If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4n^2$ unit squares by drawing $2n-1$ horizontal and $2n-1$ vertical lines one unit apart. A circle of diameter $2n-1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4n - 2$
(B) $4n$
(C) $8n - 4$
(D) $8n - 2$

An isosceles triangle with base 6 cms. and base angles $30^\circ$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3\sqrt{3}/2$
(B) $\sqrt{3}/2$
(C) $\sqrt{3}$
(D) $4/\sqrt{3}$

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players A, B, C, D, E and F play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, B has 8 points and C has 4 points. It is also known that E won against F. In the next set of games D, E and F win their games against A, B and C respectively. If A, B and D move to the final round, the final scores of E and F are, respectively,
(A) 4 and 2
(B) 2 and 4
(C) 2 and 2
(D) 4 and 4

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players A, B, C, D, E and F play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, B has 8 points and C has 4 points. It is also known that E won against F. In the next set of games D, E and F win their games against A, B and C respectively. If A, B and D move to the final round, the final scores of E and F are, respectively,
(A) 4 and 2
(B) 2 and 4
(C) 2 and 2
(D) 4 and 4

The digit in the unit's place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is
(A) 3
(B) 0
(C) 1
(D) 7

For each positive integer $n$, define a function $f_n$ on $[0,1]$ as follows: $$f_n(x) = \left\{ \begin{array}{ccc} 0 & \text{if} & x = 0 \\ \sin\frac{\pi}{2n} & \text{if} & 0 < x \leq \frac{1}{n} \\ \sin\frac{2\pi}{2n} & \text{if} & \frac{1}{n} < x \leq \frac{2}{n} \\ \sin\frac{3\pi}{2n} & \text{if} & \frac{2}{n} < x \leq \frac{3}{n} \\ \vdots & \vdots & \vdots \\ \sin\frac{n\pi}{2n} & \text{if} & \frac{n-1}{n} < x \leq 1 \end{array} \right.$$ Then, the value of $\lim_{n \rightarrow \infty} \int_0^1 f_n(x) dx$ is
(A) $\pi$
(B) 1
(C) $\frac{1}{\pi}$
(D) $\frac{2}{\pi}$

For each positive integer $n$, define a function $f _ { n }$ on $[ 0, 1 ]$ as follows: $$f _ { n } ( x ) = \left\{ \begin{array} { c c c } 0 & \text { if } & x = 0 \\ \sin \frac { \pi } { 2 n } & \text { if } & 0 < x \leq \frac { 1 } { n } \\ \sin \frac { 2 \pi } { 2 n } & \text { if } & \frac { 1 } { n } < x \leq \frac { 2 } { n } \\ \sin \frac { 3 \pi } { 2 n } & \text { if } & \frac { 2 } { n } < x \leq \frac { 3 } { n } \\ \vdots & \vdots & \vdots \\ \sin \frac { n \pi } { 2 n } & \text { if } & \frac { n - 1 } { n } < x \leq 1 \end{array} \right.$$ Then, the value of $\lim _ { n \rightarrow \infty } \int _ { 0 } ^ { 1 } f _ { n } ( x ) d x$ is
(A) $\pi$
(B) 1
(C) $\frac { 1 } { \pi }$
(D) $\frac { 2 } { \pi }$

Let $d _ { 1 } , d _ { 2 } , \ldots , d _ { k }$ be all the factors of a positive integer $n$ including 1 and $n$. If $d _ { 1 } + d _ { 2 } + \ldots + d _ { k } = 72$, then $\frac { 1 } { d _ { 1 } } + \frac { 1 } { d _ { 2 } } + \cdots + \frac { 1 } { d _ { k } }$ is:
(A) $\frac { k ^ { 2 } } { 72 }$
(B) $\frac { 72 } { k }$
(C) $\frac { 72 } { n }$
(D) none of the above

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a, b \in W$, the numbers $a - b$ and $ab$ are also in $W$. Let $S = \left\{ \left. \frac{m}{2^n} \right\rvert\, m, n \text{ integers} \right\}$ and $T = \left\{ \left. \frac{p}{q} \right\rvert\, p, q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings

A unit square has its corners chopped off to form a regular polygon with eight sides. What is the area of this polygon?
(A) $2(\sqrt{3} - \sqrt{2})$
(B) $2\sqrt{2} - 2$
(C) $\frac{\sqrt{2}}{2}$
(D) $\frac{7}{9}$.

