The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e^{-x}$ is:
(A) $1/e$
(B) 1
(C) $1/2$
(D) $e$

The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e ^ { - x }$ is:
(A) $1 / e$
(B) 1
(C) $1 / 2$
(D) $e$

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

If $f(x) = \cos(x) - 1 + \frac{x^2}{2}$, then
(A) $f(x)$ is an increasing function on the real line
(B) $f(x)$ is a decreasing function on the real line
(C) $f(x)$ is increasing on $-\infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f(x)$ is decreasing on $-\infty < x \leq 0$ and increasing on $0 \leq x < \infty$

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(A) $f ( x )$ is an increasing function on the real line
(B) $f ( x )$ is a decreasing function on the real line
(C) $f ( x )$ is increasing on $- \infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f ( x )$ is decreasing on $- \infty < x \leq 0$ and increasing on $0 \leq x < \infty$

A circular lawn of diameter 20 meters on a horizontal plane is to be illuminated by a light-source placed vertically above the centre of the lawn. It is known that the illuminance at a point $P$ on the lawn is given by the formula $I = \frac{C \sin\theta}{d^2}$ for some constant $C$, where $d$ is the distance of $P$ from the light-source and $\theta$ is the angle between the line joining the centre of the lawn to $P$ and the line joining the light-source to $P$. Then the maximum possible illuminance at a point on the circumference of the lawn is
(A) $\frac{C}{75\sqrt{3}}$
(B) $\frac{C}{100\sqrt{3}}$
(C) $\frac{C}{150\sqrt{3}}$
(D) $\frac{C}{250\sqrt{3}}$.

For a positive real number $\alpha$, let $S_\alpha$ denote the set of points $(x, y)$ satisfying $$|x|^\alpha + |y|^\alpha = 1$$ A positive number $\alpha$ is said to be good if the points in $S_\alpha$ that are closest to the origin lie only on the coordinate axes. Then
(A) all $\alpha$ in $(0,1)$ are good and others are not good.
(B) all $\alpha$ in $(1,2)$ are good and others are not good.
(C) all $\alpha > 2$ are good and others are not good.
(D) all $\alpha > 1$ are good and others are not good.

Let $f ( x )$ be a degree 4 polynomial with real coefficients. Let $z$ be the number of real zeroes of $f$, and $e$ be the number of local extrema (i.e., local maxima or minima) of $f$. Which of the following is a possible $( z , e )$ pair?
(A) $( 4,4 )$
(B) $( 3,3 )$
(C) $( 2,2 )$
(D) $( 0,0 )$

Let $S = \left\{ x - y \mid x , y \text{ are real numbers with } x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Then the maximum number in the set $S$ is
(A) 1
(B) $\sqrt { 2 }$
(C) $2 \sqrt { 2 }$
(D) $1 + \sqrt { 2 }$.

Let $A B C D$ be a rectangle with its shorter side $a > 0$ units and perimeter $2 s$ units. Let $P Q R S$ be any rectangle such that vertices $A , B , C$ and $D$ respectively lie on the lines $P Q , Q R , R S$ and $S P$. Then the maximum area of such a rectangle $P Q R S$ in square units is given by
(A) $s ^ { 2 }$
(B) $2 a ( s - a )$
(C) $\frac { s ^ { 2 } } { 2 }$
(D) $\frac { 5 } { 2 } a ( s - a )$.

