INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

Here, you will find all the questions of ISI Entrance Paper 2016 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

The largest integer $n$ for which $n+5$ divides $n^{5}+5$ is

(A) 3115

(B) 3120

(C) 3125

(D) 3130 .

**Problem 2:**

Let $p, q$ be primes and $a, b$ be integers. If $p a$ is divided by $q,$ then the remainder is $1 .$ If $q b$ is divided by $p,$ then also the remainder is $1 .$ The remainder when $p a+q b$ is divided by $p q$ is

(A) 1

(B) 0

(C) -1

(D) 2

**Problem 3: **

The polynomial $x^{7}+x^{2}+1$ is divisible by

(A) $x^{5}-x^{4}+x^{2}-x+1$

(B) $x^{5}+x^{4}+1$

(C) $x^{5}+x^{4}+x^{2}+x+1$

(D) $x^{5}-x^{4}+x^{2}+x+1$.

**Problem 4: **

Let $\alpha>0 .$ If the equation $p(x)=x^{3}-9 x^{2}+26 x-\alpha$ has three positive real roots, then $\alpha$ must satisfy

(A) $\alpha \leq 27$

(B) $\alpha>81$

(C) $27<\alpha \leq 54$

(D) $54<\alpha \leq 81$

**Problem 5: **

The largest integer which is less than or equal to $(2+\sqrt{3})^{4}$ is

(A) 192

(B) 193

(C) 194

(D) 195 .

**Problem 6: **

Consider a circle of unit radius and a chord of that circle that has unit length. The area of the largest triangle that can be inscribed in the circle with that chord as its base is

(A) $\frac{1}{2}+\frac{\sqrt{2}}{4}$

(B) $\frac{1}{2}+\frac{\sqrt{2}}{2}$

(C) $\frac{1}{2}+\frac{\sqrt{3}}{4}$

(D) $\frac{1}{2}+\frac{\sqrt{3}}{2}$

**Problem 7: **

Let $z_{1}=3+4 i$. If $z_{2}$ is a complex number such that $\left|z_{2}\right|=2,$ then the greatest and the least values of $\left|z_{1}-z_{2}\right|$ are respectively

(A) 7 and 3

(B) 5 and 1

(C) 9 and 5

(D) $4+\sqrt{7}$ and $\sqrt{7}$.

**Problem 8: **

Consider two distinct arithmetic progressions (AP) each of which has a positive first term and a positive common difference. Let $S_{n}$ and $T_{n}$ be the sums of the first $n$ terms of these AP. Then $\lim {n \rightarrow \infty} \frac{S{n}}{T_{n}}$ equals

(A) $\infty$ or 0 depending on which AP has larger first term

(B) $\infty$ or 0 depending on which AP has larger common difference

(C) the ratio of the first terms of the AP

(D) the ratio of the common differences of the AP.

**Problem 9:**

Let $f(x)=\max \{cos x, x^{2}\}, 0<x<\frac{\pi}{2} .$ If $x_{0}$ is the solution of the equation

(B) $f$ is not continuous at $x_{0}$

(C) $f$ is continuous everywhere and differentiable only at $x_{0}$

(D) $f$ is differentiable everywhere except at $x_{0}$.

**Problem 10:**

The set of all real numbers in (-2,2) satisfying

$$

2^{|x|}-\left|2^{x-1}-1\right|=2^{x-1}+1

$$ is

(A) {-1,1}

(B) {-1}$\cup[1,2)$

(C) (-2,-1]$\cup[1,2)$

(D) (-2,-1]$\cup{1}$

**Problem 11:**

Let $S(k)$ denote the set of all one-to-one and onto functions from ${1,2,3, \ldots, k}$ to itself. Let $p, q$ be positive integers. Let $S(p, q)$ be the set of all $\tau$ in $S(p+q)$ such that $\tau(1)<\tau(2)<\cdots<\tau(p)$ and $\tau(p+1)<\tau(p+2)<\cdots<\tau(p+q)$

The number of elements in the set $S(13,29)$ is

(A) $377$

(B) $(42) !$

(C) $\left(\begin{array}{l}42 \ 13\end{array}\right)$

(D) $\frac{42 !}{29 !}$

**Problem 12:**

Suppose that both the roots of the equation $x^{2}+a x+2016=0$ are positive even integers. The number of possible values of $a$ is

(A) $6$

(B) $12$

(C) $18$

(D) $24$ .