A solid cube of side five centimeters has all its faces painted. The cube is sliced into smaller cubes, each of side one centimeter. How many of these smaller cubes will have paint on exactly one of its faces?
(A) 25
(B) 54
(C) 126
(D) 150.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for any two real numbers $x$ and $y$, $$|f(x) - f(y)| \leq 7|x - y|^{201}$$ Then,
(A) $f(101) = f(202) + 8$
(B) $f(101) = f(201) + 1$
(C) $f(101) = f(200) + 2$
(D) None of the above.

In a factory, 20 workers start working on a project of packing consignments. They need exactly 5 hours to pack one consignment. Every hour 4 new workers join the existing workforce. It is mandatory to relieve a worker after 10 hours. Then the number of consignments that would be packed in the initial 113 hours is
(A) 40
(B) 50
(C) 45
(D) 52.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two functions. Consider the following two statements: $\mathbf { P ( 1 ) }$: If $\lim _ { x \rightarrow 0 } f ( x )$ exists and $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$ exists, then $\lim _ { x \rightarrow 0 } g ( x )$ must exist. $\mathbf { P ( 2 ) }$: If $f , g$ are differentiable with $f ( x ) < g ( x )$ for every real number $x$, then $f ^ { \prime } ( x ) < g ^ { \prime } ( x )$ for all $x$. Then, which one of the following is a correct statement?
(A) Both $\mathrm { P } ( 1 )$ and $\mathrm { P } ( 2 )$ are true.
(B) Both $P ( 1 )$ and $P ( 2 )$ are false.
(C) $\mathrm { P } ( 1 )$ is true and $\mathrm { P } ( 2 )$ is false.
(D) $\mathrm { P } ( 1 )$ is false and $\mathrm { P } ( 2 )$ is true.

A party is attended by twenty people. In any subset of four people, there is at least one person who knows the other three (we assume that if $X$ knows $Y$, then $Y$ knows $X$). Suppose there are three people in the party who do not know each other. How many people in the party know everyone?
(A) 16
(B) 17
(C) 18
(D) Cannot be determined from the given data.

The sides of a regular hexagon $A B C D E F$ are extended by doubling them (for example, $B A$ extends to $B A ^ { \prime }$ with $B A ^ { \prime } = 2 B A$) to form a bigger regular hexagon $A ^ { \prime } B ^ { \prime } C ^ { \prime } D ^ { \prime } E ^ { \prime } F ^ { \prime }$. Then, the ratio of the areas of the bigger to the smaller hexagon is:
(A) 2
(B) 3
(C) $2 \sqrt { 3 }$
(D) $\pi$.

Between 12 noon and 1 PM, there are two instants when the hour hand and the minute hand of a clock are at right angles. The difference in minutes between these two instants is:
(A) $32 \frac { 8 } { 11 }$
(B) $30 \frac { 8 } { 11 }$
(C) $32 \frac { 5 } { 11 }$
(D) $30 \frac { 5 } { 11 }$.

Assume that $n$ copies of unit cubes are glued together side by side to form a rectangular solid block. If the number of unit cubes that are completely invisible is 30, then the minimum possible value of $n$ is:
(A) 204
(B) 180
(C) 140
(D) 84.

You are given a $4 \times 4$ chessboard, and asked to fill it with five $3 \times 1$ pieces and one $1 \times 1$ piece. Then, over all such fillings, the number of squares that can be occupied by the $1 \times 1$ piece is
(A) 4
(B) 8
(C) 12
(D) 16 .

Consider a paper in the shape of an equilateral triangle $A B C$ with circumcenter $O$ and perimeter 9 units. If we fold the paper in such a way that each of the vertices $A , B , C$ gets identified with $O$, then the area of the resulting shape in square units is:
(A) $\frac { 3 \sqrt { 3 } } { 4 }$
(B) $\frac { 4 } { \sqrt { 3 } }$
(C) $\frac { 3 \sqrt { 3 } } { 2 }$
(D) $3 \sqrt { 3 }$.

Let $P$ be a regular twelve-sided polygon. The number of right-angled triangles formed by the vertices of $P$ is
(A) 60
(B) 120
(C) 160
(D) 220 .

A subset $S$ of the plane is called convex if given any two points $x$ and $y$ in $S$, the line segment joining $x$ and $y$ is contained in $S$. A quadrilateral is called convex if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral $Q$ of area 1, there is a rectangle $R$ of area 2 such that $Q$ can be drawn inside $R$.