Inverse Trigonometric Functions Test

Result

Inverse Trigonometric Functions Test - 5

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Question 1
1 / -0.25

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

A
0, -1

B
1, 0

C
-1, -√2

D
√2, –√2

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, 5 π/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=5 π/4

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

Maximum value of the function

y= sin π/4 + cos π/4= 2/√2=√2

Minimum value is

Sin(5 π/4)+cos (5 π/4)=-2/√2=-√2
Question 2
1 / -0.25

sin (200⁾⁰ + cos (200)⁰ is

A
Zero

B
Positive

C
Zero or positive.

D
Negative

Solution

Because both sin 200⁰ and cos 200⁰ lies in 3rd quadrant. In 3rd quadrant values of sin and cos are negative.
Question 3
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If cos⁽^-1) x + cos^(-1) y = 2 π, then the value of sin^(-1) x + sin^(-1) y is

A
−π

B
π

C
0

D
None of these.

Solution

If cos^(-1) x + cos^(-1) y = 2 π, then the value of sin^(-1) x + sin^(-1) y = π−2 π= −π.
Question 4
1 / -0.25

Domain of f(x) = sin⁻¹ x −sec⁻¹ x is

A
{-1 ,1}

B
{0, 1}

C
0 or 1

D
None of these

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 5
1 / -0.25

The value of sin is

A
7/25

B
-24/25

C
24/25

D
None of these.

Solution

Put therefore the given expressionis sin2 θ= 2sin θcos θ
Question 6
1 / -0.25

If 5 sin θ= 3, then is equal to

A
4

B
1/4

C
2

D
None of these.

Solution

= [(5/4) + (3/4)] / [(5/4) - (3/4)]
=(8/4) / (2/4)
=4
Question 7
1 / -0.25

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

A
2

B
0

C
1/2

D
1

Solution

(sinA+cosA)²
= sin² A+cos² A+2sinAcosA
=11+sin² A=1sin² A=0.
(because Sin 2A = 2sin A cos A)
Question 8
1 / -0.25

If θ= cos^-1, then tan θis equal to

A

B

C

D
None of these

Solution

Therefore, tan θ=
Question 9
1 / -0.25

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

A
One

B
Two

C
More than one

D
None of these.

Solution

As no value of x in (0, 1) can satisfy the given equation.Thus, the given equation has only one solution.
Question 10
1 / -0.25

tan (sin⁻¹ x) is equal to

A

B

C

D
None of these.

Solution
Question 11
1 / -0.25

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

A
1/2x

B
2x or 1/2x

C
2x

D
None of these.

Solution
Question 12
1 / -0.25

The value of cos 105⁰ is

A

B

C

D

Solution
Question 13
1 / -0.25

If cos(2sin⁻¹ x) = 1/9 then x =

A
2/3

B
±2/3

C
-2/3

D
None of these

Solution

Put

sin^-1 x = θ⇒x = sin θ
Question 14
1 / -0.25

A

B

C

D

Solution
Question 15
1 / -0.25

cot (cos⁻¹ x) is equal to

A

B

C

D
None of these

Solution

Put,
cos^-1 x = θ⇒x = cos θ⇒cos θ