1.5
C) 10%
D) 2/5
Answer:
B) – 1.5

Question 22.
A letter is chosen at random from the word “ROSE”. Find the probability, that the letter chosen is a vowel.
A) \(\frac{1}{4}\)
B) \(\frac{3}{4}\)
C) \(\frac{1}{2}\)
D) \(\frac{2}{5}\)
Answer:
C) \(\frac{1}{2}\)

Question 23.
Two coins are tossed simultaneously. What is the number of all outcomes?
A) 3
B) 4
C) 2
D) 1
Answer:
B) 4

Question 24.
Three coins are tossed simultaneously. What is the number of all possible outcomes?
A) 8
B) 6
C) 4
D) 2
Answer:
A) 8

Question 25.
The probability winning a prize is
A) 0
B) \(\frac{1}{2}\)
C) 2
D) 1
Answer:
B) \(\frac{1}{2}\)

Question 26.
If probability of a certain event A is P(A) = x, then P’(A) is
A) \(\frac{1}{\mathrm{x}}\) – 1
B) 1 – \(\frac{1}{\mathrm{x}}\)
C) \(\frac{1}{\mathrm{x}}\)
D) 1 – x
Answer:
D) 1 – x

Question 27.
A die is thrown once. Find the probability of getting a number less than 3.
A) \(\frac{1}{2}\)
B) \(\frac{1}{3}\)
C) \(\frac{1}{4}\)
D) \(\frac{1}{6}\)
Answer:
B) \(\frac{1}{3}\)

Question 28.
Which of the following is true?
A) 0 ≤ P(A) ≤ 1
B) P(A) > 1
C) P(A) < 0
D) -1 ≤ P(A) ≤ 1
Answer:
A) 0 ≤ P(A) ≤ 1

Question 29.
Probability of picking a two-digited number radomly for which the units and ten’s places have same digits
A) 1/10
B) 9/10
C) 9/100
D) 1/100
Answer:
A) 1/10

Question 30.
Which of the following is not to be a probability of any event ?
A) 0
B) 1
C) – 0.2
D) 0.75
Answer:
C) – 0.2

Question 31.
Which of the following is an example for impossible event?
i) Getting 7 on the top when a dice is rolled
ii) Getting head on the top while tossing a coin
iii) Picking a spade from a deck of playing cards
iv) Picking an even prime number less than 2
A) Both (i) & (ii)
B) Both (i) & (iii)
C) Both (i) & (iv)
D) Both (ii) & (iii)
Answer:
C) Both (i) & (iv)

Question 32.
In an exit poll for assembly elections winning of a person is 50%. Then the probability of his loosing is
A) 25%
B) 75%
C) 100%
D) 50%
Answer:
D) 50%

Question 33.
When a dice is rolled, total number of possible outcomes are
A) 4
B) 5
C) 7
D) 6
Answer:
D) 6

Question 34.
While tossing a coin the probability of getting head on upper side is
A) 1/4
B) 1/2
C) 1/3
D) 3/4
Answer:
B) 1/2

Question 35.
If the probability of an event is 1, then the event is called as
A) Equal likely event
B) Impossible event
C) Certain event
D) Mutually exclusive event
Answer:
A) Equal likely event

Question 36.
When a six faced dice is rolled the probability that the top face number is a perfect square and also a perfect cube is
A) 1/6
B) 1/4
C) 1/3
D) 2/3
Answer:
C) 1/3

Question 37.
The probability that random selected month to have 32 days
A) 0
B) \(\frac{32}{365}\)
C) \(\frac{1}{32}\)
D) 1
Answer:
A) 0

Question 38.
There are 5 balls iii a vessel. Out of which 2 are red, 2 are blue and ‘1’ is green. When a ball is selected at random, then probability that the ball is not a red ball …………………
A) 1/5
B) 2/5
C) 3/5
D) 4/5
Answer:
C) 3/5

Question 39.
The modal letter of the letters of the word “ASSESSMENTS”
A) A
B) E
C) S
D) T
Answer:
C) S

Question 40.
Probability of an event is P. Then ‘which of the following is TRUE ?
niafcj-r PI
A) 0 < P < 1 B) 0 > P > 1
C) 0 ≥ P ≥ 1
D) 0 ≤ P ≤ 1
Answer:
D) 0 ≤ P ≤ 1

Question 41.
In the formula P(E) = \(\frac{32}{365}\)
\(\frac{\mathbf{n}(\mathrm{E})}{\mathbf{n}(\mathrm{S})}\) ; n(E)
represents
A) No. of favourable outcomes
B) No. of not favourable outcomes
C) Total no. of outcomes
D) Probability of event
Answer:
A) No. of favourable outcomes

Practice the AP 9th Class Maths Bits with Answers Chapter 15 Proofs in Mathematics on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 9th Class Maths Bits 15th Lesson Proofs in Mathematics with Answers

Choose the correct answer:

Question 1.
The statement “the sum of two odd numbers is odd” is
A) always true
B) always false
C) ambiguous
D) none
Answer:
B) always false

Question 2.
The statement “every number can be expressed as product of primes” is
A) always true
B) always false .
C) sometimes true
D) ambiguous
Answer:
A) always true

Question 3.
A statement or an idea which gives an explanation to a series of observations is called
A) Conclusion
B) Open sentence
C) Hypothesis
D) Result
Answer:
C) Hypothesis

Question 4.
The Mathematics is mainly based on ………………. reasoning.
A) Inductive
B) Deductive
C) Both
D) None
Answer:
B) Deductive

Question 5.
Counter example to “product of two odd integers is even” is
A) 7 × 5 = 35
B) 3 × 4 = 12
C) 2 × 6 = 12
D) Not possible
Answer:
D) Not possible

Question 6.
God is immortal. Rama is a God; the conclusion based on these two statements
A) Rama is mortal
B) Rama is a God
C) God is Rama
D) Rama is immortal
Answer:
D) Rama is immortal

Question 7.
A mathematical statement whose truth has been established is called
A) Axiom
B) Conjecture
C) Theorem
D) Postulate
Answer:
C) Theorem

Question 8.
The mathematical statement which we believe to be true is called
A) postulate
B) conjecture
C) axiom
D) theorem
Answer:
B) conjecture

Question 9.
Conjecture are made based on
A) Inductive reasoning
B) Deductive reasoning
C) Proofs
D) None
Answer:
A) Inductive reasoning

Question 10.
The conjecture “If the perimeter of a rectangle increases then its area also increases” is
A) True
B) False
C) Neither true nor false
D) None
Answer:
B) False

Question 11.
A counter example to the statement “In any right triangle the square of the smallest side equals to sum of the other sides” is
A) (3, 4, 5)
B) (5, 12, 13)
C) (6, 8, 10)
D) (7, 24, 25)
Answer:
C) (6, 8, 10)

Question 12.
“A circle may be drawn with any centre and radius” is ……………….
A) Axiom
B) Conjecture
C) Theorem
D) Open sentence
Answer:
A) Axiom

Question 13.
A false axiom results into a ………………..
A) theorem
B) true statement
C) contradiction
D) none
Answer:
C) contradiction

Question 14.
If in a collection of axioms, one axiom can be used to prove other axiom, then they are said to be
A) consistent
B) inconsistent
C) false
D) true
Answer:
B) inconsistent

Question 15.
If a statement and its negation both are true, then it is a
A) Tautology
B) Contradiction
C) Conjecture
D) Postulate
Answer:
B) Contradiction

Question 16.
A process which can establish the truth of a mathematical statement based on logic is called
A) Mathematical proof
B) Disproof
C) Counter example
D) None
Answer:
A) Mathematical proof

Question 17.
The product of two consecutive even numbers is always divisible by
A) 3
B) 5
C) 4
D) 8
Answer:
C) 4

Question 18.
Counter example to”2n² + 11 is a prime” is
A) 3
B) 4
C) 5
D) 11
Answer:
D) 11

Question 19.
Counter example to the statement “a quadrilateral with all sides equal is a square” is
A) Rectangle
B) Rhombus
C) Trapezium
D) Parallelogram
Answer:
B) Rhombus

Question 20.
We prove the statement “sum of interior angles of a triangle is 180°” by
A) Counter example
B) Inductive reasoning
C) Deductive reasoning
D) None
Answer:
B) Inductive reasoning

Question 21.
Statements which are assumed to be true without proof are
A) axioms
B) counter examples
C) conjectures
D) open sentences
Answer:
A) axioms

Question 22.
If x is odd, then x² is
A) even
B) prime
C) odd
D) none of these
Answer:
C) odd

Question 23.
A sentence which is clearly true or false but not both is called a
A) Axiom
B) Theorem m
C) Conjecture
D) Statement
Answer:
D) Statement

Question 24.
Which of the following is a mathemati¬cal statement ?
A) She has blue eyes.
B) x + 7 = 18
C) Today is not Sunday.
D) What time is it ?
Answer:
B) x + 7 = 18

Question 25.
A statement of idea which gives an explanation to a sense of observation is
A) Hypothesis
B) Proof
C) Analysis
D) Conclusion
Answer:
A) Hypothesis

Question 26.
A great Indian Mathematician is
A) Euclid
B) Pythagoras
C) Srinivasa Ramanujan
D) Gold Bach
Answer:
C) Srinivasa Ramanujan

Question 27.
Which of the following subjects is based on inductive reasoning?
A) Mathematics
B) Social Studies
C) Science
D) English
Answer:
C) Science

Question 28.
Which of the following statement is always false?
A) February has only 28 days
B) Dogs fly
C) A rhombus is a parallelogram
D) For any real number x, x² ≥ 0
Answer:
B) Dogs fly

Question 29.
The very helpful technique for making conjecture is
A) Inductive reasoning
B) Deductive reasoning
C) Experimental evidence
D) Observation
Answer:
A) Inductive reasoning

Question 30.
Which of the following is a theorem?
A) A rhombus is a parallelogram.
B) Humans are meant to rule the earth.
C) The product of two odd natural numbers is odd.
D) A straight line may be drawn from any point to any other point.
Answer:
C) The product of two odd natural numbers is odd.

Question 31.
“A quadrilateral can be a rectangle” if the following condition is satisfied.
A) When diagonals are equal
B) When one angle is right angle
C) Anyone of A or B
D) Both A and B
Answer:
A) When diagonals are equal

Question 32.
Which one of the following is an example for a primary data
A) Temperatures of a place during last 10 years
B) Mid day meals records of a school in a month
C) Literacy rate of various states in the year 2001
D) List of absentee students of a day in 9th class.
Answer:
D) List of absentee students of a day in 9th class.

Practice the AP 9th 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 9th Class Maths Bits 1st Lesson Real Numbers with Answers

Choose the correct answer :

Question 1.
A rational number equivalent to \(\frac{-3}{4}\) is
A) \(\frac { -4 }{ 3 }\)
B) \(\frac { -4}{ 5 }\)
C) \(\frac { 3 }{ 4 }\)
D) \(\frac { -6 }{ 8 }\)
Answer:
D) \(\frac { -6 }{ 8 }\)

Question 2.
\(\frac { -2 }{ 3 }\) lies on …………. on the number line.
A) right side of the zero
B) left side of the zero
C) zero
D) can’t be determined
Answer:
B) left side of the zero

Question 3.
Which of the following is false?
A) Every rational number is a natural number
B) Every rational number is a whole number
C) Every rational number is an integer
D) Every integer is a rational number
Answer:
D) Every integer is a rational number

Question 4.
A rational number between 5 and 6 is
A) \(\frac { 9 }{ 2 }\)
B) \(\frac { 10}{ 2 }\)
C) \(\frac { 11 }{ 2 }\)
D) \(\frac { 12 }{ 2 }\)
Answer:
C) \(\frac { 11 }{ 2 }\)

Question 5.
\(\mathbf{0 . \overline { 3 }}\) …………….
A) \(\frac { 3 }{ 8 }\)
B) \(\frac { 2}{ 9 }\)
C) \(\frac { 3 }{ 7 }\)
D) \(\frac { 1 }{ 3 }\)
Answer:
D) \(\frac { 1 }{ 3 }\)

Question 6.
The decimal form of \(\frac { 1 }{ 18 }\) ix
A) \(0.0 \overline{5}\)
B) \(0 . \overline{05}\)
C) \(0 . \overline{5}\)
D) 0.06
Answer:
A) \(0.0 \overline{5}\)

Question 7.
1.25 in \(\frac{\mathbf{p}}{\mathbf{q}}\) form
A) \(\frac { 4 }{ 5 }\)
B) \(\frac { 5}{ 4 }\)
C) \(\frac { 5 }{ 6 }\)
D) \(\frac { 6 }{ 5 }\)
Answer:
B) \(\frac { 5}{ 4 }\)

Question 8.
If a and b are any two rational num¬bers then a rational number between
a and b is
A) a + 1
B) b -1
C) \(\frac{a+b}{2}\)
D) a . b
Answer:
C) \(\frac{a+b}{2}\)

Question 9.
If n is a natural number other than a perfect square then \(\sqrt{n}\) is number.
A) rational
B) irrational
C) natural
D) none
Answer:
B) irrational

Question 10.
If ‘x’ is an irrational number then x + 2 is ………………….. number.
A) natural
B) rational
C) irrational
D) can’t be determined
Answer:
C) irrational

Question 11.
If ‘x’ is an irrational number then x – 3 is …………. number.
A) rational
B) natural
C) irrational
D) complex
Answer:
C) irrational

Question 12.
Number which can’t be expressed in p/q form are………… numbers.
A) irrational
B) rational
C) whole
D) natural
Answer:
A) irrational

Question 13.
The combination of Q and S given the set of ……………. numbers.
A) natural
B) integers
C) whole
D) real
Answer:
D) real

Question 14.
(2 + \(\sqrt{2}\) )(2 – \(\sqrt{2}\)) is a……………. number.
A) irrational
B) rational
C) can’t be determined
D) none
Answer:
C) can’t be determined

Question 15.
\(\sqrt{\frac{a}{b}}\)

Answer:
(C)

Question 16.
\((\sqrt{\mathbf{a}}+\mathbf{b})(\sqrt{\mathbf{a}}-\mathbf{b})=\)
A) a² – b²
B) a – b
C) a² – b
D) a – b²
Answer:
D) a – b²

Question 17.
\((\sqrt{\mathbf{a}}+\mathbf{b})(\sqrt{\mathbf{a}}-\mathbf{b})\) =
A) 10
B) \(7+2 \sqrt{10}\)
C)\(7-2 \sqrt{10}\)
D) \(2 \sqrt{10}\)
Answer:
B) \(7+2 \sqrt{10}\)

Question 18.
\((\mathbf{7}+\sqrt{2})(\mathbf{7}-\sqrt{\mathbf{2}})=\)
A) 45
B) 5
C) 3
D) 47
Answer:
D) 47

Question 19.
The rationalising factor of \(\frac{1}{5 \sqrt{2}}\) is

Answer:
(C)

Question 20.
The rationalising factor of \(\frac{1}{\sqrt{27}}\) is
A) \(\frac{1}{\sqrt{27}}\)
B) \(\sqrt{27}\)
C) \(\sqrt{3}\)
D) 3
Answer:
C) \(\sqrt{3}\)

Question 21.
\(\left(\frac{3}{4}\right)^{-3} \times\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{6}=\ldots \ldots\)

Answer:
(B)

Question 22.
\(\sqrt[5]{32}\) =
A) 32⁵
B) 2
C) 4\(\sqrt{2}\)
D) 2\(\sqrt{2}\)
Answer:
B) 2

Question 23.
(128)^1/7 =
A) 2
B) 4
C) 8
D) \(8 \sqrt{2}\)
Answer:
A) 2

Question 24.
Radical form of 27^1/5 is
A) \(\sqrt{27}\)
B) \(3 \sqrt{27}\)
C) \(4\sqrt{27}\)
D) \(5\sqrt{27}\)
Answer:
D) \(5\sqrt{27}\)

Question 25.
a^1/n =
A) \(\sqrt[n]{a}\)
B) \(\frac{\mathrm{a}}{\mathrm{n}}\)
C) na
D) n + a
Answer:
A) \(\sqrt[n]{a}\)

Question 26.
The Rationalising factor of \(\frac{1}{5-\sqrt{3}}\) is
A) \(5+\sqrt{3}\)
B) \(\sqrt{3}-5\)
C) \(\frac{1}{5+\sqrt{3}}\)
D) \(\frac{1}{\sqrt{3}-5}\)
Answer:
C) \(\frac{1}{5+\sqrt{3}}\)

Question 27.
\(\sqrt[n]{a^{m}}\) =
A) a^m/n
B) a^n/m
C) a^mn
D)a^m-n
Answer:
A) a^m/n

Question 28.
If x³ = 10, then x is
A) a rational number
B) an irrational number
C) a perfect number
D) an even number
Answer:
B) an irrational number

Question 29.
If p³ = 216, then p is
A) an odd nuipber
B) an irrational number
C) a perfect number
D) a rational number
Answer:
D) a rational number

Question 30.
The radical form of 15^2/3 is
A) \(\sqrt[3]{30}\)
B) \(\sqrt[3]{15}\)
C) \(\sqrt[3]{225}\)
D) \(\sqrt[3]{45}\)
Answer:
C) \(\sqrt[3]{225}\)

Question 31.
The radical form of 6^2/3 is
A) \(\sqrt[3]{36}\)
B) \(\sqrt{36}\)
C) \(\sqrt{48}\)
D) \(\sqrt{216}\)
Answer:
A) \(\sqrt[3]{36}\)

Question 32.
The exponential form of \(\sqrt[4]{81}\) is
A) 9^1/4
B) 9^2/4
C) 3^1/4
D) 3^1/8
Answer:
B) 9^2/4

Question 33.
The exponential form of \(\sqrt[35]{105}\) is
A) 3^1/35
B) 5^1/35
C) 7^1/35
D) 105^1/35
Answer:
D) 105^1/35

Question 34.
\((\sqrt{\mathbf{a}}+\mathbf{b})(\sqrt{\mathbf{a}}-\mathbf{b})=\)
A) a² – b²
B) a – b²
C) a – b
D) a + b²
Answer:
B) a – b²

Question 35.
\((\sqrt{x}+y)^{2}=\)
A) x + y + 2\(\sqrt{xy}\)
B) \(\sqrt{x}\) + y² + 2xy
C) x + y² + 2 \(\sqrt{x}\) .y
D) x² + y² + 2 \(\sqrt{x}\) .y
Answer:
C) x + y² + 2 \(\sqrt{x}\) .y

Question 36.
\((\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})=\)
A) 1
B) 0
C) 5
D) 13
Answer:
A) 1

Question 37.
(- 8)^7/5 x (- 8)^-4/5 x (- 8)^-3/5 =
A) 0
B) – 8
C) 1
D) – 512
Answer:
C) 1

Question 38.
The decimal form of \(\mathbf{0 . \overline { 3 2 }}\) is
A) \(\frac{32}{100}\)
B) \(\frac{32}{99}\)
C) \(\frac{32}{90}\)
D) \(\frac{32}{50}\)
Answer:
B) \(\frac{32}{99}\)

Question 39.
\(\sqrt{9} \times \sqrt{16}=\)
A) \(\sqrt{25}\)
B) \(\frac{3}{4}\)
C) 12
D) 144
Answer:
C) 12

Question 40.
\(\sqrt{a} \div \sqrt{b}=\)
A) \(\sqrt{a b}\)
B) \(a \sqrt{b}\)
C) \(\sqrt{a b}\)
D) \(\sqrt{\frac{a}{b}}\)
Answer:
D) \(\sqrt{\frac{a}{b}}\)

Question 41.
\(\left(\frac{-2}{3}\right)^{2 / 7} \times\left(\frac{-2}{3}\right)^{5 / 7}=\)
A)1
B) \(\frac{-2}{3}\)
C) 0
D) \(\left(\frac{-2}{3}\right)^{2 / 5}\)
Answer:
B) \(\frac{-2}{3}\)

Question 42.
If \(\sqrt{3}\) = 1732, then \(\sqrt{27}\) =
A) 3 x 1.732
B) 9 x 1.732
C) 27 x 1.732
D) 6 x 1.732
Answer:
A) 3 x 1.732

Question 43.
Whose value is 11.18 if \(\sqrt{5}\) = 2.236 ?
A) \(\sqrt{25}\)
B) \(\sqrt{75}\)
C) \(\sqrt{125}\)
D) \(\sqrt{250}\)
Answer:
C) \(\sqrt{125}\)

Question 44.
The decimal value of \(\frac{\mathbf{2 2}}{\mathbf{7}}\) is
A) 3.421
B) 3.142
C) 3.412
D) 3.124
Answer:
B) 3.142

Question 45.
If \(\sqrt{10}\) = 3.162, then \(\sqrt{40}\) = mi i
A) 6.324
B) 9.486
C) 12.648
D) 31.62
Answer:
A) 6.324

Question 46.
\(\sqrt[5]{32^{-2}}=\)
A) 2
B) 4
C) 6
D) 1/2
Answer:
B) 4

Question 47.
Rationalising factor of \(\sqrt{5}+\sqrt{6}\) is
A) \(\sqrt{5}\) – 6
B) 5 – \(\sqrt{6}\)
C) \(\sqrt{5}-\sqrt{6}\)
D) 5 + \(\sqrt{6}\)
Answer:
C) \(\sqrt{5}-\sqrt{6}\)

Question 48.
Express 3.25 in the form of p/q
A) \(\frac { 13 }{ 4 }\)
B) \(\frac { 65 }{ 2 }\)
C) \(\frac { 13 }{ 40 }\)
D) \(\frac { 13 }{ 20 }\)
Answer:
A) \(\frac { 13 }{ 4 }\)

Question 49.
If aⁿ = b, then \(\sqrt[n]{b}\) =
A) n
B) a
C) b^1/n
D) a^1/n
Answer:
B) a

Question 50.
\(\frac{3^{1 / 5}}{3^{1 / 3}}=\)
A) 3^1/15
B) 3^2/15
C) 3^-2/15
D) 3^8/15
Answer:
C) 3^-2/15

Question 51.
The collection of negative numbers and whole numbers is denoted by
A) Q
B) W
C) Z or l
D) N
Answer:
C) Z or l

Question 52.
The rational number which lies be¬tween two rational numbers a and b is
A) a—b
B) b—a
C) \(\sqrt{a b}\)
D) \(\frac{a+b}{2}\)
Answer:
D) \(\frac{a+b}{2}\)

Question 53
2/3 =
A) \(0 . \overline{6}\)
B) 0.66
C) 0.666
D) 0.6
Answer:
A) \(0 . \overline{6}\)

Question 54.
The decimal value of \(\frac{1}{2^{3}}\) is
A) 0.5
B) 0.25
C) 0.125
D) 1.125
Answer:
C) 0.125

Question 55.
π is
A) a naturaF number
B) an irrational number
C) a rational number
D) none of these
Answer:
B) an irrational number

Question 56.
\(\sqrt{7}\) = 2.65 (approximately), then the approximate value of \(\sqrt{28}\) is
A) 2.65
B) 5.3
C) 7.95
D) 10.6
Answer:
B) 5.3

Question 57.
Cube root of \(\sqrt{4}+\sqrt{36}\) is
A) \(\sqrt[3]{144}\)
B) 8
C) 2
D) \(\sqrt[3]{40}\)
Answer:
C) 2

Question 58.
Value of \(\sqrt[4]{81}+\sqrt[5]{32}\) is
A) 5
B) 9
C) – 1
D) A or C
Answer:
A) 5

Instructions: Observe the number line and give answer from Q. No. 6-10.
In the number line at ‘O’ draw a unit square OABC with each side 1 unit in length and OB = OP, OD = OQ, OE = OR.

