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*{SECTION II}
\section*{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.

\begin{enumerate}
  \item 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 $\mathrm { w } - = 1$.
  \item Let $a , b , c d$ be real numbers in G.P. If $u , v , w$ satisfy the system of equations
\end{enumerate}

$$\begin{aligned}
& u + 2 v + 3 w = 6 \\
& 4 u + 5 v + 6 w = 12 \\
& 6 u + 9 v = 4
\end{aligned}$$

Then slow that the roots of the equation :

\section*{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.\\