Indian Statistical Institute, ISI BStat & BMath 2019 Solutions & Discussions

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ISI BMath & BSTAT 2019 Subjective Questions UGB along with hints and solutions and discussions

Note: You must try the problems on your own before looking at the solutions. I have made the solutions so that you don’t have to join any paid program to get the solutions. After you have failed several times in your attempt to solve a question, then only look at the solution of that problem. If you cannot solve a single question on your own then this exam is not for you. And if you like my work then please do like my videos and subscribe me on youtube to get more such videos in the future. 

Indian Statistical Institute,ISI BMATH/BSTAT 2021 All UGA Objective Solutions: Click Here

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.

Topic: Number Theory
Difficulty level: High

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 1 UGB Solution & Discussion Video:

Problem 2.

Let f:(0,\infty)\to \mathbb{R} be defined by

    \[f(x)=\lim_{n\to \infty}\cos^n \bigg(\frac{1}{n^x}\bigg).\]

  • Show that f has exactly one point of discontinuity.
  • Evaluate f at its point of discontinuity.

Topic: Calculus: Limits and Continuity
Difficulty Level: Medium

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 2 UGB Solution & Discussion Video:

Problem 3.

Let \Omega=\{z=x+iy\in\mathbb{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.

Topic: Complex Numbers
Difficulty Level: High

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 3 UGB Solution & Discussion Video:

Problem 4.

Let f:\mathbb{R}\to \mathbb{R} be a twice differentiable function such that

    \[\frac{1}{2y}\int^{x+y}_{x-y}f(t)dt=f(x), \mbox{ for all } x\in \mathbb{R}, y>0.\]

Show that there exist a,b\in\mathbb{R} such that f(x)=ax+b for all x\in \mathbb{R}.

Topic: Calculus: Differentiation, Definite Integral
Difficulty Level: Medium

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 4 UGB Solution & Discussion Video:

Problem 5.

A subset S of the plane is called convex if given two points x and y in S. the line segment joining x and y is containted 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.

Topic: Geometry: Construction
Difficulty Level: Very Easy

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 5 UGB Solution & Discussion Video:

Problem 6.

For all natural numbers n, let

    \[A_n=\sqrt{2-\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}} \ \ \ (n \mbox{ many radicals}).\]

  • Show that for n\geq 2,

    \[A_n=2\sin \frac{\pi}{2^{n+1}}.\]

 

  • Hence or otherwise, evaluate the limit

        \[\lim_{n\to \infty}2^nA_n.\]

Topic: Trigonometry
Difficulty Level: Medium

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 6 UGB Solution & Discussion Video:

Problem 7.

Let f be a polynomial with integer coefficients. Define

    \[a_1=f(0), a_2=f(a_1)=f(f(0)),\]

and

    \[a_n=f(a_{n-1}) \mbox{ for } n\geq 3.\]

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.

Topic: Geometry: Polynomials
Difficulty Level: Hard

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 7 UGB Solution & Discussion Video:

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

    \[d(P_0,Q_0)\leq d(P,Q) \mbox{ for all } P\in C_1 \mbox{ and } Q\in C_2.\]

Topic: Calculus, Inequality(You can even solve this using Co-ordinate Geometry)
Difficulty Level: Hard

Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 8 UGB Solution & Discussion Video:

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