17. If the solution sets of the trigonometric equations $\sin x = 0$ and $\sin 2x = 0$ are $E$ and $F$ respectively, then ( )
(A) $E \subset F$
(B) $E \supset F$
(C) $E = F$
(D) $E \cap F = \varnothing$

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 $B - A \neq \phi$

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 elements in the set $S = \left\{x \in \mathbb{R} : 2\cos\left(\frac{x^2 + x}{6}\right) = 4^x + 4^{-x}\right\}$ is
(1) 1
(2) 3
(3) 0
(4) infinite

If the solution of the equation $\log_{\cos x} \cot x + 4\log_{\sin x} \tan x = 1, \quad x \in \left(0, \frac{\pi}{2}\right)$ is $\sin^{-1}\frac{\alpha + \sqrt{\beta}}{2}$, where $\alpha, \beta$ are integers, then $\alpha + \beta$ is equal to:
(1) 3
(2) 5
(3) 6
(4) 4

The sum of the solutions $x \in R$ of the equation $\frac { 3 \cos 2 x + \cos ^ { 3 } 2 x } { \cos ^ { 6 } x - \sin ^ { 6 } x } = x ^ { 3 } - x ^ { 2 } + 6$ is
(1) 0
(2) 1
(3) - 1
(4) 3

Let $\mathrm{A} = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2 \right\}$ and $\mathrm{B} = \{ x \geqslant 0 : \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0 \}$. Then $\mathrm{n}(\mathrm{A} \cup \mathrm{B})$ is equal to:
(1) 4
(2) 8
(3) 6
(4) 2

Q83. The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$