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# ISI B.Stat 2010 Objective Paper| problems & solutions

Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

There are 8 balls numbered $1,2, \ldots, 8$ and 8 boxes numbered $1,2, \ldots, 8$. The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is

(A) $3 \times\left(\begin{array}{l}8 \\ 4\end{array}\right)$;
(B) $6 \times\left(\begin{array}{l}8 \\ 4\end{array}\right)$;
(C) $9 \times\left(\begin{array}{c}8 \\ 4\end{array}\right)$;
(D) $12 \times\left(\begin{array}{l}8 \\ 4\end{array}\right)$.

Problem 2:

Let $\alpha$ and $\beta$ be two positive real numbers. For every integer $n>0$, define

$$a_{n}=\int_{\beta}^{n} \frac{\alpha}{u\left(u^{\alpha}+2+u^{-\alpha}\right)} d u$$

Then $\lim_{n \rightarrow \infty} a_{n}$ is equal to

(A) $\frac{1}{1+\beta^{\alpha}}$
(B) $\frac{\beta^{\alpha}}{1+\beta^{-\alpha}}$
(C) $\frac{\beta^{\alpha}}{1+\beta^{\alpha}} ;$
(D) $\frac{\beta^{-\alpha}}{1+\beta^{\alpha}}$.

Problem 3:

Let $f: \mathbb{R} \rightarrow \mathbb{R}^{2}$ be a function given by $f(x)=\left(x^{m}, x^{n}\right),$ where $x \in \mathbb{R}$ and $m, n$ are fixed positive integers. Suppose that $f$ is one-one. Then

(A) both $m$ and $n$ must be odd;
(B) at least one of $m$ and $n$ must be odd;
(C) exactly one of $m$ and $n$ must be odd;
(D) neither $m$ nor $n$ can be odd.

Problem 4:

$\lim _{x \rightarrow 2} \frac{e^{x^{2}}-e^{2 x}}{(x-2) e^{2 x}}$ equals

(A) $0$ ;
(B) $1$ ;
(C) $2$;
(D) $3$

Problem 5:

A circle is inscribed in a triangle with sides $8,15$ and $17$ centimetres. The radius of the circle in centimetres is

(A) $3$;
(B) $22 / 7$;
(C) $4$ ;
(D) none of the above.

Problem 6:

Let $\alpha, \beta$ and $\gamma$ be the angles of an acute angled triangle. Then the quantity $\tan \alpha \tan \beta \tan \gamma$

(A) can have any real value;
(B) is $\leq 3 \sqrt{3}$;
(C) is $\geq 3 \sqrt{3}$;
(D) none of the above.

Problem 7:

Let $f(x)=|x| \sin x+|x-\pi| \cos x$ for $x \in \mathbb{R}$. Then

(A) $f$ is differentiable at $x=0$ and $x=\pi ;$
(B) $f$ is not differentiable at $x=0$ and $x=\pi$;
(C) $f$ is differentiable at $x=0$ but not differentiable at $x=\pi ;$
(D) $f$ is not differentiable at $x=0$ but differentiable at $x=\pi$.

Problem 8:

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) $\sqrt{5^{2}+10^{2}}$ units;
(B) $\sqrt{7^{2}+10^{2}}$ units;
(C) $4+\sqrt{3^{2}+10^{2}}$ units;
(D) $3+\sqrt{10^{2}+4^{2}}$ units.

Problem 9:

Recall that, for any non-zero complex number $w$ which does not lie on the negative real axis, arg $w$ denotes the unique real number $\theta$ in $(-\pi, \pi)$ such that
$$w=|w|(\cos \theta+i \sin \theta)$$
Let $z$ be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that $z$ satisfies the relations
$$\arg z>\arg (z+1) \text { and } \arg z>\arg (z+i)$$
Then $\cos (\arg z)$ can take

(A) any value in the set $(-1 / 2,0) \cup(0,1 / 2)$ but none from outside;
(B) any value in the interval (-1,0) but none from outside;
(C) any value in the interval (0,1) but none from outside;
(D) any value in the set (-1,0)$\cup(0,1)$ but none from outside.

Problem 10:

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 as shown in the figure below. 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 1000 metres and $500 / 3$ metres.

