Recall that $\sin^{-1}$ is the inverse function of $\sin$, as defined in the standard fashion. (Sometimes $\sin^{-1}$ is called $\arcsin$.) Let $f(x) = \sin^{-1}(\sin(\pi x))$. Write the values of the following. (Some answers may involve the irrational number $\pi$. Write such answers in terms of $\pi$.)
(i) $f(2.7)$
(ii) $f'(2.7)$
(iii) $\int_0^{2.5} f(x)\, dx$
(iv) the smallest positive $x$ at which $f'(x)$ does not exist.

The number of solutions of the equation $\sin^{-1} x = 2 \tan^{-1} x$ is
(A) 1
(B) 2
(C) 3
(D) 5

The number of solutions of the equation $\sin^{-1} x = 2 \tan^{-1} x$ is
(A) 1
(B) 2
(C) 3
(D) 5

The value of $$\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$$ is
(A) 0
(B) $\pi / 6$
(C) $\pi / 4$
(D) $\pi / 2$

The number of solutions of the equation $\sin ^ { - 1 } x = 2 \tan ^ { - 1 } x$ is
(A) 1
(B) 2
(C) 3
(D) 5

The number of solutions of the equation $\sin ^ { - 1 } x = 2 \tan ^ { - 1 } x$ is
(a) 1 .
(B) 2 .
(C) 3 .
(D) 5 .

11. The number of real solutions of $\tan - 1 \sqrt { } ( x ( x + 1 ) ) + \sin - 1 \sqrt { } ( x 2 + x + 1 ) = \pi / 2$ is:
(A) zero
(B) one
(C) two
(D) infinite

34. $\sin - 1 ( x - x 2 / 2 + x 3 / 4 - \ldots ) + \cos - 1 ( x 2 - x 4 / 2 + x 6 / 4 - \ldots ) = n / 2$ for $0 < | x | < \sqrt { } ( 2$, ) then $x$ equals :
(A) $\frac { 1 } { 2 }$
(B) 1
(C) $- 1 / 2$
(D) - 1

Considering only the principal values of the inverse trigonometric functions, the value of
$$\frac { 3 } { 2 } \cos ^ { - 1 } \sqrt { \frac { 2 } { 2 + \pi ^ { 2 } } } + \frac { 1 } { 4 } \sin ^ { - 1 } \frac { 2 \sqrt { 2 } \pi } { 2 + \pi ^ { 2 } } + \tan ^ { - 1 } \frac { \sqrt { 2 } } { \pi }$$
is $\_\_\_\_$.

Let $\tan ^ { - 1 } ( x ) \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$, for $x \in \mathbb { R }$. Then the number of real solutions of the equation $\sqrt { 1 + \cos ( 2 x ) } = \sqrt { 2 } \tan ^ { - 1 } ( \tan x )$ in the set $\left( - \frac { 3 \pi } { 2 } , - \frac { \pi } { 2 } \right) \cup \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right)$ is equal to

For any $y \in \mathbb { R }$, let $\cot ^ { - 1 } ( y ) \in ( 0 , \pi )$ and $\tan ^ { - 1 } ( y ) \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the sum of all the solutions of the equation $\tan ^ { - 1 } \left( \frac { 6 y } { 9 - y ^ { 2 } } \right) + \cot ^ { - 1 } \left( \frac { 9 - y ^ { 2 } } { 6 y } \right) = \frac { 2 \pi } { 3 }$ for $0 < | y | < 3$, is equal to
(A) $2 \sqrt { 3 } - 3$
(B) $3 - 2 \sqrt { 3 }$
(C) $4 \sqrt { 3 } - 6$
(D) $6 - 4 \sqrt { 3 }$

The total number of real solutions of the equation
$$\theta = \tan ^ { - 1 } ( 2 \tan \theta ) - \frac { 1 } { 2 } \sin ^ { - 1 } \left( \frac { 6 \tan \theta } { 9 + \tan ^ { 2 } \theta } \right)$$
is (Here, the inverse trigonometric functions $\sin ^ { - 1 } x$ and $\tan ^ { - 1 } x$ assume values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ and ( $- \frac { \pi } { 2 } , \frac { \pi } { 2 }$ ), respectively.)

