41. The equation of a plane passing through the line of intersection of the planes $x + 2 y + 3 z = 2$ and $x - y + z = 3$ and at a distance $\frac { 2 } { \sqrt { 3 } }$ from the point $( 3,1 , - 1 )$ is\\
(A) $5 x - 11 y + z = 17$\\
(B) $\sqrt { 2 } x + y = 3 \sqrt { 2 } - 1$\\
(C) $x + y + z = \sqrt { 3 }$\\
(D) $x - \sqrt { 2 } y = 1 - \sqrt { 2 }$

\section*{ANSWER : A}
\begin{enumerate}
  \setcounter{enumi}{41}
  \item Let $P Q R$ be a triangle of area $\triangle$ with $a = 2 , b = \frac { 7 } { 2 }$ and $c = \frac { 5 } { 2 }$, where $a , b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P , Q$ and $R$ respectively. Then $\frac { 2 \sin P - \sin 2 P } { 2 \sin P + \sin 2 P }$ equals\\
(A) $\frac { 3 } { 4 \Delta }$\\
(B) $\frac { 45 } { 4 \Delta }$\\
(C) $\left( \frac { 3 } { 4 \Delta } \right) ^ { 2 }$\\
(D) $\left( \frac { 45 } { 4 \Delta } \right) ^ { 2 }$
\end{enumerate}

\section*{ANSWER : C}
\begin{enumerate}
  \setcounter{enumi}{42}
  \item If $\vec { a }$ and $\vec { b }$ are vectors such that $| \vec { a } + \vec { b } | = \sqrt { 29 }$ and $\vec { a } \times ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) = ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) \times \vec { b }$, then a possible value of $( \vec { a } + \vec { b } ) \cdot ( - 7 \hat { i } + 2 \hat { j } + 3 \hat { k } )$ is\\
(A) 0\\
(B) 3\\
(C) 4\\
(D) 8
  \item If $P$ is a $3 \times 3$ matrix such that $P ^ { T } = 2 P + I$, where $P ^ { T }$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = \left[ \begin{array} { c } x \\ y \\ z \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$ such that\\
(A) $P X = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$\\
(B) $P X = X$\\
(C) $P X = 2 X$\\
(D) $P X = - X$
\end{enumerate}

\section*{ANSWER : D}
\begin{enumerate}
  \setcounter{enumi}{44}
  \item Let $\alpha$ (a) and $\beta$ (a) be the roots of the equation\\
$( \sqrt [ 3 ] { 1 + a } - 1 ) x ^ { 2 } + ( \sqrt { 1 + a } - 1 ) x + ( \sqrt [ 6 ] { 1 + a } - 1 ) = 0$ where $a > - 1$.\\
Then $\lim _ { a \rightarrow 0 ^ { + } } \alpha ( a )$ and $\lim _ { a \rightarrow 0 ^ { + } } \beta ( a )$ are\\
(A) $- \frac { 5 } { 2 }$ and 1\\
(B) $- \frac { 1 } { 2 }$ and - 1\\
(C) $- \frac { 7 } { 2 }$ and 2\\
(D) $- \frac { 9 } { 2 }$ and 3
\end{enumerate}

\section*{ANSWER : B}
\begin{enumerate}
  \setcounter{enumi}{45}
  \item Four fair dice $D _ { 1 } , D _ { 2 } , D _ { 3 }$ and $D _ { 4 }$, each having six faces numbered $1,2,3,4,5$ and 6 , are rolled simultaneously. The probability that $D _ { 4 }$ shows a number appearing on one of $D _ { 1 } , D _ { 2 }$ and $D _ { 3 }$ is\\
(A) $\frac { 91 } { 216 }$\\
(B) $\frac { 108 } { 216 }$\\
(C) $\frac { 125 } { 216 }$\\
(D) $\frac { 127 } { 216 }$
\end{enumerate}

\section*{ANSWER : A}
\begin{enumerate}
  \setcounter{enumi}{46}
  \item The value of the integral\\
$\int _ { - \pi / 2 } ^ { \pi / 2 } \left( x ^ { 2 } + \ln \frac { \pi + x } { \pi - x } \right) \cos x \mathrm {~d} x$ is\\
(A) 0\\
(B) $\frac { \pi ^ { 2 } } { 2 } - 4$\\
(C) $\frac { \pi ^ { 2 } } { 2 } + 4$\\
(D) $\frac { \pi ^ { 2 } } { 2 }$
\end{enumerate}

\section*{ANSWER : B}
\begin{enumerate}
  \setcounter{enumi}{47}
  \item Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be in harmonic progression with $a _ { 1 } = 5$ and $a _ { 20 } = 25$. The least positive integer $n$ for which $a _ { n } < 0$ is\\
(A) 22\\
(B) 23\\
(C) 24\\
(D) 25
\end{enumerate}

\section*{SECTION II : Paragraph Type}
This section contains $\mathbf { 6 }$ multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

\section*{Paragraph for Questions 49 and 50}
Let $a _ { n }$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let $b _ { n } =$ the number of such $n$-digit integers ending with digit 1 and $c _ { n } =$ the number of such $n$-digit integers ending with digit 0 .\\