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QUESTION: 1

If ƒ(x) = √(2tan(x)),then f^{-1}(√(2)) =

Solution:

**Correct Answer :- b**

**Explanation : **ƒ(x) = √(2tan(x))

x = √(2tan(x))

x^{2} = (2tan(x))

(x^{2})/2 = tan(x)

tan^{-1}(x^2)/2 = x

By putting x = √2, we get

x = 1

QUESTION: 2

The value of cos15^{º}− sin15^{º} is

Solution:

QUESTION: 3

If 2tan^{−1}(cos x) = tan^{−1}(2cosec x) , then x =

Solution:

If 2 tan^{-1 }(cos x) = tan ^{-1}(2 cosec x),

2tan^{-1}(cos x) = tan^{-1} (2 cosec x)

= tan^{-1}(2 cosec x)

= cot x cosec x = cosec x = x = π/4

QUESTION: 4

tan^{−1}(−2) + tan^{−1}(−3) is equal to

Solution:

tan^{-1}(-2) + tan^{-1}(-3)

QUESTION: 5

The values of x which satisfy the trigonometric equation are :

Solution:

QUESTION: 6

The maximum value of sin x + cos x is

Solution:

QUESTION: 7

The value of tan15^{0 }+ cot15^{0 }is

Solution:

The value of tan15^{0} + cot 15^{0}

QUESTION: 8

is equal to

Solution:

QUESTION: 9

The number of solutions of the equation sin^{-1} x - cos^{-1} x = sin^{-1}(1/2) is

Solution:

Hence , the given equation has only one solution.

QUESTION: 10

Solution:

cot^{-1}a - cot^{-1}b + cot^{-1} b - cot^{-1} c - cot^{-1 }a = 0

QUESTION: 11

What is the maximum and minimum value of sin x +cos x?

Solution:

Let y= sin x + cos x

dy/dx=cos x- sin x

For maximum or minimum dy/dx=0

Setting cosx- sin x=0

We get cos x = sin x

x= π/4, 5π/4———-

Whether these correspond to maximum or minimum, can be found from the sign of second derivative.

d^2y/dx^2=-sin x - cos x=-1/√2–1/√2 (for x=π/4) which is negative. Hence x=π/4 corresponds to maximum.For x=5π/4

d^2y/dx^2=-(-1/√2)-(-1/√2)=2/√2 a positive quantity. Hence 5π/4 corresponds to minimum

Maximum value of the function

y= sin π/4 + cos π/4= 2/√2=√2

Minimum value is

Sin(5π/4)+cos (5π/4)=-2/√2=-√2

QUESTION: 12

sin (200^{)0 }+ cos (200)^{0} is

Solution:

Because both sin 200^{0} and cos 200^{0} lies in 3rd quadrant. In 3rd quadrant values of sin and cos are negative.

QUESTION: 13

If cos^{(}^{-1)}x + cos^{(-1)}y = 2π, then the value of sin^{(-1)}x + sin^{(-1)}y is

Solution:

If cos^{(-1)}x + cos^{(-1)} y = 2π, then the value of sin^{(-1)}x + sin^{(-1)}y = π−2π = −π.

QUESTION: 14

Domain of f(x) = sin^{−1}x−sec^{−1}x is

Solution:

Since sin^{−1}x is defined for |x|⩽1, and sec^{−1}x is defined for |x|⩾1,therefore,f(x) is defined only when|x|=1.so, D_{f} = {−1,1}.

QUESTION: 15

The value of sin is

Solution:

Put therefore the given expressionis sin2θ = 2sinθcosθ

QUESTION: 16

If 5 sin θ = 3, then is equal to

Solution:

= [(5/4) + (3/4)] / [(5/4) - (3/4)]

=(8/4) / (2/4)

=4

QUESTION: 17

If sin A + cos A = 1, then sin 2A is equal to

Solution:

(sinA+cosA)^{2}

= sin^{2}A+cos^{2}A+2sinAcosA

=11+sin^{2}A=1sin^{2}A=0.

(because Sin 2A = 2sin A cos A)

QUESTION: 18

If θ = cos^{-1}, then tan θ is equal to

Solution:

Therefore, tanθ =

QUESTION: 19

The number of solutions of the equation cos^{-1}(1-x) - 2cos^{-1 }x = π/2 is

Solution:

As no value of x in (0, 1) can satisfy the given equation.Thus, the given equation has only one solution.

QUESTION: 20

tan (sin^{−1}x) is equal to

Solution:

QUESTION: 21

If x ∈ R, x ≠ 0, then the value of sec θ + tan θ is

Solution:

QUESTION: 22

The value of cos 105^{0} is

Solution:

QUESTION: 23

If cos(2sin^{−1}x) = 1/9 then x =

Solution:

Put

sin^{-1 }x = θ ⇒ x = sin θ

QUESTION: 24

Solution:

QUESTION: 25

cot (cos^{−1}x) is equal to

Solution:

Put,

cos^{-1}x = θ ⇒ x = cos θ ⇒ cos θ

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