Inverse Trigonometric Functions Test

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Inverse Trigonometric Functions Test - 22

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Weekly Quiz Competition

Question 1
1 / -0

If $$f(x)=sin^{-1}(sinx)+cos^{-1}(sinx)$$ and $$\phi (x)=f(f(f(x))),$$ then $$\phi '(x)=$$

A
1

B
sin x

C
0

D
None of these

Solution
Question 2
1 / -0

$$\tan^{-1}\dfrac {1}{5}+\tan^{-1}\dfrac {1}{7}+\tan^{-1}\dfrac {1}{3}+\tan^{-1}\dfrac {1}{8}$$ is equal to

A
$$\dfrac {\pi}{3}$$

B
$$\dfrac {\pi}{4}$$

C
$$\dfrac {\pi}{2 }$$

D
$$\pi$$

Solution

$$\tan^{-1}\dfrac{1}{5}+\tan^{-1}\dfrac{1}{7}+\tan^{-1}\dfrac{1}{3}+\tan^{-1}\left(\dfrac{1}{8}\right)$$
$$\tan^{-1}\left(\dfrac{1}{5}\right)+\tan^{-1}\left(\dfrac{1}{3}\right)+\tan^{-1}\left(\dfrac{1}{7}\right)+\tan^{-1}\left(\dfrac{1}{8}\right)$$
$$\Rightarrow \tan^{-1}\left(\dfrac{8}{14}\right)+\tan^{-1}\left(\dfrac{15}{55}\right)$$
$$\Rightarrow \tan^{-1}\left(\dfrac{4}{7}\right)+\tan^{-1}\left(\dfrac{3}{11}\right)$$
$$\Rightarrow \tan^{-1}\left(\dfrac{44+21}{77-12}\right)=\tan^{-1}\left(\dfrac{65}{65}\right)$$
$$\Rightarrow \tan^{-1}(1)=\dfrac{\pi}{4}$$.
Question 3
1 / -0

If $$n - 1\sum\limits_{}^\infty {{{\cot }^{ - 1}}\left( {{{{n^2}} \over 8}} \right) = \pi .} $$ where $${a \over b}$$ is rational number in its lowest, then correct option is/are

A
$$a - b = 3$$

B
$$a + b = 11$$

C
$$a + b = 10$$

D
$$a - b = 4$$
Question 4
1 / -0

If $$\sin (\sin^{-1}\dfrac{1}{5}+\cos ^{-1}x)=1$$, then find the value of x.

A
$$-1$$

B
$$\dfrac { 1 }{ 3 } $$

C
$$\dfrac { 1 }{ 5 } $$

D
$$\dfrac { 1 }{ 2 } $$

Solution

$$\sin(\sin^{-1}\dfrac{1}{5}+\cos^{-1}x)=1$$

$$\sin^{-1}\dfrac{1}{5}+\cos^{-1}x =\sin^{-1} 1$$

$$\sin^{-1}\dfrac{1}{5}+\cos ^{-1}x =\dfrac{\pi}{2}$$

$$\cos ^{-1} x =\dfrac{\pi}{2}-\sin^{-1}\dfrac{1}{5}$$

$$\cos ^{-1} x =\cos ^{-1}\dfrac{1}{5}$$

$$x=\dfrac{1}{5}$$
Question 5
1 / -0

Find the value of :
$$\sec^2 (\tan^{-1} 2) +\csc^2 (\cot^{-1} 3)$$

A
$$11$$

B
$$15$$

C
$$17$$

D
$$21$$

Solution

$$\sec^2 (\tan^{-1} 2) +\csc^2 (\cot^{-1} 3)$$
$$=1+\tan^2 (\tan^{-1} 2) +1+\cot^2 (\cot^{-1} 3)$$
$$=1+[\tan (\tan^{-1} 2)]^2 +1+[\cot (\cot^{-1} 3)]^2=1+2^2+1+3^2=15$$
Question 6
1 / -0

