A father wants to distribute a certain sum of money between his daughter and son in such a way that if both of them invest their shares in the scheme that offers compound interest at $\frac { 25 } { 3 } \%$ per annum, for $t$ and $t + 2$ years respectively, then the two shares grow to become equal. If the son's share was rupees 4320, then the total money distributed by the father was
(A) rupees 9360
(B) rupees 9390
(C) rupees 16, 590
(D) rupees 16, 640.

For each natural number $k$, choose a complex number $z _ { k }$ with $\left| z _ { k } \right| = 1$ and denote by $a _ { k }$ the area of the triangle formed by $z _ { k } , i z _ { k } , z _ { k } + i z _ { k }$. Then, which of the following is true for the series below?
$$\sum _ { k = 1 } ^ { \infty } \left( a _ { k } \right) ^ { k }$$
(A) It converges only if every $z _ { k }$ lies in the same quadrant.
(B) It always diverges.
(C) It always converges.
(D) none of the above.

The function $y = e ^ { k x }$ satisfies
$$\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } \right) \left( \frac { d y } { d x } - y \right) = y \frac { d y } { d x }$$
for
(A) exactly one value of $k$.
(B) two distinct values of $k$.
(C) three distinct values of $k$.
(D) infinitely many values of $k$.

A particle is allowed to move in the $XY$-plane by choosing any one of the two jumps:

1. move two units to right and one unit up, i.e., $( a , b ) \mapsto ( a + 2 , b + 1 )$ or
2. move two units up and one unit to right, i.e., $( a , b ) \mapsto ( a + 1 , b + 2 )$.

Let $P = ( 30,63 )$ and $Q = ( 100,100 )$. If the particle starts at the origin, then
(A) $P$ is reachable but not $Q$.
(B) $Q$ is reachable but not $P$.
(C) both $P$ and $Q$ are reachable.
(D) neither $P$ nor $Q$ is reachable.

For a real polynomial in one variable $P$, let $Z ( P )$ denote the locus of points ( $x , y$ ) in the plane such that $P ( x ) + P ( y ) = 0$. Then,
(A) there exist polynomials $Q _ { 1 }$ and $Q _ { 2 }$ such that $Z \left( Q _ { 1 } \right)$ is a circle and $Z \left( Q _ { 2 } \right)$ is a parabola.
(B) there does not exist any polynomial $Q$ such that $Z ( Q )$ is a circle or a parabola.
(C) there exists a polynomial $Q$ such that $Z ( Q )$ is a circle but there does not exist any polynomial $P$ such that $Z ( P )$ is a parabola.
(D) there exists a polynomial $Q$ such that $Z ( Q )$ is a parabola but there does not exist any polynomial $P$ such that $Z ( P )$ is a circle.

Three children and two adults want to cross a river using a rowing boat. The boat can carry no more than a single adult or, in case no adult is in the boat, a maximum of two children. The least number of times the boat needs to cross the river to transport all five people is:
(A) 9
(B) 11
(C) 13
(D) 15 .

Let a real-valued sequence $\left\{x_{n}\right\}_{n \geq 1}$ be such that
$$\lim_{n \rightarrow \infty} n x_{n} = 0$$
Find all possible real values of $t$ such that $\lim_{n \rightarrow \infty} x_{n}(\log n)^{t} = 0$.

There are 128 numbers $1,2 , \ldots , 128$ which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number 2, then skip the next available number (which is 3 ) and delete 4 . Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number remains. What is the last number left ?
(A) 1
(B) 63
(C) 127
(D) None of the above.

Let $A , B , C$ be finite subsets of the plane such that $A \cap B , B \cap C$ and $C \cap A$ are all empty. Let $S = A \cup B \cup C$. Assume that no three points of $S$ are collinear and also assume that each of $A , B$ and $C$ has at least 3 points. Which of the following statements is always true?
(A) There exists a triangle having a vertex from each of $A , B , C$ that does not contain any point of $S$ in its interior.
(B) Any triangle having a vertex from each of $A , B , C$ must contain a point of $S$ in its interior.
(C) There exists a triangle having a vertex from each of $A , B , C$ that contains all the remaining points of $S$ in its interior.
(D) There exist 2 triangles, both having a vertex from each of $A , B , C$ such that the two triangles do not intersect.

Let $f ( x ) , g ( x )$ be functions on the real line $\mathbb { R }$ such that both $f ( x ) + g ( x )$ and $f ( x ) g ( x )$ are differentiable. Which of the following is FALSE ?
(A) $f ( x ) ^ { 2 } + g ( x ) ^ { 2 }$ is necessarily differentiable.
(B) $f ( x )$ is differentiable if and only if $g ( x )$ is differentiable.
(C) $f ( x )$ and $g ( x )$ are necessarily continuous.
(D) If $f ( x ) > g ( x )$ for all $x \in \mathbb { R }$, then $f ( x )$ is differentiable.

Let $S$ be the set consisting of all those real numbers that can be written as $p - 2 a$ where $p$ and $a$ are the perimeter and area of a right-angled triangle having base length 1 . Then $S$ is
(A) $( 2 , \infty )$
(B) $( 1 , \infty )$
(C) $( 0 , \infty )$
(D) the real line $\mathbb { R }$.

Let $S = \{ 1,2 , \ldots , n \}$. For any non-empty subset $A$ of $S$, let $l ( A )$ denote the largest number in $A$. If $f ( n ) = \sum _ { A \subseteq S } l ( A )$, that is, $f ( n )$ is the sum of the numbers $l ( A )$ while $A$ ranges over all the nonempty subsets of $S$, then $f ( n )$ is
(A) $2 ^ { n } ( n + 1 )$
(B) $2 ^ { n } ( n + 1 ) - 1$
(C) $2 ^ { n } ( n - 1 )$
(D) $2 ^ { n } ( n - 1 ) + 1$.