Then $\alpha$ is equal to

(A) $\tan ^{-1}\left(\frac{2-\sqrt{3}}{2 \sqrt{3}-1}\right)$;
(B) $\tan ^{-1}\left(\frac{2 \sqrt{3}-3}{4-2 \sqrt{3}}\right)$;
(C) $\tan ^{-1}\left(\frac{2 \sqrt{3}-2}{5-\sqrt{3}}\right)$;
(D) $\tan ^{-1}\left(\frac{6-\sqrt{3}}{6 \sqrt{3}-1}\right)$.

Problem 11:

The sum of all even positive divisors of $1000$ is

(A) $2170$ ;
(B) $2184$ ;
(C) $2325$;
(D) $2340$ .

Problem 12:

The equation $x^{2}+\frac{b}{a} x+\frac{c}{a}=0$ has two real roots $\alpha$ and $\beta .$ If $a>0,$ then the area under the curve $f(x)=x^{2}+\frac{b}{a} x+\frac{c}{a}$ between $\alpha$ and $\beta$ is

(A) $\frac{b^{2}-4 a c}{2 a}$;
(B) $\frac{\left(b^{2}-4 a c\right)^{3 / 2}}{6 a^{3}}$
(C) $-\frac{\left(b^{2}-4 a c\right)^{3 / 2}}{6 a^{3}}$
(D) $-\frac{b^{2}-4 a c}{2 a}$.

Problem 13:

The minimum value of $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$ subject to $x_{1}+x_{2}+x_{3}+x_{4}=a$ and $x_{1}-x_{2}+x_{3}-x_{4}=b$ is

(A) $\frac{a^{2}+b^{2}}{4}$
(B) $\frac{a^{2}+b^{2}}{2}$;
(C) $\frac{(a+b)^{2}}{4}$;
(D) $\frac{(a+b)^{2}}{2}$.

Problem 14:

The value of

$$\lim_{n\to\infty}\frac{\sum_{r=0}^{n} \left(\begin{array} {c} 2n\\2r\end{array}\right)3^{r}}{\sum_{r=0}^{n-1} \left(\begin{array} {c}2n\\2r+1\end{array}\right)3^{r}}$$ is

(A) $0 ;$
(B) $1$ ;
(C) $\sqrt{3}$;
(D) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$.

Problem 15:

For any real number $x$, let $\tan ^{-1}(x)$ denote the unique real number $\theta$ in $(-\pi / 2, \pi / 2)$ such that $\tan \theta=x$. Then

$$\lim _{n \rightarrow \infty} \sum_{m=1}^{n} \tan ^{-1} \frac{1}{1+m+m^{2}}$$

(A) is equal to $\pi / 2$;
(B) is equal to $\pi / 4$;
(C) does not exist;
(D) none of the above

Problem 16:

Let $n$ be an integer. The number of primes which divide both $n^{2}-1$ and $(n+1)^{2}-1$ is

(A) at most one;
(B) exactly one;
(C) exactly two;
(D) none of the above.

Problem 17:

The value of

$$\lim_{n \rightarrow \infty} \sum_{r=1}^{n} \frac{6 n}{9 n^{2}-r^{2}}$$ is

(A) $0$ ;
(B) $\log \frac{3}{2}$;
(C) $\log \frac{2}{3}$;
(D) $\log 2$.

Problem 18:

A person $X$ standing at a point $P$ on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps $X$ comes back to the original position $P$. Then the number of distinct paths that $X$ can take is

(A) $196$;
(B) $256$ ;
(C) $344$;
(D) $400$ .

Problem 19:

Consider the branch of the rectangular hyperbola $x y=1$ in the first quadrant. Let $P$ be a fixed point on this curve. The locus of the mid-point of the line segment joining $P$ and an arbitrary point $Q$ on the curve is part of

(A) a hyperbolas;

(B) a parabola;

(C) an ellipse;
(D) none of the above.

Problem 20:

The digit at the unit place of $(1 !-2 !+3 !-\cdots+25 !)^{(1!-2!+3!-\cdots+25!)}$ is

(A) $0$ ;
(B) $1 ;$
(C) $5$;
(D) $9$ .

Problem 21:

Let $A_{1}, A_{2}, \ldots, A_{n}$ be the vertices of a regular polygon and $A_{1} A_{2}, A_{2} A_{3} \ldots \ldots$ $A_{n-1} A_{n}, A_{n} A_{1}$ be its $n$ sides. If

$$\frac{1}{A_{1} A_{2}}-\frac{1}{A_{1} A_{4}}=\frac{1}{A_{1} A_{3}}$$

then the value of $n$ is

(A) $5$ ;
(B) $6$ ;
(C) $7$ ;
(D) $8$ .

