Let $f : [ 0,2 ] \rightarrow \mathbb { R }$ be the function defined by
$$f ( x ) = ( 3 - \sin ( 2 \pi x ) ) \sin \left( \pi x - \frac { \pi } { 4 } \right) - \sin \left( 3 \pi x + \frac { \pi } { 4 } \right)$$
If $\alpha , \beta \in [ 0,2 ]$ are such that $\{ x \in [ 0,2 ] : f ( x ) \geq 0 \} = [ \alpha , \beta ]$, then the value of $\beta - \alpha$ is $\_\_\_\_$

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 $\_\_\_\_$.

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

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 $x, y, z$ are in AP and $\tan^{-1}x$, $\tan^{-1}y$ and $\tan^{-1}z$ are also in AP, then
(1) $x = y = z$
(2) $2x = 3y = 6z$
(3) $6x = 3y = 2z$
(4) $6x = 4y = 3z$

The number of solutions of the equation $\sin 2 x - 2 \cos x + 4 \sin x = 4$ in the interval $[ 0,5 \pi ]$ is :
(1) 3
(2) 5
(3) 4
(4) 6

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

If $12 \cot^2\theta - 31 \csc\theta + 32 = 0$, then the value of $\sin\theta$ is:
(1) $\frac{3}{5}$ or $1$
(2) $\frac{2}{3}$ or $-\frac{2}{3}$
(3) $\frac{4}{5}$ or $\frac{3}{4}$
(4) $\pm\frac{1}{2}$

The number of distinct real roots of the equation $\tan^{2}x - \sec^{10}x + 1 = 0$ in the interval $\left(0, \frac{\pi}{3}\right)$ is: (1) 0 (2) 1 (3) 2 (4) 3

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) $-\dfrac{7}{9}$

If an angle $A$ of a $\triangle A B C$ satisfies $5 \cos A + 3 = 0$, then the roots of the quadratic equation $9 x ^ { 2 } + 27 x + 20 = 0$ are
(1) $\sec A , \cot A$
(2) $\sec A , \tan A$
(3) $\tan A , \cos A$
(4) $\sin A , \sec A$

If sum of all the solutions of the equation $8 \cos x \cdot \left( \cos \left( \frac { \pi } { 6 } + x \right) \cdot \cos \left( \frac { \pi } { 6 } - x \right) - \frac { 1 } { 2 } \right) = 1$ in $[ 0 , \pi ]$ is $k \pi$, then $k$ is equal to:
(1) $\frac { 20 } { 9 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 13 } { 9 }$
(4) $\frac { 8 } { 9 }$

Let $S = \left\{ \theta \in [ - 2 \pi , 2 \pi ] : 2 \cos ^ { 2 } \theta + 3 \sin \theta = 0 \right\}$. Then the sum of the elements of $S$ is:
(1) $\pi$
(2) $\frac { 13 \pi } { 6 }$
(3) $\frac { 5 \pi } { 3 }$
(4) $2 \pi$

If $0 \leq x < \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$, is:
(1) 4
(2) 3
(3) 2
(4) 1

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) $7\pi$

Let $\alpha$ and $\beta$ be two real roots of the equation $(k + 1) \tan ^ { 2 } x - \sqrt { 2 } \cdot \lambda \tan x = (1 - k)$, where $k (\neq -1)$ and $\lambda$ are real numbers. If $\tan ^ { 2 } (\alpha + \beta) = 50$, then a value of $\lambda$ is
(1) $10 \sqrt { 2 }$
(2) 10
(3) 5
(4) $5 \sqrt { 2 }$

The number of solutions of the equation $32 ^ { \tan ^ { 2 } x } + 32 ^ { \sec ^ { 2 } x } = 81 , \quad 0 \leq x \leq \frac { \pi } { 4 }$ is :
(1) 0
(2) 2
(3) 1
(4) 3

The number of solutions of $\sin ^ { 7 } x + \cos ^ { 7 } x = 1 , x \in [ 0,4 \pi ]$ is equal to
(1) 11
(2) 7
(3) 5
(4) 9

The sum of solutions of the equation $\frac { \cos x } { 1 + \sin x } = | \tan 2 x | , x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) - \left\{ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right\}$

All possible values of $\theta \in [ 0,2 \pi ]$ for which $\sin 2 \theta + \tan 2 \theta > 0$ lie in :
(1) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \pi , \frac { 3 \pi } { 2 } \right)$
(2) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 7 \pi } { 6 } \right)$
(3) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 5 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 } \right)$
(4) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 } \right)$