49. The value of $b _ { 6 }$ is
(A) 7
(B) 8
(C) 9
(D) 11

ANSWER : B

1. Which of the following is correct?
   (A) $a _ { 17 } = a _ { 16 } + a _ { 15 }$
   (B) $c _ { 17 } \neq c _ { 16 } + c _ { 15 }$
   (C) $b _ { 17 } \neq b _ { 16 } + c _ { 16 }$
   (D) $a _ { 17 } = c _ { 17 } + b _ { 16 }$

ANSWER:A

Paragraph for Questions 51 and 52

Let $f ( x ) = ( 1 - x ) ^ { 2 } \sin ^ { 2 } x + x ^ { 2 }$ for all $x \in \mathbb { R }$, and let $g ( x ) = \int _ { 1 } ^ { x } \left( \frac { 2 ( t - 1 ) } { t + 1 } - \ln t \right) f ( t ) d t$ for all $x \in ( 1 , \infty )$. 51. Which of the following is true?
(A) $g$ is increasing on $( 1 , \infty )$
(B) $g$ is decreasing on $( 1 , \infty )$
(C) $g$ is increasing on $( 1,2 )$ and decreasing on $( 2 , \infty )$
(D) $g$ is decreasing on $( 1,2 )$ and increasing on $( 2 , \infty )$

ANSWER : B

1. Consider the statements : $\mathbf { P }$ : There exists some $x \in \mathbb { R }$ such that $f ( x ) + 2 x = 2 \left( 1 + x ^ { 2 } \right)$ Q: There exists some $x \in \mathbb { R }$ such that $2 f ( x ) + 1 = 2 x ( 1 + x )$ Then
   (A) both $\mathbf { P }$ and $\mathbf { Q }$ are true
   (B) $\mathbf { P }$ is true and $\mathbf { Q }$ is false
   (C) $\mathbf { P }$ is false and $\mathbf { Q }$ is true
   (D) both $\mathbf { P }$ and $\mathbf { Q }$ are false

ANSWER : C

MATHEMATICS

Paragraph for Questions 53 and 54

A tangent $P T$ is drawn to the circle $x ^ { 2 } + y ^ { 2 } = 4$ at the point $P ( \sqrt { 3 } , 1 )$. A straight line $L$, perpendicular to $P T$ is a tangent to the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 1$. 53. A possible equation of $L$ is
(A) $x - \sqrt { 3 } y = 1$
(B) $x + \sqrt { 3 } y = 1$
(C) $x - \sqrt { 3 } y = - 1$
(D) $x + \sqrt { 3 } y = 5$

ANSWER : A

1. A common tangent of the two circles is
   (A) $x = 4$
   (B) $y = 2$
   (C) $x + \sqrt { 3 } y = 4$
   (D) $x + 2 \sqrt { 2 } y = 6$

ANSWER : D

SECTION III : Multiple Correct Answer(s) Type

This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 55. For every integer $n$, let $a _ { n }$ and $b _ { n }$ be real numbers. Let function $f : \mathbb { R } \rightarrow \mathbb { R }$ be given by $f ( x ) = \left\{ \begin{array} { l l } a _ { n } + \sin \pi x , & \text { for } x \in [ 2 n , 2 n + 1 ] \\ b _ { n } + \cos \pi x , & \text { for } x \in ( 2 n - 1,2 n ) \end{array} \right.$, for all integers $n$. If $f$ is continuous, then which of the following hold(s) for all $n$ ?
(A) $a _ { n - 1 } - b _ { n - 1 } = 0$
(B) $a _ { n } - b _ { n } = 1$
(C) $a _ { n } - b _ { n + 1 } = 1$
(D) $a _ { n - 1 } - b _ { n } = - 1$

ANSWER : BD

1. If $f ( x ) = \int _ { 0 } ^ { x } e ^ { t ^ { 2 } } ( t - 2 ) ( t - 3 ) d t$ for all $x \in ( 0 , \infty )$, then
   (A) $f$ has a local maximum at $x = 2$
   (B) $f$ is decreasing on $( 2,3 )$
   (C) there exists some $c \in ( 0 , \infty )$ such that $f ^ { \prime \prime } ( c ) = 0$
   (D) $f$ has a local minimum at $x = 3$

ANSWER : ABCD 57. If the straight lines $\frac { x - 1 } { 2 } = \frac { y + 1 } { k } = \frac { z } { 2 }$ and $\frac { x + 1 } { 5 } = \frac { y + 1 } { 2 } = \frac { z } { k }$ are coplanar, then the plane(s) containing these two lines is(are)
(A) $y + 2 z = - 1$
(B) $y + z = - 1$
(C) $y - z = - 1$
(D) $y - 2 z = - 1$

