(Calculus) For $0 \leqq x < 2\pi$, the sum of all distinct values of $x$ satisfying the equation $\sin 2x = 2\cos x - 2\cos^2 x$ is? [3 points]
(1) $\pi$
(2) $\frac{5}{4}\pi$
(3) $\frac{3}{2}\pi$
(4) $\frac{7}{4}\pi$
(5) $2\pi$

When $0 \leq x < 2 \pi$, what is the sum of all solutions to the equation $$\cos ^ { 2 } x = \sin ^ { 2 } x - \sin x$$ ? [3 points]
(1) $2 \pi$
(2) $\frac { 5 } { 2 } \pi$
(3) $3 \pi$
(4) $\frac { 7 } { 2 } \pi$
(5) $4 \pi$

When $0 \leq \theta < 2 \pi$, the range of all values of $\theta$ such that the quadratic equation in $x$ $$6 x ^ { 2 } + ( 4 \cos \theta ) x + \sin \theta = 0$$ has no real roots is $\alpha < \theta < \beta$. What is the value of $3 \alpha + \beta$? [3 points]
(1) $\frac { 5 } { 6 } \pi$
(2) $\pi$
(3) $\frac { 7 } { 6 } \pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$

For $0 \leq x < 4 \pi$, what is the sum of all solutions to the equation $$4 \sin ^ { 2 } x - 4 \cos \left( \frac { \pi } { 2 } + x \right) - 3 = 0$$ ? [4 points]
(1) $5 \pi$
(2) $6 \pi$
(3) $7 \pi$
(4) $8 \pi$
(5) $9 \pi$

The set of all solutions of the equation $\cos 2\theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n\pi + \frac{\pi}{2}$, where $n$ is any integer
(C) $\theta = 2n\pi$ or $\theta = 2n\pi - \frac{\pi}{2}$ or $\theta = n\pi - \frac{\pi}{4}$, where $n$ is any integer
(D) $\theta = 2n\pi$ or $\theta = n\pi + \frac{\pi}{4}$, where $n$ is any integer

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer
(C) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer
(D) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer

The number of roots of the equation $x^2 + \sin^2 x = 1$ in the closed interval $\left[ 0, \frac{\pi}{2} \right]$ is
(A) 0
(B) 1
(C) 2
(D) 3

The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is
(A) 0
(B) 1
(C) 2
(D) 3

For $0 \leq x < 2 \pi$, the number of solutions of the equation $$\sin ^ { 2 } x + 2 \cos ^ { 2 } x + 3 \sin x \cos x = 0$$ is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(a) $\theta = 0$.
(b) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer.
(c) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer.
(d) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer.

The number of solutions of the pair of equations $$2\sin^2\theta - \cos 2\theta = 0$$ $$2\cos^2\theta - 3\sin\theta = 0$$ in the interval $[0, 2\pi]$ is
(A) 0
(B) 1
(C) 2
(D) 4

The number of values of $\theta$ in the interval $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ such that $\theta \neq \frac { \mathrm { n } \pi } { 5 }$ for $\mathrm { n } = 0 , \pm 1 , \pm 2$ and $\tan \theta = \cot 5 \theta$ as well as $\sin 2 \theta = \cos 4 \theta$ is

The maximum value of the expression $\frac { 1 } { \sin ^ { 2 } \theta + 3 \sin \theta \cos \theta + 5 \cos ^ { 2 } \theta }$ is

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

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

The sum of all values of $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$ satisfying $\sin ^ { 2 } 2 \theta + \cos ^ { 4 } 2 \theta = \frac { 3 } { 4 }$ is
(1) $\frac { \pi } { 2 }$
(2) $\frac { 3 \pi } { 8 }$
(3) $\frac { 5 \pi } { 4 }$
(4) $\pi$

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) $5 \sqrt { 2 }$

If the equation $\cos ^ { 4 } \theta + \sin ^ { 4 } \theta + \lambda = 0$ has real solutions for $\theta$ then $\lambda$ lies in interval
(1) $\left( - \frac { 5 } { 4 } , - 1 \right)$
(2) $\left[ - 1 , - \frac { 1 } { 2 } \right]$
(3) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 4 } \right]$
(4) $\left[ - \frac { 3 } { 2 } , - \frac { 5 } { 4 } \right]$

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

If $n$ is the number of solutions of the equation $2 \cos x \left(4 \sin \frac { \pi } { 4 } + x \sin \frac { \pi } { 4 } - x\right) - 1 = 1 , x \in [0 , \pi]$ and $S$ is the sum of all these solutions, then the ordered pair $(n , S)$ is :
(1) $\left(2 , \frac { 8 \pi } { 9 }\right)$
(2) $\left(3 , \frac { 13 \pi } { 9 }\right)$
(3) $\left(2 , \frac { 2 \pi } { 3 }\right)$
(4) $\left(3 , \frac { 5 \pi } { 3 }\right)$

The solutions of the equation $\left| \begin{array} { c c c } 1 + \sin ^ { 2 } x & \sin ^ { 2 } x & \sin ^ { 2 } x \\ \cos ^ { 2 } x & 1 + \cos ^ { 2 } x & \cos ^ { 2 } x \\ 4 \sin 2 x & 4 \sin 2 x & 1 + 4 \sin 2 x \end{array} \right| = 0 , ( 0 < x < \pi )$, are
(1) $\frac { \pi } { 12 } , \frac { \pi } { 6 }$
(2) $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$
(3) $\frac { 5 \pi } { 12 } , \frac { 7 \pi } { 12 }$
(4) $\frac { 7 \pi } { 12 } , \frac { 11 \pi } { 12 }$

The number of distinct real roots of $\left| \begin{array} { c c c } \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{array} \right| = 0$ in the interval $- \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 4 }$ is:
(1) 4
(2) 1
(3) 2
(4) 3

The number of solutions of the equation $4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$; $x \in [-2\pi, 2\pi]$ is:
(1) 1
(2) 3
(3) 2
(4) 0

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