Observe above number line and answer the following:

Question 59.
The length of \(\overline{\mathrm{OP}}\) is
A) \(\frac{3}{2}\)Units
B) \(\sqrt{2}\) Units
C) \(\sqrt{3}\) Units
D) 1 Unit
Answer:
B) \(\sqrt{2}\) Units

Question 60.
The length of \(\overline{\mathrm{AQ}}\) is
A) \(\sqrt{3}\) – 1 Units
B) \(\sqrt{3}\) + \(\sqrt{2}\) Units
C) \(\sqrt{3}\) – \(\sqrt{3}\) Units
D) \(\sqrt{3}\) + 1 Units
Answer:
A) \(\sqrt{3}\) – 1 Units

Question 61.
Perimeter of the rectangle OCSQ
A) 2(\(\sqrt{3}\) + 1) Units
B) 6 Units
C) 2(\(\sqrt{2}\) + \(\sqrt{3}\)) Units
D) 2(\(\sqrt{2}\) + 1) Units
Answer:
B) 6 Units

Question 62.
Area of rectangle PQRS in sq. units
A) \(\sqrt{3}\) – 1
B) \(\sqrt{3}\) + \(\sqrt{2}\)
C) \(\sqrt{3}\) – \(\sqrt{2}\)
D) \(\sqrt{2}\) – 1
Answer:
C) \(\sqrt{3}\) – \(\sqrt{2}\)

Question 63.
\(\sqrt{5}\) lie between
A) 1 and 2
B) 2 and 3
C) 3 and 4
D) 0 and 1
Answer:
B) 2 and 3

Question 64.
A Rectangular park dimensions are (3 + \(\sqrt{2}\)) and (2 + \(\sqrt{2}\)) units, then the area of that park in square unit.
A) 8 + 5\(\sqrt{2}\)
B) 5 + 2\(\sqrt{2}\)
C) 13\(\sqrt{2}\)
D) 5 + \(\sqrt{2}\)
Answer:
A) 8 + 5\(\sqrt{2}\)

Question 65.
The value of 1.999 ……………… in the form of \(\frac{\mathbf{p}}{\mathbf{q}}\) (p,q are integers, q ≠ 0 )
A) \(\frac{1999}{1000}\)
B) 2
C) \(\frac{1}{9}\)
D) \(\frac{19}{10}\)

Question 66.
Observe the successive magnification of 2.8746 on the number line.

Arrange these steps orderly.
A) 1,3, 4, 2
B) 3, 4, 2,1
C) 3, 4, 1, 2
D) 1, 2, 4, 3
Answer:
A) 1,3, 4, 2

Question 67.
If ‘x’ is a positive real number and x² = 2, then the value of x³ is
A) 2\(\sqrt{2}\)
B)3\(\sqrt{2}\)
C) 4
D) \(\sqrt{2}\)
Answer:
A) 2\(\sqrt{2}\)

Question 68.
If \(\sqrt{10}\) = 3.162, then the value of \(\frac{1}{\sqrt{10}}\) is
A) 31.62
B) 3.162
C) 0.3162
D) 316.2
Answer:
C) 0.3162

Question 69.
If x = \(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\) and y = \(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\) then the value of x + y is
A) 5
B) 5 + 2\(\sqrt{6}\)
C) 10
D) 5 – 2\(\sqrt{6}\)
Answer:
C) 10

Question 70.
If x = 2 + 7\(\sqrt{3}\) , then the value of x – \(\frac{1}{x}\) is
A) 4
B) \(\sqrt{3}\).
C) 2 \(\sqrt{3}\).
D) 2 + \(\sqrt{3}\).
Answer:
B) \(\sqrt{3}\).

Question 71.
A rational number equivalent to \(\frac{5}{7}\) is
A) \(\frac { 15 }{ 17 }\)
B) \(\frac { 25 }{ 27 }\)
C) \(\frac { 10 }{ 14 }\)
D) \(\frac { 10 }{ 27 }\)
Answer:
C) \(\frac { 10 }{ 14 }\)

Question 72.
If x = \(\sqrt{5}\) + 2, then the value of x – \(\frac{1}{x}\) is
A) 2\(\sqrt{5}\)
B) 4
C) -4
D) -2\(\sqrt{5}\)

Question 73.
The value of \(\left(\frac{1}{2}\right)^{3}+\left(\frac{1}{3}\right)^{3}-\left(\frac{5}{6}\right)^{3}\) is
A) \(\frac { 5 }{ 6 }\)
B) \(\frac { 5 }{ 12 }\)
C) \(\frac { -5 }{ 36 }\)
D) \(\frac { -5 }{ 12 }\)
Answer:
D) \(\frac { -5 }{ 12 }\)

Question 74.
If the length and breadth of a rectangular sheet are \(\sqrt{5}\) + \(\sqrt{2}\) and \(\sqrt{5}\) – \(\sqrt{2}\) units, then it’s area in sq. units
A) \(\sqrt{3}\)
B) 2\(\sqrt{5}\)
C) 3
D) 7
Answer:
C) 3

Question 75.
Ail irrational number between 4 and 5
A) \(\sqrt{20}\)
B) \(\sqrt{9}\)
C) \(\sqrt{4.5}\)
D) 4.5
Answer:
A) \(\sqrt{20}\)

Question 76.
Rationalizing Factor of 3\(\sqrt{9}\)
A) \(\sqrt[3]{3}\)
B) \(\sqrt[3]{6}\)
C) \(\sqrt[3]{9}\)
D) \(\sqrt[3]{27}\)
Answer:
A) \(\sqrt[3]{3}\)

Question 77.
There are three odd numbers and one even number. What type of number is their sum ?
A) Even number
B) Neither even nor odd number
C) Cannot be determined as actual numbers are not given
D) Odd number.
Answer:
D) Odd number.

Question 78.
\(\sqrt[5]{32^{2}}\) = …………..
A) 1
B) 2
C) 4
D) 5
Answer:
C) 4

Question 79.
The product of first 2019 whole numbers……
A) 0
B) 2019
C) 1.2.3.4 …………..2018.2019
D) 1 + 2 + 3 + ………………+ 2019
Answer:
A) 0

Question 80.
Express 3.25 in the form of \(\frac{\mathbf{p}}{\mathbf{q}}\).
A) \(\frac{13}{4}\)
B) \(\frac{65}{2}\)
C) \(\frac{13}{40}\)
D) \(\frac{13}{20}\)
Answer:
A) \(\frac{13}{4}\)

Question 81.
Radical form of \(3^{\frac{1}{5}}\) is
A) \(\sqrt[5]{3^{1}}\)
B) \(\sqrt[3]{5^{1}}\)
C) \(\sqrt[5]{1^{3}}\)
D) \(\sqrt[3]{1^{5}}\)
Answer:
A) \(\sqrt[5]{3^{1}}\)

Question 82.
Which of the following is the \(\frac{\mathbf{p}}{\mathbf{q}}\) form of 3.72 ? (q ≠ 0 and p and q are coprimes)
A) \(\frac{372}{100}\)
B) \(\frac{31}{5}\)
C) \(\frac{93}{25}\)
D) \(\frac{93}{5}\)
Answer:
A) \(\sqrt[5]{3^{1}}\)

Question 83.
Which of the following rational numbers lies between 2 and 3 ?
A) \(\frac{2}{3}\)
B) \(\frac{3}{2}\)
C) \(\frac{5}{2}\)
D) \(\frac{5}{3}\)
Answer:
C) \(\frac{5}{2}\)

Question 84.
1.121231234………….. is not a rational number because it is
A) Recurring Decimal
B) Non terminating Decimal
C) Non recurring Decimal
D) Both B and C
Answer:
D) Both B and C

Question 85.
The exponential form of \(\sqrt[5]{3^{2}}\) is
A) \(3^{5 / 2}\)
B) \(3^{2 / 5}\)
C) \(3^{5 /times 2}\)
D) \(3^{5 – 2}\)
Answer:
B) \(3^{2 / 5}\)

Question 86.
The square root of which number is a rational number ?
A) 14
B) 80
C) 1.96
D) 0.004
Answer:
C) 1.96

Question 87.
The value of “P” on the given number line.

A) \(\frac{-7}{5}\)
B) \(-\frac{5}{7}\)
C) \(\frac{7}{5}\)
D) \(\frac{5}{7}\)
Answer:
A) \(\frac{-7}{5}\)

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

AP State Syllabus 10th Class Maths Bits 10th Lesson Mensuration with Answers

Question 1.
The total surface area of a cube is 54 cm², then find its side.
Answer:
3 cm
Explanation:
TSA = 6s² = 54
⇒ side² = 9 ⇒ side = 3 cm.

Question 2.
Base area of a regular cylinder is 154 cm², then find its radius.
Answer:
7 cm
Explanation:
πr² = 154
⇒ r² = 154 × \(\frac{7}{22}=\frac{14 \times 7}{2}\) × 7 × 7
⇒ r = 7 cm

Question 3.
If the height and radius of a cone are 15 cm and 8 cm, then find its slant height
Answer:
17 cm
Explanation:

Question 4.
Write a formula to find curved surface area of a hemisphere.
Answer:
2πr²

Question 5.
How much the volume of a cube having 1 cm side ?
Answer:
1 cm³

Question 6.
Ratio of volumes of two spheres is 8 : 27, then find ratio of their curved surface area.
Answer:
4 : 9
Explanation:

Question 7.
Football is in a model of …………..
Answer:
sphere

Question 8.
If the volume of a cube is 216 cm³, then find its side.
Answer:
6 cm
Explanation:
S³ = 216 ⇒ Side = \(\sqrt[3]{216}\) = 6 cm

Question 9.
Find the curved surface area of a right circular cylinder.
Answer:
2πrh

Question 10.
Find the curved surface area of a sphere will be……………. whose radius
is 10 cm.
Answer:
400 π

Question 11.
Find the volume of a cube will be ………….. (in cm³), whose total surface area is 216 cm².
Answer:
216
Explanation:
6S² = 216
⇒ S² = \(\frac{216}{6}\) = 36
⇒ S = \(\sqrt{36}\) = 6
∴ Volume = S³ = 6³ = 216

Question 12.
Write the name of a famous book writ-ten by ancient mathematician Aryabhatta.
Answer:
Aryabhatteeyam.

Question 13.
Which of the following vessel can be filled with more water (A, B are in cylindrical shape)?

Answer:
B)

Question 14.
Find the volume of right circular cylinder with radius 6 cm and height 7 cm.
Answer:
792 cm³

Question 15.
A sphere of radius ‘r’ inscribed in a cylinder. The surface area of the sphere …………… of the cylinder.
Answer:
Curved surface area.

Question 16.
How much the maximum length of the stick that can be placed in a cuboid, whose measurements are 8 × 4 × 1?
Answer:
9

Question 17.
A cylinder and cone have bases of equal radii and are of equal heights, then find their volumes are in the ratio.
Answer:
3 : 1

Question 18.
Find the total surface area of a solid hemisphere of radius 7 m.
Answer:
147 π sq.m.
Explanation:
3πr² = 7² × 3 × π = 147π sq.m.

Question 19.
Radius of a cone is ‘r’, height is ‘h’ and its slant height is ‘l’, then which of the following is false ?
Answer:
Always r > p

Question 20.
Radius, height, slant height of a cone are r, h, ‘l’, then 7′ value in terms of r and h.
Answer:
\(\sqrt{r^{2}+h^{2}}\)

Question 21.
Volumes of two spheres are in the ratio of 8:27. Find the ratio of their surface areas.
Answer:
4 : 9

Question 22.
A solid ball is exactly fitted inside the cubical box of side ‘a’. Find the volume of the ball.
Answer:
\(\frac{1}{6}\) πa³

Question 23.
If the total surface area of cube is 96 cm³, then find side of cube.
Answer:
4 cm

Question 24.
Base area of the prism is 30 cm² and its height is 10 cm. Then find the volume of the prism.
Answer:
300 cm³

Question 25.
The volume of a cone with base radius 7 cm is 462 c.c., find its height.
Answer:
9 cm

Question 26.
If total surface area of a cube is 96 cm², then find its volume.
Answer:
64 cm³

Question 27.
Find the volume of cone, whose radius is 3 cm and height is 8 cm.
Answer:
24 π

Question 28.
Write a formula to find total surface
area of cone in sq. units. f
Answer:
πr² + πrl

Question 29.
Find the volume of a hemisphere of radius 3.5 cm is …………… cm³.
Answer:
89.83

Question 30.
In the above problem find TSA = ………………… cm².
Answer:
115.5

Question 31.
Write a combination of a shuttle cock.
Answer:
Hemisphere, frustum cone

Question 32.
The volume of cone is 462 cm³, r = 7 cm, then find h is ……………… cm.
Answer:
9

Question 33.
10³ (cm)³ = ……………. litre.
Answer:
1

Question 34.
In l² = h² + r², h = 15, r = 8, then l = ………………
Answer:
17

Question 35.
The perimeter of an equilateral triangle is 60 cm, then find its area (in cm²).
Answer:
173.2

Question 36.
Write the number of faces of a cuboid.
Answer:
8

Question 37.
If the ratio of radii of two spheres is 2:3, then find the ratio of their surface areas.
Answer:
4 : 9

Question 38.
Write a formula to find volume of sphere in …………….. cu. units.
Answer:
\(\frac{4}{3}\)πr³

Question 39.
In a hollow cuboid box of size 4 × 3 × 2 m, find the number of solid iron spherical balls of radius 0.5 m that can be packed.
Answer:
24

Question 40.
In a cone, r = 7 cm, h = 10 cm, then find l = cm.
Answer:
12.2

Question 41.
Find T.S.A of a solid hemisphere whose radius is 7 cm.
Answer:
147π

Question 42.
Find the total surface area of hemisphere of radius ’r’.
Answer:
3πr²

Question 43.
If the length of each diagonal of a cube is doubled, then how many times its volume becomes.
Answer:
8

Question 44.
Who gave the symbol π?
Answer:
Euler

Question 45.
Find the surface area of a cube, whose side is 27 cm.
Answer:
4374 cm³

Question 46.
Find the volume of cone if r = 2 cm, h= 4 cm.
Answer:
\(\frac{16}{3}\)π cm³
Explanation:
Volume of cone = \(\frac{1}{3}\) πr²h
\(\frac{1}{3}\) × π × 4 × 4 = \(\frac{16}{3}\)π cm³

Question 47.
Write the number of edges of a cuboid has.
Answer:
12

Question 48.
Find the volume of a right circular cone with radius 6 cm and height 7 cm is …………….. cm³.
Answer:
264
Explanation:
Volume of right circular cone V = \(\frac{1}{3}\) πr²h
= \(\frac{1}{3}\) . π . 6² . 7 = 264 cm³.

Question 49.
If a right angled triangle is revolved about its hypotenuse, then find it will form a ……………..
Answer:
Double cone

Question 50.
Write a formula to find CSA of cylinder ……………… sq. units.
Answer:
2πrh

Question 51.
Volume of cuboid = ………….. cu.units.
Answer:
lbh

Question 52.
Write a formula to find surface area of sphere ……………. in sq. units.
Answer:
4πr²

Question 53.
Total surface area of a cube is 216 cm², then find its volume.
Answer:
216

Question 54.
If the radii of circular ends of a frustum of a cone are 20 cm and 12 cm and its height is 6 cm, then find the slant height of the frustum.
Answer:
10

Question 55.
If the external and internal radii of a hollow hemispherical bowl are R and r, then find its total surface area.
Answer:
π (3R² + r²)

Question 56.
Rational value of π = …………………
Answer:
22/7

Question 57.
A cylinder, a cone and a hemisphere are of equal base and have the same height, then find the ratio of their volumes.
Answer:
3 : 1 : 2
Explanation:
Cylinder, Cone, Hemisphere have equal base and same height. So, the height will become radius.
= Volume of cylinder: Volume of cone : Volume of hemisphere
= πr²h : \(\frac{1}{3}\) πr²h : \(\frac{2}{3}\) πr²h
= \(\frac{1}{3}\) πr³ : \(\frac{2}{3}\) πr³ : πr³
= 1 : \(\frac{1}{3}\) : \(\frac{2}{3}\) = 3 : 1 : 2

Question 58.
A solid sphere of radius 10 cm is moulded into 8 spherical solid balls of equal radius, then find radius of small spherical balls.
Answer:
5
Explanation:
Volume of sphere = 8 (volume of one spherical ball)

Question 59.
Write a formula to find CSA of cone = ………………… sq. units.
Answer:
πrl

Question 60.
Find the total surface area of a solid hemisphere of radius 7 cm.
Answer:
239π cm²

Question 61.
The volume of a vessel in the form of a right circular cylinder is 448π cm³ and its height is 7 cm, then find the radius of the base.
Answer:
8 cm

Question 62.
If the diameter of a sphere is’d’, then find its volume.
Answer:
\(\frac{1}{6}\) πd²

Question 63.
Write a formula to find total surface area of cylinder in ……………….. sq. units.
Answer:
2πrh+ 2πr²

Question 64.
In a cylinder, r= 7 m, h=15 m, then find V.
Answer:
2310 m³.