ANSWER : BC

1. Let $X$ and $Y$ be two events such that $P ( X \mid Y ) = \frac { 1 } { 2 } , P ( Y \mid X ) = \frac { 1 } { 3 }$ and $P ( X \cap Y ) = \frac { 1 } { 6 }$. Which of the following is (are) correct?
   (A) $P ( X \cup Y ) = \frac { 2 } { 3 }$
   (B) $X$ and $Y$ are independent
   (C) $X$ and $Y$ are not independent
   (D) $P \left( X ^ { \mathrm { c } } \cap Y \right) = \frac { 1 } { 3 }$

ANSWER : AB

MATHEMATICS

1. If the adjoint of a $3 \times 3$ matrix $P$ is $\left[ \begin{array} { l l l } 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{array} \right]$, then the possible value(s) of the determinant of $P$ is (are)
   (A) - 2
   (B) - 1
   (C) 1
   (D) 2

ANSWER : AD

1. Let $f : ( - 1,1 ) \rightarrow \mathbb { R }$ be such that $f ( \cos 4 \theta ) = \frac { 2 } { 2 - \sec ^ { 2 } \theta }$ for $\theta \in \left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$. Then the value(s) of $f \left( \frac { 1 } { 3 } \right)$ is (are)
   (A) $1 - \sqrt { \frac { 3 } { 2 } }$
   (B) $1 + \sqrt { \frac { 3 } { 2 } }$
   (C) $1 - \sqrt { \frac { 2 } { 3 } }$
   (D) $1 + \sqrt { \frac { 2 } { 3 } }$

Zero Marks to all

Let $p , q$ be integers and let $\alpha , \beta$ be the roots of the equation, $x ^ { 2 } - x - 1 = 0$, where $\alpha \neq \beta$. For $n = 0,1,2 , \ldots$, let $a _ { n } = p \alpha ^ { n } + q \beta ^ { n }$.
FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt { 5 } = 0$, then $a = 0 = b$.
$a _ { 12 } =$
[A] $a _ { 11 } - a _ { 10 }$
[B] $a _ { 11 } + a _ { 10 }$
[C] $2 a _ { 11 } + a _ { 10 }$
[D] $a _ { 11 } + 2 a _ { 10 }$

Let $p , q$ be integers and let $\alpha , \beta$ be the roots of the equation, $x ^ { 2 } - x - 1 = 0$, where $\alpha \neq \beta$. For $n = 0,1,2 , \ldots$, let $a _ { n } = p \alpha ^ { n } + q \beta ^ { n }$.
FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt { 5 } = 0$, then $a = 0 = b$.
If $a _ { 4 } = 28$, then $p + 2 q =$
[A] 21
[B] 14
[C] 7
[D] 12

Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - x - 1 = 0$, with $\alpha > \beta$. For all positive integers $n$, define $$\begin{aligned} & a _ { n } = \frac { \alpha ^ { n } - \beta ^ { n } } { \alpha - \beta } , \quad n \geq 1 \\ & b _ { 1 } = 1 \text { and } \quad b _ { n } = a _ { n - 1 } + a _ { n + 1 } , \quad n \geq 2 . \end{aligned}$$ Then which of the following options is/are correct?
(A) $\quad a _ { 1 } + a _ { 2 } + a _ { 3 } + \cdots + a _ { n } = a _ { n + 2 } - 1$ for all $n \geq 1$
(B) $\quad \sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { 10 ^ { n } } = \frac { 10 } { 89 }$
(C) $b _ { n } = \alpha ^ { n } + \beta ^ { n }$ for all $n \geq 1$
(D) $\quad \sum _ { n = 1 } ^ { \infty } \frac { b _ { n } } { 10 ^ { n } } = \frac { 8 } { 89 }$

For positive integer $n$, define
$$f ( n ) = n + \frac { 16 + 5 n - 3 n ^ { 2 } } { 4 n + 3 n ^ { 2 } } + \frac { 32 + n - 3 n ^ { 2 } } { 8 n + 3 n ^ { 2 } } + \frac { 48 - 3 n - 3 n ^ { 2 } } { 12 n + 3 n ^ { 2 } } + \cdots + \frac { 25 n - 7 n ^ { 2 } } { 7 n ^ { 2 } }$$
Then, the value of $\lim _ { n \rightarrow \infty } f ( n )$ is equal to
(A) $3 + \frac { 4 } { 3 } \log _ { e } 7$
(B) $4 - \frac { 3 } { 4 } \log _ { e } \left( \frac { 7 } { 3 } \right)$
(C) $4 - \frac { 4 } { 3 } \log _ { e } \left( \frac { 7 } { 3 } \right)$
(D) $3 + \frac { 3 } { 4 } \log _ { e } 7$