**Problem 13:**

Let $b \neq 0$ be a fixed real number. Consider the family of parabolas given by the equations

$$

y^{2}=4 a x+b, \quad \text { where } a \in \mathbb{R} \text { . }

$$

The locus of the points on the parabolas at which the tangents to the parabolas make $45^{\circ}$ angle with the $x$ -axis is

(A) a straight line

(B) a pair of straight lines

(C) a parabola

(D) a hyperbola.

**Problem 14:**

Consider the curve represented by the equation

$$

a x^{2}+2 b x y+c y^{2}+d=0

$$

in the plane, where $a>0, c>0$ and $a c>b^{2}$. Suppose that the normals to the curve drawn at 5 distinct points on the curve all pass through the origin. Then

(A) $a=c$ and $b>0$

(B) $a \neq c$ and $b=0$

(C) $a \neq c$ and $b<0$

(D) None of the above.

**Problem 15:**

Let $P$ be a 12-sided regular polygon and $T$ be an equilateral triangle with its incircle having radius 1 . If the area of $P$ is the same as the area of $T$, then the length of the side of $P$ is

(A) $\sqrt{\sqrt{3} \cot 15^{\circ}}$

(B) $\sqrt{\sqrt{3} \tan 15^{\circ}}$

(C) $\sqrt{3 \sqrt{2} \tan 15^{\circ}}$

(D) $\sqrt{3 \sqrt{2} \cot 15^{\circ}}$

**Problem 16:**

Let $A B C$ be a right-angled triangle with $\angle A B C=90^{\circ} .$ Let $P$ be the midpoint of $B C$ and $Q$ be a point on $A B .$ Suppose that the length of $B C$ is $2 x, \angle A C Q=$ $\alpha,$ and $\angle A P Q=\beta .$ Then the length of $A Q$ is

(A) $\frac{3 x}{2 \cot \alpha-\cot \beta}$

(B) $\frac{2 x}{3 \cot \alpha-2 \cot \beta}$

(C) $\frac{3 x}{\cot \alpha-2 \cot \beta}$

(D) $\frac{2 x}{2 \cot \alpha-3 \cot \beta}$

**Problem 17:**

Let $[x]$ denote the greatest integer less than or equal to $x .$ The value of the

integral

is equal to

(A) $1+\frac{2^{3}}{\log {e} 2}-\frac{2^{2}}{\log {e} 2}+\frac{3^{4}}{\log {e} 3}-\frac{3^{3}}{\log {e} 3}+\cdots+\frac{(n-1)^{n}}{\log {e}(n-1)}-\frac{(n-1)^{n-1}}{\log {e}(n-1)}$

(B) $1+\frac{1}{\log {e} 2}+\frac{2}{\log {e} 3}+\cdots+\frac{n-2}{\log _{e}(n-1)}$

(C) $\frac{1}{2}+\frac{2^{2}}{3}+\cdots+\frac{n^{n+1}}{n+1}$

(D) $\frac{2^{3}-1}{3}+\frac{3^{4}-2^{3}}{4}+\cdots+\frac{n^{n+1}-(n-1)^{n}}{n+1}$

**Problem 18:**

Let $\alpha>0, \beta \geq 0$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous at 0 with $f(0)=\beta .$ If $g(x)=|x|^{\alpha} f(x)$ is differentiable at $0,$ then

(A) $\alpha=1$ and $\beta=1$

(B) $0<\alpha<1$ and $\beta=0$

(C) $\alpha \geq 1$ and $\beta=0$

(D) $\alpha>0$ and $\beta>0$

**Problem 19:**

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable and strictly decreasing function such that $f(0)=1$ and $f(1)=0 .$ For $x \in \mathbb{R},$ let

$$

F(x)=\int_{0}^{x}(t-2) f(t) d t

$$

Then

(A) $F$ is strictly increasing in [0,3]

(B) $F$ has a unique minimum in (0,3) but has no maximum in (0,3)

(C) $F$ has a unique maximum in (0,3) but has no minimum in (0,3)

(D) $F$ has a unique maximum and a unique minimum in (0,3) .

