Consider a rectangular cardboard box of height 3, breadth 4 and length 10 units. There is a lizard in one corner $A$ of the box and an insect in the corner $B$ which is farthest from $A$. The length of the shortest path between the lizard and the insect along the surface of the box is
(a) $\sqrt{5^{2} + 10^{2}}$
(b) $\sqrt{7^{2} + 10^{2}}$
(c) $4 + \sqrt{3^{2} + 10^{2}}$
(d) $3 + \sqrt{4^{2} + 10^{2}}$

An aeroplane $P$ is moving in the air along a straight line path which passes through the points $P_1$ and $P_2$, and makes an angle $\alpha$ with the ground. Let $O$ be the position of an observer. When the plane is at the position $P_1$ its angle of elevation is $30^\circ$ and when it is at $P_2$ its angle of elevation is $60^\circ$ from the position of the observer. Moreover, the distances of the observer from the points $P_1$ and $P_2$ respectively are 100 metres and $500/3$ metres.
Then $\alpha$ is equal to
(a) $\tan^{-1}\{(2-\sqrt{3})/(2\sqrt{3}-1)\}$
(b) $\tan^{-1}\{(2\sqrt{3}-3)/(4-2\sqrt{3})\}$
(c) $\tan^{-1}\{(2\sqrt{3}-2)/(5-\sqrt{3})\}$
(d) $\tan^{-1}\{(6-\sqrt{3})/(6\sqrt{3}-1)\}$

The sum of all even positive divisors of $1000$ is
(a) $2170$
(b) $2184$
(c) $2325$
(d) $2340$

The digit at the unit place of $$(1! - 2! + 3! - \ldots + 25!)^{(1! - 2! + 3! - \ldots + 25!)}$$ is
(a) 0
(b) 1
(c) 5
(d) 9

Consider an equilateral triangle $ABC$ with side 2.1 cm. You want to place a number of smaller equilateral triangles, each with side 1 cm, over the triangle $ABC$, so that the triangle $ABC$ is fully covered. What is the minimum number of smaller triangles that you need?
(a) 4
(b) 5
(c) 6
(d) 7.

A regular tetrahedron has all its vertices on a sphere of radius $R$. Then the length of each edge of the tetrahedron is
(a) $( \sqrt{2} / \sqrt{3} ) R$
(b) $( \sqrt{3} / 2 ) R$
(c) $( 4 / 3 ) R$
(d) $( 2 \sqrt{2} / \sqrt{3} ) R$

Consider the L-shaped brick in the diagram below. If an ant starts from $A$, find the minimum distance it has to travel along the surface to reach $B$.
(a) $\sqrt{5}$
(b) $2\sqrt{5}$
(c) $( 3 / 2 ) \sqrt{5}$
(d) $3\sqrt{5}$

A room is in the shape of a rectangular box. The shortest path along the surface from one corner $A$ to the opposite corner $B$ has length $\sqrt{29}$ (given the relevant dimensions are $5$ and $2$). Find this shortest distance.

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.

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

Find the number of six-digit numbers using digits from $\{2, 3, 9\}$ (repetition allowed) that are divisible by 6.
(A) 80 (B) 82 (C) 81 (D) 83

For what values of $a$ does $\displaystyle\lim_{x \to 0} \frac{\sin^a x}{x} = 0$?
(A) $a \geq 1$ (B) $a > 1$ (C) $a \leq 1$ (D) All real $a$

Let $f$ be a differentiable function with $f(3) \neq 0$. Evaluate $\displaystyle\lim_{x \to \infty} \left(\frac{f(3 + 1/x)}{f(3)}\right)^x$.
(A) $e^{f'(3)/f(3)}$ (B) $e^{f(3)}$ (C) $e^{f'(3)}$ (D) 1

If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then
(a) $n$ must be odd
(b) $n$ cannot be a perfect square
(c) $2 n$ cannot be a perfect square
(d) none of the above.

The digit in the units' place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is
(a) 3
(b) 0
(c) 1
(d) 7.

The digit in the units' place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is
(a) 3
(b) 0
(c) 1
(d) 7.

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 $a b$ 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.

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

The limit $$\lim _ { x \rightarrow 0 } \frac { \left( e ^ { x } - 1 \right) \tan ^ { 2 } x } { x ^ { 3 } }$$ (A) does not exist
(B) exists and equals 0
(C) exists and equals $2/3$
(D) exists and equals 1

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$