In the interval $( 0,2 \pi )$, the function $\sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(a) never changes sign
(b) changes sign only once
(c) changes sign more than once, but finitely many times
(d) changes sign infinitely many times.

In the interval $( 0,2 \pi )$, the function $\sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(a) never changes sign
(b) changes sign only once
(c) changes sign more than once, but finitely many times
(d) changes sign infinitely many times.

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

Find all pairs $( x , y )$ with $x , y$ real, satisfying the equations: $$\sin \left( \frac { x + y } { 2 } \right) = 0 , \quad | x | + | y | = 1$$

For a real number $\alpha$, let $S _ { \alpha }$ denote the set of those real numbers $\beta$ that satisfy $\alpha \sin ( \beta ) = \beta \sin ( \alpha )$. Then which of the following statements is true?
(A) For any $\alpha , S _ { \alpha }$ is an infinite set.
(B) $S _ { \alpha }$ is a finite set if and only if $\alpha$ is not an integer multiple of $\pi$.
(C) There are infinitely many numbers $\alpha$ for which $S _ { \alpha }$ is the set of all real numbers.
(D) $S _ { \alpha }$ is always finite.

The number of solutions of the equation $\sin ( 7 x ) + \sin ( 3 x ) = 0$ with $0 \leq x \leq 2 \pi$ is
(A) 9
(B) 12
(C) 15
(D) 18.

In the range $0 \leq x \leq 2 \pi$, the equation $\cos ( \sin ( x ) ) = \frac { 1 } { 2 }$ has
(A) 0 solutions.
(B) 2 solutions.
(C) 4 solutions.
(D) infinitely many solutions.

The locus of points ( $x , y$ ) in the plane satisfying $\sin ^ { 2 } ( x ) + \sin ^ { 2 } ( y ) = 1$ consists of
(A) A circle that is centered at the origin.
(B) infinitely many circles that are all centered at the origin.
(C) infinitely many lines with slope $\pm 1$.
(D) finitely many lines with slope $\pm 1$.

Let $S$ be the set of those real numbers $x$ for which the identity $$\sum _ { n = 2 } ^ { \infty } \cos ^ { n } x = ( 1 + \cos x ) \cot ^ { 2 } x$$ is valid, and the quantities on both sides are finite. Then
(A) $S$ is the empty set.
(B) $S = \{ x \in \mathbb { R } : x \neq n \pi$ for all $n \in \mathbb { Z } \}$.
(C) $S = \{ x \in \mathbb { R } : x \neq 2 n \pi$ for all $n \in \mathbb { Z } \}$.
(D) $S = \{ x \in \mathbb { R } : x \neq ( 2 n + 1 ) \pi$ for all $n \in \mathbb { Z } \}$.

For $x \in (0, \pi)$, the equation $\sin x + 2\sin 2x - \sin 3x = 3$ has
(A) infinitely many solutions
(B) three solutions
(C) one solution
(D) no solution

The number of distinct solutions of the equation $$\frac { 5 } { 4 } \cos ^ { 2 } 2 x + \cos ^ { 4 } x + \sin ^ { 4 } x + \cos ^ { 6 } x + \sin ^ { 6 } x = 2$$ in the interval $[ 0,2 \pi ]$ is

Let $-\frac{\pi}{6} < \theta < -\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2 - 2x\sec\theta + 1 = 0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2 + 2x\tan\theta - 1 = 0$. If $\alpha_1 > \beta_1$ and $\alpha_2 > \beta_2$, then $\alpha_1 + \beta_2$ equals
(A) $2(\sec\theta - \tan\theta)$
(B) $2\sec\theta$
(C) $-2\tan\theta$
(D) $0$

Consider the following lists:
List-I (I) $\left\{ x \in \left[ - \frac { 2 \pi } { 3 } , \frac { 2 \pi } { 3 } \right] : \cos x + \sin x = 1 \right\}$ (II) $\left\{ x \in \left[ - \frac { 5 \pi } { 18 } , \frac { 5 \pi } { 18 } \right] : \sqrt { 3 } \tan 3 x = 1 \right\}$ (III) $\left\{ x \in \left[ - \frac { 6 \pi } { 5 } , \frac { 6 \pi } { 5 } \right] : 2 \cos ( 2 x ) = \sqrt { 3 } \right\}$ (IV) $\left\{ x \in \left[ - \frac { 7 \pi } { 4 } , \frac { 7 \pi } { 4 } \right] : \sin x - \cos x = 1 \right\}$
List-II (P) has two elements (Q) has three elements (R) has four elements (S) has five elements (T) has six elements
The correct option is:
(A) $( \mathrm { I } ) \rightarrow ( \mathrm { P } ) ; ( \mathrm { II } ) \rightarrow ( \mathrm { S } ) ; ( \mathrm { III } ) \rightarrow ( \mathrm { P } ) ; ( \mathrm { IV } ) \rightarrow ( \mathrm { S } )$
(B) (I) → (P); (II) → (P); (III) → (T); (IV) → (R)
(C) (I) → (Q); (II) → (P); (III) → (T); (IV) → (S)
(D) (I) → (Q); (II) → (S); (III) → (P); (IV) → (R)

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) $6 - 4 \sqrt { 3 }$

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

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \frac { \sin x } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) } + \frac { 2 } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) }$$
Then the number of solutions of $f ( x ) = 0$ in $\mathbb { R }$ is $\_\_\_\_$ .

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

$$\lim_{x\rightarrow 2}\left(\frac{\sqrt{1-\cos\{2(x-2)\}}}{x-2}\right)$$
(1) equals $\sqrt{2}$
(2) equals $-\sqrt{2}$
(3) equals $\frac{1}{\sqrt{2}}$
(4) does not exist

Let $\mathrm { A } = \{ \theta : \sin ( \theta ) = \tan ( \theta ) \}$ and $\mathrm { B } = \{ \theta : \cos ( \theta ) = 1 \}$ be two sets. Then:
(1) $\mathrm { A } = \mathrm { B }$
(2) $A \not\subset B$
(3) $B \not\subset A$
(4) $A \subset B$ and $B - A \neq \phi$

The number of values of $\alpha$ in $[ 0,2 \pi ]$ for which $2 \sin ^ { 3 } \alpha - 7 \sin ^ { 2 } \alpha + 7 \sin \alpha = 2$, is:
(1) 3
(2) 1
(3) 6
(4) 4

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) $- \frac { 3 \pi } { 4 }$

Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt { 2 } \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt { 2 } \sin \theta \}$, be two sets. Then
(1) $P \subset Q$ and $Q - P \neq \phi$
(2) $Q \not \subset P$
(3) $P = Q$
(4) $P \not \subset Q$

If $0 \leq x < 2\pi$, then the number of real values of $x$, which satisfy the equation $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is: (1) 3 (2) 5 (3) 7 (4) 9

If $5\tan^2 x - \cos^2 x = 2\cos 2x + 9$, then the value of $\cos 4x$ is
(1) $-\dfrac{3}{5}$
(2) $\dfrac{1}{3}$
(3) $\dfrac{2}{9}$
(4) $-\dfrac{7}{9}$