(A)

1

(B)

2

(C)

3

(D)

5

If $\sin ^ { - 1 } \left( \frac { x } { 5 } \right) + \operatorname { cosec } ^ { - 1 } \left( \frac { 5 } { 4 } \right) = \frac { \pi } { 2 }$ then a value of $x$ is
(1) 1
(2) 3
(3) 4
(4) 5

The principal value of $\tan ^ { - 1 } \left( \cot \frac { 43 \pi } { 4 } \right)$ is
(1) $\frac { \pi } { 4 }$
(2) $- \frac { \pi } { 4 }$
(3) $\frac { 3 \pi } { 4 }$
(4) $- \frac { 3 \pi } { 4 }$

The value of $\tan ^ { - 1 } \left[ \frac { \sqrt { 1 + x ^ { 2 } } + \sqrt { 1 - x ^ { 2 } } } { \sqrt { 1 + x ^ { 2 } } - \sqrt { 1 - x ^ { 2 } } } \right] , | x | < \frac { 1 } { 2 } , x \neq 0$, is equal to:
(1) $\frac { \pi } { 4 } + \frac { 1 } { 2 } \cos ^ { - 1 } x ^ { 2 }$
(2) $\frac { \pi } { 4 } - \cos ^ { - 1 } x ^ { 2 }$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 2 } \cos ^ { - 1 } x ^ { 2 }$
(4) $\frac { \pi } { 4 } + \cos ^ { - 1 } x ^ { 2 }$

If $x = \sin^{-1}(\sin 10)$ and $y = \cos^{-1}(\cos 10)$, then $y - x$ is equal to:
(1) 10
(2) $\pi$
(3) 0
(4) $7\pi$

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy $\sin ^ { - 1 } \left( \frac { 3x } { 5 } \right) + \sin ^ { - 1 } \left( \frac { 4x } { 5 } \right) = \sin ^ { - 1 } x$ is equal to:
(1) 2
(2) 1
(3) 3
(4) 0

The sum of possible values of $x$ for $\tan ^ { - 1 } ( x + 1 ) + \cot ^ { - 1 } \left( \frac { 1 } { x - 1 } \right) = \tan ^ { - 1 } \left( \frac { 8 } { 31 } \right)$ is:
(1) $- \frac { 32 } { 4 }$
(2) $- \frac { 31 } { 4 }$
(3) $- \frac { 30 } { 4 }$
(4) $- \frac { 33 } { 4 }$

$\operatorname { cosec } \left[ 2 \cot ^ { - 1 } ( 5 ) + \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right]$ is equal to:
(1) $\frac { 65 } { 56 }$
(2) $\frac { 75 } { 56 }$
(3) $\frac { 65 } { 33 }$
(4) $\frac { 56 } { 33 }$

Let $S _ { k } = \sum _ { r = 1 } ^ { k } \tan ^ { - 1 } \left( \frac { 6 ^ { r } } { 2 ^ { 2 r + 1 } + 3 ^ { 2 r + 1 } } \right)$, then $\lim _ { k \rightarrow \infty } S _ { k }$ is equal to :
(1) $\tan ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
(2) $\frac { \pi } { 2 }$
(3) $\cot ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
(4) $\tan ^ { - 1 } ( 3 )$

The number of solutions of the equation $\sin ^ { - 1 } \left[ x ^ { 2 } + \frac { 1 } { 3 } \right] + \cos ^ { - 1 } \left[ x ^ { 2 } - \frac { 2 } { 3 } \right] = x ^ { 2 }$ for $x \in [ - 1,1 ]$, and $[ x ]$ denotes the greatest integer less than or equal to $x$, is:
(1) 2
(2) 0
(3) 4
(4) Infinite

Let $x \times y = x ^ { 2 } + y ^ { 3 }$ and $( x \times 1 ) \times 1 = x \times ( 1 \times 1 )$. Then a value of $2 \sin ^ { - 1 } \left( \frac { x ^ { 4 } + x ^ { 2 } - 2 } { x ^ { 4 } + x ^ { 2 } + 2 } \right)$ is
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 6 }$
(4) $\pi$

The value of $\tan^{-1}\left(\frac{\cos\frac{15\pi}{4} - 1}{\sin\frac{\pi}{4}}\right)$ is equal to
(1) $-\frac{\pi}{4}$
(2) $-\frac{\pi}{8}$
(3) $-\frac{5\pi}{12}$
(4) $-\frac{4\pi}{9}$

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}x - 2\sin^{-1}x = \cos^{-1}(2x)$ is equal to
(1) 0
(2) 1
(3) $\frac{1}{2}$
(4) $-\frac{1}{2}$

$\lim _ { x \rightarrow \frac { 1 } { \sqrt { 2 } } } \frac { \sin \left( \cos ^ { - 1 } x \right) - x } { 1 - \tan \left( \cos ^ { - 1 } x \right) }$ is equal to
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { - 1 } { \sqrt { 2 } }$
(3) $\sqrt { 2 }$
(4) $- 1$