For any real number $x$, let $[ x ]$ be the greatest integer $m$ such that $m \leq x$. Then the number of points of discontinuity of the function $g ( x ) = \left[ x ^ { 2 } - 2 \right]$ on the interval $( - 3,3 )$ is
(A) 5
(B) 9
(C) 13
(D) 16 .

Define $a = p ^ { 3 } + p ^ { 2 } + p + 11$ and $b = p ^ { 2 } + 1$, where $p$ is any prime number. Let $d = \operatorname { gcd } ( a , b )$. Then the set of possible values of $d$ is
(A) $\{ 1,2,5 \}$.
(B) $\{ 2,5,10 \}$.
(C) $\{ 1,5,10 \}$.
(D) $\{ 1,2,10 \}$.

Let us denote the fractional part of a real number $x$ by $\{ x \}$ (note: $\{ x \} = x - [ x ]$ where $[ x ]$ is the integer part of $x$ ). Then, $$\lim _ { n \rightarrow \infty } \left\{ ( 3 + 2 \sqrt { 2 } ) ^ { n } \right\}$$ (A) equals 0 .
(B) equals 1 .
(C) equals $\frac { 1 } { 2 }$.
(D) does not exist.

Let $$\begin{gathered} p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2 x , x \in \mathbb { R } \\ f _ { 0 } ( x ) = \begin{cases} \int _ { 0 } ^ { x } p ( t ) d t , & x \geq 0 \\ - \int _ { x } ^ { 0 } p ( t ) d t , & x < 0 \end{cases} \\ f _ { 1 } ( x ) = e ^ { f _ { 0 } ( x ) } , \quad f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) } , \quad \ldots \quad , f _ { n } ( x ) = e ^ { f _ { n - 1 } ( x ) } \end{gathered}$$ How many roots does the equation $\frac { d f _ { n } ( x ) } { d x } = 0$ have in the interval $( - \infty , \infty ) ?$
(A) 1 .
(B) 3 .
(C) $n + 3$.
(D) $3n$.

The sides of a regular hexagon $A B C D E F$ is extended by doubling them to form a bigger hexagon $A ^ { \prime } B ^ { \prime } C ^ { \prime } D ^ { \prime } E ^ { \prime } F ^ { \prime }$ as in the figure below. Then the ratio of the areas of the bigger to the smaller hexagon is:
(A) $\sqrt { 3 }$
(B) 3
(C) $2 \sqrt { 3 }$
(D) 4

A $3 \times 3$ magic square is a $3 \times 3$ rectangular array of positive integers such that the sum of the three numbers in any row, any column or any of the two major diagonals, is the same. For the following incomplete magic square

27	36
31

the column sum is
(A) 90
(B) 96
(C) 94
(D) 99

The number of triples $( a , b , c )$ of positive integers satisfying the equation $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = 1 + \frac { 2 } { a b c }$$ and such that $a < b < c$, equals:
(A) 3
(B) 2
(C) 1
(D) 0

In the following diagram, four triangles and their sides are given. Areas of three of them are also given. Find the area $x$ of the remaining triangle. The four triangles have areas 4, 5, $x$, and 13 respectively.
(A) 12
(B) 13
(C) 14
(D) 15

Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ be a continuous function which is differentiable on $( 0,1 )$. Prove that either $f$ is a linear function $f ( x ) = a x + b$ or there exists $t \in ( 0,1 )$ such that $| f ( 1 ) - f ( 0 ) | < \left| f ^ { \prime } ( t ) \right|$.

Let $$S = \left\{ \left( \theta \sin \frac { \pi \theta } { 1 + \theta } , \frac { 1 } { \theta } \cos \frac { \pi \theta } { 1 + \theta } \right) : \theta \in \mathbb { R } , \theta > 0 \right\}$$ and $$T = \left\{ ( x , y ) : x \in \mathbb { R } , y \in \mathbb { R } , x y = \frac { 1 } { 2 } \right\}$$ How many elements does $S \cap T$ have?
(A) 0
(B) 1
(C) 2
(D) 3

The limit $$\lim _ { n \rightarrow \infty } n ^ { - \frac { 3 } { 2 } } \left( ( n + 1 ) ^ { ( n + 1 ) } ( n + 2 ) ^ { ( n + 2 ) } \ldots ( 2 n ) ^ { ( 2 n ) } \right) ^ { \frac { 1 } { n ^ { 2 } } }$$ equals
(A) 0.
(B) 1.
(C) $e ^ { - \frac { 1 } { 4 } }$.
(D) $4 e ^ { - \frac { 3 } { 4 } }$.

Suppose $x$ and $y$ are positive integers. If $4 x + 3 y$ and $2 x + 4 y$ are divided by 7, then the respective remainders are 2 and 5. If $11 x + 5 y$ is divided by 7, then the remainder equals
(A) 0.
(B) 1.
(C) 2.
(D) 3.

The limit $$\lim _ { x \rightarrow 0 } \frac { 1 } { x } \left( \cos ( x ) + \cos \left( \frac { 1 } { x } \right) - \cos ( x ) \cos \left( \frac { 1 } { x } \right) - 1 \right)$$ (A) equals 0.
(B) equals $\frac { 1 } { 2 }$.
(C) equals 1.
(D) does not exist.