Question 65.
If the diagonals of a rhombus are 10 cm and 24 cm, then find area in ………………… cm².
Answer:
120

Question 66.
Find the curved surface area of a right circular cone of height 15 cm and base diameter is 16 cm.
Answer:
136π cm²

Question 67.
Find the number of balls, each of radius 1 cm that can be made from a solid sphere of radius 8 cm.
Answer:
512

Question 68.
Volume of hemisphere is19404 cm³, then find its TSA (in cm²).
Answer:
4158

Question 69.
r³ = 1728, then find ‘r’.
Answer:
12

Question 70.
In a cone, d = 14 cm, l = 10 cm, then find CSA = ……………….. cm².
Answer:
220

Question 71.
In the figure, P = …………………

Answer:
h² + r²

Question 72.
The surface area of a sphere is 616 sq.cm, then find its radius in …………. cm.
Answer:
7

Question 73.
In a hemisphere, r = 7 cm, then find CSA (in cm²)
Answer:
308

Question 74.
Write a formula to find volume of hollow cylinder.
Answer:
(πR² – r²)

Question 75.
In a cube, a = 4 cm, then find TSA (in cm²).
Answer:
90

Question 76.
In a cylinder, h=14 cm, V= 176 cm³, r = …………….. cm.
Answer:
2

Question 77.
In a hemisphere, r = 1.75 cm, then find CSA (in cm²).
Answer:
38.5

Question 78.
TSA of a cylinder is 1188 cm², h = 20 cm (in cm), then find its volume.
Answer:
3080

Question 79.
Heap of stones is an example of ………………
Answer:
Cone

Question 80.
Write a formula to find diagonal of rectangle.
Answer:
\(\sqrt{l^{2}+b^{2}}\)

Question 81.
The area of the base of a right circular cone is 78.5 cm². If its height is 12 cm, then find its volume (in cm³).
Answer:
314

Question 82.
Surface area of a sphere and cube are equal, then find the ratio of their volumes.
Answer:
√π : √6
Explanation:

Question 83.
A conical flask is full of water. The flask has base radius r and height h. The water is poured into a cylindrical flask of base radius mr. Find the height of water in the cylindrical flask.
Answer:
\(\frac{\mathrm{h}}{3}\) m²

Question 84.
Find the volume of the greatest cylinder that can be cut from a solid wooden cube of length of edge 14 cm.
Answer:
2156 cm³

Question 85.
The area of equilateral triangle is 36√3 cm², then find the perimeter (in etn).
Answer:
36
Explanation:
Area 36√3 cm² ⇒ \(\frac{\sqrt{3} a^{2}}{4}\) = 36√3
⇒ a² = 36 × 4
⇒ a = 6 × 2 = 12 cm
Perimeter = 3a = 3 × 12 = 36 cm.

Question 86.
Base circumference of a cylinder is 220 cm and height is 63 cm, then find CSA (in cm²).
Answer:
13860

Question 87.
Write a formula to find area of equi¬lateral triangle of side ‘a’ units (in sq. units).
Answer:
\(\frac{\sqrt{3}}{4}\)a²

Question 88.
A solid iron cuboid of dimensions 49 × 33 × 24 cm is melted to form a solid sphere, then find its radius.
Answer:
21 cm

Question 89.
A cone and a hemisphere have equal bases and equal volumes, then find the ratio of their heights.
Answer:
2 : 1

Question 90.
Laddu is an example of ……………
Answer:
Sphere

Question 91.
Write a formula to find total surface area of a cube (in sq. units).
Answer:
6l²

Question 92.
The height of a cylinder is doubled and radius is tripled, then how many times its curved surface area will become.
Answer:
6 times
Explanation:
CSA of cylinder is 2πrh, if radius is tripled and height is doubled, then CSA = 2π . 3r . 2h = 12πrh = 6(2πrh)

Question 93.
Write a formula to find volume of hemisphere (in cu. units).
Answer:
\(\frac{2}{3}\)πr³

Question 94.
The surface areas of two spheres are in the ratio 1 : 4, then find ratio of their volumes.
Answer:
1 : 64

Question 95.
Write the diameter of a sphere which can inscribe a cube of edge ‘x’ cm.
Answer:
x

Question 96.
The volume of a cube is 216 cm³, then find its edge.
Answer:
6

Question 97.
Write a formula find volume of cylinder (in cu. units).
Answer:
πr²h

Question 98.
Find the ratio of volume of a cone and cylinder of equal diameter and height.
Answer:
1 : 3

Question 99.
Write a formula to find volume of a cube (in cu. units).
Answer:
a³

Question 100.
The sphere of radius 2.1 cm, then find its volume (in cm³).
Answer:
38.08

Question 101.
A sphere, a cylinder and a cone have the same radius, then find the ratio of their curved surface areas.
Answer:
4 : 4 : √5

Question 102.
If the diagonal of a cube is 2.5 m, then find its volume (in m³).
Answer:
\(\frac{5.2}{\sqrt{3}}\)
Explanation:
Applying Pythagoras theorem,
(2.5)² = [a² + (√2a)²]
⇒ 6.25 = 3a²
⇒ a² = \(\frac{6.25}{3}\) ⇒ a = \(\frac{2.5}{\sqrt{3}}\) m³
Volume of cube = a³
\(\left(\frac{2.5}{\sqrt{3}}\right)^{3}\) = \(=\frac{15.625}{3 \sqrt{3}}=\frac{5.2}{\sqrt{3}}\) m³

Question 103.
A heap of rice is in the form of a cone of diameter 12 m and height 8 m, then find the volume (in m³).
Answer:
301.71

Question 104.
Perimeter of square is 20 cm, find then area (in cm²).
Answer:
25

Question 105.
CSA of a cone is 4070 cm² and its diameter is 70 cm, then find slant height.
Answer:
37

Question 106.
The volume of a cuboid is 3,36,000 cm³. If its area is 5,600 cm², then find h. (in cm).
Answer:
60

Question 107.
Write a formula to find diagonal of a cuboid.
Answer:
\(\sqrt{l^{2}+b^{2}+h^{2}}\)

Question 108.
Find the volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm.
Answer:
19.4 cm³

Question 109.
The diameter of a metallic sphere is 6 cm and melted to draw a wire of diameter 2 cm, then find the length of the wire.
Answer:
9 cm
Explanation:
Volume of sphere = Volume of cylinder
\(\frac{4}{3}\)πr³ = πr²h
⇒ \(\frac{4}{3}\) × 3³ = h × 2²
⇒ 4 × 9 = h × 4 ⇒ h = 9 cm

Question 110.
The volume and surface area of a sphere are numerically equal. Then find the volume of the smallest cylinder in which the sphere is exactly kept.
Answer:
54π

Question 111.
Write a formula to find volume of cone with’d’ as diameter and ’h’ as height is ………………. cu. units.
Answer:
\(\frac{\pi \mathrm{d}^{2} \mathrm{~h}}{12}\)

Question 112.
An iron cylindrical rod has a height 4 times its radius is melted and cast into spherical balls of the same radius. Find the number of balls cast.
Answer:
6
Explanation:
Volume of cylinder = n × Volume of sphere
πr²h = n × \(\frac{4}{3}\)πr³
πr²(8r) = n × \(\frac{4}{3}\)πr³
n = \(\frac{24}{4}\) = 6
∴ Number of balls = 6.

Question 113.
If a cone is cut into two parts by a horizontal plane passing through the mid point of the axis, find the ratio of the volumes of the upper part and the cone.
Answer:
1 : 8

Question 114.
In a cylinder, r = 8 cm, h = 10 cm, then CSA = ……………….. cm³
Answer:
\(\frac{3520}{7}\)

Question 115.
In a cone, (l + r)(l – r) = …………..
Answer:
h²

Question 116.
A solid sphere of radius r melted and recast into the shape of a solid cone of height r, then find radius of the base of the cone.
Answer:
2r

Question 117.
If the radius of base of a cylinder is doubled and the height remains un-changed, its C.S.A becomes.
Answer:
3 times

Question 118.
Write a formula to find volume of cone, (in cu. units).
Answer:
\(\frac{1}{3}\)πr²h

Question 119.
Write a formula to find diagonal of a cube (in units).
Answer:
a√3

Question 120.
The ratio of volume of two cones is 4 : 5 and the ratio of the radii of their base is 2 : 3, then find ratio of their vertical heights.
Answer:
9 : 5

Question 121.
Find the number of cubes of side 2 cm which can be cut from a cube of side 6 cm.
Answer:
27

Question 122.
A cuboid has dimensions 10 × 8 × 6 cm, then find its volume (in cm³).
Answer:
480

❖ Choose the correct answer satisfying the following statements.
Question 123.
Statement (A): The slant height of the frustum of a cone is 5 cm and the difference between the radii of its two circular ends is 4 cm. Than the height of the frustum is 3 cm.
Statement (B) : Slant height of the frustum of the cone is given by
l = \(\sqrt{(\mathrm{R}-\mathrm{r})^{2}+\mathrm{h}^{2}}\)
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.
Explanation:
We have, L = 5 cm, R – r = 4 cm
∴ 5 = \(\sqrt{(4)^{2}+\mathrm{h}^{2}}\)
⇒ 16 + h² = 25
⇒ h² = 25 – 16 = 9
⇒ h = 3 cm
Hence, (i) is the correct option.

Question 124.
Statement (A) : If the volumes of two spheres are in the ratio 27 : 8. Then their surface areas’are in the ratio 3 : 2.
Statement (B) : Volume of the sphere
= \(\frac{4}{3}\) πr³ and its surface area = 4πr³.
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) Both A and B are false.
Explanation:

Hence, (iv) is the correct option.

Question 125.
Statement (A) : Two identical solid cube of side 5 cm are joined end to end. Then total surface area of the resulting cuboid is 300 cm².
Statement (B): Total surface area of a cuboid is 2(lb + bh + lh).
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:
When cubes are joined end to end, it will form a cuboid.
∴ l = 2 × 5 = 10 cm, b = 5 cm and h = 5 cm
∴ Total surface area = 2 (lb + bh + lh) = 2(10 × 5 + 5 × 5 + 10 × 5)
= 2 × 125 = 250 cm²
Hence, (iii) is the correct option.

Question 126.
Statement (A) : The number of 90ms 1.75 cm in diameter and 2 mm thick if formed from a melted cuboid 10 cm × 5.5 cm × 3.5 cm is 400.
Statement (B) : Volume of a cylinder = πr²h cubic units and area of cuboid = (l × b × h) cu. units.
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:

Hence, (i) is the correct option.

Question 127.
Statement (A) : The radii of two cones are in the ratio 2 : 3 and their volumes in the ratio 1 : 3. Then the ratio of their height is 3 : 2.
Statement (B) :
Volume of the cone = \(\frac{1}{3}\)πr²h
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:
We have, ratio of volume

Hence, (iii) is the correct option.

Question 128.
Statement (A) : The curved surface area of a cone of base radius 3 cm and height 4 cm is 15 π cm².
Statement (B): Volume of a cone = \(\frac{1}{3}\)πr²h
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 129.
Statement (A): If the surface area of a sphere is 616 cm². Then its radius 6 cm.
Statement (B): Surface area of sphere = 4πr²sq. units.
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 130.
Statement (A) : A hemisphere of radius 7 cm is to be painted outside on the surface of it. The total cost of painting at ₹ 5 per cm2 is ₹ 2300.
Statement (B): The total surface area of a hemisphere is 3πr²sq. units.
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 131.
Statement (A) : Total surface area of the cylinder having radius of the base 14 cm and height 30 cm is 3872 cm².
Statement (B): If r be the radius and ‘h’ be the height of the cylinder, then total surface area= (2πrh + 2πr²)
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:
A and B both are correct and B is the correct explanation of the A.
Total surface area = 2πrh × 2πr²
= 2πrh × 2πr²
= 2πr(h + r)
= 2 × \(\frac{22}{7}\) × 14 (30 + 14)
= 88 (44) = 3872 cm²
Hence, (i) is the correct’option,

Question 132.
Statement (A): If the height of a cone is 24 cm and diameter of the base is 14 cm, then the slant height of the cone is 15 cm.
Statement (B) : If r be the radius and h the slant height of the cone, then slant
height = \(\sqrt{\mathrm{h}^{2}+\mathrm{r}^{2}}\)
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) A is false, B is true.
Explanation:
A is incorrect here, but B is correct.
Slant height = \(\sqrt{\left(\frac{14}{2}\right)^{2}+(24)^{2}}\)
= \(\sqrt{49+576}\)
= \(\sqrt{625}\) = 25
Hence, (iii) is the correct option.

Question 133.
Statement (A): If the radius of a cone is halved and volume is not changed, then height remains same.
Statement (B): If the radius of a cone is halved and volume is not changed then height must become four times of the original height.
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:
A is incorrect and B is correct.

as V₁ = V₂
∴ 4h₁ = h₂
Hence, (iii) is the correct option.

Question 134.
Statement (A): If a ball in the shape of a sphere has a surface area of 221.76 cm2, then its diameter is 8.4 cm.
Statement (B) : If the radius of the sphere be r, then surface area S = 4πr²,
i.e., r = \(\frac{1}{2} \sqrt{\frac{\mathrm{s}}{\pi}}\)
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 135.
Statement (A) : Number of spherical balls that can be made out of a solid cube of lead whose edge is 44 cm, each ball being 4 cm in diameter is 2541.
Statement (B) : Number of balls = \(\frac{\text { Volume of one ball }}{\text { Volume of lead }}\)
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 136.
Statement (A) : If the base area and height of a prism be 25 cm² and 6 cm respectively, then its volume is 150 cm³.
Statement (B): Volume of a pyramid = \(\frac{\text { Basearea } \times \text { height }}{3}\)
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:
A and B both are correct, but B is not the correct explanation of the A.
Volume of a prism = Base area × height
= 25 × 6 = 150 cm₃
Hence, (i) is the correct option.

❖ Read the below passages and answer to the following questions.
A tent is in the form of a right circular cylinder, surmounted by a cone. The diameter of the cylinder is 24 m. The height of the cylindrical portion is 11m, while the vertex of the cone is 16 m above the ground.

Question 137.
The curved surface area of the cylindrical portion is
Answer:
(264π) m²
Explanation:
R = Radius = \(\frac{24}{2}\) = 12 m.
H = Height = 11 m.
Curved surface area of the cylindrical portion = 2πRH
= 2π(12)(11) = (264 π)m₂.

Question 138.
The slant height of the cone is
Answer:
13 m
Explanation:

h = Height of the cylindrical portion
= 16 – 11 = 5m .
Slant height,
L = \(\sqrt{h^{2}+R^{2}}\) =\(\sqrt{25+144}\) = 13 m

Question 139.
The area of the canvas required for the tent is
Answer:
1320 m²
Latha said “Cuboid is one of right prism”.
Explanation:
Area of canvas required for the tent = Curved surface area of the cylindrical portion + Curved surface area of the cone
Surface area = 2πrh + πrl
= πr(2h + l)
= \(\frac{22}{7}\) × 12 (22 + 13)
\(\frac{264}{7}\) (22 + 13)
\(\frac{264}{7}\) × 35
= 132 cm ₂

Question 140.
Is Latha right or wrong?
Answer:
Yes.

Question 141.
Which concept is used from your text-book to support Latha?
Answer:
Mensuration
A toy is in the form of a cone mounted on a hemisphere. The radius of the base and the height of the cone are 7 Cm and 8 cm respectively.

Question 142.
What is the common measure in the toy of two situations?
Answer:
Radius of hemisphere = Radius of cone.

Question 143.
Find the slant height of cone.
Answer:
l = \(\sqrt{113} \mathrm{~cm}\)cm

Question 144.
For finding surface area of the toy, what are required?
Answer:
C.S. A of cone and surface area of hemisphere.
An ice-cream cone full of ice-cream having radius 5 cm and height 10 cm.

Question 145.
Write the combinations of given solid figure.
Answer:
Cone + hemisphere

Question 146.
What are the radii of cone and hemisphere?
Answer:
Hemisphere radius = 5 cm and cone radius = 5 cm

Question 147.
How much the volume of ice-cream contained in conical part?
Answer:
V = 130.95 cm³

Question 148.
How much the volume of ice-cream contained in hemisphere part?
261.90 cm³

Question 149.
For figure shown, match the column.

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

Question 150.
For a wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in fig. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm.

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

Question 151.
From a solid cylinder of height 2.4 cm and diameter 1.4 cm., a conical cavity of the same height and some diameter is followed out them match the column.

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

Question 152.
The capacity of an oil drum is 10litres then what is its volume? (in cm³)
AP LModel Paper I
Answer:
10,000 cm³

Question 153.
Food grains are to be stored in containers of the same base length and height. Which type of containers are required less in number to store a fixed quantity of grains?
i) Right Circular Cylinder
ii) Cube
iii) Right Circular Cone
Answer:
ii) Cube

Question 154.
An open water tank is in the shape of a Cuboid with outer dimensions – length V units, breadth ‘y’ units and height ‘z’ units. If the thickness of the wall is ‘a’ units, express the inner dimensions.
Solution:
Inner Dimensions :
Length = x – a – a = x – 2a units
(both side wall thickness reduced).
Breadth = y – a – a = y – 2a units
(both side wall thickness reduced).
Height = z – a units (as open from top so only bottom thickness reduced)

Question 155.
Choose the correct answer satisfying the following statements.
Statement (A) : The ratio of volumes of cone and cylinder of same base and same height is 3 : 1 Statement (B) : The ratio of volumes of sphere and cone of same radius and same height is 2 : 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) Both A and B are false

Practice the AP 10th Class Maths Bits with Answers Chapter 9 Tangents and Secants to a Circle on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 9th Lesson Tangents and Secants to a Circle with Answers

Question 1.
Find the number of tangents drawn at the end points of the diameter.
Answer:
2 tangents

Question 2.
Find the area of a sector, whose radius is 7 cm and the angle is 120°.
Answer:
51.3 sq.cm
Explanation:
r = 7 cm, θ = 120°
Area of sector = \(\frac{120}{360} \times \frac{22}{7}\) x 7 x 7
= \(\frac{1}{3} \times \frac{22}{7}\) x 7 x 7
= 51.3 sq.cm.

Question 3.
If ‘r’ is the radius of a semi-circle, then find its perimeter.
Answer:
P = πr + 2r (or) r[π + 2] (or) \(\frac{36}{7}\) r

Question 4.
Write the number of parallel tangents of a circle with a given tangent.
Answer:
1 tangent

Question 5.
Write the number of secant that can be drawn to a circle.
Answer:
Infinity

Question 6.
\(\overline{\mathbf{A B}}\) is a tangent drawn to a circle with centre “O” from an external point A and B is a point of contact, then which of the following is always true ?
(i) OA > OB (ii) OA > AB (iii) AB > OB
Answer:
OA > OB and OA > AB

Question 7.
Write the angle in a semi-circle.
Answer:
90°

Question 8.
The diameter of a circle is 10.2 cm, then find its radius.
Answer:
5.1 cm
Explanation:
d = 10.2 cm, r = \(\frac{\mathrm{d}}{2}=\frac{10.2}{2}\) = 5.1 cm

Question 9.
In the given figure, ∠APB = 60° and OP = 10 cm, then find PAnswer:

Answer:
\(5 \sqrt{3}\) cm
Explanation:
In right angled triangle side QP =10 cm.
∠APO = 30°,
cos 30° = \(\frac{AP}{OP} \Rightarrow \frac{AP}{10}=\frac{\sqrt{3}}{2}\)
∴ AP = \(5 \sqrt{3}\) cm

Question 10.
PA and PB are two tangents drawn to a circle with centre ‘O’ from an exter¬nal point P. If ∠APB = 30°, then find ∠AOB.
Answer:
∠AOB = 150°

Question 11.
In the below figure,
AC = 5 cm. Find BC.

Answer:
BC = 2.5 cm

Question 12.
If \(\overline{\mathbf{A P}}\) and \(\overline{\mathbf{A Q}}\) are two tangents to a circle with centre O; such that ∠POQ = 105°, then find ∠PAQ.

Answer:
∠PAQ = 75°
Explanation:
∠POQ + ∠PAQ = 180°
⇒ 105° + ∠PAQ = 180°
⇒ ∠PAQ = 180°- 105° = 75°

Question 13.
Write the maximum number of possible tangents that can be drawn to a circle.
Answer:
Infinity

Question 14.
In the given figure,
∠AOB = 120°, then find ∠APO.

Answer:
30°
Explanation:
∠APB = 180°- 120° = 60°
∠APO = \(\frac{\angle \mathrm{APB}}{2}=\frac{60^{\circ}}{2}\) = 30°

Question 15.
If a circle is inscribed in a Quadri-lateral, then find AB + CD.
Answer:
BC + DA

Question 16.
From the given figure,
∠APB = 40°, then find ∠AOB.

Answer:
140°

Question 17.
Radius of a circle with centre ‘O’ is 5 cm. P is a point at a distance of 3 cm from ‘O’. Then with the number of tan-gents that can be drawn to the circle from the point.
Answer:
No tangents are drawn from that point (or) zero tangent.

Question 18.
Find the angle between the tangent and the radius drawn at the point of contact.
Answer:
90°

Question 19.
The centre of the circle is (2, 1) and one end of the diameter is (3, -4). Find other end of the diameter.
Answer:
(1, 6)
Explanation:
\(\frac{3+\mathrm{x}}{2}\) = 2
x =4 – 3 = 1
\(\frac{-4+y}{2}\) = 1
y = 2 + 4
y = 6
∴ Other end = (x, y) = (1, 6)

Question 20.
Find the angle made at the centre of a circle.
Answer:
360°

Question 21.
The length of a tangent to a circle from a point P is 12 cm and the radius of the circle is 5 cm, then find the distance from point P to the centre of the circle.
Answer:
13 cm
Explanation:

In right angled triangle,
OP² = OA² + AP²
OP² = 5² + 12²
= 25 + 144 = 169
OP = \(\sqrt{169}\) = 13 cm

Question 22.
Which of the following is not correct?
i) Maximum possible tangents that can be drawn to a circle from a point ‘p’ is 2.
ii) The number of secants drawn to a circle from a point at exterior is 2.
Answer:
Maximum possible tangents that can be drawn to a circle from a point ‘p’ is 2.