The value of the expression $$2\sec^{-1} 2 +\sin^{-1} \dfrac{1}{2}$$ is

A
$$\dfrac{\pi}{6}$$

B
$$\dfrac{5\pi}{6}$$

C
$$\dfrac{7\pi}{6}$$

D
$$1$$

Solution

$$2\sec^{-1} 2 +\sin^{-1} \dfrac{1}{2}=\dfrac{2\pi}{3}+\dfrac{\pi}{6}=\dfrac{5\pi}{6}$$
Question 7
1 / -0

Number of solutions of the equation $$3{\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \pi $$ is

A
Zero

B
$$2$$

C
$$3$$

D
$$1$$

Solution
Question 8
1 / -0

If $$x,y,z \in [-1,1]$$ such that $$\cos^{-1}x +\cos^{-1}y +\cos^{-1}z=0$$, find $$x+y+z$$.

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

We have,
$$x,y,z \in [-1,1]$$
$$-1 \leq x \leq 1, -1\leq y \leq 1, -1\leq z \leq 1$$
$$0 \leq \cos ^{-1}x \leq \pi, 0 \leq \cos^{-1} y \leq \pi, 0 \leq \cos^{-1} z \leq \pi$$
Therefore,
$$\cos^{-1}x +\cos^{-1} y +\cos^{-1} z =0$$
$$\implies \cos^{-1} x=0, \cos^{-1} y=0, \cos^{-1} z=0$$
$$x=y=z=1$$
Hence, $$x+y+z=3$$
Question 9
1 / -0

Find the value of $$\sin^{-1}(2\cos^{2}x-1)+\cos^{-1}(1-2\sin^{2}x)$$.

A
$$\dfrac {\pi}{2}$$

B
$$\dfrac {\pi}{3}$$

C
$$\dfrac {\pi}{4}$$

D
$$\dfrac {\pi}{6}$$

Solution

As we know that
$$\cos 2 A=2\cos^2 A-1=1-2\sin^2 A$$
And also $$\text{sin}^{-1} y+\text{cos}^{-1} y=\dfrac{\pi}{2}$$
So $$\text{sin}^{-1}(2\cos^2 x-1)+\text{cos}^{-1}(1-2\sin^2 x)=\text{sin}^{-1} (\cos 2 x)+\text{cos}^{-1} (\cos 2 x)=\dfrac{\pi}{2}$$
Question 10
1 / -0

Find the value of $$\cot (\tan^{-1} a +\cot^{-1} a)$$.

A
$$0$$

B
$$-1$$

C
$$2$$

D
$$1$$

Solution

We know,
$$\tan^{-1} a +\cot^{-1} a=\dfrac{\pi}{2}$$

Therefore,
$$\cot (\tan^{-1} a +\cot^{-1} a )=\cot \dfrac{\pi}{2}=0$$

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Re-Attempt Weekly Quiz Competition

Inverse Trigonometric Functions Test - 22

Result

Inverse Trigonometric Functions Test - 22

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SHARING IS CARING

Question 1

1

sin x

0

None of these

Solution

Question 2

$$\dfrac {\pi}{3}$$

$$\dfrac {\pi}{4}$$

$$\dfrac {\pi}{2 }$$

$$\pi$$

Solution

Question 3

$$a - b = 3$$

$$a + b = 11$$

$$a + b = 10$$

$$a - b = 4$$

Question 4

$$-1$$

$$\dfrac { 1 }{ 3 } $$

$$\dfrac { 1 }{ 5 } $$

$$\dfrac { 1 }{ 2 } $$

Solution

Question 5

$$11$$

$$15$$

$$17$$

$$21$$

Solution

Question 6

$$\dfrac{\pi}{6}$$

$$\dfrac{5\pi}{6}$$

$$\dfrac{7\pi}{6}$$

$$1$$

Solution

Question 7

Zero

$$2$$

$$3$$

$$1$$

Solution

Question 8

$$0$$

$$1$$

$$2$$

$$3$$

Solution

Question 9

$$\dfrac {\pi}{2}$$

$$\dfrac {\pi}{3}$$

$$\dfrac {\pi}{4}$$

$$\dfrac {\pi}{6}$$

Solution

Question 10

$$0$$

$$-1$$

$$2$$

$$1$$

Solution

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