If
$$\frac { \sin ^ { 4 } x } { 2 } + \frac { \cos ^ { 4 } x } { 3 } = \frac { 1 } { 5 } ,$$
then
(A) $\quad \tan ^ { 2 } x = \frac { 2 } { 3 }$
(B) $\quad \frac { \sin ^ { 8 } x } { 8 } + \frac { \cos ^ { 8 } x } { 27 } = \frac { 1 } { 125 }$
(C) $\quad \tan ^ { 2 } x = \frac { 1 } { 3 }$
(D) $\frac { \sin ^ { 8 } x } { 8 } + \frac { \cos ^ { 8 } x } { 27 } = \frac { 2 } { 125 }$

If $A=\sin^{2}x+\cos^{4}x$, then for all real $x$
(1) $\frac{13}{16}\leq\mathrm{A}\leq 1$
(2) $1\leq A\leq 2$
(3) $\frac{3}{4}\leq\mathrm{A}\leq\frac{13}{16}$
(4) $\frac{3}{4}\leq\mathrm{A}\leq 1$

The expression $\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}$ can be written as:
(1) $\tan A + \cot A$
(2) $\sec A + \operatorname{cosec} A$
(3) $\sin A \cos A + 1$
(4) $\sec A \operatorname{cosec} A + 1$

If $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3 x ^ { 2 } - 10 x - 25 = 0$ then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cdot \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is
(1) 25
(2) - 25
(3) - 10
(4) 10

If $\sin ^ { 4 } \alpha + 4 \cos ^ { 4 } \beta + 2 = 4 \sqrt { 2 } \sin \alpha \cos \beta , \alpha , \beta \in [ 0 , \pi ]$, then $\cos ( \alpha + \beta ) - \cos ( \alpha - \beta )$ is equal to
(1) - 1
(2) $- \sqrt { 2 }$
(3) $\sqrt { 2 }$
(4) 0

The value of $\cos ^ { 2 } 10 ^ { \circ } - \cos 10 ^ { \circ } \cos 50 ^ { \circ } + \cos ^ { 2 } 50 ^ { \circ }$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 4 } + \cos 20 ^ { \circ }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 3 } { 2 } \left( 1 + \cos 20 ^ { \circ } \right)$

The value of $\sin 10 ^ { \circ } \sin 30 ^ { \circ } \sin 50 ^ { \circ } \sin 70 ^ { \circ }$ is:
(1) $\frac { 1 } { 36 }$
(2) $\frac { 1 } { 16 }$
(3) $\frac { 1 } { 18 }$
(4) $\frac { 1 } { 32 }$

If $15 \sin ^ { 4 } \alpha + 10 \cos ^ { 4 } \alpha = 6$, for some $\alpha \in R$, then the value of $27 \sec ^ { 6 } \alpha + 8 \operatorname { cosec } ^ { 6 } \alpha$ is equal to:
(1) 350
(2) 500
(3) 400
(4) 250

If $\sin ^ { 2 } \left( 10 ^ { \circ } \right) \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 50 ^ { \circ } \right) \sin \left( 70 ^ { \circ } \right) = \alpha - \frac { 1 } { 16 } \sin \left( 10 ^ { \circ } \right)$, then $16 + \alpha ^ { - 1 }$ is equal to $\_\_\_\_$.

$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$ is equal to
(1) 3
(2) 1
(3) 4
(4) 2

Let $f ( \theta ) = 3 \left( \sin ^ { 4 } \left( \frac { 3 \pi } { 2 } - \theta \right) + \sin ^ { 4 } ( 3 \pi + \theta ) \right) - 2 \left( 1 - \sin ^ { 2 } 2 \theta \right)$ and $S = \left\{ \theta \in [ 0 , \pi ] : f ^ { \prime } ( \theta ) = - \frac { \sqrt { 3 } } { 2 } \right\}$. If $4 \beta = \sum _ { \theta \in S } \theta$ then $f ( \beta )$ is equal to
(1) $\frac { 11 } { 8 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 9 } { 8 }$
(4) $\frac { 3 } { 2 }$

If $\sin x = - \frac { 3 } { 5 }$, where $\pi < x < \frac { 3 \pi } { 2 }$, then $80 \left( \tan ^ { 2 } x - \cos x \right)$ is equal to
(1) 108
(2) 109
(3) 18
(4) 19

Between the following two statements: Statement I : Let $\vec { a } = \hat { i } + 2 \hat { j } - 3 \hat { k }$ and $\vec { b } = 2 \hat { i } + \hat { j } - \hat { k }$. Then the vector $\vec { r }$ satisfying $\vec { a } \times \vec { r } = \vec { a } \times \vec { b }$ and $\vec { a } \cdot \vec { r } = 0$ is of magnitude $\sqrt { 10 }$. Statement II : In a triangle $A B C , \cos 2 A + \cos 2 B + \cos 2 C \geq - \frac { 3 } { 2 }$.
(1) Statement I is incorrect but Statement II is correct.
(2) Both Statement I and Statement II are correct.
(3) Statement I is correct but Statement II is incorrect.
(4) Both Statement I and Statement II are incorrect.