Question 23.
In the figure, AP and BP are two tangents drawn to a circle with centre ‘O’. If ∠OAB = 30°, then find ∠APB.

Answer:
60°

Question 24.
In the figure PQ and PR are tangents to the circle with centre ‘O’. Then find ‘x’.

Answer:
40°

Question 25.
How much the angle in a major seg¬ment ?
Answer:
An acute angle.

Question 26.
Find the area of a circle that can be inscribed in a square of side 6 cm.
Answer:
9π sq. cm

Question 27.
A tangent to a circle intersects it in …………… point (s).
Answer:
Only one point.

Question 28.
To draw a pair of tangents to a circle which are inclined to each other at an angle of 60° it is required to draw the tangents at the end points of two radii inclined at an angle of ………………
Answer:
120°

Question 29.
From the figure find ‘x’.

Answer:
120° = x

Question 30.
Write the number of parallel tangents to a circle with a given tangent.
Answer:
1

Question 31.
Find the area of a square inscribed in a circle of radius 8 cm.
Answer:
128 cm²
Explanation:
Area of square inscribed in a circle having radius ‘x’ is 2x²
= 2.(8)²
= 2 x 64 = 128 cm²

Question 32.
How many tangent lines can be drawn to a circle from a point outside the circle ?
Answer:
2 tangents

Question 33.
The length of the tangents from a point ‘A’ to a circle of radius 3 cm is 4 cm, then find the distance between A and the centre of the circle.
Answer:
5 cm

Question 34.
PQ is the chord of a circle.The tangent XR drawn at X meets PQ at R when produced. IfXR =12 cm, PQ = Xcm, QR = (x – 2) cm, then find x.
Answer:
10 cm

Question 35.
Name the common point to a tangent and a circle is called.
Answer:
Point of contact.

Question 36.
If tangents PA and PB from a point ‘P’ to a circle with centre O are inclined to each other at an angle of 110°, then find ∠PAO.
Answer:
35°

Question 37.
The circumference of a circle is 100 cm. Find the side of a square inscribed in the circle.
Answer:
\(\frac{50 \sqrt{2}}{\pi}\) cm
Explanation:

Diagonal of a square = diameter By Pythagoras theorem,

Question 38.
If two tangents inclined at an angle of 60° are drawn to circle of radius 3 cm, find the length of each tangent.
Answer:
\(3 \sqrt{3}\) cm.

Question 39.
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a points Q so that OQ = 12 cm, then find PQ.
Answer:
\(\sqrt{119}\) cm

Question 40.
Write the length of the tangent drawn to a circle with radius ‘r’ from a point ‘P’ which is’d’ units from the centre.
Answer:
\(\sqrt{\mathrm{d}^{2}-r^{2}}\)

Question 41.
A circle may have …………….. parallel tangents atmost.
Answer:
2

Question 42.
In the figure PT is a tangent drawn from P. If the radius is 7 cm and OP is 25 cm, then find the length of the tangents.

Answer:
24cm
Explanation:
OP² = OT² + PT²
⇒ 25² = 7² + PT²
⇒ PT² = 25² – 7² = 24²
⇒ PT = 24 cm

Question 43.
Write a line which intersects the given circle at two distinct points.
Answer:
Secant

Question 44.
Find the length of the tangent drawn from a point 8 cm away from the cen¬tre of a circle with radius 6 cm.
Answer:
\(2 \sqrt{7}\)cm

Question 45.
In a right triangle ABC, right angled at B, BC = 15 cm and AB = 8 cm. A circle is inscribed in the triangle ABC. Find the radius of the circle.
Answer:
3 cm
Explanation:

15² + 8² = 225 + 64 = 289
∴ AC = \(\sqrt{289}\) = 17 cm
BDIF is a square 2 (a + b + c) =15 + 17 + 8
= \(\frac{40}{2}\) = 20
∴ a = 3, b = 12, c = 5
Radius = IF = ID = IE = 3 cm.

Question 46.
If radii of two concentric circles are 6 cm and 10 cm, then find length of chord of the larger circle which is tan-gent to other is ……………..
Answer:
16 cm

Question 47.
How much the angle between the tan-gent and radius drawn through the point of contact ?
Answer:
90°

Question 48.
The semi perimeter of ΔABC = 28 cm, then find AF + BD + CE.

Answer:
28 cm
Explanation:
S = 28 cm ⇒ 2S = 56 cm
Semi perimeter = AF + BD + CE
= 28 cm

Question 49.
Two circles intersect at A, B. PS, PT are two tangents drawn from P which lies on AB to the two circles, then write the relation between A, B, PS, PT.

Answer:
PS = PT

Question 50.
Find the length of the tangent drawn from an exterior points is 8 cm away from the centre of a circle of radius 6 cm.
Answer:
10 cm

Question 51.
Write a line segment joining any point on a circle is called ………………..
Answer:
Chord of that circle.

Question 52.
In the figure O is the centre of the circle and PA, PB are tangents, then find their lengths.

Answer:
12 cm, 12 cm

Question 53.
Find the radius of a circle is equal to the sum of the circumferences of two circles of diameters 36 cm and 20 cm.
Answer:
28 cm = r
Explanation:
r₁ = 18 cm, r₂ = 10 cm, r₃ = ?
C₁ + C₂ = C₃ ⇒ 2π(r₁ + r₂) = 2πr₃
⇒ r₁ + r₂ = r₃
⇒ 18 + 10 = r₃ = 28 cm

Question 54.
In the figure, AB is a diameter and AC is chord of the circle such that ∠BAC = 30°. If DC is a tangent, then which type of ΔBCD ?
Answer:
Isosceles triangle

Question 55.
If the radii of two concentric circles are 5 cm and 13 cm, then find the length of the chord of one circle which is tangent to the other circle.
Answer:
24 cm

Question 56.
Two concentric circles of radii a and b(a>b) are given. The chord AB of larger circle touches the smaller circle at C, find the length of AB.

Answer:
\(2 \sqrt{a^{2}-b^{2}}\)

Question 57.
Three circles are drawn with the ver-tices of a triangle as centres such that each circle touches the other two. If the sides of the triangle are 2 cm, 3 cm, 4 cm find the diameter of the smallest circle.
Answer:
2 cm
Explanation:

Diameter of the smallest circle is 2 cm.

Question 58.
In the figure find the value of x.

Answer:
x = 5 cm

Question 59.
Find the perimeter of a quadrant of a circle of radius \(\frac { 7 }{ 2 }\) cm.
Answer:
5.5 cm
Explanation:
Perimeter of a quadrant is = \(\frac{2 \pi r}{4}\)
= \(\frac{2 \times \frac{22}{7} \times \frac{7}{2}}{4}=\frac{22}{4}=\frac{11}{2}\) = 5.5 cm.

Question 60.
Write area of circle interms of diameter.
Answer:
\(\frac{\pi \mathrm{d}^{2}}{4}\) sq.umts

Question 61.
How much the value of each angle in a square ?
Answer:
Right angle.

Question 62.
Write the sum of the central angles in a circle.
Answer:
360°

Question 63.
What do you observe from the below figure ?

Answer:
PA = PB

Question 64.
Write a formula to find area of sector.
Answer:
\(\frac{x^{0}}{360} \times \pi r^{2}\)

Question 65.
A line which is perpendicular to the radius of the circle through the point of contact is called a
Answer:
tangent

Question 66.
How much angle in a minor segment?
Answer:
Obtuse angle

Question 67.
Number of tangents drawn to a circle.
Answer:
Infinite

Question 68.
A tangent to a circle is a line which ……………. the circle exactly at one point.
Answer:
Touches

Question 69.
In how many situations a secant meets a circle ?
Answer:
At ‘2’ places

Question 70.
Find the angle between a tangent to a circle and the radius drawn at the point of contact.
Answer:
90°

Question 71.
Write a formula to find area of circle.
Answer:
πr² sq. units

Question 72.
Side of a square is 4 cm, then find
Answer:
16

Question 73.
In the figure, how much the ” value of ‘x’.

Answer:
60°

Question 74.
Write a formula to find area of regu¬lar hexagon of side ‘a’ units is …………… sq. units.
Answer:
\(\frac{6 \sqrt{3}}{4} a^{2}\)

Question 75.
In the figure, AB = 6.2 cm, then find CD.

Answer:
6.2 cm

Question 76.
The radius of a circle is doubled, then its area becomes …………… times.
Answer:
4

Question 77.
Angle described by hour hand in 12 hours.
Answer:
360°

Question 78.
Write a formula of area of semi-circle.
Answer:
\(\frac{\pi r^{2}}{2}\)

Question 79.
Angle made by minute hand in 1 minute.
Answer:
6°
Explanation:
360° in 1 hour, so in 60 minutes 360°
Angle in 1 minute = \(\frac{360}{60}\) = 6°

Question 80.
In the figure, how much the value of x°.

Answer:
x° = 50°
Explanation:
∠ADB = ∠ACB, so x ° = 50°

Question 81.
If AP and AQ are the two tangents to a circle with centre ‘O’. So that ∠POQ = 110°, then find ∠PAQ.

Answer:
70°

Question 82.
In the figure,
BC = ………….. cm.

Answer:
2.5 cm

Question 83.
The longest chord in a circle
Answer:
Diameter

Question 84.
The shaded portion in the figure represents.

Answer:
Minor segment

Question 85.
Write a formula to find area of ring.
Answer:
π(R² – r²)

Question 86.
The area of square is 49 cm², then find its side.
Answer:
7 cm

Question 87.
How the angles in the same segment of the circle ?
Answer:
Equal

Question 88.
In the figure find the area of shaded region.

Answer:
42 cm²

Question 89.
Write a formula to find perimeter of semicircle.
Answer:
\(\frac{36 r}{7}\) units

Question 90.
The given figure represents.

Answer:
Isosceles trapezium

Question 91.
ABCD is a cyclic quadrilateral, then find the value of ∠A +∠C.
Answer:
180°

Question 92.
Number of chords of a circle have
Answer:
Infinite

Question 93.
In the figure, ∠ACB.

Answer:
90°
Explanation:
Angle in a semi cirlce = ∠ACB = 90°

Question 94.
x° = 60°, r = 14 cm, then find area of sector.
Answer:
102.66 sq. cm

Question 95.
Find the number of tangents at one point of a circle.
Answer:
1

Question 96.
In the figure, ∠OAB.

Answer:
90°

Question 97.
How much angle in a semi circle ?
Answer:
90°

Question 98.
How many tangents are drawn at the ends of a diameter of a circle ?
Answer:
Two parallel lines.

Question 99.
From the figure, then find x.

Answer:
4.8 cm

Question 100.
In the figure, P is called

Answer:
Secant

Question 101.
Write the number of tangents to a circle which are parallel to secant.
Answer:
2 only

Question 102.
A tangent meets a circle in …………. points.
Answer:
1

Question 103.
How much the angle in a at the centre semi circle.
Answer:
180°

Question 104.
How the tangent drawn at the end point of radius ?
Answer:
Perpendicular to radius.

Question 105.
How many tangents can be drawn from a point inside a circle ?
Answer:
Not possible.

Question 106.
In the’figure write the relation among a, b and c.

Answer:
c² = a² + b²

Question 107.
A bicycle wheel makes 75 revolutions per minute to maintain a speed of 8.91km per hour find diameter of the wheel.
Answer:
0.63 cm

Question 108.
In the figure, how much the value of ’a’.

Answer:
a = 80°
Explanation:
∠AOB = 2∠ACB
⇒ ∠AOB = 2 x 40° = 80°

Question 109.
In the figure, x = …………… cm.

Answer:
x = 8

Question 110.
In a circle, d = 10.2 cm, then find r = ……………. cm.
Answer:
5.1 cm

Question 111.
In the figure,
∠BAC = ………..?

Answer:
30°
Explanation:
∠BAC = \(\frac { 1 }{ 2 }\) ∠BOC = \(\frac { 60 }{ 2 }\) = 30°

Question 112.
Diameter of a circle passes through, …………..
Answer:
Centre

Question 113.
The shaded portion of the figure represents.

Answer:
Major segment

Question 114.
Write a formula to find area of triangle.
Answer:
\(\frac { 1 }{ 2 }\) bh

Question 115.
In the figure, XY and X’Y1 are two par-allel tangents to a circle with centre ‘O’ and another tangent AB with point of contact C intersecting XY at A and X¹Y¹ at B, then find ∠AOB.

Answer:
90°

Question 116.
In the figure, AP = 12 cm, PB = 16 cm and π = 3.14, then find perimeter of shaded region.

Answer:
58

Question 117.
How many tangents are drawn from an exterior point of a circle ?
Answer:
Two tangents

Question 118.
How much the sum of opposite angles in a cyclic quadrilateral ?
Answer:
180°

Question 119.
Circles having same centre are called …………… circles.
Answer:
Concentric

Question 120.
How many circles passing through 3 collinear points in a plane ?
Answer:
No circle. .

Choose the correct answer satisfying the following statements.

Question 121.
Statement (A) : The two tangents are drawn to a circle from an external point, than they subtend equal angles at the centre.
Statement (B) : A parallelogram cir-cumscribing a circle is a rhombus.
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 an external point the two tan¬gents drawn subtend equal angles at the centre. So A is true. Also, a paral¬lelogram circumscribing a circle is a rhombus, so B is also true but B is not correct explanation of Answer:
Hence, (i) is the correct option.

Question 122.
Statement (A): PA and PB are two tan-gents to a circle with centre O, such that ∠AOB =110°, then ∠APB = 90°.

Statement (B) : The length of two tan-gents drawn from an external point 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 B are false.
Answer:
(iv)

Question 123.
Statement (A) : In the given figure, XA + AR = XB + BR, where XP, XQ and AB are tangents.

Statement (B): A tangent to the circle can be drawn from a point inside the circle.
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 XP = XQ
⇒ XA + AP = XB + BQ
⇒ XA + AR = XB + BR
[ ∵ PA =-AR and BQ = BR]
(The length of tangents drawn from in external point are equal).
So, A is correct but B is incorrect. Hence, (ii) is the correct option.

Question 124.
Statement (A) : In the given figure, a quadrilateral ABCD is drawn to cir-cumscribe a given circle, as shown. Then AB + BC = AD + DC.

Statement (B) : In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

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 two concentric circles (shown in fig. (b)) O is the centre of concen¬tric circles and AB is the tangent.
∴ OM ⊥ AB ⇒ AM = MB

(Perpendicular from centre ‘O’ to the chord AB bisect the chord AB)
So, A is incorrect but B is correct. Hence, (iii) is the correct option.

Question 125.
Statement (A): In the given figure, O is the centre of a circle and AT is a tan-gents at point A, then ∠BAT = 60°.

Statement (B) : A straight line can meet a circle at one point only.
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:

(Angle in the semi-circle)
In AABC
∠ABC + ∠ACB + ∠CAB = 180°
⇒ 90° + 60° + ∠CAB = 180°
⇒ ∠CAB = 30°
Now, OA ⊥ AT
∵ ∠BAT = 90° – 30° = 60°
So, A is correct but B is incorrect.
Hence, (ii) is the correct option.

Question 126.
Statement (A) : If in a circle, the ra¬dius of the circle in 3 cm and distance of a point from the centre of a circle is 5 cm, then length of the tangent will be 4 cm.
Statement (B) : In a right triangle (hypotenuse)² = (base)² + (height)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:

(OA)² = (AB)² + (OB)²
AB = \(\sqrt{25-9}\) = 4 cm
Both statement (A) and statement (B) are correct. Also, statement (B) is the correct explanation of the statement (A). Option (i) is correct.

Question 127.
Statement (A) : If in a cyclic quadri¬lateral, one angle is 40°, then the opposite angle is 140°.
Statement (B): Sum of opposite angles in a cyclic quadrilateral is equal is 360°.
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:
Angle + 40° = 180°
Angle = 180°-40° = 140°
∴ A is true, B is false. Option (ii) is cor-rect.

Question 128.
Statement (A) : If length of a tangent from an external point to a circle is 8cm, then length of the other tangent from the same point is 8 cm.
Statement (B): Length of the tangents drawn from an external point to a circle 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 B are false.
Answer:
(i)

Question 129.
Statement (A) : If the circumference of a circle is 176 cm, then its radius is 28 cm.
Statement (B) : Circumference = 2πr radius.
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:
Both statement (A) and statement (B) are correct. Also statement (B) is the correct explanation of the statement (A).

Option (i) is correct.

Question 130.
Statement (A) : If the outer and inner diameter of a circular path is 10m and 6m, then area of the path is 1671 m².
Statement (B) : If R and r be the ra¬dius of outer and inner circular path respectively, then area of path = π(R² – r²)
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:
Both statement (A) and statement (B) are correct. Also, statement (B) is the correct explanation of the statement (A).
Area of the path = π \(\left[\left(\frac{10}{2}\right)^{2}-\left(\frac{6}{2}\right)^{2}\right]\)
= π(25 – 9) = 16π
Hence, (i) is the correct option.

Question 131.
Statement (A): If a wire of length 22 cm is bent in the shape of a circle, then area of the circle so formed is 40 cm².
Statement (B) : Circumference of the circle = length of the wire.
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) is pot correct, but state-ment (B) is correct.
2πr = 22 ⇒ r = 3.5 cm
∴ Area of the circle = \(\frac{22}{7}\) x 3.5 x 3.5 = 38.5 cm²
Hence, option (iii) is correct.

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

In the above given figure, a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre.

Question 132.
The area of the sector A’OB’is
Answer:
6π cm²
Explanation:
∠AOB = 60
Area of the sector A’OB’
= \(\frac{60}{360}\) π(6)² = 6π cm²

Question 133.
The area of the shaded region is
Answer:
156.552 cm²
Explanation:
Area of shaded region = Area of circle + Area of ΔOAB – Area of sector A’OB’ R
= π(6)² + \(\frac{\sqrt{3}}{4}\) (12)² – 6π
= 36π + \(\frac{\sqrt{3}}{4}\) (144) – 6π
= 94.2 + 62.352
= 156.552 cm²

The length of two parallel chords of a circle are 6 cm and 8 cm. The smaller chord is at a distance of 4 cm from the centre.

Question 134.
The radius of the circle is
Answer:
5 cm
Explanation:

AB = 6 cm, CD = 8 cm, OM = 4 cm
AM = \(\frac { 1 }{ 2 }\) x (AB) = \(\frac { 1 }{ 2 }\) x (6) = 3 cm
In ΔOAM, By Pythagoras theorem,
OA² = OM² + AM²
OA = \(\sqrt{16+9}\) = 5 cm.
∴ Radius = OA = 5 cm.

Question 135.
The distance of the other chord from the centre is
Answer:
3 cm
Explanation:
CN = \(\frac { 1 }{ 2 }\) x CD = \(\frac { 1 }{ 2 }\) x 8 = 4 cm.
In ΔCON, By Pythagoras theorem,
(OC)² = (ON)² + (CN)²
(ON)² = (OC)² – (CN)²
ON = \(\sqrt{(5)^{2}-(4)^{2}}\) = 3 cm

Sohan made the following pictures also with wash basin.

Question 136.
Split the shape (i) into solid combina¬tion.
Answer:
2 hemispheres + 1 square

Question 137.
Split the shape (ii) into solid figure combination.
Answer:
1 circle + 1 cylinder + 1 triangle

Question 138.
Which mathematical concept was Sohan used to find the area of objects?
Answer:
Area of circles.

Question 139.
Observe the figure, match the column.

Answer:
A – (ii), B – (iii).

Question 140.
Observe the figure match the column.

Answer:
A – (i), B – (iv).

Question 141.
Two circular flower beds have been shown on two sides of a square lawn ABCD of side 56m. If the centre of each circular flowered bed is the point of intersection O of the diagonals of the square lawn, then match the column.

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

Question 142.
Choose correct matching.

Answer:
A – (i), B – (ii).

Question 143.
For a circle which is inscribed in a ΔABC having sides 8 cm, 10 cm and 12 cm. Then match the column.

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

Question 144.
If two tangents PA and PB are drawn to a circle with centre ‘O’ from an ex¬ternal point P (figure), then match the column.

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

Question 145.
If AB is a chord of length 6 cm, of a circle of radius 5 cm, the tangents at A and B intersects at a point X, then match the column.

