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

The value of $\cos \frac { \pi } { 2 ^ { 2 } } \cdot \cos \frac { \pi } { 2 ^ { 3 } } \cdot \ldots \cdot \cos \frac { \pi } { 2 ^ { 10 } } \cdot \sin \frac { \pi } { 2 ^ { 10 } }$ is:
(1) $\frac { 1 } { 1024 }$
(2) $\frac { 1 } { 512 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 256 }$

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 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) $\pi$

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

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) $\frac { 7 \pi } { 12 } , \frac { 11 \pi } { 12 }$

Let $\tan \alpha , \tan \beta$ and $\tan \gamma ; \alpha , \beta , \gamma \neq \frac { ( 2 n - 1 ) \pi } { 2 } , n \in N$ be the slopes of the three line segments $O A , O B$ and $O C$, respectively, where $O$ is origin. If circumcentre of $\Delta A B C$ coincides with origin and its orthocentre lies on $y$-axis, then the value of $\left( \frac { \cos 3 \alpha + \cos 3 \beta + \cos 3 \gamma } { \cos \alpha \cdot \cos \beta \cdot \cos \gamma } \right) ^ { 2 }$ is equal to:

Let $S = \left\{ \theta \in \left( 0 , \frac { \pi } { 2 } \right) : \sum _ { m = 1 } ^ { 9 } \sec \left( \theta + ( m - 1 ) \frac { \pi } { 6 } \right) \sec \left( \theta + \frac { m \pi } { 6 } \right) = - \frac { 8 } { \sqrt { 3 } } \right\}$. Then
(1) $\mathrm { S } = \left\{ \frac { \pi } { 12 } \right\}$
(2) $S = \left\{ \frac { 2 \pi } { 3 } \right\}$
(3) $\sum _ { \theta \in S } \theta = \frac { \pi } { 2 }$
(4) $\sum _ { \theta \in S } \theta = \frac { 3 \pi } { 4 }$

Let $AB$ and $PQ$ be two vertical poles, 160 m apart from each other. Let $C$ be the middle point of $B$ and $Q$, which are feet of these two poles. Let $\frac{\pi}{8}$ and $\theta$ be the angles of elevation from $C$ to $P$ and $A$, respectively. If the height of pole $PQ$ is twice the height of pole $AB$, then $\tan^2\theta$ is equal to
(1) $\frac{3-2\sqrt{2}}{2}$
(2) $\frac{3+\sqrt{2}}{2}$
(3) $\frac{3-2\sqrt{2}}{4}$
(4) $\frac{3-\sqrt{2}}{4}$

If $0 < x < \frac { 1 } { \sqrt { 2 } }$ and $\frac { \sin ^ { - 1 } x } { \alpha } = \frac { \cos ^ { - 1 } x } { \beta }$, then a value of $\sin \frac { 2 \pi \alpha } { \alpha + \beta }$ is
(1) $4 \sqrt { 1 - x ^ { 2 } } \left( 1 - 2 x ^ { 2 } \right)$
(2) $4 x \sqrt { 1 - x ^ { 2 } } \left( 1 - 2 x ^ { 2 } \right)$
(3) $2 x \sqrt { 1 - x ^ { 2 } } \left( 1 - 4 x ^ { 2 } \right)$
(4) $4 \sqrt { 1 - x ^ { 2 } } \left( 1 - 4 x ^ { 2 } \right)$

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 $\_\_\_\_$.

The value of $36 \left( 4 \cos ^ { 2 } 9 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 27 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 81 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 243 ^ { \circ } - 1 \right)$ is
(1) 54
(2) 18
(3) 27
(4) 36

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) $\frac { 3 } { 2 }$

The value of $\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}$ is $\_\_\_\_$.

Consider the function $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathrm { R }$ defined by $f ( x ) = 4 \sqrt { 2 } x ^ { 3 } - 3 \sqrt { 2 } x - 1$. Consider the statements (I) The curve $y = f ( x )$ intersects the $x$-axis exactly at one point (II) The curve $y = f ( x )$ intersects the $x$-axis at $x = \cos \frac { \pi } { 12 }$ Then
(1) Only (II) is correct
(2) Both (I) and (II) are incorrect
(3) Only (I) is correct
(4) Both (I) and (II) are correct

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

Q63. Suppose $\theta \epsilon \left[ 0 , \frac { \pi } { 4 } \right]$ is a solution of $4 \cos \theta - 3 \sin \theta = 1$. Then $\cos \theta$ is equal to :
(1) $\frac { 4 } { ( 3 \sqrt { 6 } + 2 ) }$
(2) $\frac { 6 + \sqrt { 6 } } { ( 3 \sqrt { 6 } + 2 ) }$
(3) $\frac { 4 } { ( 3 \sqrt { 6 } - 2 ) }$
(4) $\frac { 6 - \sqrt { 6 } } { ( 3 \sqrt { 6 } - 2 ) }$

Q64. 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

The value of $\operatorname{cosec} 10^{\circ} - \sqrt{3} \sec 10^{\circ}$
(A) 4 (B) 2 (C) 1 (D) None of these