Let $S$ be the set of all $( \alpha , \beta ) \in \mathbb { R } \times \mathbb { R }$ such that
$$\lim _ { x \rightarrow \infty } \frac { \sin \left( x ^ { 2 } \right) \left( \log _ { e } x \right) ^ { \alpha } \sin \left( \frac { 1 } { x ^ { 2 } } \right) } { x ^ { \alpha \beta } \left( \log _ { e } ( 1 + x ) \right) ^ { \beta } } = 0 .$$
Then which of the following is (are) correct?
(A) $( - 1,3 ) \in S$
(B) $( - 1,1 ) \in S$
(C) $( 1 , - 1 ) \in S$
(D) $( 1 , - 2 ) \in S$

Let $\alpha$ and $\beta$ be the roots of equation $x ^ { 2 } - 6 x - 2 = 0$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n } , \forall n \geq 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 2 a _ { 9 } }$ is equal to
(1) - 3
(2) 6
(3) - 6
(4) 3

The sum $\sum _ { r = 1 } ^ { 10 } \left( r ^ { 2 } + 1 \right) \times ( r ! )$, is equal to:
(1) $11 \times ( 11 ! )$
(2) $10 \times ( 11 ! )$
(3) $(11)!$
(4) $101 \times ( 10 ! )$

The value of $4 + \frac { 1 } { 5 + \frac { 1 } { 4 + \frac { 1 } { 5 + \frac { 1 } { 4 + \ldots . . \infty } } } }$ is:
(1) $2 + \frac { 2 } { 5 } \sqrt { 30 }$
(2) $2 + \frac { 4 } { \sqrt { 5 } } \sqrt { 30 }$
(3) $4 + \frac { 4 } { \sqrt { 5 } } \sqrt { 30 }$
(4) $5 + \frac { 2 } { 5 } \sqrt { 30 }$

The value of $3 + \frac { 1 } { 4 + \frac { 1 } { 3 + \frac { 1 } { 4 + \frac { 1 } { 3 + \ldots . \infty } } } }$ is equal to
(1) $1.5 + \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $3 + 2 \sqrt { 3 }$
(4) $4 + \sqrt { 3 }$

Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - 6 x - 2 = 0$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n }$ for $n \geqslant 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 3 a _ { 9 } }$ is:
(1) 1
(2) 3
(3) 2
(4) 4

Consider an arithmetic series and a geometric series having four initial terms from the set $\{ 11,8,21,16,26,32,4 \}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to $\_\_\_\_$.

Let $\left\{ a _ { n } \right\} _ { n = 1 } ^ { \infty }$ be a sequence such that $a _ { 1 } = 1 , a _ { 2 } = 1$ and $a _ { n + 2 } = 2 a _ { n + 1 } + a _ { n }$ for all $n \geq 1$. Then the value of $47 \sum _ { n = 1 } ^ { \infty } \left( \frac { a _ { n } } { 2 ^ { 3 n } } \right)$ is equal to $\underline{\hspace{1cm}}$.

Let $\left\{ a _ { n } \right\} _ { n = 0 } ^ { \infty }$ be a sequence such that $a _ { 0 } = a _ { 1 } = 0$ and $a _ { n + 2 } = 2 a _ { n + 1 } - a _ { n } + 1$ for all $n \geq 0$. Then, $\sum _ { n = 2 } ^ { \infty } \frac { a _ { n } } { 7 ^ { n } }$ is equal to
(1) $\frac { 6 } { 343 }$
(2) $\frac { 7 } { 216 }$
(3) $\frac { 8 } { 343 }$
(4) $\frac { 49 } { 216 }$

Consider the sequence $a_1, a_2, a_3, \ldots$ such that $a_1 = 1$, $a_2 = 2$ and $a_{n+2} = \frac{2}{a_{n+1}} + a_n$ for $n = 1, 2, 3, \ldots$ If $\frac{a_1 + \frac{1}{a_2}}{a_3} \cdot \frac{a_2 + \frac{1}{a_3}}{a_4} \cdot \frac{a_3 + \frac{1}{a_4}}{a_5} \cdots \frac{a_{30} + \frac{1}{a_{31}}}{a_{32}} = 2^\alpha \binom{61}{31}$ then $\alpha$ is equal to
(1) $-30$
(2) $-31$
(3) $-60$
(4) $-61$