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

Question 146.
The length of the minutes hand of a clock is 7 cm then how much distance does it cover in one hour ?
Answer:
44 cm

Question 147.
Draw a rough diagram of minor seg¬ment of a circle and shade it.
Answer:

Question 148.
How many tangents can be drawn on a circle from a point outside the circle?
Solution:

Only two tangents can be drawn from an external point to the circle i.e., PA, PB are the two tangents to the circle.

Question 149.
What is the angle between the radius and tangent at the point of contact ?
Answer:
90°

Question 150.
What is the measure of angle at the centre in a semi circle ?
Answer:
180°

Practice the AP 10th Class Maths Bits with Answers Chapter 8 Similar Triangles on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 8th Lesson Similar Triangles with Answers

Question 1.
Write maximum possible tangents that can drawn to a circle.
Answer:
Infinity

Question 2.
If ΔABC ~ ΔDEF and area (ΔABC): area (ΔDEF) = 49 :100. Then find DE : AB.
Answer:
10:7

Question 3.
Name the theorem applied to divide the line segment in the given ratio.
Answer:
Thales theorem.

Question 4.
Write an example for the side of a right angled triangle.
Answer:
5, 12, 13.

Question 5.
If ΔPQR ~ ΔXYZ and ∠X = 30°, ∠Q = 50°, then find ∠Z.
Answer:
∠Z = ∠R = 100°
Explanation:

∠X = ∠P = 30°, ∠Q = ∠Y = 50°, then ∠Z = ∠R = 180° – (30° + 50°) = 100°

Question 6.
The perimeters of two similar triangles are in 4 : 9 ratio, find the ratio of their’ corresponding sides.
Answer:
4:9.
Explanation:
Ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides.
∴ 4 : 9.

Question 7.
ΔABC ~ ΔDEF and area of ΔABC, ΔDEF are 64 cm2 and 121 cm2, then find the ratio of corresponding sides.
Answer:
8:11
Explanation:
Ratio of corresponding sides
= \(\sqrt{64: 121}\) = 8 : 11

Question 8.
In ΔABC, AC = 12 cm, AB = 5 cm and ∠BAC = 30°, find the area of ΔABC,
Answer:
15 cm²

Question 9.
From the given figure, find ‘x’.

Answer:
3
Explanation:
\(\frac{3}{x}=\frac{5}{5} \Rightarrow \frac{3}{x}=1 \Rightarrow x=3\)

Question 10.
If a man walks 6 m to east and 8 m to north. Now he is at a distance of from origin point.
Answer:
10m
Explanation:

Question 11.
If ΔPQR ~ ΔXYZ, QR = 3 cm, YZ = 4 cm, ΔPQR area = 54 cm2. Then find ΔXYZ areAnswer:
Answer:
96 cm²

Question 12.
Write height of an equilateral triangle whose sides is ‘a’ cm.
Answer:
\(\frac{\sqrt{3}}{2} \mathrm{a}\)

Question 13.
Find are of a regular hexagon whose side is ‘a’ cm.
Answer:
\(6\left(\frac{\sqrt{3}}{4} \mathrm{a}^{2}\right)\)

Question 14.
In the figure, find ∠BDE.

Answer:
45°
Explanation:
From figure, ∠B = 60°,
∠E = ∠C = 75°, then ∠BDE = 180°- 135° = 45°

Question 15.
ΔABC ~ ΔPQR and ∠A + ∠B = 115°, then find ∠R.
Answer:
65°

Question 16.
In ΔABC, E and F are the points on the sides AB and AC respectively. If AE = 2 cm, EB = 2.5 cm, AF = 4 cm and FC = 5 cm, then write the relation.
Answer:
EF || BC
Explanation:

Question 17.
In the figure ΔABC, DE || BC, AD = 1.5 cm, DB = 6 cm, AE = x cm, EC = 8 cm, then find ‘x’.

Answer:
2 cm
Explanation:

Question 18.
Find ∠CAD in the given figure.

Answer:
50°

Question 19.
In a right angled triangle with inte¬gral sides at least one of its measure¬ments must be
Answer:
Multiple of 3 and multiple of 2.

Question 20.
ΔABC ~ ΔXYZ, ∠C = 60°, ∠B = 70°, then find ∠X.
Answer:
∠X = 50°

Question 21.
Express ‘x’ in terms of a, b and c in the following figure.

Answer:
x = \(\frac{a c}{b+c}\)

Question 22.
Write the altitude of an equilateral tri¬angle of side ‘x’ cm.
Answer:
\(\frac{\sqrt{3}}{2} \mathrm{x}\) cm

Question 23.
When we construct a triangle similar to a given triangle as per given scale factor, on which similarity we con-struct.
Answer:
Basic proportionality theorem.

Question 24.
In Heron’s formula, area of triangle =\(\sqrt{s(s-a)(s-b)(s-c)}\), write ‘s’ is of the triangle.
Answer:
Half of perimeter (or) s = \(\frac{a+b+c}{2}\)

Question 25.
In ΔABC, ∠C = 90°, BC = a, AB = c, AC = b and ‘p’ is length of height drawn from ‘C’ to AB, then write the correct relation between a, b, c, p.
Answer:
\(\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}\)

Question 26.
ΔABC ~ ΔDEF and their areas are re-spectively 64 cm² and 121 cm², then \(\frac{\mathbf{B C}}{\mathbf{E F}}\)
Answer:
\(\frac{8}{11}\)
Explanation:
\(\frac{\mathrm{BC}}{\mathrm{EF}}=\sqrt{\frac{\mathrm{ar} \Delta \mathrm{ABC}}{\mathrm{ar} \Delta \mathrm{DEF}}}=\sqrt{\frac{64}{121}}=\frac{8}{11}\)

Question 27.
From the given figure, find ar (ΔADE) : ar (ΔABC)

Answer:
9.25
Explanation:
\(\frac{{ar} \Delta \mathrm{ADE}}{{ar} \Delta \mathrm{ABC}}=\left(\frac{3}{5}\right)^{2}=\frac{9}{25}=9: 25\)

Question 28.
ΔABC ~ ΔDEF is given, then draw rough diagram.
Answer:

Question 29.
Areas of 2 similar triangles are 100cm² and 64 cm². If the median of bigger
triangle is 10 cm, then find the me dian of the smaller triangle.
Answer:
8cm

Question 30.
From the given figure, find the value represented by x.

Answer:
x = 13 cm
Explanation:
From figure, AC = 5 cm
(by Pythagoras triplet)
In ∆ABC, AB = x = 13 cm
(by Pythagoras triplet)

Question 31.
In the figure, D, E are mid-points of AB and AC, then find ΔADE: 10 BCED.

Answer:
1:3

Question 32.
In a right triangle, write the relation between sides.
Answer:
Hypotenuse² = (Adj. side)² + (Opp. side)²

Question 33.
In an isosceles ΔPQR, PR = QR and PQ² = 2PR², then find ∠R.
Answer:
90°

Question 34.
In ΔPQR; PQ = 6\(\sqrt{3}\) cm, PR = 12 cm and QR = 6 cm, then find ∠Q.
Answer:
90°
Explanation:

Phthagoras theorem is satisfied.
∴ ∠Q = 90°.

Question 35.
If ΔABC ~ ΔXYZ; ∠C = 60°, ∠B = 75°, then find ∠Z.
Answer:
60°

Question 36.
If ΔABC ~ ΔDEF, BC = 4 cm, EF = 5 cm and ΔABC = 80 cm², then find ΔDEF.
Answer:
125 cm²

Question 37.
The sides PQ and PR of right triangle PQR are such that PQ = 5 cm, PR = 13 cm. If ∠Q = 90°, then find QR.
Answer:
12 cm

Question 38.
From the figure, find ∠DAC.

Answer:
35°

Question 39.
If in two triangles, corresponding sides are in the same ratio, then the two triangles are similar, which crite¬rion it relates ?
Answer:
S.S.S. criterion.

Question 40.
In the figure DE // BC and AD : DB =1:2, then find ΔADE: ΔABC.
Answer:
1 : 9
Explanation:

AD : DB = 1 : 2 ⇒ AD : AB = 1 : 3
AE : EC = 1 : 2 ⇒ AE : AC = 1 : 3
ar ∆ADE : ar ∆ABC = 1² : 3² = 1 : 9

Question 41.
ΔABC and ΔBDE are two equilateral triangles such that ‘D’ is the midpoint of BC. Find the ratio of the areas of triangles AABC and ABDE.
Answer:
4:1.

Question 42.
From the figure, find ‘x’.

Answer:
x = 15 cm

Question 43.
In the following figure, DE // BC, then find x’.

Answer:
± \(\sqrt{7}\)
Explanation:
\(\frac{A D}{D B}=\frac{A E}{E C}\)
\(\frac{x+4}{x+3}=\frac{2 x-1}{x+1}\)
(x + 4) (x + 1) = (2x – 1) (x + 3)
x² + x + 4x + 4 = 2x² + 6x – x – 3
x² + 5x + 4 = 2x² + 5x – 3
=> x² – 2x² + 4 + 3 = 0
x² – 7 = 0 x = \(\pm \sqrt{7}\)

Question 44.
The ratio of the areas of two similar triangles is 1 : 4, then write the ratio of their corresponding sides.
Answer:
1 : 2
Explanation:
Ratio of corresponding sides
= \(\sqrt{1}: \sqrt{4}\) =1:2

Question 45.
The perimeter of ΔABC ~ ΔLMN are 60 cm and 48 cm of LM = 8 cm, then find AB.
Answer:
10 cm
Explanation:

Question 46.
ΔABC ~ ΔPQR, ∠A = 32°, ∠R = 65°, then find ∠B.
Answer:
83°

Question 47.
All circles are …………….
Answer:
Similar

Question 48.
All squares are …………..
Answer:
Similar

Question 49.
All……………triangles are similar.
Answer:
Equilateral.

Question 50.
Write the longest side in a right tri-angle.
Answer:
Hypotenuse.

Question 51.
In the figure, AB = 2.5 cm, A AC = 3.5 cm. If AD is the bisector of ∠BAC, then find BD : DC. 8

Answer:
5 : 7
Explanation:
According to the angle bisector theorem
\(\frac{B D}{D C}=\frac{A B}{A C} \Rightarrow \frac{B D}{D C}=\frac{25}{35}=\frac{5}{7}\)
∴ BD : DC = 5 : 7

Question 52.
The ratio of corresponding sides of two similar triangles is 3 : 2, then find the ratio of their corresponding heights.
Answer:
3 : 2

Question 53.
The diagonals of a rhombus are 24 cm and 32 cm, then find its perimeter.
Answer:
80 cm

Question 54.
In ΔABC, BC² + AB² = AC², then where is the right angle ?
Answer:
∠B

Question 55.
Write each angle in an equilateral tri-angle.
Answer:
60°

Question 56.
ΔABC ~ ΔDEF if DE : AB = 2 : 3 and ar ΔDEF = 44 sq. units, then find ar (ΔABC).
Answer:
99
Explanation:

Question 57.
In an equilateral triangle ABC, AD⊥BC meeting BC in D, then find AD².
Answer:
3 BD²

Question 58.
ΔABC ~ ΔDEF and 2AB = DE and BC = 8 cm, then find EF = ……………..cm.
Answer:
16

Question 59.
From the figure, find AD.

Answer:
2.4 cm

Question 60.
Side of rhombus is 4 cm, then find its perimeter.
Answer:
16 cm

Question 61.
Write any one example for the sides of, a right triangle.
Answer:
10 cm, 8 cm, 6 cm.

Question 62.
In the figure, ΔABC, DE//BC and \(\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{3}{5}\) AC = 5.6, then find AE.

Answer:
2.1 cm

Question 63.
If 82 + 152 = k2, then find k.
Answer:
k = 17.

Question 64.
In the figure PQR, ∠QPR = 90°, PQ = 24 cm and QR = 26 cm and in ΔPKR, ∠PKR = 90° and KR = 8 cm, then find PK.

Answer:
PK 6 cm.
Explanation:
By Pythagoras theorem,
In APQR,
QR² = QP² + PR²
⇒ PR = 10 cm
In ∆PKR, PK = 6 cm.

Question 65.
In the figure, DE // BC if AD = x, AE = x + 2, DB = x – 2 and CE = x – 1, then find ‘x’.

Answer:
4

Question 66.
The ratio of areas of two similar triangles is equal to the ratio of the squares of corresponding ……….
Answer:
Sides

Question 67.
If the diagonal of a square is \(7 \sqrt{2}\) cm, then find its areAnswer:
Answer:
49 cm²

Question 68.
In the figure, AC = ……………..cm.

Answer:
12 cm

Question 69.
Write the height of an equilateral tri-angle whose side is ‘a’ units.
Answer:
\(\frac{\sqrt{3}}{2} \mathrm{a}\)

Question 70.
The ratio of the corresponding sides of two similar triangles is 5 : 3. Then find the ratio of their areas.
Answer:
25:9

Question 71.
The areas of two similar triangles are 36 cm2 and 64 cm². If one side of the first triangle is 6 cm, then find the cor-responding side of the later triangle.
Answer:
8 cm
Explanation:
Ratio of area of two Similar triangles is equal to the ratio of their corre¬sponding sides.
\(\frac{6}{x}=\sqrt{\frac{36}{64}} \Rightarrow \frac{6}{x}=\frac{6}{8}\)

Question 72.
The lengths of diagonals of a rhombus are 24 cm and 32 cm, then find the pe-rimeter of the rhombus.
Answer:
80 cm

Question 73.
In the given figure, DE // BC and AD : DB = 5 : 4, then find \(\frac{\Delta \text { DEF }}{\Delta \text { CFB }}\)

Answer:
\(\frac{25}{81}\)

Question 74.
Write each exterior angle of an equi-lateral triangle.
Answer:
120°

Question 75.
ΔABC ~ ΔPQR, ∠A = 50°, then find ∠Q + ∠R.
Answer:
130°

Question 76.
ΔABC ~ ΔPQR; M is the midpoint of BC and N is the midpoint of QR. If ΔABC = 100 cm2 and ΔPQR = 144 cm² and AM = 4 cm, then find PN.
Answer:
PN 4.8 cm

Question 77.
In the figure, PQ // MN, \(\frac{\mathbf{K P}}{\mathbf{P M}}=\frac{4}{13}\) and
KN = 20.4cm, them find KQ.

Answer:
4.8 cm.
Explanation:
PQ // MN, by Thales theorem KP

⇒ 4(20.4-KQ) = 13 KQ
⇒ 81.6-4 KQ – 13 KQ .
⇒ 13 KQ + 4 KQ = 81.6
⇒ KQ = 4.8 cm.

Question 78.
The diagonal of a trapezium ABCD in which AB // CD intersect at ‘O’. If AB = 2CD, then find the ratio of ar¬eas of triangles AOB and COD.
Answer:
4:1

Question 79.
In a trapezium how diagonals divide each other ?
Answer:
Proportionally to each other.

Question 80.
In the figure, \(\frac{\mathbf{P S}}{\mathrm{SQ}}=\frac{\mathbf{P T}}{\mathbf{T R}}\)
∠PST = ∠PRQ, then find ∆PQR is which type of triangle ?
Answer:
Isosceles

Question 81.
From the figure, y = …………….. cm.

y = 15 cm

Question 82.
∆ABC is an isosceles right triangle, ∠C= 90°, then find AB².
Answer:
AC² + BC²

Question 83.
In a square, the diagonal is ……………… time of its side.
Answer:
\(\sqrt{2}\)

Question 84.
In the figure, CD =……………….cm.

Answer:
\(6 \sqrt{3}\) cm.

Question 85.
In the figure, AABC is an isosceles tri¬angle right angled at B. Two equilat¬eral triangles are constructed with sides AC and BC. Then find ∆BCD.
I’m 67
Answer:
∆ABC

Question 86.
The areas of two similar triangles are 25 m2 and 36 m2. If the median of smaller triangle is 10 m, then find the median of the larger triangle.
Answer:
12 m
Explanation:
(The ratio of sides)² = (ratio of medians)²
= Ratio of areas
∴ Ratio of medians = \(\sqrt{\text { Ratio of areas }}\)
\(\frac{x}{10}=\sqrt{\frac{25}{36}} \Rightarrow \frac{x \cdot}{10}=\frac{5}{6}\)
⇒ x = 12 m

Question 87.
∆ABC ~ ∆PQR, then find AB : PQ.
Answer:
AC : PR

Question 88.
In ∆LMN, ∠L = 60°, ∠M = 50° and ∆LMN ~ ∆PQR, then find ∠R.
Answer:
∠R = 70°

Question 89.
In the figure, ∠BAD = ∠CAD; AB = 3.4 cm, BD = 4 cm, BC = 10 cm, then find AC.

Answer:
5.1 cm

Question 90.
Two sides of a right triangle are 3 cm and 4 cm, then find the third side.
Answer:
5 cm.

Question 91.
∆ABC ~ ∆DEF, BC = 4 cm, EF = 5 cm and area of ∆ABC = 80 cm2, then find area of ∆DEF.
Answer:
125 cm²

Question 92.
In the figure, find x.

Answer:
x = 8 cm

Question 93.
In the figure, if AB // CD, then find x.

Answer:
7 cm

Question 94.
When two polygons are similar ? Write similarity conditions.
Answer:
If corresponding angles are equal & corresponding sides are equal.

Question 95.
The side of an equilateral triangle is ‘a’ units, then write its height in terms of side.
Answer:
\(\frac{\sqrt{3} a}{2}\)

Question 96.
The angles of a triangle are in the ra¬tio 1 :2 : 3, then find the largest angle.
Answer:
90°

Question 97.
∆ABC ~ ∆DEF and their areas are re-spectively 64 cm² and 121 cm2 if EF = 15.4 cm, then find BC.
Answer:
11.2 cm

Question 98.
In the figure, DE // AB and FE // DB, then find DC².

Answer:
CF x AC

Question 99.
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are pro-portional, the two triangles are simi-lar. This property is
Answer:
S.Answer:S. criterion.

Question 100.
In ∆ABC, the midpoints are D, E and F of the sides, AB, BC and CA, then find ∆DEF : ∆ABC.
Answer:
1:4

Question 101.
In ∆ABC, XY || BC, AX : XB = 2 : 1, then find ∆AXY : ∆ABC.
Answer:
4 : 9

Question 102.
In the figure, ∠ABC.

Answer:
60°.

Question 103.
In the figure, in ∆PQR, QR // ST, \(\frac{\mathbf{P S}}{\mathbf{S Q}}=\frac{3}{5}\) and PR = 28 cm, then find PT.

Answer:
10.5 cm

Question 104.
∆ABC ~ ∆PQR, AB : PQ = 3 : 4, then find ar (∆ABC) far (∆PQR).
Answer:
9 : 16

Question 105.
The bisector of ∠A of ∆ABC intersects BC at D. If BD : DC = 4 : 7 and AC = 3.5. Then find AB.
Answer:
2 cm
Explanation:

Question 106.
A perpendicular is drawn from the vertex of a right angle to the hypot-enuse, then the triangles on each side of the perpendicular are
Answer:
Similar.

Question 107.
In the figure, D, E are the midpoints of the sides AB and AC. If DE = 4 cm, then find BC.

Answer:
BC = 8 cm.
Explanation:
By B.P.T. BC = 2 x DE
⇒ BC = 2 x 4 = 8 cm.

Question 108.
In ∆ABC, AD bisects ∠Answer: AB = 6 cm, BD = 8 cm and DC = 6 cm, then find AC.
Answer:
4.5 cm

Question 109.

If ∆ABC ~ ∆PQR, then find y + z.
Answer:
4 + 3\(\sqrt{3}\)

Question 110.
In the figure, find ‘x’.

Answer:
∠x = 135°.

Question 111.
In ∆ABC, AB = BC = AC, then find ∠A = ∠B = ∠C.
Answer:
60°

Question 112.
If the diagonals in a quadrilateral di-vide each other proportionally then it is which type of polygon ?
Answer:
Trapezium.

Question 113.
∆ABC ~ ∆PQR, ∠A + ∠B = 100°, then find ∠R.
Answer:
80°

Question 114.
If the sides of two similar triangles are in the ratio 7 : 2, then find the ratio of their areas.
Answer:
49 : 4

Question 115.
In the figure, QA⊥AB and PB ⊥ AB, if AO = 20 cm, BO = 12 cm, PB = 18 cm, then find AQ.

Answer:
AQ = 30 cm

Question 116.
D, E and F are the mid-points of the sides BC, CA and AB respectively of ∆ABC, then find the ratio of the areas of ∆DEFand ∆ABC.
Answer:
1 : 4

Question 117.
In the figure, DE divides AB and AC in the ratio 1 : 3. If DE = 2.4 cm, then find BC.

Answer:
7.2 cm

Question 118.
In the figure, APQR and ASQR are two triangles on the same base QR. If PS intersects QR at ‘O’, then find ∆PQR : ∆SQR.

Answer:
PO : SO.