Consider the following subsets of the plane: $$C_{1} = \left\{(x, y) : x > 0,\ y = \frac{1}{x}\right\}$$ and $$C_{2} = \left\{(x, y) : x < 0,\ y = -1 + \frac{1}{x}\right\}$$ Given any two points $P = (x, y)$ and $Q = (u, v)$ of the plane, their distance $d(P, Q)$ is defined by $$d(P, Q) = \sqrt{(x - u)^{2} + (y - v)^{2}}$$ Show that there exists a unique choice of points $P_{0} \in C_{1}$ and $Q_{0} \in C_{2}$ such that $$d(P_{0}, Q_{0}) \leq d(P, Q) \quad \text{for all } P \in C_{1} \text{ and } Q \in C_{2}.$$

Let $f$ be a real-valued differentiable function defined on the real line $\mathbb { R }$ such that its derivative $f ^ { \prime }$ is zero at exactly two distinct real numbers $\alpha$ and $\beta$. Then,
(A) $\alpha$ and $\beta$ are points of local maxima of the function $f$.
(B) $\alpha$ and $\beta$ are points of local minima of the function $f$.
(C) one must be a point of local maximum and the other must be a point of local minimum of $f$.
(D) given data is insufficient to conclude about either of them being local extrema points.

Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius 1 is regular (i.e., has equal sides).

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 $f ( x ) = \sin x + \alpha x , x \in \mathbb { R }$, where $\alpha$ is a fixed real number. The function $f$ is one-to-one if and only if
(A) $\alpha > 1$ or $\alpha < - 1$.
(B) $\alpha \geq 1$ or $\alpha \leq - 1$.
(C) $\alpha \geq 1$ or $\alpha < - 1$.
(D) $\alpha > 1$ or $\alpha \leq - 1$.

The polynomial $x ^ { 4 } + 4 x + c = 0$ has at least one real root if and only if
(A) $c < 2$.
(B) $c \leq 2$.
(C) $c < 3$.
(D) $c \leq 3$.

Consider the function $$f(x) = \sum_{k=1}^{m} (x-k)^4, \quad x \in \mathbb{R}$$ where $m > 1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.

If $x , y$ are positive real numbers such that $3 x + 4 y < 72$, then the maximum possible value of $12 x y ( 72 - 3 x - 4 y )$ is:
(A) 12240
(B) 13824
(C) 10656
(D) 8640

The function $x ^ { 2 } \log _ { e } x$ in the interval $( 0,2 )$ has:
(A) exactly one point of local maximum and no points of local minimum.
(B) exactly one point of local minimum and no points of local maximum.
(C) points of local maximum as well as local minimum.
(D) neither a point of local maximum nor a point of local minimum.

Suppose $F : \mathbb { R } \rightarrow \mathbb { R }$ is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) $F$ cannot have a local minimum.
(B) $F$ must have exactly one local minimum.
(C) $F$ must have at least two local minima.
(D) $F$ must have either a global maximum or a local minimum.

Suppose that $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ where $a , b , c , d$ are real numbers with $a \neq 0$. The equation $f ( x ) = 0$ has exactly two distinct real solutions. If $f ^ { \prime } ( x )$ is the derivative of $f ( x )$, then which of the following is a possible graph of $f ^ { \prime } ( x )$?
(A), (B), (C), (D) [graphs as provided in the figure]

Let $f(x) = x^x$ for $x > 0$. Then $f$ is
(A) increasing on $(0, \infty)$
(B) decreasing on $(0, \infty)$
(C) increasing on $(0, 1/e)$ and decreasing on $(1/e, \infty)$
(D) decreasing on $(0, 1/e)$ and increasing on $(1/e, \infty)$

Let $f(x) = \frac{x}{\sqrt{a^2+x^2}} - \frac{d-x}{\sqrt{b^2+(d-x)^2}}$, where $a$, $b$, and $d$ are positive constants. Then
(A) $f$ is an increasing function of $x$
(B) $f$ is a decreasing function of $x$
(C) $f$ is neither increasing nor decreasing function of $x$
(D) $f'$ is not a monotonic function of $x$

Let $f(x) = 2x^3 - 3x^2 - 12x + 4$. Then
(A) $f$ has a local maximum at $x = -1$ and a local minimum at $x = 2$
(B) $f$ has a local minimum at $x = -1$ and a local maximum at $x = 2$
(C) $f$ has local minima at $x = -1$ and at $x = 2$
(D) $f$ has local maxima at $x = -1$ and at $x = 2$