In a right angle triangle with sides $a < b < c$, where $\angle ACB = \theta$ is the smallest angle, show that $\sin^2\theta - \sqrt{5}\sin\theta + 1 = 0$, given that $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$ (i.e., the reciprocals of the sides also form a right triangle).

Show that $\sin^5 x + \cos^3 x \geq \sin^3 x + \cos^2 x$ implies the expression equals $1$, and find when equality holds.

Let $ABC$ be a triangle $A \neq B$ and let $P \in (AB)$ be a point for which denote $m(\widehat{ACP}) = x$ and $m(\widehat{BCP}) = y$. Prove that $\frac{\sin A \sin B}{\sin(A-B)} = \frac{\sin x \sin y}{\sin(x-y)}$ if and only if $PA = PB$.

Let $\alpha$, $\beta$ and $\gamma$ be the angles of an acute angled triangle. Then the quantity $\tan\alpha\tan\beta\tan\gamma$
(a) Can have any real value
(b) Is $\leq 3\sqrt{3}$
(c) Is $\geq 3\sqrt{3}$
(d) None of the above.

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

Suppose $ABCD$ is a quadrilateral such that $\angle BAC = 50^\circ, \angle CAD = 60^\circ, \angle CBD = 30^\circ$ and $\angle BDC = 25^\circ$. If $E$ is the point of intersection of $AC$ and $BD$, then the value of $\angle AEB$ is
(A) $75^\circ$
(B) $85^\circ$
(C) $95^\circ$
(D) $110^\circ$

Suppose $x, y \in (0, \pi/2)$ and $x \neq y$. Which of the following statement is true?
(A) $2\sin(x + y) < \sin 2x + \sin 2y$ for all $x, y$.
(B) $2\sin(x + y) > \sin 2x + \sin 2y$ for all $x, y$.
(C) There exist $x, y$ such that $2\sin(x + y) = \sin 2x + \sin 2y$.
(D) None of the above.

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

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.

2) For the bicycle to proceed smoothly on the platform it is necessary that: – to the left and right of the points of non-differentiability the sections of the graph are orthogonal; – the length of the side of the square wheel equals the length of a "bump", that is, the arc of the curve with equation $y = f(x)$ for $x \in [-a; a]$.
Establish whether these conditions are satisfied.${ } ^ { 1 }$

4. Let $A B C$ be a triangle having $O$ and $I$ as its circumcentre and incentre respectively. If $R$ and $r$ are the circumradius and the inradius, respectively, then prove that (IO) $2 = R 2 - 2 R r$. Further show that the triangle BIO is a right-angled triangle if and only if $b$ is the arithmetic mean of a and c.

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.

3. (a) In any triangle ABC , prove that
$$\operatorname { Cot } \mathrm { A } / \mathrm { b } + \cot \mathrm { B } / 2 + \cot \mathrm { C } / 2 = \mathrm { A } / 2 \cot \mathrm {~B} / 2 \cot \mathrm { C } / 2$$
(b) Let ABC be a triangle with incentre I and inradius r . Let $\mathrm { D } , \mathrm { E } , \mathrm { F }$ be the feet of the perpendiculars from I to the sides $\mathrm { BC } , \mathrm { CA }$ and AB respectively. If $\mathrm { r } 1 , \mathrm { r } 2$ and B are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that r1/ ( $\mathrm { r } - \mathrm { r } 1$ ) + உ / ( $\mathrm { r } - \mathrm { r } 2$ ) + r3 / ( $\mathrm { r } - \mathrm { r } 3$ ) = ( r 1 B B ) / ( $\mathrm { r } - \mathrm { r } 1$ ) ( $\mathrm { r } - \mathrm { r } 2$ ) ( $\mathrm { r } - \mathrm { r } 3$ )

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) $\frac { \sin ^ { 8 } x } { 8 } + \frac { \cos ^ { 8 } x } { 27 } = \frac { 2 } { 125 }$

Column I
(A) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $2 \left( a ^ { 2 } - b ^ { 2 } \right) = c ^ { 2 }$ and $\lambda = \frac { \sin ( X - Y ) } { \sin Z }$, then possible values of $n$ for which $\cos ( n \pi \lambda ) = 0$ is (are)
(B) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $1 + \cos 2 X - 2 \cos 2 Y = 2 \sin X \sin Y$, then possible value(s) of $\frac { a } { b }$ is (are)
(C) In $\mathbb { R } ^ { 2 }$, let $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\beta \hat { i } + ( 1 - \beta ) \hat { j }$ be the position vectors of $X , Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow { O X }$ with $\overrightarrow { O Y }$ is $\frac { 3 } { \sqrt { 2 } }$, then possible value(s) of $| \beta |$ is (are)
(D) Suppose that $F ( \alpha )$ denotes the area of the region bounded by $x = 0 , x = 2 , y ^ { 2 } = 4 x$ and $y = | \alpha x - 1 | + | \alpha x - 2 | + \alpha x$, where $\alpha \in \{ 0,1 \}$. Then the value(s) of $F ( \alpha ) + \frac { 8 } { 3 } \sqrt { 2 }$, when $\alpha = 0$ and $\alpha = 1$, is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5

The value of $\sum _ { k = 1 } ^ { 13 } \frac { 1 } { \sin \left( \frac { \pi } { 4 } + \frac { ( k - 1 ) \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { k \pi } { 6 } \right) }$ is equal to
(A) $3 - \sqrt { 3 }$
(B) $2 ( 3 - \sqrt { 3 } )$
(C) $2 ( \sqrt { 3 } - 1 )$
(D) $2 ( 2 + \sqrt { 3 } )$

If the triangle $P Q R$ varies, then the minimum value of
$$\cos ( P + Q ) + \cos ( Q + R ) + \cos ( R + P )$$
is
[A] $- \frac { 5 } { 3 }$
[B] $- \frac { 3 } { 2 }$
[C] $\frac { 3 } { 2 }$
[D] $\frac { 5 } { 3 }$

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) $\sin(7\cos^{-1}f(5)) = 0$

Let $\frac { \pi } { 2 } < x < \pi$ be such that $\cot x = \frac { - 5 } { \sqrt { 11 } }$. Then
$$\left( \sin \frac { 11 x } { 2 } \right) ( \sin 6 x - \cos 6 x ) + \left( \cos \frac { 11 x } { 2 } \right) ( \sin 6 x + \cos 6 x )$$
is equal to
(A) $\frac { \sqrt { 11 } - 1 } { 2 \sqrt { 3 } }$
(B) $\frac { \sqrt { 11 } + 1 } { 2 \sqrt { 3 } }$
(C) $\frac { \sqrt { 11 } + 1 } { 3 \sqrt { 2 } }$
(D) $\frac { \sqrt { 11 } - 1 } { 3 \sqrt { 2 } }$

If $n$ is a positive integer, then $(\sqrt{3}+1)^{2n} - (\sqrt{3}-1)^{2n}$ is
(1) an irrational number
(2) an odd positive integer
(3) an even positive integer
(4) a rational number other than positive integers

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

Let $f _ { k } ( x ) = \frac { 1 } { k } \left( \sin ^ { k } x + \cos ^ { k } x \right)$ where $x \in R$ and $k \geq 1$. Then $f _ { 4 } ( x ) - f _ { 6 } ( x )$ equals
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 3 }$

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) $\sqrt { \mathrm { pq } }$