Question 119.
In the figure, LM // CB and LN // CD, then find \(\frac{\mathbf{A M}}{\mathbf{A B}}\)

Answer:
\(\frac{AN}{AD}\)

Question 120.
In the figure, two triangles are siml Iar, then find ‘x’.
Answer:
x = 7.5 cm

Question 121.
In the figure, ∠A = ∠B and AD = BE, then write the relation.

Answer:
DE//AB

Choose the correct answer satisfying the following statements.

Question 122.
Statement (A) : ∆ABC ~ ∆DEF such that ar(∆ABC) = 36 cm² and ar (∆DEF) = 49 cm², then AB : DE = 6 : 7.
Statement (B): If ∆ABC ~ ∆DEF, then \(\frac{{ar}(\Delta \mathrm{ABC})}{{ar}(\Delta \mathrm{DEF})}=\frac{\mathrm{AB}^{2}}{\mathrm{DE}^{2}}=\frac{\mathrm{BC}^{2}}{\mathrm{EF}^{2}}=\frac{\mathrm{AC}^{2}}{\mathrm{DF}^{2}}\)
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:

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

Question 123.
Statement (A) : In AABC, DE // BC such that AD — (7x – 4) cm,
AE = (5x – 2) cm, DB = (3x + 4) cm and EC = 3x cm, then x equal to 5.
Statement (B): If a line is drawn par¬allel to one side of a triangle to inter¬sect the other two sides in distant point, then the other two sides are divided in the same ratio.
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, \(\frac{A D}{D B}=\frac{A E}{E C} \Rightarrow \frac{7 x-4}{3 x+4}=\frac{5 x-2}{3 x}\)
⇒ 21x² – 12x = 15x² + 20x – 6x – 8
⇒ 6x² – 26x + 8 = 0
⇒ 3x² – 13x + 4 = 0
⇒ 3x² – 12x – x + 4 = 0
⇒ 3x (x – 4) – 1 (x – 4) = 0
⇒ (x-4) (3x – 1) = 0
x = 4, \(\frac { 1 }{ 3 }\)
So, A is incorrect but B is correct. Hence, (iii) is the correct option.

Question 124.
Statement (A) : ∆ABC is an isosceles triangle right angled of C, then
AB² = 2AC².
Statement (B) : In right∆ABC, right angled at B, AC² = AB² + BC².
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 an isosceles AABC, right angled at C is AB² = AC² + BC²
⇒ AB² = AC² + AC²
⇒ AB² = 2AC²
So, both A and B are correct and B explains Answer:
Hence, (i) is the correct option.

Question 125.
Statement (A): In the ∆ABC, AB = 24 cm, BC = 10 cm and AC = 26 cm, then ∆ABC is a right angle triangle.
Statement (B) : If in two triangles, their corresponding angles are equal, then the triangles are similar.
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:
We have, AB² + BC² = (24)² + (10)²
= 576 + 100 = 676 = AC²
AB² + BC² = AC²
∴ ABC is a right angled triangle. Also, two triangles are similar, if their corresponding angles are equal.
So, both A and B are correct.
Hence, (i) is the correct option.

Question 126.
Statement (A) : Two similar triangles are always congruent.
Statement (B): If the areas of two simi-lar triangles are equal, then the tri¬angles are congruent.
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:
Two similar triangles are not congru¬ent generally.
So, A is incorrect, but B is correct. Hence, (iii) is the correct option.

Question 127.
Statement (A) : If in a AABC, a line DE // BC, intersects, AB in D and AC in E, then \(\frac{AD}{DB}=\frac{AE}{EC}\)
Statement (B): If a line is drawn paral¬lel to one side of a triangle intersecting the other two sides, then the other two sides ore divided in the same ratio.
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 (B) is true [This is Thale’s Theorem]
For statement (A),
Since DE // BC

∴ By Thales’ Theorem
\(\frac{AD}{DB}=\frac{AE}{EC} \Rightarrow \frac{DB}{AD}=\frac{EC}{AE}\)

∴ Statement (A) is true.
Since statement (B) gives statement (A).
∴ Option (i) is correct.

Question 128.
Statement (A) : ABC is an isosceles, right triangle, right angled at C. Then AB² = 3 AC².
Statement (B): In an isosceles triangle ABC if AC = BC and AB² = 2AC², then ∠C .= 90°.
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:
In right angled ΔABC,
AB² = AC² + BC²
(By Pythagoras Theorem)
AB² = AC² + AC² (∵ BC = AC)
AB² = 2AC²
Statement (A) is false.
Again since AB² = 2AC² = AC² + AC²
= AC² + BC² (∵ AC = BC given)
∠C = 90° (By converse of Pythagoras
Theorem)
∴ Statement (B) is true.
Hence, (iii) is the correct option.

Question 129.
Statement (A): ABC and DEF are two similar-triangles such that BC = 4 cm, EF = 5 cm and area of AABC 64 cm2, then area of ∆DEF =100 cm². Statement (B): The areas of two simi¬lar triangles are in the ratio of the squares of the corresponding altitudes.
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 (B) is true.
[∵ It is Standard result]
For statement (A),
since ΔABC ~ ΔDEF

( ∵ Ratio of areas of two similar tri¬angles is equal to the ratio of squares of corresponding sides)

∴ Statement (A) is true. But statement (B) is not the correct explanation for statement (A).
∴ Option (i) is correct.

In figure, AD is a median of a triangle ABC and AM ⊥ BC.

Question 130.
AD² + BC . DM = \(\left(\frac{B C}{2}\right)^{2}\) is equal to
Answer:
AC²
Explanation:
AD is the median, so D is the mid point of BC.
So, BD = DC = \(\frac { 1 }{ 2 }\)BC …………. (1)
In right angled AAMC,
AC² = AM² + MC² ………..(2)
In right angled AAMD,
AM² = AD² – MD² ……………….(3)
Putting AM² from (3) in (2),
We get AC² = AD² – MD² + MC²
= AD² – MD² + (MD + DC)²
= AD² + 2DM + \(\frac{BC}{2}+\left(\frac{BC}{2}\right)^{2}\)
So, AC² = AD² + BC . DM + \(\left(\frac{BC}{2}\right)^{2}\)

Question 131.
AD² – BC . DM = \(\left(\frac{B C}{2}\right)^{2}\) is equal to
Answer:
AB²
Explanation:
In right angled AABM,
AB² = AM² + BM²
From AAMD, AM² = AD² – MD²
So, AB² = AD² – MD² + BM²
= AD² – MD² + (BD – MD)²
= AD² – MD² + BD² – 2BD.MD+MD²
=» AB² = AD² – BC . DM + \(\left(\frac{BC}{2}\right)^{2}\) proved.

Question 132.
2AD² + \(\frac { 1 }{ 2 }\) BC² is equal to
Answer:
AC² + AB²
Explanation:
From the solution of above two ques-tions
AC² = AD² + BC . DM + \(\left(\frac{BC}{2}\right)^{2}\) …. (i)
and AB² – AD² – BC . DM + \(\frac { 1 }{ 2 }\) BC² …. (ii)
Adding results of (i) and (ii) we get
AC² + AB² = AD² + BC . DM + \(\left(\frac{BC}{2}\right)^{2}\) + AD² – BC . DM + \(\left(\frac{BC}{2}\right)^{2}\)
AC² + AB² = 2 AD² + \(\frac { 1 }{ 2 }\) (BC)²

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

Raju said that the triangle with sides 25 cm, 5 cm and 24 cm a right tri¬angle.

Question 133.
Are you support with Raju ? (or) not ?
Answer:
No, I didn’t.

Question 134.
Why you support / does not support with him ? Give reason.
Answer:
Given measurements are not support¬ing the Pythagoras theorem.

Question 135.
Which mathematical concept was used to check your answer from textbook ?
Answer:
Similar Triangles.

If ∆ABC ~ ∆DEF, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm.

Question 136.
Find AC.
Answer:
8 cm

Question 137.
Find perimeter of ∆ABC.
Answer:
Perimeter of ∆ABC =18 cm.

∆ABC ~ ∆EDF such that AB = 5 cm, AC = 7 cm, DF = 15 cm and DE = 12 cm.

Question 138.
Which property was used to solve the given problem ?
Answer:
Similar triangles.

Question 139.
Find “EF”.
Answer:
EF = 16.8 cm.

Question 140.
Find “BC”.
Answer:
BC = 6.25 cm

Write the correct matching options.

Question 141.
If in ∆ABC, DE // BC and intersects AB in D and AC in E, then

Answer:
A – (ii), B – (i).

Question 142.
If in ∆ABC, DE // BC and intersects AB in D and AC in E, then

Answer:
A – (iv), B – (iii).

Question 143.
Match the correct option.

Answer:
A-(ii), B – (i)

Question 144.
Match the correct option

Answer:
A – (iii), B – (iv).

Question 145.
Write correct matching

Answer:
A – (ii), B – (i).

Question 146.
Write correct matching.

Answer: A – (iii), B – (iv).

Question 147.
In figure, the line segment XY // AC of side ∆ABC and it divides the triangle into two parts of equal areas, then

Answer:
A – (i), B – (ii).

Question 148.
In figure, the line segment XY // AC of side AABC and it divides the triangle into two parts of equal areas, then

Answer:
A – (iii), B – (iv).

Question 149.
The perimeters of two similar triangles are 24 cm and 18 cm respectively. If one side of the first triangle is 8 cm then what is the corresponding side of second triangle ?
Answer:
6.

Question 150.
In ∆ABC, ∠B = 90°, BD ⊥ AC.
If AD = 8 cm and BD = 4 cm, then what is the length of CD ?
Answer:
2 cm.

Question 151.
Among “Circles”, “Squares” and “Tri¬angles”, which are always similar ?
Answer:
Circles and squares are always similar.

Question 152.
In AABC and ADEF,
if ∠B = ∠E, ∠C = ∠F, then which of the following is a true statement ?

Answer:
(c)

Question 153.
Given DE || BC, AD = 4.5 cm, BD = 9 cm, and EC = 8 cm. What is the length of AE?

Sol.
AD = 4.5 cm, BD = 9 cm, EC = 8 cm
\(\frac{AD}{DB}=\frac{AE}{EC} \Rightarrow \frac{4.5}{9}=\frac{AE}{8}\)
⇒ 9 AE = 36
∴ AE = 4 cm

Question 154.
The sides of two equilateral triangles ABC and DEF are 4 cm. and 5 cm. respectively. Find \(\frac{{ar}(\Delta D E F)}{{ar}(\Delta A B C)}\)
Solution:
Ratio of areas of Δ = (Ratio of corresponding sides)²
\(\frac{(\Delta {DEF})}{(\Delta \mathrm{ABC})}=\frac{5^{2}}{4^{2}}=\frac{25}{16}\)

Question 155.
ΔABC ~ ΔDEF and ∠A = ∠D = 90°, find the measure of ∠B + ∠F.
Solution:
ΔABC ~ ΔDEF
⇒ ∠A = ∠D = 90°
∠B = ∠E;∠C = ∠F
Now in ΔABC, ∠A + ∠B + ∠C = 180°
Now putting ∠A = ∠D = 90°
⇒ 90° + ∠B + ∠C = 180°
⇒ ∠B + ∠C = 90°
Now putting ∠C = ∠F ⇒ ∠B + ∠F = 90°

Question 156.
Given AB || CD, identify the pair of similar triangles in this image and justify.

Solution:
∠AEB = ∠CED (Vertically opp. angle)
∴ ∠BEA and ∠CDE are similar by
AAA – Similarity rule
ΔBEA ~ ΔCDE

Practice the AP 10th Class Maths Bits with Answers Chapter 11 Trigonometry on a regular basis so that you can attempt exams with utmost confidence.

AP State Syllabus 10th Class Maths Bits 11th Lesson Trigonometry with Answers

Question 1.
Find the value of cos 12° – sin 78°
Answer:
0
Explanation:
cos 12°- sin(90°- 12°)
⇒ cos 12° – cos 12° = 0

Question 2.
If x = cosec θ + cot θ and y = cosec θ – cot θ, then write the relation between ‘x’ and ‘y’
Answer:
xy = 1
Explanation:
xy = (cosec θ + cot θ) (cosec θ – cot θ)
⇒ cosec² θ – cot² θ = 1

Question 3.
Write a formula to cos (A – B).
Answer:
cos A cos B + sin A sin B

Question 4.
The value of cos (90 – θ).
Answer:
sin θ

Question 5.
In Δ ABC sin C = \(\frac {3}{5}\),then find cos A.
Answer:
\(\frac {3}{5}\)

Question 6.
Complete the value tan² θ – sec² θ.
Answer:
-1
Explanation:
-(sec² θ – tan² θ) = – 1

Question 7.
The value of sec (90 – A).
Answer:
cosec A

Question 8.
If cosec θ + cot θ = 5, then cosec θ – cot θ.
Answer:
\(\frac {1}{5}\)

Question 9.
If x = 2 sec θ; y = tan θ,then the value of x² – y².
Answer:
4
Explanation:
sec² θ = \(\left(\frac{x}{2}\right)^{2}\),tan² θ = \(\left(\frac{y}{2}\right)^{2}\)
⇒ sec² θ – tan² θ = \(\frac{x^{2}}{4}-\frac{y^{2}}{4}\)
⇒ 4 = x² – y²

Question 10.
If √3 tan θ = 1,then the value of θ.
A.
30°

Question 11.
The value of (sec 60)(cos 60).
Answer:
1

Question 12.
How much the value of sin (60 + 30)?
Answer:
1

Question 13.
If sec θ + tan θ = \(\frac{1}{2}\) then find sec θ – tan θ.
Answer:
2

Question 14.
The value of cos(90 – θ).
Answer:
sin θ

Question 15.
Simplify: tan 26°. tan 64°
Answer:
1
Explanation:
tan 26° . tan 64°
⇒ tan 26° . tan (90° – 26°)
⇒ tan 26° . cot 26° = 1

Question 16.
How much value of 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 17.
Find the value of \(\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}\).
Answer:
0
Explanation:
\(\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}\) = \(\frac{1-1}{1+1}=\frac{0}{2}\) = 0

Question 18.
If sin x = \(\frac{5}{7}\), then find the value of cosec x.
Answer:
\(\frac{7}{5}\)
Explanation:
sin x = \(\frac{5}{7}\) ⇒ cosec x = \(\frac{7}{5}\)

Question 19.
Given ∠A = 75°, ∠B = 30°,then find the value of tan (A – B).
Answer:
1
Explanation:
tan (75° – 30°) = tan 45° = 1

Question 20.
If sec θ + tan θ = \(\frac{1}{3}\),then find the value of sec θ – tan θ.
Answer:
3
Explanation:
⇒ sec θ + tan θ = \(\frac{1}{3}\)
⇒ sec θ – tan θ = 3

Question 21.
If cosec θ + cot θ = 2,then find the value of cos θ.
Answer:
\(\frac{3}{5}\)
Explanation:

Question 22.
If cos (A + B) = 0, cos B = \(\frac{\sqrt{3}}{2}\),then
how much value of A?
Answer:
60°
Explanation:
cos (A + B) = 0, cos B = \(\frac{\sqrt{3}}{2}\)
⇒ cos B = cos 30°
⇒ A + B = 90°
⇒ A + 30° = 90°
∴ A = 90°- 30° = 60°

Question 23.
If sec θ – tan θ = 3, then find the value of sec θ + tan θ.
Answer:
\(\frac{1}{3}\)

Question 24.
If sin 2θ = cos 3θ, then how the value of θ.
Answer:
18°
Explanation:
sin 2θ = cos 3θ
⇒ 2θ + 3θ = 90°
⇒ 5θ = 90°
θ = 18°

Question 25.
If cos θ = \(\frac{3}{5}\), then find the value of sin θ.
Answer:
\(\frac{4}{5}\)
Explanation:
cos θ = \(\frac{3}{5}\) ⇒ sin θ = \(\frac{4}{5}\)

Question 26.
Simplify : cos 60° + sin 30°.
Answer:
1
Explanation:
cos 60° + sin 30° = \(\frac{1}{2}+\frac{1}{2}=\frac{2}{2}\) = 1

Question 27.
If sec A + tan A = \(\frac{1}{5}\), then find the value of sec A – tan A.
Answer:
5

Question 28.
sin (90 – A) = \(\frac{1}{2}\) , then how much the value A?
A.
60°

Question 29.
If cot A = \(\frac{5}{12}\), then find the value of sin A + cos A.
Answer:
\(\frac{17}{13}\)

Question 30.
Write any value which is not possible for sin x?
Answer:
\(\frac{5}{4}\)
Explanation:
sin x = \(\frac{5}{4}\) > 1, so it is not possible.

Question 31.
If sin θ = cos θ (0 < θ < 90), then the value of tan θ + cot θ.
Answer:
2
Explanation:
sin θ = cos θ
⇒ 0 = 45°
⇒ tan 45° + cot 45° = 2

Question 32.
If sec θ + tan θ = 3, then find the value of sec θ – tan θ.
Answer:
\(\frac{1}{3}\)

Question 33.
In ΔABC; AB = c, BC = a, AC = b and ∠BAC = 0, then calculate the value of area of ΔABC is ………………(θ is acute)
Answer:
\(\frac{1}{2}\) bc sin θ

Question 34.
Write the value of tan θ in terms of cosec θ.
Answer:
\(\frac{1}{\sqrt{\operatorname{cosec}^{2} \theta-1}}\)

Question 35.
Observe the following:
(I) sin² 20° + sin² 70° = 1
(II) log₂ (sin 90°) = 1
Which one is CORRECT?
Answer:
(I) only

Question 36.
Simplify : tan 36°. tan 54° + sin 30°
Answer:
\(\frac{3}{2}\)
Explanation:
tan 36° . tan (90 – 36°) + sin 30°
⇒ tan 36° . cot 36° + \(\frac{1}{2}\)
⇒ 1 + \(\frac{1}{2}=\frac{3}{2}\)

Question 37.
If sin A = \(\frac{24}{25}\), then find the value of sec A.
Answer:
\(\frac{25}{7}\)

Question 38.
Which one of the following is NOT defined?
sin 90°, cos 0°, sec 90°, cos 90°.
Answer:
sec 90°
Explanation:
sec 90° is not defined

Question 39.
Simplify:\(\sqrt{\frac{1-\cos ^{2} A}{1+\cot ^{2} A}}\)
Answer:
sin²A
Explanation:
\(\sqrt{\frac{\sin ^{2} A}{\operatorname{cosec}^{2} A}}\) = \(\sqrt{\sin ^{4} A}\) = sin²A

Question 40.
If cot θ – cosec θ = p, then find cot θ + cosec θ.
Answer:
\(\frac{-1}{\mathrm{p}}\)

Question 41.
Express tan θ in terms of cos θ.
Answer:
\(\frac{\sqrt{1-\cos ^{2} \theta}}{\cos \theta}\)

Question 42.
Who was introduced Trigonometry?
Answer:
Hipparchus

Question 43.
\(\frac{\sin ^{4} \theta-\cos ^{4} \theta}{\sin ^{2} \theta-\cos ^{2} \theta}\) equal to
Answer:
1
Explanation:

Question 44.
sin²47° + sin²43° equal to
Answer:
1

Question 45.
sin 2A equal to
Answer:
2sin A cos A

Question 46.
sin 30° + cos 60° equal to
Answer:
1

Question 47.
sec⁴A – sec²A equal to
Answer:
tan⁴ A – tan² A
Explanation:
sec⁴ A – sec²A
⇒1 + tan⁴ A – (1 + tan² A)
⇒ tan⁴ A – tan² A

Question 48.
Find the value of \(\frac{\tan ^{2} \theta}{1+\tan ^{2} \theta}\).
Answer:
sin² θ

Question 49.
2sin 45°. cos 45° equal to
Answer:
1

Question 50.
sin 81° equal to
Answer:
cos 9°
1

Question 51.
tan θ = \(\frac{1}{\sqrt{3}}\), then find cos θ.
Answer:
\(\frac{\sqrt{3}}{2}\)

Question 52.
If cos θ = \(\frac{1}{2}\); then find cos\(\frac{\theta}{2}\)
Answer:
\(\frac{\sqrt{3}}{2}\)
Explanation:
cos θ = \(\frac{1}{2}\)
⇒ cos θ = cos 60° ⇒ θ = 60°
⇒ cos\(\frac{\theta}{2}\) ⇒ cos 30° = \(\frac{\sqrt{3}}{2}\)

Question 53.
\(\frac{\cos \theta}{1-\tan \theta}+\frac{\sin \theta}{1-\cot \theta}\) equal to
Answer:
cos θ + sin θ

Question 54.
cos 300° equal to
Answer:
\(\frac{1}{2}\)
Explanation:
cos (270° + 30°) = sin 30° =\(\frac{1}{2}\)

Question 55.
\(\frac{\sin \theta}{\sqrt{1-\sin ^{2} \theta}}\) equal to
Answer:
tan θ
Explanation:

Question 56.
(1 + tan² θ)cos² θ equal to
Answer:
1

Question 57.
If 3 tan θ = 1; then find θ.
Answer:
30°

Question 58.
\(\frac{\sqrt{\sec ^{2} \theta-1}}{\sec \theta}\) equal to
Answer:
sin θ
Explanation:

Question 59.
\(\frac{1}{\sqrt{1+\tan ^{2} \theta}}\) equal to
Answer:
cos θ

Question 60.
Find the value of cosec 60° × cos 90°.
Answer:
0

Question 61.
sec² θ + cosec² θ equal to
Answer:
sec² θ.cosec² θ
Explanation:

Question 62.
sec² 33° – cot² 57° equal to
Answer:
1

Question 63.
If sin θ = \(\frac{11}{15}\), then find cos θ.
Answer:
\(\frac{2 \sqrt{26}}{15}\)

Question 64.
equal to \(\frac{\tan \theta}{\sqrt{1+\tan ^{2} \theta}}\)
Answer:
sin θ

Question 65.
If sin θ = \(\frac{\mathbf{a}}{\mathbf{b}}\); cos θ = \(\frac{\mathbf{c}}{\mathbf{d}}\); then find tan θ.
Answer:
\(\frac{\mathrm{ad}}{\mathrm{bc}}\)
Explanation:

Question 66.
tan (A+B) equal to
Answer:
\(\frac{\tan A+\tan B}{1-\tan A \cdot \tan B}\)

Question 67.
If sin A = \(\frac{1}{\sqrt{2}}\) ; then find tan A.
Answer:
1

Question 68.
sin\(\frac{\pi}{6}\) + cos\(\frac{\pi}{3}\) equal to
Answer:
1

Question 69.
Find the value of cos 75°.
Answer:
sin 15°
Explanation:
cos 75° = cos (90° – 15°) = sin 15°

Question 70.
If sin 0 = \(\frac{1}{2}\); then find cos \(\frac{3 \theta}{2}\).
Answer:
\(\frac{1}{\sqrt{2}}\)

Question 71.
\(\frac{1}{\sec ^{2} A}+\frac{1}{\operatorname{cosec}^{2} A}\) equal to
Answer:
1
Explanation:

Question 72.
sin 240° equals to
Answer:
–\(\frac{\sqrt{3}}{2}\)
Explanation:
sin 240° = sin (270° – 30°)
= -cos 30°= \(\frac{-\sqrt{3}}{2}\)

Question 73.
If sec θ + tan θ = \(\frac{1}{5}\) ,then find sin θ.
Answer:
\(\frac{12}{13}\)

Question 74.
tan 240° equal to
Answer:
√3

Question 75.
Find the value of
sin 60° cos 30°+ cos 60°. sin 30°.
Answer:
1
Explanation:

Question 76.
\(\frac{1}{1-\sin \theta}+\frac{1}{1+\sin \theta}\) equal to
Answer:
2sec² θ
Explanation:

Question 77.
tan 0° equal to
Answer:
0

Question 78.
\(\frac{\sqrt{1+\tan ^{2} \theta}}{\sqrt{1+\cot ^{2} \theta}}\) equal to
Answer:
tan θ

Question 79.
\(\frac{\sin 18^{\circ}}{\cos 72^{\circ}}\) equal to
Answer:
1

Question 80.
If sin θ = cos θ, then find θ.
Answer:
45°

Question 81.
In right triangle ΔABC; ∠B= 90°; tan C = \(\frac{5}{12}\) , then find the length of hypotenuse.
Answer:
13
Explanation:

By Py thagoras theorem,
AC² = AB² + BC²
AC² = 5² + 12²
= 25 + 144= 169
⇒ AC = hypotenuse = 13

Question 82.
If A, B are acute angles ;
sin(A – B)= \(\frac{1}{2}\); cos (A + B) = \(\frac{1}{2}\), then find B.
Answer:
15° (or) \(\frac{\pi}{12}\)
Explanation:
sin (A – B) = \(\frac{1}{2}\) = sin30°
A – B = 30°

Question 83.
cos (270° – θ) equal to
Answer:
-sin θ

Question 84.
Find the value of
cos 0° + sin 90° + √2sin 45°.
Answer:
3
Explanation:
1 + 1 + √2.\(\frac{1}{\sqrt{2}}\) = 1 + 1 + 1 = 3

Question 85.
If tan θ = \(\frac{1}{\sqrt{3}}\); then find cos θ.
Answer:
\(\frac{\sqrt{3}}{2}\)
Explanation:
tan θ = \(\frac{1}{\sqrt{3}}\) = tan 30°
⇒ θ = 30°
∴ cos θ = cos 30° = \(\frac{\sqrt{3}}{2}\)

Question 86.
Equal to cosec (90 + θ).
Answer:
sec θ

Question 87.
sin θ. sec θ equals to
Answer:
tan θ

Question 88.
Find the value of 3sin² 45°+2cos² 60°.
Answer:
2
Explanation:

Question 89.
Find the value of tan² 30° + 2 cot² 60°.
Answer:
1
Explanation:

Question 90.
Find the value of secA.\(\sqrt{1-\sin ^{2} A}\)
Answer:
1

Question 91.
If 5 sin A= 3; then find sec² A – tan² A.
Answer:
1
Explanation:
sin A = \(\frac{3}{5}\) ⇒ sec²A = tan²A = 1

Question 92.
Find the value of cos 240°.
Answer:
–\(\frac{1}{2}\)

Question 93.
If sin θ. cosec θ = x; then find x.
Answer:
1

Question 94.
sin(45°+ θ) – cos(45°- θ).
Answer:
0

Question 95.
Find the value of cos² 17° – sin² 73°.
Answer:
0
Explanation:
cos² 17°- sin² 73°
= cos² (90 – 73) – sin² 73°
= sin² 73° – sin² 73° = 0

Question 96.
Find the value of sin² 60° – sin² 30°.
Answer:
\(\frac{1}{2}\)

Question 97.
ten θ is not defined when θ is equal to
Answer:
90°

Question 98.
Find the value of sin 45° + cos 45°.
Answer:
√2
Explanation:

Question 99.
Simplify: \(\frac{1-\sec ^{2} A}{\operatorname{cosec}^{2} A-1}\)
Answer:
– tan⁴A

Question 100.
Find the value of sin θ. cosec θ+ cos θ. sec θ + tan θ. cot θ.
Answer:
3
Explanation:

= 1 + 1 + 1 = 3

Question 101.
If A = 30°, then sin 2A equals to
Answer:
\(\frac{\sqrt{3}}{2}\)

Question 102.
If sec θ = 3k and tan θ = \(\frac{3}{\mathbf{k}}\), then find the value of (k² – \(\frac{1}{\mathbf{k}^{2}}\))
Answer:
\(\frac{1}{9}\)
Explanation:

Question 103.
If tan θ + sec θ = 8, then find sec θ – tan θ.
Answer:
\(\frac{1}{8}\)

Question 104.
If sin θ = \(\frac{12}{13}\), then find tan θ.
Answer:
\(\frac{12}{5}\)

Question 105.
In a right angled ΔABC, right angle at C if tan A = \(\frac{8}{15}\), then find the value of cosec² A – 1.
Answer:
\(\frac{225}{64}\)
Explanation:
tan A = \(\frac{8}{15}\),
cosec ² A – 1 = cot²A = \(\left(\frac{15}{8}\right)^{2}=\frac{225}{64}\)

Question 106.
Find the value of \(\frac{2 \tan 30^{\circ}}{1+\tan ^{2} 30^{\circ}}\)
Answer:
sin 60°

Question 107.
If sin A = ;\(\frac{1}{\sqrt{2}}\) then find tan A
Answer:
1

Question 108.
In ΔABC, sin \(\left(\frac{B+C}{2}\right)\) equal to .
Answer:
A. cos\(\frac{A}{2}\)

Question 109.
tan 26°. tan 64° equal to
Answer:
1
Explanation:
tan 26° . tan 64°
= tan 26° . tan (90° – 26°)
= tan 26° . cot 26° = 1

Question 110.
cos² θ equal to
Answer:
1 – sin² θ

Question 111.
If tan A = \(\frac{3}{4}\) then find sec² A – tan² A.
Answer:
1

Question 112.
sin⁴ θ – cos⁴ θ equal to
Answer:
2sec² θ – 1

Question 113.
\(\frac{\sqrt{1-\cos ^{2} \theta}}{\cos \theta}\) equal to
Answer:
tan θ
Explanation:

Question 114.
sin(90 – Φ) equal to
Answer:
cos Φ

Question 115.
sec θ – tan θ = \(\frac{1}{n}\), then find sec θ + tan θ.
Answer:
n

Question 116.
x = 2 cosec θ, y = 2 cot θ; find x² – y².
Answer:
4

Question 117.
tan θ is not defined if θ.
Answer:
90°

Question 118.
sec θ is not defined if θ.
Answer:
90°

Question 119.
tan² Φ – sec² Φ equal to
Answer:
-1

Question 120.
\(\left|\begin{array}{ll}
\tan \theta & \sec \theta \\
\sec \theta & \tan \theta
\end{array}\right|\)
Answer:
1
Explanation:
\(\left|\tan ^{2} \theta-\sec ^{2} \theta\right|=|-1|\) = 1

Question 121.
sin 225° equal to
Answer:
\(\frac{-1}{\sqrt{2}}\)

Question 122.
cos (x – y) equal to
Answer:
cos x cos y + sin x sin y

Question 123.
\(\frac{\sec 35^{\circ}}{\operatorname{cosec} 55^{\circ}}\) equal to
Answer:
1
Explanation:

Question 124.
\(\frac{1}{\sec ^{2} A}+\frac{1}{\operatorname{cosec}^{2} A}\) equal to
Answer:
1

Question 125.
sin (-θ) equal to
Answer:
-sin θ

Question 126.
cosec (270 – θ) equal to
Answer:
-sec θ

Question 127.
sec (90 + θ) equal to
Answer:
-cosec θ

Question 128.
tan (360 – θ) equal to
Answer:
-tan θ

Question 129.
cos (-θ) equal to
Answer:
cos θ

Question 130.
sin (180 – θ) equal to
Answer:
sin θ

Question 131.
cos (270 – θ) equal to
Answer:
-sin θ

Question 132.
Find the maximum value of cos θ.
Answer:
1

Question 133.
Find the minimum and maximum values of tan θ.
Answer:
(- ∞ ,∞ )

Question 134.
sin 420° equal to
Answer:
\(\frac{\sqrt{3}}{2}\)

Question 135.
sec 240° equal to
Answer:
-2

Question 136.
cos 0° + sin 90° + √3 cosec 60° equal to
Answer:
4
Explanation:
1 + 1 + 3 . \(\frac{2}{\sqrt{3}}\) = 4

Question 137.
sec θ + tan θ = \(\frac{1}{2}\); then find sin θ.
Answer:
\(\frac{12}{13}\)

Question 138.
sin 45°.cos 45° + √3 sin 60° equal to
Answer:
2

Question 139.
tan 30° + cot 30° equal to
Answer:
\(\frac{4}{\sqrt{3}}\)

Question 140.
tan (A – B) equal to
Answer:
\(\frac{\tan A-\tan B}{1+\tan A \tan B}\)

Question 141.
If sin A = \(\frac{3}{5}\); then find sin (90 + A).
Answer:
\(\frac{4}{5}\)
Explanation:
sin A = \(\frac{3}{5}\), sin (90 + A) = cos A = \(\frac{4}{5}\)

Question 142.
Find the value of cos 1°.cos 2°.cos 3°……………, cos 180°.
Answer:
0
Explanation:
cos 1° × cos 2° × cos 3° × …………….. × cos 90° × ……….. × cos 180°
cos 1° × cos 2° × cos 3° × …………. × 0 × …………. × (- 1) = 0

Question 143.
Find the value of cos²17° – sin² 73°.
Answer:
0

Question 144.
If cosec θ + cot θ = 2; then find cosec θ – cot θ.
Answer:
\(\frac{1}{2}\)

Question 145.
cosec 60°. sec 60° equal to
Answer:
\(\frac{4}{\sqrt{3}}\)

Question 146.
sin (A – B) equal to
Answer:
sin A cos B – cos A sin B

Question 147.
If tan (15°+ B)= √3 ; then find B.
Answer:
45°
Explanation:
tan (15° + B) = √3 = tan 60°
15 + B = 60 ⇒B = 60°- 15° = 45°

Question 148.
Simplify: \(\frac{\sin (90-\theta) \sin \theta}{\tan \theta}\) – 1
Answer:
-sin² θ

Question 149.
sin 450° equal to
Answer:
1

Question 150.
cos 150° equal to
Answer:
–\(\frac{\sqrt{3}}{2}\)

Question 151.
If sin θ = \(\frac{1}{2}\) ; then find cot θ.
Answer:
√3

Question 152.
Find the value of sin 29° – cos 61°
Answer:
0

Question 153.
If tan θ = 1; then find cos θ.
Answer:
\(\frac{1}{\sqrt{2}}\)

Question 154.
cos (A + B) equal to
Answer:
cos A cos B – sin A sin B

Question 155.
Express tan θ, in terms of sin θ.
Answer:
\(\frac{\sin \theta}{\sqrt{1-\sin ^{2} \theta}}\)

Question 156.
sin θ + sin² θ = 1, then find cos² θ + cos⁴ θ.
Answer:
1

Question 157.
in ΔABC,write tan(\(\frac{B+C}{2}\))equal to
Answer:
Cot (\(\frac{\mathrm{A}}{2},\))
Explanation:
A + B + C = 180°

Question 158.
(cos A + sin A)² + (cos A – sin A)² equal to
Answer:
2

Question 159.
cos(180 – θ) equal to
Answer:
– cos θ

Question 160.
Find the value of (cosec θ – cot θ).
Answer:
\(\frac{1-\cos \theta}{\sin \theta}\)
Explanation:
cosec θ – cot θ
= \(\frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}=\frac{1-\cos \theta}{\sin \theta}\)

Question 161.
sin² 75° + cos² 75° equal to
Answer:
1

Question 162.
If cos θ. sin θ = \(\frac{1}{2}\) ; then find tan θ.
Answer:
1
Explanation:
cos θ . sin θ = cos 45° . sin 45°
= \(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}=\frac{1}{2}\)
Then tan 45° = 1.

Question 163.
For which value of cosine equal to sin 81°.
Answer:
cos 9°.

Question 164.
sin 750° equal to
Answer:
\(\frac{1}{2}\)

Question 165.
\(\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}\) equal to
Answer:
tan θ
Explanation:

Question 166.
If sin(A+B) = 1; sin B = \(\frac{1}{2}\) ; then find A.
Answer:
60°
Explanation:
sin (A + B) = 1 = sin 90°
A + B = 90°
sin B = sin 30° ⇒ B = 30°
A + 30° = 90° ⇒ A = 60°

Question 167.
If tan θ = √3 , then find sec θ.
Answer:
2
Explanation:
tan θ = √3 = tan 60° ⇒ θ = 60°
sec 60° = 2

Question 168.
Find the value of
cos 0°+ sin 90° + √3 cosec 60°.
Answer:
4

Question 169.
\(\sqrt{\sec ^{2} \theta+\operatorname{cosec}^{2} \theta}\) equal to
Answer:
sec θ . cosec θ
Explanation:

Question 170.
Find the value of
cos 60°. cos 30° – sin 60°. sin30°.
Answer:
0

Question 171.
cot²θ – \(\frac{1}{\sin ^{2} \theta}\) equal to
Answer:
-1
Explanation:
cot² θ – cosec² θ = – 1

Question 172.
π radians equal into degrees.
Answer:
180°

Question 173.
sin (A + B). cos(A – B) + sin (A – B). cos (A + B) equal to
Answer:
sin 2A

Question 174.
If cos (A+B) = 0, cos B = \(\frac{\sqrt{3}}{2}\) ;then find A.
Answer:
60°

Question 175.
cos⁶ θ + sin⁶ θ equal to .
Answer:
1 – 3 sin² θ.cos² θ
Explanation:
cos⁶ θ + sin⁶ θ = (cos² θ)³ + (sin² θ)³
a³ + b³ = (a + b) – 3ab (a + b)
= (sin² θ + cos² θ)³
– 3 sin² θ cos² θ (sin² θ + cos² θ)
= 1 – 3 sin² θ cos² θ (1)
= 1 – 3 sin² θ cos² θ

Question 176.
sin 225° equal to
Answer:
–\(\frac{1}{\sqrt{2}}\)

Question 177.
sin 180° equal to
Answer:
0

Question 178.
If x = 2 cosec θ; y = 2 cot θ; then find x² – y².
Answer:
4
Explanation:
\(\frac{\mathrm{x}}{2}\) = cosec θ, \(\frac{\mathrm{y}}{2}\) = cot θ
cosec² θ – cot² θ = \(\left(\frac{x}{2}\right)^{2}-\left(\frac{y}{2}\right)^{2}\)
1 = \(\frac{x^{2}}{4}-\frac{y^{2}}{4}\)
x² – y² = 4

Question 179.
cos θ. tan θ equal to
Answer:
sin θ
Explanation:
cos θ = \(\frac{\sin \theta}{\cos \theta}\) = sin θ

Question 180.
If cot² θ = 3; then find cosec θ.
Answer:
2
Explanation:
cot θ = √3 = cot 30° ⇒ θ = 30°
∴ cosec 30° = 2.

Question 181.
If sec θ = cosec θ; then find the value of θ.
Answer:
\(\frac{\pi}{4}\)

Question 182.
\(\frac{\tan \theta \cdot \sqrt{1-\sin ^{2} \theta}}{\sqrt{1-\cos ^{2} \theta}}\) equal to
Answer:
1
Explanation:

Question 183.
cos(x – y) equal to
Answer:
cos x cos y + sin x sin y

Question 184.
Simplify : cosec 31° – sec 59°
Answer:
0
Explanation:
cosec 31° – sec (90° – 31°)
[∵ sec (90 – θ) = cosec θ]
= cosec 31° – cosec 31°
= 0

Question 185.
(sec 45° + tan 45°) (sec 45° – tan 45°) equal to
Answer:
1

Question 186.
If sin θ = \(\frac{\mathbf{a}}{\mathbf{b}}\); cos θ = \(\frac{\mathbf{c}}{\mathbf{d}}\), then find cot θ.
Answer:
\(\frac{\mathrm{bc}}{\mathrm{ad}}\)

Question 187.
\(\sqrt{\operatorname{cosec}^{2} \theta-\cot ^{2} \theta}\) equal to
Answer:
1

Question 188.
sin² 45° + cos² 45° + tan² 45° equal to
Answer:
2
Explanation:

Question 189.
sec (360° – θ) equal to
Answer:
sec θ

Question 190.
tan θ. cot θ = sec θ. x ; then find x.
Answer:
cos θ
Explanation:
tan θ . cot θ = sec θ . x
\(\frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta}=\frac{x}{\cos \theta}\)
cos θ = x

Question 191.
If 4 sin 30°. sec 60° = x tan 45°; then find x.
Answer:
4
Explanation:
4 . sin 30° – sec 60° = x . tan 45°
4 . \(\frac{1}{2}\) . 2 = x . 1
⇒ x = 4

Question 192.
In the following which are in geometric progression?
A) sin 30°, sin 45°, sin 60°
B) sec 30°, sec 45°, sec 60°
C) tan 30°, tan 45°, tan 60°
D) cos 45°, cos 60°, cos 90°
Answer:
C) tan 30°, tan 45°, tan 60°

Question 193.
(1 + tan² A) (1 – sin² A)
equal to
Answer:
1
Explanation:
sec² A × cos² A = 1

Question 194.
Find the value of
cos 60° cos 30° + sin 60°. sin 30°.
Answer:
\(\frac{\sqrt{3}}{2}\)
Explanation:
\(\frac{1}{2} \cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\)
= \(\frac{\sqrt{3}}{4}+\frac{\sqrt{3}}{4}=\frac{2 \sqrt{3}}{4}=\frac{\sqrt{3}}{2}\)

Question 195.
\(\frac{1-\tan ^{2} 30^{\circ}}{1+\tan ^{2} 30^{\circ}}\) equal to
Answer:
\(\frac{1}{2}\)

Question 196.
cos(\(\frac{3 \pi}{2}\) + θ) equal to
Answer:
sin θ
Explanation:
cos (270 + θ) = sin θ

Question 197.
\(\sqrt{1+\cot ^{2} \theta}\) equal to
Answer:
cosec θ

Question 198.
If tan θ + cot θ = 2; then find tan² θ + cot² θ.
Answer:
2
Explanation:
tan θ + cot θ = 2
⇒ tan² θ + cot² θ + 2 . tan θ . cot θ = 4
⇒ tan² θ + cot² θ = 4 – 2 = 2

Question 199.
If sec θ = \(\frac{13}{12}\), then find sin θ.
Answer:
\(\frac{5}{13}\)

