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

## Problem 1.

Prove that the positive integers that cannot be written as a sum of consecutive positive integers, with are of the form for some ## Problem 2.

Let be defined by • Show that has exactly one point of discontinuity.
• Evaluate at its point of discontinuity.

## Problem 3.

Let If then draw a sketch of ## Problem 4.

Let be a twice differentiable function such that Show that there exist such that for all ## Problem 5.

A subset of the plane is called if given two points and in the line segment joining and is containted in 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 of area 1, there is a rectangle of area such that can be drawn inside ## Problem 6.

For all natural numbers let • Show that for  • Hence or otherwise, evaluate the limit ## Problem 7.

Let be a polynomial with integer coefficients. Define and If there exists a natural number such that then prove that either or ## Problem 8.

Consider the following subsets of the plane: and Given any two points and of the plane, their distance is defined by Show that there exists a unique choice of points and such that ### Indian Statistical Institute, ISI Subjective BStat & BMath 2019 Problem 8 UGB Solution & Discussion Video:

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Thank you
Fractions

• Like!! I blog frequently and I really thank you for your content. The article has truly peaked my interest.

• Thank you.

• Jatin

Hello sir thanks for making YouTube videos for isi b math solution.

I am preparing for isi b math can you tell how much complex numbers and matrices and determinants should we practice and till what level.

• You are welcome Jatin. For complex numbers try to follow the book “Complex Numbers from A to …Z
Book by Dorin Andrica and Titu Andreescu” and for matrices and determinants just study the basic properties.

• Yamun

Sir how much probability and coordinate geometry weigh in isi b math exam how much to practice.

• You can expect around two questions from probability and coordinate geometry each. Try to follow “Coordinate Geometry.
Book by S. L. Loney” and for probability, you can check this out, “https://projects.iq.harvard.edu/stat110/youtube”.
How much to practice depends on you. If you are not confident enough to solve any problem on a particular topic, then do keep practicing.

• Yamun

Thank you sir

• Sougata

sir I have solved Q1 by contradiction Q3 in another way and Q8 in another way..but I am not sure that my solutions are right or wrong.. what should I do..and also I found recent ISI subjective polynomial problems hard but I am able to solve other past paper polynomial problems with ease …what should I do??

• Okay, you can type your solutions here in the comment box. If you are finding them hard then don’t worry about it, even I find them hard sometimes, just keep on practicing for polynomials practice from Problem solving strategies.

• Aary sajal bhiwankar

Sir do you provide doubt solving sessions in isi and cmi entrance exam course

• We provide personal one to one classes for doubt clearing.

• ajitesh das

sir , in an hour i was able to solve only 3 questions , kindly suggest how to gain approach to crack the solutions for the hard questions and where should i study induction from

• Three questions in an hour are considered to be good. You can still try to do better. There is no such algorithm or formula to get the problems done. You have to practice as much as possible. For mathematical induction you can follow this notes: https://cseweb.ucsd.edu/classes/sp14/cse20-a/InductionNotes.pdf

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