Given $\alpha \in ( 0 , \pi )$ and $3 \cos 2 \alpha - 8 \cos \alpha = 5$, then $\sin \alpha =$
A. $\frac { \sqrt { 5 } } { 3 }$
B. $\frac { 2 } { 3 }$
C. $\frac { 1 } { 3 }$
D. $\frac { \sqrt { 5 } } { 9 }$

``$\sin^{2} \alpha + \sin^{2} \beta = 1$'' is ``$\cos \alpha + \cos \beta = 0$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition

In the triangle $ABC$, the angle $\angle BAC$ is a root of the equation $$\sqrt { 3 } \cos x + \sin x = 1 / 2$$ Then the triangle $ABC$ is
(A) obtuse angled
(B) right angled
(C) acute angled but not equilateral
(D) equilateral

If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C = 2 \left( \cos ^ { 2 } A + \cos ^ { 2 } B + \cos ^ { 2 } C \right) ,$$ prove that the triangle must have a right angle.

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 $\operatorname { cosec } \theta = \frac { \mathrm { p } + \mathrm { q } } { \mathrm { p } - \mathrm { q } } ( \mathrm { p } \neq \mathrm { q } , \mathrm { p } \neq 0 )$, then $\left| \cot \left( \frac { \pi } { 4 } + \frac { \theta } { 2 } \right) \right|$ is equals to:
(1) $p q$
(2) $\sqrt { \frac { p } { q } }$
(3) $\sqrt { \frac { q } { p } }$
(4) $\sqrt { \mathrm { pq } }$

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

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

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:

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

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

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.

$$3\sin x - 4\cos x = 0$$
Given this, what is the value of $|\cos 2x|$?
A) $\frac{3}{4}$
B) $\frac{3}{5}$
C) $\frac{4}{5}$
D) $\frac{7}{25}$
E) $\frac{9}{25}$

In Figure 1, a parallelogram with two sides of lengths 4 units and 8 units and the angle between these sides measuring $x$ degrees is given. In Figure 2, a parallelogram with two sides of lengths 4 units and 6 units and the angle between these sides measuring $2x$ degrees is given.
If the area of the parallelogram in Figure 1 is 24 square units, what is the area of the parallelogram in Figure 2 in square units?
A) $6\sqrt{7}$ B) $7\sqrt{7}$ C) $8\sqrt{7}$ D) $9\sqrt{7}$ E) $10\sqrt{7}$