Question 200.
How much the value of
(sin θ + cos θ)² + (sin θ – cos θ)² ?
Answer:
2
Explanation:
(a + b)² + (a – b)² = 2(a² + b²)
= 2 (sin² θ + cos² θ) = 2

Question 201.
(1 + tan θ)² equal to
Answer:
sec² θ + 2 tan θ

Question 202.
sin(A – B) = \(\frac{1}{2}\); cos (A+B) = \(\frac{1}{2}\). So
find A.
Answer:
45°
Explanation:

⇒ A = 45°

Question 203.
Find the value of tan 135°.
Answer:
-1

Question 204.
Find the value of \(\sqrt{1+\sin A} \cdot \sqrt{1-\sin A}\)
Answer:
cos A
Explanation:

Question 205.
Find the value of tan 60° – tan 30°.
Answer:
\(\frac{2 \sqrt{3}}{3}\)
Explanation:
tan 60° – tan 30°

Question 206.
In ΔABC, a = 3; b = 4; c = 5, then find cos A.
Answer:
4/5
Explanation:

Question 207.
sin³ θ cos θ.cos³ θ sin θ equals to
Answer:
sin θ cos θ

Question 208.
If tan θ \(\frac{1}{\sqrt{3}}\) , then find the value of 7 sin² θ + 3 cos² θ.
Answer:
4
Explanation:

Question 209.
cot (270° – θ) equal to
Answer:
tan θ

Question 210.
(1 + tan² 60°)² equals to
Answer:
16
Explanation:
[1 + (√3)²]² = (1 + 3)² = 4² = 16

Question 211.
sin (270° + θ) equal to
Answer:
– cos θ

Question 212.
If tan² 60° + 2 tan² 45° = x tan 45°, then find x.
Answer:
5
Explanation:
(√3)² + 2(1)² = x . 1
⇒ 3 + 2 = x ⇒ x = 5

Question 213.
cos² 0° + cos² 60° equal to
Answer:
5/4

Question 214.
Simplify : sin⁴ θ – cos⁴ θ
Answer:
2sin² θ – 1

Question 215.
If α + β = 90° and α = 2β, then find cos² β + sin² β.
Answer:
1
Explanation:
α = 90 – β
⇒ 90 – β = 2β ⇒ 3β = 90° ⇒ β = 30°
∴ cos² 30° + sin² 30° = (\(\frac{\sqrt{3}}{2}\))² + (\(\frac{1}{2}\))²
= \(\frac{3}{4}+\frac{1}{4}=\frac{4}{4}\) = 1

Question 216.
If sin θ = \(\frac{\mathbf{a}}{\mathbf{b}}\) , then find cos θ.
Answer:
\(\frac{\sqrt{b^{2}-a^{2}}}{b}\)

Question 217.
2sin θ = sin² θ is true for the value of θ is
Answer:
0°

Question 218.
If sin θ = \(\frac{\mathbf{a}}{\mathbf{b}}\) , then find tan θ.
Answer:
\(\frac{a}{\sqrt{b^{2}-a^{2}}}\)

Question 219.
\(\frac{\sin \theta}{1+\cos \theta}\) is equal to
Answer:
\(\frac{1-\cos \theta}{\sin \theta}\)

Question 220.
If sin θ = cos θ, then find the value of 2 tan θ + cos² θ.
Answer:
\(\frac{5}{2}\)
Explanation:
sin θ = cos θ ⇒ θ = 45°
2 tan 45° + cos² 45° = 2 + \(\frac{1}{2}=\frac{5}{2}\)

Question 221.
If sin x = cos x, 0 ≤ x ≤ 90°, then find x.
Answer:
45°

Question 222.
How much the maximum value of sin θ?
Answer:
1

Question 223.
In the figure find tan x.

Answer:
\(\frac{15}{8}\)

Question 224.
sin A = cos B, then find A + B.
Answer:
90°

Question 225.
If cosec θ + cot θ = 3, then find cosec θ – cot θ.
Answer:
\(\frac{1}{3}\)

Question 226.
If tan 2A = cot (A – 18°) where 2A is an acute angle, then find A.
Answer:
36°
Explanation:
90 – 2A = A – 18°
⇒ 3A = 108° ⇒ A = 36°

Question 227.
sec 0° equal to
Answer:
1

Question 228.
cosec 300° equal to
Answer:
\(\frac{-2}{\sqrt{3}}\)

Question 229.
cos 240° equal to
Answer:
\(\frac{-1}{2}\)

Question 230.
\(\frac{\operatorname{cosec}^{2} \theta}{\cot \theta}\) – cot θ equal to
Answer:
tan θ

Question 231.
Find the minimum value of cos θ.
Answer:
-3

Question 232.
If sec θ = cosec θ; then find the value of θ in radians.
Answer:
\(\frac{\pi^{\mathrm{c}}}{4}\)
Explanation:
sec θ = cosec θ ⇒ θ = 45° = \(\frac{\pi^{\mathrm{c}}}{4}\)

Question 233.
Reciprocal of cot A.
Answer:
tan A

Question 234.
sin \(\frac{\pi^{\mathrm{c}}}{4}\) + cos 45° equal to
Answer:
√2

Question 235.

Answer:
\(\frac{\mathrm{x}}{\mathrm{z}}\)

Question 236.
If cosec θ = \(\frac{25}{7}\), then find cot θ.
Answer:
\(\frac{24}{7}\)

Question 237.
If sin (A + B) = \(\frac{\sqrt{3}}{2}\); cos B = \(\frac{\sqrt{3}}{2}\) , then find A.
Answer:
30°

Question 238.
tan 750° equal to
Answer:
\(\frac{1}{\sqrt{3}}\)
Question 239.
(1 – sec² θ) (1 – cosec² θ) equal to
Answer:
1

Question 240.
If cosec θ – cot θ = 4, then find cosec θ + cot θ.
Answer:
\(\frac{1}{4}\)

Question 241.
\(\sqrt{\tan ^{2} \theta+\cot ^{2} \theta+2}\) equal to
Answer:
tan θ + cot θ
Explanation:
= \(\sqrt{1+\tan ^{2} \theta+1+\cot ^{2} \theta}\)
= \(\sqrt{\sec ^{2} \theta+\operatorname{cosec}^{2} \theta}\)
= tan θ + cot θ

Question 242.
If cos θ = \(\frac{3}{5}\); then cos (-θ) equal to
Answer:
\(\frac{3}{5}\)

Question 243.
\(\frac{\sqrt{\operatorname{cosec}^{2} \theta-1}}{\operatorname{cosec} \theta}\)
equal to
Answer:
cos θ

Question 244.
If 5 sin A = 3, then find sec² A – tan² A.
Answer:
1

Question 245.
In the figure, find AB.

Answer:
20√3 (or) \(\frac{60}{\sqrt{3}}\)

Question 246.
If cot θ = x; then find cosec θ.
Answer:
\(\sqrt{\mathrm{x}_{1}^{2}+1}\)
Explanation:
cot θ = x ⇒ cosec θ = \(\sqrt{x^{2}+1}\)

Question 247.
\(\sqrt{\operatorname{cosec}^{2} \theta-\sin ^{2} \theta-\cos ^{2} \theta}\) equal to
Answer:
cot θ

Question 248.
Find the value of \(\frac{1}{\sec \theta-\tan \theta}\)
Answer:
sec θ + tan θ

Question 249.
sin (A + B) equal to
Answer:
sin A cos B + cos A sin B

Question 250.
\(\sqrt{(\sec \theta+1)(\sec \theta-1)}\) equal to
Answer:
tan θ

Question 251.
Find the value of tan 5° × tan 30° × 4 tan 85°.
Answer:
\(\frac{4}{\sqrt{3}}\)

Question 252.
cos 110°.cos 70° – sin 110°.sin 70° equal to ,
Answer:
-1
Explanation:
cos A . cos B – sin A . sin B
= cos (A + B)
= cos (110 + 70) = cos 180° = – 1

Question 253.
Find the value of tan 1°.tan 2°.tan 3°………….tan 89°.
Answer:
1

Question 254.
If cos θ = -cos θ; then Write θ in radian measure.
Answer:
\(\frac{\pi^{\mathrm{c}}}{3}\)

Question 255.
sec A = cosec B, then write A and B are ……….. angles.
Answer:
Complementary.

Question 256.
Find the value of tan 75°.
Answer:
2 + √3

Question 257.
\(\sqrt{\sec ^{2} \theta-\tan ^{2} \theta+\cot ^{2} \theta}\) equal to
Answer:
cosec θ

Question 258.
sin 240° + sin 120° equal to
Answer:
0

Question 259.
Find the value of
sec 70°. sin 20° + cos 20°. cosec 70°.
Answer:
2

Question 260.
If sec A + tan A = \(\frac{1}{3}\); then find sec A – tan A.
Answer:
3

Question 261.
(sec² θ – 1)(1 – cosec² θ)equal to
Answer:
-1

Question 262.
cos θ equal to
Answer:
\(\frac{\cot \theta}{\operatorname{cosec} \theta}\)

Question 263.
The radius of a circle is ‘r’; an arc of length ‘L’ is making an angle θ, at the centre of the circle, then find θ.
Answer:
L/r

Question 264.
cos (A – B) = \(\frac{1}{2}\); sin B = \(\frac{1}{\sqrt{2}}\), find measure of A.
Answer:
105°

Question 265.
If sec θ + tan θ = 4; then find cos θ.
Answer:
\(\frac{8}{17}\)

Question 266.
sec (360° – θ) equals to
Answer:
sec θ

Question 267.
If A is acute and tan A = \(\frac{1}{\sqrt{3}}\); then find sin ‘A’.
Answer:
\(\frac{1}{2}\)

Question 268.
(1 +cot² 45°)² equal to
Answer:
4

Question 269.
\(\frac{\tan 45^{\circ}}{\operatorname{cosec} 30^{\circ}}+\frac{\sec 60^{\circ}}{\cot 45^{\circ}}\) equal to
Answer:
2\(\frac{1}{2}\)

Question 270.
\(\sqrt{\frac{\sec x+\tan x}{\sec x-\tan x}}\) equal to
Answer:
sec x + tan x

Question 271.
\(\frac{\sin ^{4} A-\cos ^{4} A}{\sin ^{2} A-\cos ^{2} A}\) equal to
Answer:
1

Question 272.
If the angle in a triangle are in the ratio of 1:2:3, then find the smallest angle in radins.
Answer:
π/6

Question 273.
If sin θ + cos θ = √2; then find the value of ‘θ’.
Answer:
45°

Question 274.
If cosec θ = 2 and cot θ = √3 P; where θ is an acute angle, then find the value of ‘P’.
Answer:
1

Question 275.
If cos 2θ = sin 4θ; here 2θ and 4θ are acute angles, then find the value of θ.
Answer:
15°
Explanation:
2θ + 4θ = 90°
⇒ 6θ = 90°⇒ θ = \(\frac{90^{\circ}}{6}\) = 15°

Question 276.
If sin 45°.cos 45°+cos 60° = tan θ, then find the value of θ.
Answer:
45°

Question 277.
If P, Q and R are interior angles of a ΔPQR, then tan \(\left(\frac{\mathbf{P}+\mathbf{Q}}{2}\right)\) equals
Answer:
cot (\(\frac{\mathrm{R}}{2}\))

Question 278.
If tan θ = 1, then find the value of \(\frac{5 \sin \theta+4 \cos \theta}{5 \sin \theta-4 \cos \theta}\)
Answer:
9
Explanation:
tan θ = 1 = tan 45° ⇒ θ = 45°

Question 279.
If sec 2A = cosec (A – 27°), where 2A is an acute angle, then find the measure of ∠A.
Answer:
39°
Explanation:
90 – 2A = A – 27°
⇒ 117° = 3A ⇒ A = \(\frac{117^{\circ}}{3}\) = 39°

Question 280.
If sin C = \(\frac{3}{5}\); then find cos A.

Answer:
3/5

Question 281.
Expressing tan θ, interms of sec θ.
Answer:
\(\sqrt{\sec ^{2} \theta-1}\)

Question 282.
If sin θ. cos θ = k; then find sin θ + cos θ.
Answer:
\(\sqrt{1+2 \mathrm{k}}\)

Question 283.
If \(\frac{1}{2}\) tan² 45° = sin2 A and ’A’ is acute, then find the value of ‘A’.
Answer:
45°

Question 284.
Find the value of (\(\frac{11}{\cot ^{2} \theta}-\frac{11}{\cos ^{2} \theta}\))
Answer:
-11
Explanation:
11 (tan² θ – sec² θ) = 11 (-1) = -11

Question 285.
Find the maximum value of \(\frac{1}{\sec \theta}\)
0° ≤ θ ≤ 90°.
Answer:
1

Question 286.
If π < θ < \(\frac{3 \pi}{2}\), then θ lies in which quadrant?
Answer:
Third quadrant

Question 287.
If cos θ = \(\frac{\sqrt{3}}{2}\) and’θ’is acute, then find the value of 4sin² θ + tan² θ.
Answer:
4/3

Question 288.
If tan θ = \(\frac{7}{8}\), then find the value of \(\frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}\) ,
Answer:
\(\frac{64}{49}\)

Question 289.
When 0° ≤ 0 ≤ 90°; find the maximum value of sin θ + cos θ.
Answer:
√2

Question 290.
In ΔABC, ∠B = 90° ; ∠C = θ. From the figure find tan θ.

Answer:
\(\frac{15}{8}\)

Question 291.
If sin (x – 20°) = cos(3x – 10°), then find x’.
Answer:
15°

Question 292.
If sin (A – B)= \(\frac{1}{2}\); cos (A + B)= \(\frac{1}{2}\),
then find ‘B’.
Answer:
15°

Question 293.
If 5 tan θ = 4, then find die value of
\(\frac{5 \sin \theta-3 \cos \theta}{5 \sin \theta+3 \cos \theta}\)
Answer:
\(\frac{1}{7}\)

Question 294.
If 4 cos2 θ – 3 = 0, then find the value of sin θ.
Answer:
\(\frac{1}{2}\)

Choose the correct answer satisfying the following statements.
Question 295.
Statement (A): sin² 67° + cos² 67° = 1
Statement (B) : For any value of θ,
sin² θ + cos² θ = 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
Explanation:
sin² θ + cos² θ = 1
⇒ sin² 67° + cos² 67° = 1
Hence, (i) is the correct option.

Question 296.
Statement (A): If cos A + cos² A = 1,
then sin² A + sin⁴ A = 2
Statement (B): 1 – sin² A = cos² A, for any value of 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) A is false, B is true
Explanation:
cos A + cos² A = 1
cos A = 1 – cos² A = sin² A
∴ sin² A + sin⁴ A = cos A + cos² A = 1
⇒ sin² A + sin⁴ A = 1
Hence, (iii) is the correct option.

Question 297.
Statement (A):
The value of sec² 10° – cot2 80° is 1.
Statement (B):
The value of sin 30° = \(\frac{1}{2}\)
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
Explanation:
We have sec² 10° – cot² 80°
= sec² 10°-cot² (90°- 10°)
= sec² 10° – tan² 10° = 1
Also, sin 30° = \(\frac{1}{2}\).
Hence, (i) is the correct option.

Question 298.
Statement (A) : The value of sin θ cos (90 – θ) + cos θ sin(90 – θ) ‘ equal to 1.
Statement (B): tan θ = sec(90 – θ)
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:
sin θ cos (90 – θ) + Cos θ sin (90 – θ)
= sin θ . sin θ + cos θ – cos θ
= sin² θ + cos² θ = 1 and tan θ = cot (90 – θ)
Hence, (ii) is the correct option.

Question 299.
Statement (A) : The value of sin θ = \(\frac{4}{3}\) is not possible.
Statement (B): Hypotenuse is the largest side in any right angled triangle.
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:
sin ² = \(\frac{\mathrm{P}}{\mathrm{H}}=\frac{4}{3}\)
Here, perpendicular is greater than the hypotenuse which is not possible in any right triangle.
Hence, (i) is the correct option.

Question 300.
Statement (A) : In a right angled triangle, if tan θ = \(\frac{3}{4}\), the greatest side of
the triangle is 5 units.
Statement (B) : (Greatest side hypotenuse)² = (perpendicular)² – (base)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
Explanation:
Both A and B are correct and B is the correct explanation of the A.
Greatest side = \(\sqrt{(3)^{2}+(4)^{2}}\) = 5 units.
Hence, (i) is the correct option.

Question 301.
Statement (A) : In a right angled triangle, if cos θ \(\frac{1}{2}\) = and sin θ = \(\frac{\sqrt{3}}{2}\) then tan θ = √3
‘Statement (B) : tan θ = \(\frac{\sin \theta}{\cos \theta}\)
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 correct and B is the correct explanation of the A.
tan θ = \(\frac{\sqrt{3}}{2}\) × 2 = √3
Hence, (i) is the correct option.

Question 302.
Statement (A) : sin 47° cos 43°.
Statement (B) : sin θ = cos(90 + θ),
where θ is an acute angle.
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:
A is correct, but B is not correct,
sin θ = cos (90 – θ)
sin 47° = cos (90 – 47)
= cos 43°
Hence, (ii) is the correct option.

❖ Study the given information and answer to the following questions.
In ΔABC, right angled at B.

AB + AC = 9 cm and BC = 3 cm

Question 303.
The value of cot C is
Answer:
\(\frac{3}{4}\)
Explanation:
cot C = \(\frac{\mathrm{BC}}{\mathrm{AB}}=\frac{3}{4}\)
[∴ In ΔABC, By Pythagoras theorem,
AC² = AB² + BC²
AB = 4 cm, AC = 5 cm]

Question 304.
The value of sec C is
Answer:
\(\frac{5}{3}\)
Explanation:
sec C = \(\frac{A C}{B C}=\frac{5}{3}\)

Question 305.
sin² C + cos² C is equal to
Answer:
1
In figure, ΔABC has a right angle at B. If AB = BC = 1cm and AC = √2 cm.

Explanation:
sin C= \(\frac{4}{5}\) ;cosC = \(\frac{3}{5}\)
L.H.S. = sin² C + cos² C
= \(\left(\frac{4}{5}\right)^{2}+\left(\frac{3}{5}\right)^{2}\)
= \(\frac{16+9}{25}\) = 1 = R.H.S.

Question 306.
Find sin C.
Answer:
\(\frac{1}{\sqrt{2}}\)
Explanation:
sin C = \(\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{1}{\sqrt{2}}\)

Question 307.
Find cos C.
Answer:
\(\frac{1}{\sqrt{2}}\)
Explanation:
cos C =\(\frac{\mathrm{BC}}{\mathrm{AC}}=\frac{1}{\sqrt{2}}\)

Question 308.
Find tan C.
Answer:
1
The length of a pendulum is 80 cm. Its end describes an arc of length 16 cm.
Explanation:
tan C = \(\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{1}{1}\) = 1

Question 309.
To find the length of the arc which formula is useful?
Answer:
l = r0

Question 310.
Calculate the angle of arc makes at centre.
Answer:
θ = \(\frac{1}{r}=\frac{16}{80}=\frac{1}{5}\)
In ΔPQR, right angled at Q,
PR + QR = 25 cm and PQ = 5 cm

Question 311.
Determine the value of “QR”.
Answer:
QR = 12 cm

Question 312.
Determine the value of “PR”.
Answer:
PR = 13 cm

Question 313.
Find the value of sin P.
Answer:
\(\frac{12}{13}\) cm

Question 314.
Find the value of cos P.
Answer:
\(\frac{5}{13}\) cm

Question 315.
Find the value of tan P.
Answer:
\(\frac{12}{5}\) cm

Question 316.
In ΔABC, ∠B = 90°, AB = 3 cm and BC = 4 cm, then match the column.
A) sin C [ ] i) 3/5
B) tanA [ ] ii) 4/5
C) cos C [ ] iii) 5/3
D) sec A [ ] iv) 4/3
Answer:
A – (i), B – (iv), C – (ii), D – (iii)

Question 317.
Match the following.

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

Question 318.
Match the following.

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

Question 319.
If sin θ = \(\frac{7}{25}\), then

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

Question 320.
Match the following.

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

Question 321.
What is the value of
sec 16°- cosec 74° – cot 74° • tan 16° ?
Answer:
1 (one)

Question 322.
If x = 2019°, then what is the value of sin² x + cos² x ?
Solution:
If x = 2019°, then
sin²x + cos²x = sin²2019° + cos²2019° = 1 [∵ sin²θ + cos²θ = 1]

Question 323.
If x is in first quadrant and sin x = cos x, then what is the value of x?
Solution:
Given, sin x = cos x
We know, sin (90°- θ) = cos θ
So, cos x = sin(90° – x)
⇒ sin x = sin(90° – x)
[note : If sin A = sin B, then A = B]
⇒ x = 90° – x
⇒2x = 90°
∴ x = 45°