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 $4$.
