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I.S.I B.Stat Paper 2019 Subjective| problems & solutions

Here, you will find all the questions of ISI Entrance Paper 2019 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r>1$, are of the form $n=2^{l}$ for some $l \geq 0$.

Problem 2:

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be defined by
$$
f(x)=\lim _{n \rightarrow \infty} \cos ^{n}\left(\frac{1}{n^{x}}\right)
$$
(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.

Problem 3:

Let $\Omega={z=x+i y \in \mathbf{C}:|y| \leq 1} .$ If $f(z)=z^{2}+2,$ then draw a
sketch of
$$
f(\Omega)={f(z): z \in \Omega}
$$
Justify your answer.

Problem 4:

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that
$$
\frac{1}{2 y} \int_{x-y}^{x+y} f(t) d t=f(x), \quad \text { for all } x \in \mathbf{R}, y>0
$$
Show that there exist $a, b \in \mathbf{R}$ such that $f(x)=a x+b$ for all $x \in \mathbb{R}$

Problem 5:

A subset $S$ of the plane is called convex if given any two points $x$ and $y$ in $S,$ the line segment joining $x$ and $y$ is contained in $S .$ A quadrilateral is called convex if the region enclosed by the edges of the quadrilateral is a convex set.

Show that given a convex quadrilateral $Q$ of area $1,$ there is a rectangle $R$ of area 2 such that $Q$ can be drawn inside $R$.

Problem 6:

For all natural numbers $n$, let
$$
A_{n}=\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}(n \text { many radicals })
$$
(a) Show that for $n \geq 2$.
$$
A_{n}=2 \sin \frac{\pi}{2^{n+1}}
$$
(b) Hence, or otherwise, evaluate the limit
$\lim_{n \to \infty} 2^{n}A_{n}$

Problem 7:

Let $f$ be a polynomial with integer coefficients. Define
$$
a_{1}=f(0), a_{2}=f\left(a_{1}\right)=f(f(0))
$$
and
$$
a_{n}=f\left(a_{n-1}\right) \quad \text { for } n \geq 3
$$
If there exists a natural number $k \geq 3$ such that $a_{k}=0,$ then prove that either $a_{1}=0$ or $a_{2}=0$.

Problem 8:

Consider the following subsets of the plane:
$C_{1}=\{(x, y): x>0, y=\frac{1}{x}\}$
and
$C_{2}=\{(x, y): x<0, y=-1+\frac{1}{x}\}$
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
$d\left(P_{0}, Q_{0}\right) \leq d(P, Q)$ for all $P \in C_{1}$ and $Q \in C_{2}$

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