We are to find the maximum and the minimum values of the function
$$f ( x ) = 4 \sin ^ { 3 } x + 4 \cos ^ { 3 } x - 8 \sin 2 x - 7$$
where $0 \leqq x \leqq \pi$.
Set $t = \sin x + \cos x$. Since
$$\sin x + \cos x = \sqrt { \mathbf { A } } \sin \left( x + \frac { \mathbf { B } } { \mathbf { C } } \pi \right) , \quad ( \text { note: have } \mathbf { B } < \mathbf { C } )$$
the range of values which $t$ takes is $- \mathbf { D } \leqq t \leqq \sqrt { \mathbf{E} }$. Next, since
$$\sin 2 x = t ^ { 2 } - \mathbf { F }$$
and
$$4 \sin ^ { 3 } x + 4 \cos ^ { 3 } x = - \mathbf { G } t ^ { 3 } + \mathbf { H } t ,$$
we have
$$f ( x ) = - \mathbf { G } t ^ { 3 } - \mathbf { I } t ^ { 2 } + \mathbf { H } t + \mathbf { J } . \tag{1}$$
When we set the right side of (1) as $g ( t )$ and differentiate with respect to $t$, we have
$$g ^ { \prime } ( t ) = - \mathbf { K } ( \mathbf { L } t - \mathbf { M } ) \left( t + \mathbf { N } \right) .$$
Hence at $t = \frac { \mathbf { O } } { \mathbf { P } } , g ( t ) ( = f ( x ) )$ takes the maximum value $\frac { \mathbf { Q R } } { \mathbf{S} }$, and at $t = \sqrt { \mathbf { U } }$, it takes the minimum value $\mathbf { V } \sqrt { \mathbf { W } } - \mathbf { X Y }$.

2. For ALL APPLICANTS.

(i) Show, with working, that
$$x ^ { 3 } - ( 1 + \cos \theta + \sin \theta ) x ^ { 2 } + ( \cos \theta \sin \theta + \cos \theta + \sin \theta ) x - \sin \theta \cos \theta ,$$
equals
$$( x - 1 ) \left( x ^ { 2 } - ( \cos \theta + \sin \theta ) x + \cos \theta \sin \theta \right)$$
Deduce that the cubic in (1) has roots
$$1 , \quad \cos \theta , \quad \sin \theta$$
(ii) Give the roots when $\theta = \frac { \pi } { 3 }$.
(iii) Find all values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ such that two of the three roots are equal.
(iv)What is the greatest possible difference between two of the roots, and for what values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ does this greatest difference occur?
Show that for each such $\theta$ the cubic (1) is the same.

As shown in the figure, $\triangle A B C$ is an acute triangle, $P$ is a point outside the circumcircle $\Gamma$ of $\triangle A B C$, and both $\overline { P B }$ and $\overline { P C }$ are tangent to circle $\Gamma$. Let $\angle B P C = \theta$. What is the value of $\cos A$?
(1) $\sin 2 \theta$
(2) $\frac { \sin \theta } { 2 }$
(3) $\sin \frac { \theta } { 2 }$
(4) $\frac { \cos \theta } { 2 }$
(5) $\cos \frac { \theta } { 2 }$

Let $f(x) = \sin x + \sqrt{3} \cos x$. Select the correct options.
(1) The vertical line $x = \frac{\pi}{6}$ is an axis of symmetry of the graph of $y = f(x)$
(2) If the vertical lines $x = a$ and $x = b$ are both axes of symmetry of the graph of $y = f(x)$, then $f(a) = f(b)$
(3) In the interval $[0, 2\pi)$, there is only one real number $x$ satisfying $f(x) = \sqrt{3}$
(4) In the interval $[0, 2\pi)$, the sum of all real numbers $x$ satisfying $f(x) = \frac{1}{2}$ does not exceed $2\pi$
(5) The graph of $y = f(x)$ can be obtained from the graph of $y = 4\sin^{2}\frac{x}{2}$ by appropriate (left-right, up-down) translation

Find the value of
$$\sin ^ { 2 } 0 ^ { \circ } + \sin ^ { 2 } 1 ^ { \circ } + \sin ^ { 2 } 2 ^ { \circ } + \sin ^ { 2 } 3 ^ { \circ } + \cdots + \sin ^ { 2 } 87 ^ { \circ } + \sin ^ { 2 } 88 ^ { \circ } + \sin ^ { 2 } 89 ^ { \circ } + \sin ^ { 2 } 90 ^ { \circ }$$
A 0.5
B 1
C 1.5
D 45
E 45.5
F 46

This question is about pairs of functions f and g that satisfy
$$\begin{aligned} f ( x ) - g ( x ) & = 2 \sin x \\ f ( x ) g ( x ) & = \cos ^ { 2 } x \end{aligned}$$
for all real numbers $x$.
Across all solutions for $\mathrm { f } ( x )$, what is the minimum value that $\mathrm { f } ( x )$ attains for any $x$ ?