Problem 22:

Suppose that $\alpha$ and $\beta$ are two distinct numbers in the interval $(0, \pi)$. If

$$\sin \alpha+\sin \beta=\sqrt{3}(\cos \alpha-\cos \beta)$$

then the value of $\sin 3 \alpha+\sin 3 \beta$ is

(A) $0$;
(B) $2 \sin \frac{3(\alpha+\beta)}{2}$;
(C) $2 \cos \frac{3(\alpha-\beta)}{2}$
(D) $\cos \frac{3(\alpha-\beta)}{2}$

Problem 23:

Consider the function $h(x)=x^{2}-2 x+2+\frac{4}{x^{2}-2 x+2}, x \in \mathbb{R} .$ Then $h(x)-5=0$ has

(A) no solution;
(B) only one solution;
(C) exactly two solutions;
(D) exactly three solutions.

Problem 24:

Consider the quadratic equation $x^{2}+b x+c=0 .$ The number of pairs $(b, c)$ for which the equation has solutions of the form $\cos \alpha$ and $\sin \alpha$ for some $\alpha$ is

(A) $0 ;$
(B) $1$ ;
(C) $2$;
(D) infinite.

Problem 25:

Let $\theta_{1}=\frac{2 \pi}{3}, \theta_{2}=\frac{4 \pi}{7}, \theta_{3}=\frac{7 \pi}{12} .$ Then

(A) $\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{2}\right)^{\sin \theta_{2}}<\left(\sin \theta_{3}\right)^{\sin \theta_{3}}$
(B) $\left(\sin \theta_{2}\right)^{\sin \theta_{2}}<\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{3}\right)^{\sin \theta_{3}}$
(C) $\left(\sin \theta_{3}\right)^{\sin \theta_{3}}<\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{2}\right)^{\sin \theta_{2}}$
(D) $\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{3}\right)^{\sin \theta_{3}}<\left(\sin \theta_{2}\right)^{\sin \theta_{2}}$.

Problem 26:

Consider the following two curves on the interval (0,1) :

$$C_{1}: y=1-x^{4} \text { and } C_{2}: y=\sqrt{1-x^{2}}$$

Then on $(0,1)$

(A) $C_{1}$ lies above $C_{2}$;
(B) $C_{2}$ lies above
(C) $C_{1}$ and $C_{2}$ intersect at exactly one point;
(D) none of the above.

Problem 27:

Let $f$ be a real valued function on $\mathbb{R}$ such that $f$ is twice differentiable. Suppose that $f^{\prime}$ vanishes only at 0 and $f^{\prime \prime}$ is everywhere negative. Define a function $h$ by $h(x)=(x-a)^{2}-f(x),$ where $a>0 .$ Then

(A) $h$ has a local minima in $(0, a)$;
(B) $h$ has a local maxima in $(0, a)$;
(C) $h$ is monotonically increasing in $(0, a)$;
(D) $h$ is monotonically decreasing in $(0, a)$.

Problem 28:

Consider the triangle with vertices $(1,2),(-5,-1)$ and $(3,-2)$ . Let $\Delta$ denote the region enclosed by the above triangle. Consider the function $f: \Delta \rightarrow \mathbb{R}$ defined by $f(x, y)=|10 x-3 y| .$ Then the range of $f$ is the interval

(A) $[0,36]$ ;
(B) $[0,47]$;
(C) $[4,47]$ ;
(D) $[36,47]$.

Problem 29:

For every positive integer $n$, let $\langle n\rangle$ denote the integer closest to $\sqrt{n}$. Let $A_{k}=\{n>0:\langle n\rangle=k\} .$ The number of elements in $A_{49}$ is

(A) $97$ ;
(B) $98 ;$
(C) $99$ ;
(D) $100$ .

Problem 30:

Consider a square $A B C D$ inscribed in a circle of radius $1 .$ Let $A^{\prime}$ and $C^{\prime}$ be two points on the (smaller) arcs $A D$ and $C D$ respectively, such that $A^{\prime} A B C C^{\prime}$ is a pentagon in which $A A^{\prime}=C C^{\prime}$. If $P$ denotes the area of the pentagon $A^{\prime} A B C C^{\prime}$ then

(A) $P$ can not be equal to $2$ ;
(B) $P$ lies in the interval $(1,2]$ ;
(C) $P$ is greater than or equal to $2$ ;
(D) none of the above.

Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

There are 8 balls numbered $1,2, \ldots, 8$ and 8 boxes numbered $1,2, \ldots, 8$. The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is

(A) $3 \times\left(\begin{array}{l}8 \\ 4\end{array}\right)$;
(B) $6 \times\left(\begin{array}{l}8 \\ 4\end{array}\right)$;
(C) $9 \times\left(\begin{array}{c}8 \\ 4\end{array}\right)$;
(D) $12 \times\left(\begin{array}{l}8 \\ 4\end{array}\right)$.

Problem 2:

Let $\alpha$ and $\beta$ be two positive real numbers. For every integer $n>0$, define

$$a_{n}=\int_{\beta}^{n} \frac{\alpha}{u\left(u^{\alpha}+2+u^{-\alpha}\right)} d u$$

Then $\lim_{n \rightarrow \infty} a_{n}$ is equal to

(A) $\frac{1}{1+\beta^{\alpha}}$
(B) $\frac{\beta^{\alpha}}{1+\beta^{-\alpha}}$
(C) $\frac{\beta^{\alpha}}{1+\beta^{\alpha}} ;$
(D) $\frac{\beta^{-\alpha}}{1+\beta^{\alpha}}$.

Problem 3:

Let $f: \mathbb{R} \rightarrow \mathbb{R}^{2}$ be a function given by $f(x)=\left(x^{m}, x^{n}\right),$ where $x \in \mathbb{R}$ and $m, n$ are fixed positive integers. Suppose that $f$ is one-one. Then

(A) both $m$ and $n$ must be odd;
(B) at least one of $m$ and $n$ must be odd;
(C) exactly one of $m$ and $n$ must be odd;
(D) neither $m$ nor $n$ can be odd.

Problem 4:

$\lim _{x \rightarrow 2} \frac{e^{x^{2}}-e^{2 x}}{(x-2) e^{2 x}}$ equals

(A) $0$ ;
(B) $1$ ;
(C) $2$;
(D) $3$

Problem 5:

A circle is inscribed in a triangle with sides $8,15$ and $17$ centimetres. The radius of the circle in centimetres is

(A) $3$;
(B) $22 / 7$;
(C) $4$ ;
(D) none of the above.

Problem 6:

Let $\alpha, \beta$ and $\gamma$ be the angles of an acute angled triangle. Then the quantity $\tan \alpha \tan \beta \tan \gamma$

(A) can have any real value;
(B) is $\leq 3 \sqrt{3}$;
(C) is $\geq 3 \sqrt{3}$;
(D) none of the above.

Problem 7:

Let $f(x)=|x| \sin x+|x-\pi| \cos x$ for $x \in \mathbb{R}$. Then

(A) $f$ is differentiable at $x=0$ and $x=\pi ;$
(B) $f$ is not differentiable at $x=0$ and $x=\pi$;
(C) $f$ is differentiable at $x=0$ but not differentiable at $x=\pi ;$
(D) $f$ is not differentiable at $x=0$ but differentiable at $x=\pi$.

Problem 8:

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) $\sqrt{5^{2}+10^{2}}$ units;
(B) $\sqrt{7^{2}+10^{2}}$ units;
(C) $4+\sqrt{3^{2}+10^{2}}$ units;
(D) $3+\sqrt{10^{2}+4^{2}}$ units.

Problem 9:

Recall that, for any non-zero complex number $w$ which does not lie on the negative real axis, arg $w$ denotes the unique real number $\theta$ in $(-\pi, \pi)$ such that
$$w=|w|(\cos \theta+i \sin \theta)$$
Let $z$ be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that $z$ satisfies the relations
$$\arg z>\arg (z+1) \text { and } \arg z>\arg (z+i)$$
Then $\cos (\arg z)$ can take

(A) any value in the set $(-1 / 2,0) \cup(0,1 / 2)$ but none from outside;
(B) any value in the interval (-1,0) but none from outside;
(C) any value in the interval (0,1) but none from outside;
(D) any value in the set (-1,0)$\cup(0,1)$ but none from outside.

Problem 10:

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 as shown in the figure below. 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 1000 metres and $500 / 3$ metres.

Then $\alpha$ is equal to

(A) $\tan ^{-1}\left(\frac{2-\sqrt{3}}{2 \sqrt{3}-1}\right)$;
(B) $\tan ^{-1}\left(\frac{2 \sqrt{3}-3}{4-2 \sqrt{3}}\right)$;
(C) $\tan ^{-1}\left(\frac{2 \sqrt{3}-2}{5-\sqrt{3}}\right)$;
(D) $\tan ^{-1}\left(\frac{6-\sqrt{3}}{6 \sqrt{3}-1}\right)$.

