For any integer $n \geq 1$, define $a_n = \frac{1000^n}{n!}$. Then the sequence $\{a_n\}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

Let the sequence $\left\{ a _ { n } \right\} _ { n \geq 1 }$ be defined by
$$a _ { n } = \tan ( n \theta )$$
where $\tan ( \theta ) = 2$. Show that for all $n , a _ { n }$ is a rational number which can be written with an odd denominator.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function such that its derivative $f ^ { \prime }$ is a continuous function. Moreover, assume that for all $x \in \mathbb { R }$, $$0 \leq \left| f ^ { \prime } ( x ) \right| \leq \frac { 1 } { 2 }$$ Define a sequence of real numbers $\left\{ a _ { n } \right\} _ { n \in \mathbb { N } }$ by: $$\begin{gathered} a _ { 1 } = 1 , \\ a _ { n + 1 } = f \left( a _ { n } \right) \text { for all } n \in \mathbb { N } . \end{gathered}$$ Prove that there exists a positive real number $M$ such that for all $n \in \mathbb { N }$, $$\left| a _ { n } \right| \leq M$$

Consider the real-valued function $h : \{ 0,1,2 , \ldots , 100 \} \rightarrow \mathbb { R }$ such that $h ( 0 ) = 5 , h ( 100 ) = 20$ and satisfying $h ( i ) = \frac { 1 } { 2 } ( h ( i + 1 ) + h ( i - 1 ) )$, for every $i = 1,2 , \ldots , 99$. Then, the value of $h ( 1 )$ is:
(A) 5.15
(B) 5.5
(C) 6
(D) 6.15.

Consider a sequence $P_1, P_2, \ldots$ of points in the plane such that $P_1, P_2, P_3$ are non-collinear and for every $n \geq 4$, $P_n$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_1$ and $P_5$. Prove the following:
(a) The area of the triangle formed by the points $P_n, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity.
(b) The point $P_9$ lies on $L$.

Let $a _ { 0 } = \frac { 1 } { 2 }$ and $a _ { n }$ be defined inductively by
$$a _ { n } = \sqrt { \frac { 1 + a _ { n - 1 } } { 2 } } , n \geq 1 .$$
(a) Show that for $n = 0,1,2 , \ldots$,
$$a _ { n } = \cos \theta _ { n } \text { for some } 0 < \theta _ { n } < \frac { \pi } { 2 } ,$$
and determine $\theta _ { n }$.
(b) Using (a) or otherwise, calculate
$$\lim _ { n \rightarrow \infty } 4 ^ { n } \left( 1 - a _ { n } \right)$$

There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t _ { n }$ denote the number of ways this can be done. For example, clearly $t _ { 1 } = 2$ because we can have either a red or a blue tile. Also, $t _ { 2 } = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
(a) Prove that $t _ { 2 n + 1 } = t _ { n } \left( t _ { n - 1 } + t _ { n + 1 } \right)$ for all $n > 1$.
(b) Prove that $t _ { n } = \sum _ { d \geq 0 } \binom { n - d } { d } 2 ^ { n - 2 d }$ for all $n > 0$.
Here,
$$\binom { m } { r } = \begin{cases} \frac { m ! } { r ! ( m - r ) ! } , & \text { if } 0 \leq r \leq m , \\ 0 , & \text { otherwise } , \end{cases}$$
for integers $m , r$.

Find, with proof, all possible values of $t$ such that
$$\lim _ { n \rightarrow \infty } \left\{ \frac { 1 + 2 ^ { 1/3 } + 3 ^ { 1/3 } + \cdots + n ^ { 1/3 } } { n ^ { t } } \right\} = c$$
for some real number $c > 0$. Also find the corresponding values of $c$.

Let $S_n$ be the set of all $n$-digit numbers whose digits are all 1 or 2 and there are no consecutive 2's. (Example: 112 is in $S_3$ but 221 is not in $S_3$). Then the number of elements in $S_{10}$ is
(A) 512
(B) 256
(C) 144
(D) 89

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(a) 0 .
(B) 1 .
(C) $e$.
(D) $\sqrt { e } / 2$.

28. For a positive integer n , let $\mathrm { a } ( \mathrm { n } ) = 1 + 1 / 2 + 1 / 3 + 1 / 4 + \ldots \ldots + 1 / ( ( 2 \mathrm { n } ) - 1 )$. Then :
(A) a (100) £ 100
(B) a $( 100 ) > ( 100 )$
(C) a (200) $\pounds 100$
(D) a $( 200 ) > 100$

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 $S _ { n } = \sum _ { k = 1 } ^ { 4 n } ( - 1 ) ^ { \frac { k ( k + 1 ) } { 2 } } k ^ { 2 }$. Then $S _ { n }$ can take value(s)
(A) 1056
(B) 1088
(C) 1120
(D) 1332

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

The number of real solutions of the equation $$\sin ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } x ^ { i + 1 } - x \sum _ { i = 1 } ^ { \infty } \left( \frac { x } { 2 } \right) ^ { i } \right) = \frac { \pi } { 2 } - \cos ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } \left( - \frac { x } { 2 } \right) ^ { i } - \sum _ { i = 1 } ^ { \infty } ( - x ) ^ { i } \right)$$ lying in the interval $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ is $\_\_\_\_$. (Here, the inverse trigonometric functions $\sin ^ { - 1 } x$ and $\cos ^ { - 1 } x$ assume values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ and $[ 0 , \pi ]$, respectively.)

For each positive integer $n$, let $$y _ { n } = \frac { 1 } { n } ( ( n + 1 ) ( n + 2 ) \cdots ( n + n ) ) ^ { \frac { 1 } { n } }$$ For $x \in \mathbb { R }$, let $[ x ]$ be the greatest integer less than or equal to $x$. If $\lim _ { n \rightarrow \infty } y _ { n } = L$, then the value of $[ L ]$ is $\_\_\_\_$.

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 $a \in \mathbb{R}$, $|a| > 1$, let $$\lim_{n\rightarrow\infty}\left(\frac{1 + \sqrt[3]{2} + \cdots + \sqrt[3]{n}}{n^{7/3}\left(\frac{1}{(an+1)^2} + \frac{1}{(an+2)^2} + \cdots + \frac{1}{(an+n)^2}\right)}\right) = 54$$
Then the possible value(s) of $a$ is/are
(A) $-9$
(B) $-6$
(C) $7$
(D) $8$

Let $A P ( a ; d )$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If $$A P ( 1 ; 3 ) \cap A P ( 2 ; 5 ) \cap A P ( 3 ; 7 ) = A P ( a ; d )$$ then $a + d$ equals $\_\_\_\_$

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) $3 + \frac { 3 } { 4 } \log _ { e } 7$

The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots$ is
(1) $\frac{7}{81}(179-10^{-20})$
(2) $\frac{7}{9}(99-10^{-20})$
(3) $\frac{7}{81}(179+10^{-20})$
(4) $\frac{7}{9}(99+10^{-20})$

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 of the following series $1 + 6 + \frac{9\left(1^2 + 2^2 + 3^2\right)}{7} + \frac{12\left(1^2 + 2^2 + 3^2 + 4^2\right)}{9} + \frac{15\left(1^2 + 2^2 + \ldots + 5^2\right)}{11} + \ldots$ up to 15 terms, is:
(1) 7520
(2) 7510
(3) 7830
(4) 7820