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence such that $a_0 = a_1 = 0$ and $a_{n+2} = 3a_{n+1} - 2a_n + 1, \forall n \geq 0$. Then $a_{25}a_{23} - 2a_{25}a_{22} - 2a_{23}a_{24} + 4a_{22}a_{24}$ is equal to
(1) 483
(2) 528
(3) 575
(4) 624

If $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } - n - 1 } + n\alpha + \beta \right) = 0$ then $8\alpha + \beta$ is equal to
(1) 4
(2) - 8
(3) - 4
(4) 8

Let $\beta = \lim _ { x \rightarrow 0 } \frac { \alpha x - \left( e ^ { 3 x } - 1 \right) } { \alpha x \left( e ^ { 3 x } - 1 \right) }$ for some $\alpha \in \mathbb { R }$. Then the value of $\alpha + \beta$ is:
(1) $\frac { 14 } { 5 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 1 } { 2 }$

If $\lim _ { n \rightarrow \infty } \frac { (n+1)^{k-1} } { n ^ { k + 1 } } \left[ (nk+1) + (nk+2) + \ldots + (nk+n) \right] = 33 \cdot \lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { k + 1 } } \cdot \left( 1 ^ { k } + 2 ^ { k } + 3 ^ { k } + \ldots + n ^ { k } \right)$, then the integral value of $k$ is equal to $\_\_\_\_$.

$\lim _ { t \rightarrow 0 } 1 ^ { \frac { 1 } { \sin ^ { 2 } t } } + 2 ^ { \frac { 1 } { \sin ^ { 2 } t } } + 3 ^ { \frac { 1 } { \sin ^ { 2 } t } } \ldots \ldots n ^ { \frac { 1 } { \sin ^ { 2 } t } } \sin ^ { 2 } t$ is equal to
(1) $n ^ { 2 } + n$
(2) $n$
(3) $\frac { n n + 1 } { 2 }$
(4) $n ^ { 2 }$

Let $a _ { 1 } = b _ { 1 } = 1$ and $a _ { n } = a _ { n - 1 } + ( n - 1 ) , b _ { n } = b _ { n - 1 } + \mathrm { a } _ { n - 1 } , \forall n \geq 2$. If $\mathrm { S } = \sum _ { \mathrm { n } = 1 } ^ { 10 } \left( \frac { b _ { n } } { 2 ^ { n } } \right)$ and $\mathrm { T } = \sum _ { n = 1 } ^ { 8 } \frac { \mathrm { n } } { 2 ^ { n - 1 } }$ then $2 ^ { 7 } ( 2 S - T )$ is equal to $\_\_\_\_$

Let $\left\langle a _ { n } \right\rangle$ be a sequence such that $a _ { 1 } + a _ { 2 } + \ldots + a _ { n } = \frac { n ^ { 2 } + 3 n } { ( n + 1 ) ( n + 2 ) }$. If $28 \sum _ { k = 1 } ^ { 10 } \frac { 1 } { a _ { k } } = p _ { 1 } p _ { 2 } p _ { 3 } \ldots p _ { m }$, where $p _ { 1 } , p _ { 2 } , \ldots p _ { m }$ are the first $m$ prime numbers, then $m$ is equal to
(1) 5
(2) 8
(3) 6
(4) 7

The value of $\lim _ { n \rightarrow \infty } \frac { 1 + 2 - 3 + 4 + 5 - 6 + \ldots + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n } { \sqrt { 2 n ^ { 4 } + 4 n + 3 } - \sqrt { n ^ { 4 } + 5 n + 4 } }$ is
(1) $\frac { \sqrt { 2 } + 1 } { 2 }$
(2) $3 ( \sqrt { 2 } + 1 )$
(3) $\frac { 3 } { 2 } ( \sqrt { 2 } + 1 )$
(4) $\frac { 3 } { 2 \sqrt { 2 } }$

$\lim_{n \rightarrow \infty} \left\{\left(2^{\frac{1}{2}} - 2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}} - 2^{\frac{1}{5}}\right) \ldots \left(2^{\frac{1}{2}} - 2^{\frac{1}{2n+1}}\right)\right\}$ is equal to
(1) 1
(2) 0
(3) $\sqrt{2}$
(4) $\frac{1}{\sqrt{2}}$

If $a _ { \alpha }$ is the greatest term in the sequence $a _ { n } = \frac { n ^ { 3 } } { n ^ { 4 } + 147 } , n = 1 , 2 , 3 \ldots$, then $\alpha$ is equal to $\_\_\_\_$