**Problem 20:**

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a nonzero function such that $\lim {x \rightarrow \infty} \frac{f(x y)}{x^{3}}$ exists for all $y>0$ Let $g(y)=\lim {x \rightarrow \infty} \frac{f(x y)}{x^{3}} .$ If $g(1)=1,$ then for all $y>0$

(A) $g(y)=1$

(B) $g(y)=y$

(C) $g(y)=y^{2}$

(D) $g(y)=y^{3}$.

**Problem 21:**

Let $D=\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leq 1\}$. Then the maximum number of points in $D$ such that the distance between any pair of points is at least 1 will be

(A) $5$

(B) $6$

(C) $7$

(D)$8$.

**Problem 22:**

The number of 3 -digit numbers $a b c$ such that we can construct an isosceles triangle with sides $a, b$ and $c$ is

(A) $153$

(B) $163$

(C) $165$

(D) $183$ .

**Problem 23:**

The function

$$

f(x)=x^{1 / 2}-3 x^{1 / 3}+2 x^{1 / 4}, \quad x \geq 0

$$

(A) has more than two zeros

(B) is always nonnegative

(C) is negative for $0<x<1$

(D) is one-to-one and onto.

**Problem 24:**

Let $X={a+\sqrt{-5} b: a, b \in \mathbb{Z}}$. An element $x \in X$ is called special if there exists $y \in X$ such that $x y=1 .$ The number of special elements in $X$ is

(A) $2$

(B) $4$

(C) $6$

(D) $8$ .

**Problem 25:**

For a set $X$, let $P(X)$ denote the set of all subsets of $X$. Consider the following statements.

(1)$P(A) \cap P(B)=P(A \cap B)$

(2) $P(A) \cup P(B)=P(A \cup B)$

(3)$P(A)=P(B) \Longrightarrow A=B$

(4) $P(\emptyset)=\emptyset$

Then

(A) All the statements are true

(B) (1) (2) (3) are true and (4) is false

(C) (1) (3) are true and (2) (4) are false

(D) (2) (3) (4) are true and (1) is false.

**Problem 26:**

(A) $a<0$ and $c<0$

(B) $a<0$ and $c>0$

(C) $a>0$ and $c<0$

(D) $a>0$ and $c>0$

**Problem 27:**

For $\alpha \in\left(0, \frac{3}{2}\right),$ define $x_{n}=(n+1)^{\alpha}-n^{\alpha} .$ Then $\lim {n \rightarrow \infty} x{n}$ is

(A) 1 for all $\alpha$

(B) 1 or 0 depending on the value of $\alpha$

(C) 1 or $\infty$ depending on the value of $\alpha$

(D) $1,0,$ or $\infty$ depending on the value of $\alpha$.

**Problem 28:**

Let $f$ be a continuous strictly increasing function from $[0, \infty)$ onto $[0, \infty)$ and $g=f^{-1}$ (that is, $f(x)=y$ if and only if $g(y)=x$ ). Let $a, b>0$ and $a \neq b$ Then

$$

\int_{0}^{a} f(x) d x+\int_{0}^{b} g(y) d y

$$

is

(A) greater than or equal to $a b$

(B) less than $a b$

(C) always equal to $a b$

(D) always equal to $\frac{a f(a)+b g(b)}{2}$.

**Problem 29:**

The sum of the series $\sum_{n=1}^{\infty} n^{2} e^{-n}$ is

(A) $\frac{e^{2}}{(e-1)^{3}}$

(B) $\frac{e^{2}+e}{(e-1)^{3}}$

(C) $\frac{3}{2}$

(D) $\infty$.

**Problem 30:**

Let $f:[0,1] \rightarrow[-1,1]$ be a non-zero function such that

Then $\lim _{x \rightarrow 0+} f(x)$ is equal to

(A) $\frac{1}{2}$

(B) $\frac{1}{3}$

(C) $\frac{2}{3}$

(D) $0$ .

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