QUESTION 180
The value of $\cos^2 30^\circ + \sin^2 30^\circ$ is
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) 1
(D) $\sqrt{3}$
(E) 2

In a right triangle, $\sin(\theta) = \dfrac{3}{5}$. What is the value of $\cos(\theta)$?
(A) $\dfrac{1}{5}$
(B) $\dfrac{2}{5}$
(C) $\dfrac{3}{5}$
(D) $\dfrac{4}{5}$
(E) $\dfrac{5}{4}$

Find the value of the following sum of 100 terms. (Possible hint: also consider the same sum with $\sin^{2}$ instead of $\cos^{2}$.)
$$\cos^{2}\left(\frac{\pi}{101}\right) + \cos^{2}\left(\frac{2\pi}{101}\right) + \cos^{2}\left(\frac{3\pi}{101}\right) + \cdots + \cos^{2}\left(\frac{99\pi}{101}\right) + \cos^{2}\left(\frac{100\pi}{101}\right)$$

19. (Total Score: 12 points) Given $0 < x < \frac { \pi } { 2 }$ , simplify: $\lg \left( \cos x \cdot \tan x + 1 - 2 \sin ^ { 2 } \frac { x } { 2 } \right) + \lg \left[ \sqrt { 2 } \cos \left( x - \frac { \pi } { 4 } \right) \right] - \lg ( 1 + \sin 2 x )$ .

16. Given the function $f ( x ) = ( \sin x + \cos x ) ^ { 2 } + \cos 2 x$
(1) Find the minimum positive period of $f ( x )$;
(2) Find the maximum and minimum values of $f ( x )$ on the interval $\left[ 0 , \frac { \pi } { 2 } \right]$.

15. (This question is worth 13 points) Given the function $f ( x ) = \sqrt { 2 } \sin \frac { x } { 2 } \cos \frac { x } { 2 } - \sqrt { 2 } \sin ^ { 2 } \frac { x } { 2 }$. (I) Find the minimum positive period of $f ( x )$; (II) Find the minimum value of $f ( x )$ on the interval $[ - \pi , 0 ]$.

Given $\sin \theta + \sin \left( \theta + \frac { \pi } { 3 } \right) = 1$, then $\sin \left( \theta + \frac { \pi } { 6 } \right) =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 3 } } { 3 }$
C. $\frac { 2 } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x).$$

Let $f(x) = \pi \operatorname{cotan}(\pi x)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x)$$

35. For a positive integer n ., let $\mathrm { f } \_ \mathrm { n } ( \theta ) = ( \tan \theta / 2 ) ( 1 + \sec \theta ) ( 1 + \sec 2 \theta ) ( 1 + \sec 4 \theta )$... $( 1 + \sec 2 \mathrm { n } \theta )$. Then
(A) $\mathrm { f } 2 ( \Pi / 16 ) = 1$;
(B) $f 3 ( \pi / 32 ) = 1$
(C) $\mathrm { f } 4 ( \pi / 16 ) = 1$
(D) f5 $( \sqcap / 128 ) = 1$

SECTION II

Instructions

There 12 questions in the section. Attempt ALL questions. At the end of the anwers to a question, draw a horizontal line and start answer to the next question. The corresponding question number must be written in the left margin. Answer all parts of a question at one place only. The use of Arabic numerals ( $0,1,2 , \ldots \ldots . .9$ ) only is allowed in answering the questions irrespective of the language in which you answer.

1. For complex numbers z and q , prove that $| \mathrm { z } | 2 \mathrm { w } - | \mathrm { w } | 2 \mathrm { z } = \mathrm { z } - \mathrm { w }$ if and only if $\mathrm { z } = \mathrm { w }$ or z $\mathrm { w } - = 1$.
2. Let $a , b , c d$ be real numbers in G.P. If $u , v , w$ satisfy the system of equations

$$\begin{aligned} & u + 2 v + 3 w = 6 \\ & 4 u + 5 v + 6 w = 12 \\ & 6 u + 9 v = 4 \end{aligned}$$
Then slow that the roots of the equation :

III askllTians ||

... Powered By IITians $( 1 / u + 1 / v + 1 / w ) \times 2 + [ ( b - c ) 2 + ( c - a ) 2 + ( d - b ) 2 ] x + u + v + w = 0$ and $20 \times 2 + 10 ( a - d ) 2 x - 9 = 0$ are reciprocals of each other.

For non-negative integers $n$, let $$f(n) = \frac{\sum_{k=0}^{n} \sin\left(\frac{k+1}{n+2}\pi\right)\sin\left(\frac{k+2}{n+2}\pi\right)}{\sum_{k=0}^{n} \sin^2\left(\frac{k+1}{n+2}\pi\right)}$$
Assuming $\cos^{-1}x$ takes values in $[0, \pi]$, which of the following options is/are correct?
(A) $f(4) = \frac{\sqrt{3}}{2}$
(B) $\lim_{n\rightarrow\infty} f(n) = \frac{1}{2}$
(C) If $\alpha = \tan(\cos^{-1}f(6))$, then $\alpha^2 + 2\alpha - 1 = 0$
(D) $\sin(7\cos^{-1}f(5)) = 0$

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) $\sec A \operatorname{cosec} A + 1$

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) $\frac { 1 } { 32 }$

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

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 $\_\_\_\_$.

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) $\frac { 3 } { 2 }$

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

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) $3/2$

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

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

$$\frac{(\sin x - \cos x)^{2}}{\cos x} + 2\sin x$$
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) $\arccos x$

$$\frac { \cot \left( 34 ^ { \circ } \right) \cdot \sin \left( 44 ^ { \circ } \right) } { \sin \left( 22 ^ { \circ } \right) \cdot \sin \left( 56 ^ { \circ } \right) }$$
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) $\sec x$