Problem 11:

The sum of all even positive divisors of $1000$ is

(A) $2170$ ;
(B) $2184$ ;
(C) $2325$;
(D) $2340$ .

Problem 12:

The equation $x^{2}+\frac{b}{a} x+\frac{c}{a}=0$ has two real roots $\alpha$ and $\beta .$ If $a>0,$ then the area under the curve $f(x)=x^{2}+\frac{b}{a} x+\frac{c}{a}$ between $\alpha$ and $\beta$ is

(A) $\frac{b^{2}-4 a c}{2 a}$;
(B) $\frac{\left(b^{2}-4 a c\right)^{3 / 2}}{6 a^{3}}$
(C) $-\frac{\left(b^{2}-4 a c\right)^{3 / 2}}{6 a^{3}}$
(D) $-\frac{b^{2}-4 a c}{2 a}$.

Problem 13:

The minimum value of $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$ subject to $x_{1}+x_{2}+x_{3}+x_{4}=a$ and $x_{1}-x_{2}+x_{3}-x_{4}=b$ is

(A) $\frac{a^{2}+b^{2}}{4}$
(B) $\frac{a^{2}+b^{2}}{2}$;
(C) $\frac{(a+b)^{2}}{4}$;
(D) $\frac{(a+b)^{2}}{2}$.

Problem 14:

The value of

$$\lim_{n\to\infty}\frac{\sum_{r=0}^{n} \left(\begin{array} {c} 2n\\2r\end{array}\right)3^{r}}{\sum_{r=0}^{n-1} \left(\begin{array} {c}2n\\2r+1\end{array}\right)3^{r}}$$ is

(A) $0 ;$
(B) $1$ ;
(C) $\sqrt{3}$;
(D) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$.

Problem 15:

For any real number $x$, let $\tan ^{-1}(x)$ denote the unique real number $\theta$ in $(-\pi / 2, \pi / 2)$ such that $\tan \theta=x$. Then

$$\lim _{n \rightarrow \infty} \sum_{m=1}^{n} \tan ^{-1} \frac{1}{1+m+m^{2}}$$

(A) is equal to $\pi / 2$;
(B) is equal to $\pi / 4$;
(C) does not exist;
(D) none of the above

Problem 16:

Let $n$ be an integer. The number of primes which divide both $n^{2}-1$ and $(n+1)^{2}-1$ is

(A) at most one;
(B) exactly one;
(C) exactly two;
(D) none of the above.

Problem 17:

The value of

$$\lim_{n \rightarrow \infty} \sum_{r=1}^{n} \frac{6 n}{9 n^{2}-r^{2}}$$ is

(A) $0$ ;
(B) $\log \frac{3}{2}$;
(C) $\log \frac{2}{3}$;
(D) $\log 2$.

Problem 18:

A person $X$ standing at a point $P$ on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps $X$ comes back to the original position $P$. Then the number of distinct paths that $X$ can take is

(A) $196$;
(B) $256$ ;
(C) $344$;
(D) $400$ .

Problem 19:

Consider the branch of the rectangular hyperbola $x y=1$ in the first quadrant. Let $P$ be a fixed point on this curve. The locus of the mid-point of the line segment joining $P$ and an arbitrary point $Q$ on the curve is part of

(A) a hyperbolas;

(B) a parabola;

(C) an ellipse;
(D) none of the above.

Problem 20:

The digit at the unit place of $(1 !-2 !+3 !-\cdots+25 !)^{(1!-2!+3!-\cdots+25!)}$ is

(A) $0$ ;
(B) $1 ;$
(C) $5$;
(D) $9$ .

Problem 21:

Let $A_{1}, A_{2}, \ldots, A_{n}$ be the vertices of a regular polygon and $A_{1} A_{2}, A_{2} A_{3} \ldots \ldots$ $A_{n-1} A_{n}, A_{n} A_{1}$ be its $n$ sides. If

$$\frac{1}{A_{1} A_{2}}-\frac{1}{A_{1} A_{4}}=\frac{1}{A_{1} A_{3}}$$

then the value of $n$ is

(A) $5$ ;
(B) $6$ ;
(C) $7$ ;
(D) $8$ .

