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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2018 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Find all pairs $(x,y)$ with $x,y$ real, satisfying the equations:

$\sin(\frac{x+y}{2})=0,\vert x\vert+\vert y\vert=1$

**Problem 2:**

Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$. It is given that $PO=3 \mathrm{cm}$ and $SO=4 \mathrm{cm}$. Moreover, the area of the triangle $POR$ is $7 \mathrm{cm}^2$. Find the area of the triangle $QOS$.

**Problem 3:**

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that for all $x \in \mathbb{R}$ and for all $t \geq 0$, $f(x)=f(e^{t}x)$. Show that $f$ is a constant function.

**Problem 4:**

Let $f:(0,\infty)\to \mathbb{R}$ be a continuous function such that for all $x \in(0,\infty)$, $f(2x)=f(x)$. Show that the function $g$ defined by the equation $g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}$ for $x>0$ is a constant function.

**Problem 5:**

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x \in\mathbb{R}$, $0 \leq \vert f'(x)\vert\leq \frac{1}{2}$. Define a sequence of real numbers $ \{a_n\}_{n\in\mathbb{N}}$ by : $a_1=1$ and $a_{n+1}=f(a_n)$ for all $n\in\mathbb{N}$. Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$,

**Problem 6:**

Let, $a\geq b\geq c >0$ be real numbers such that for all natural number $n$, there exist triangles of side lengths $a^{n} , b^{n} ,c^{n}$. Prove that the triangles are isosceles.

**Problem 7:**

Let $a, b, c$ are natural numbers such that $a^{2}+b^{2}=c^{2}$ and $c-b=1$

Prove that,

(i) $a$ is odd,

(ii) $b$ is divisible by $4$,

(iii) $a^{b}+b^{a}$ is divisible by $c$.

**Problem 8:**

Let $n\geq 3$. Let $A=((a_{ij}))_{1\leq i,j\leq n}$ be an $n\times n$ matrix such that $a_{ij}\in\{-1,1\}$ for all $1\leq i,j\leq n$. Suppose that $a_{k1}=1$ for all $1\leq k\leq n$ and $\sum_{k=1}^n a_{ki}a_{kj}=0$ for all $i\neq j$. Show that $n$ is a multiple of $4$.

Here, you will find all the questions of ISI Entrance Paper 2018 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Find all pairs $(x,y)$ with $x,y$ real, satisfying the equations:

$\sin(\frac{x+y}{2})=0,\vert x\vert+\vert y\vert=1$

**Problem 2:**

Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$. It is given that $PO=3 \mathrm{cm}$ and $SO=4 \mathrm{cm}$. Moreover, the area of the triangle $POR$ is $7 \mathrm{cm}^2$. Find the area of the triangle $QOS$.

**Problem 3:**

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that for all $x \in \mathbb{R}$ and for all $t \geq 0$, $f(x)=f(e^{t}x)$. Show that $f$ is a constant function.

**Problem 4:**

Let $f:(0,\infty)\to \mathbb{R}$ be a continuous function such that for all $x \in(0,\infty)$, $f(2x)=f(x)$. Show that the function $g$ defined by the equation $g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}$ for $x>0$ is a constant function.

**Problem 5:**

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x \in\mathbb{R}$, $0 \leq \vert f'(x)\vert\leq \frac{1}{2}$. Define a sequence of real numbers $ \{a_n\}_{n\in\mathbb{N}}$ by : $a_1=1$ and $a_{n+1}=f(a_n)$ for all $n\in\mathbb{N}$. Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$,

**Problem 6:**

Let, $a\geq b\geq c >0$ be real numbers such that for all natural number $n$, there exist triangles of side lengths $a^{n} , b^{n} ,c^{n}$. Prove that the triangles are isosceles.

**Problem 7:**

Let $a, b, c$ are natural numbers such that $a^{2}+b^{2}=c^{2}$ and $c-b=1$

Prove that,

(i) $a$ is odd,

(ii) $b$ is divisible by $4$,

(iii) $a^{b}+b^{a}$ is divisible by $c$.

**Problem 8:**

Let $n\geq 3$. Let $A=((a_{ij}))_{1\leq i,j\leq n}$ be an $n\times n$ matrix such that $a_{ij}\in\{-1,1\}$ for all $1\leq i,j\leq n$. Suppose that $a_{k1}=1$ for all $1\leq k\leq n$ and $\sum_{k=1}^n a_{ki}a_{kj}=0$ for all $i\neq j$. Show that $n$ is a multiple of $4$.

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I have the objective qus paper and I wish to see ur solution on those qus..

Can I have ur fb account or whatsApp no. I can send u pics of qus paper

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Please someone provide me with the objective question paper of this entrance test without solution

plz provide the obj. question paper

Sir in question 4 isi subjective paper it is a constant function

Sir isi question 4 subjective is a constant function

Please give me the solution of problem no 5

Regarding the hint for Problem 8, det(n I) = n^n, not n. Is this an intentional mistake?

Yes that is a typo. We will post a complete discussion soon.