The value of $\left( \sin 70 ^ { \circ } \right) \left( \cot 10 ^ { \circ } \cot 70 ^ { \circ } - 1 \right)$ is
(1) $2/3$
(2) 1
(3) 0
(4) $3/2$

If $\sin x + \sin ^ { 2 } x = 1 , x \in \left( 0 , \frac { \pi } { 2 } \right)$, then $\left( \cos ^ { 12 } x + \tan ^ { 12 } x \right) + 3 \left( \cos ^ { 10 } x + \tan ^ { 10 } x + \cos ^ { 8 } x + \tan ^ { 8 } x \right) + \left( \cos ^ { 6 } x + \tan ^ { 6 } x \right)$ is equal to :
(1) 4
(2) 1
(3) 3
(4) 2

Given $45^{\circ} < \theta < 50^{\circ}$, and let $a = 1 - \cos^{2}\theta$, $b = \frac{1}{\cos\theta} - \cos\theta$, $c = \frac{\tan\theta}{\tan^{2}\theta + 1}$. Regarding the relative sizes of the three values $a$, $b$, $c$, select the correct option.
(1) $a < b < c$
(2) $a < c < b$
(3) $b < a < c$
(4) $b < c < a$
(5) $c < a < b$

$$\frac{(\sin x - \cos x)^{2}}{\cos x} + 2\sin x$$
Which of the following is this expression equal to?
A) $\frac{1}{\cos x}$
B) $\frac{1}{\sin x}$
C) $1$
D) $\arcsin x$
E) $\arccos x$

In the regular pentagon ABCDE shown, K and L are the midpoints of line segments AB and DA, respectively.
Given this, what is the measure of angle LKB in degrees?
A) 105
B) 108
C) $\mathbf { 1 2 0 }$
D) 126
E) 135

ABC is an isosceles triangle $\mathrm { D } \in [ \mathrm { BC } ] , \mathrm { B } \in [ \mathrm { AE } ]$ $| \mathrm { AB } | = | \mathrm { BC } |$ $| \mathrm { AC } | = | \mathrm { AD } | = | \mathrm { DE } |$ $\mathrm { m } ( \widehat { \mathrm { ADB } } ) = 111 ^ { \circ }$ $m ( \widehat { B D E } ) = x$
Accordingly, how many degrees is $x$?
A) 15 B) 18 C) 21 D) 24 E) 27

$$\frac { \cot \left( 34 ^ { \circ } \right) \cdot \sin \left( 44 ^ { \circ } \right) } { \sin \left( 22 ^ { \circ } \right) \cdot \sin \left( 56 ^ { \circ } \right) }$$
What is the equivalent of this expression?
A) $2 \cot \left( 22 ^ { \circ } \right)$ B) $2 \cos \left( 56 ^ { \circ } \right)$ C) $4 \sin \left( 44 ^ { \circ } \right)$ D) $4 \cos \left( 34 ^ { \circ } \right)$ E) $4 \tan \left( 56 ^ { \circ } \right)$

Below are shown a semicircle with center O and radius 1 unit, and right triangles OAB and ODC. Points A and C lie on both the triangle OAB and the semicircle.
Accordingly, $$\frac { | \mathrm { AB } | + | \mathrm { BC } | } { | \mathrm { CD } | + | \mathrm { DA } | }$$
What is the equivalent of this ratio in terms of x?
A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\csc x$ E) $\sec x$

Two right triangles $A B C$ and $B C D$ with one side coinciding are drawn as shown in the figure, and the resulting two regions are painted yellow and blue.
$$\mathrm { m } ( \widehat { \mathrm { DCA } } ) = \mathrm { m } ( \widehat { \mathrm { BAC } } ) = \mathrm { x }$$
Accordingly, what is the expression in terms of x for the ratio of the area of the yellow region to the area of the blue region?
A) $\sin 2 x$
B) $\cos 2 x$
C) $\sin ^ { 2 } x$
D) $\cot ^ { 2 } x$
E) $\csc ^ { 2 } x$

$$\frac{2\tan x - \sin(2x)}{\sin^2 x}$$
What is the simplified form of this expression?
A) $2\tan x$
B) $\tan(2x)$
C) $2\cos x$
D) $\cos(2x)$
E) 1

In the figure, the line segments $[OA]$ and $[OD]$ intersect perpendicularly.
Accordingly, the ratio of the area of triangle OAB to the area of triangle OCD in terms of $\alpha$ is which of the following?
A) $\tan \alpha$
B) $\cot \alpha$
C) $\tan^2 \alpha$
D) $\cot^2 \alpha$
E) $\sec^2 \alpha$

Let $a \in \left(\frac{3\pi}{4}, \pi\right)$,
$$\begin{aligned} & x = \sin(2a) \cdot \tan(a) \\ & y = \cos(2a) \cdot \cot(2a) \\ & z = \sin(a) \cdot \cot(2a) \end{aligned}$$
Given these equalities.
Accordingly, what are the signs of $\mathbf{x}$, $y$ and $\mathbf{z}$ respectively?
A) $+, +, -$ B) $+, -, -$ C) $-, -, -$ D) $-, +, +$ E) $-, -, +$