Problem 22:

Suppose that $\alpha$ and $\beta$ are two distinct numbers in the interval $(0, \pi)$. If

$$\sin \alpha+\sin \beta=\sqrt{3}(\cos \alpha-\cos \beta)$$

then the value of $\sin 3 \alpha+\sin 3 \beta$ is

(A) $0$;
(B) $2 \sin \frac{3(\alpha+\beta)}{2}$;
(C) $2 \cos \frac{3(\alpha-\beta)}{2}$
(D) $\cos \frac{3(\alpha-\beta)}{2}$

Problem 23:

Consider the function $h(x)=x^{2}-2 x+2+\frac{4}{x^{2}-2 x+2}, x \in \mathbb{R} .$ Then $h(x)-5=0$ has

(A) no solution;
(B) only one solution;
(C) exactly two solutions;
(D) exactly three solutions.

Problem 24:

Consider the quadratic equation $x^{2}+b x+c=0 .$ The number of pairs $(b, c)$ for which the equation has solutions of the form $\cos \alpha$ and $\sin \alpha$ for some $\alpha$ is

(A) $0 ;$
(B) $1$ ;
(C) $2$;
(D) infinite.

Problem 25:

Let $\theta_{1}=\frac{2 \pi}{3}, \theta_{2}=\frac{4 \pi}{7}, \theta_{3}=\frac{7 \pi}{12} .$ Then

(A) $\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{2}\right)^{\sin \theta_{2}}<\left(\sin \theta_{3}\right)^{\sin \theta_{3}}$
(B) $\left(\sin \theta_{2}\right)^{\sin \theta_{2}}<\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{3}\right)^{\sin \theta_{3}}$
(C) $\left(\sin \theta_{3}\right)^{\sin \theta_{3}}<\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{2}\right)^{\sin \theta_{2}}$
(D) $\left(\sin \theta_{1}\right)^{\sin \theta_{1}}<\left(\sin \theta_{3}\right)^{\sin \theta_{3}}<\left(\sin \theta_{2}\right)^{\sin \theta_{2}}$.

Problem 26:

Consider the following two curves on the interval (0,1) :

$$C_{1}: y=1-x^{4} \text { and } C_{2}: y=\sqrt{1-x^{2}}$$

Then on $(0,1)$

(A) $C_{1}$ lies above $C_{2}$;
(B) $C_{2}$ lies above
(C) $C_{1}$ and $C_{2}$ intersect at exactly one point;
(D) none of the above.

Problem 27:

Let $f$ be a real valued function on $\mathbb{R}$ such that $f$ is twice differentiable. Suppose that $f^{\prime}$ vanishes only at 0 and $f^{\prime \prime}$ is everywhere negative. Define a function $h$ by $h(x)=(x-a)^{2}-f(x),$ where $a>0 .$ Then

(A) $h$ has a local minima in $(0, a)$;
(B) $h$ has a local maxima in $(0, a)$;
(C) $h$ is monotonically increasing in $(0, a)$;
(D) $h$ is monotonically decreasing in $(0, a)$.

Problem 28:

Consider the triangle with vertices $(1,2),(-5,-1)$ and $(3,-2)$ . Let $\Delta$ denote the region enclosed by the above triangle. Consider the function $f: \Delta \rightarrow \mathbb{R}$ defined by $f(x, y)=|10 x-3 y| .$ Then the range of $f$ is the interval

(A) $[0,36]$ ;
(B) $[0,47]$;
(C) $[4,47]$ ;
(D) $[36,47]$.

Problem 29:

For every positive integer $n$, let $\langle n\rangle$ denote the integer closest to $\sqrt{n}$. Let $A_{k}=\{n>0:\langle n\rangle=k\} .$ The number of elements in $A_{49}$ is

(A) $97$ ;
(B) $98 ;$
(C) $99$ ;
(D) $100$ .

Problem 30:

Consider a square $A B C D$ inscribed in a circle of radius $1 .$ Let $A^{\prime}$ and $C^{\prime}$ be two points on the (smaller) arcs $A D$ and $C D$ respectively, such that $A^{\prime} A B C C^{\prime}$ is a pentagon in which $A A^{\prime}=C C^{\prime}$. If $P$ denotes the area of the pentagon $A^{\prime} A B C C^{\prime}$ then

(A) $P$ can not be equal to $2$ ;
(B) $P$ lies in the interval $(1,2]$ ;
(C) $P$ is greater than or equal to $2$ ;
(D) none of the above.

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