Section 1: Algebra View the other sections of this test. Geometry || Analysis Try a online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at helpdesk@cheenta.com Which of the following statements are true? Every group of order 11 is cyclic. Every group of order 111 is cyclic. Every group of […]
Any normal subgroup of order 2 is contained in the center of the group. True Discussion: Center of a group Z(G) is the sub group of elements that commute with all members of the group. A subgroup of order two has two elements: identity element and another element, say x, which is self inverse. Since […]
There is an element of order 51 in the multiplicative group (Z/103Z) True Discussion: First note that (Z/103Z) has 102 elements as 103 is a prime (in fact one of the twin primes of 101, 103 pair). Also 102 = 2317. So it has Sylow-3 subgroup of order 3 (prime order hence it is cyclic […]
All non-trivial proper subgroups of (R, +) are cyclic. False Discussion: There is a simple counter example: (Q, +) (the additive group of rational numbers). We also note that every additive subgroup of integers is cyclic (in fact they are of the for nZ). Cyclic groups have exactly one generator. We can construct numerous counter […]
The equation $latex x^3 + 10x^2 - 100x + 1729 $ has at least one complex root α such that |α| > 12. False ** Discussion: A fun fact : 1729 is the Ramanujan Number; it is the smallest number expressible as the sum of two cubes in two different ways We conduct normal extrema tests. First […]
The equation $latex x^3 + 3x - 4 $ has exactly one real root. True Discussion: Consider the derivative of the function $latex f(x) = x^3 + 3x - 4 = 0 $ . It is $latex 3x^2 + 3 $ . Note that the derivative is strictly positive ( positive times square + positive is […]
Problem: Every differentiable function f: (0, 1) --> [0, 1] is uniformly continuous. Discussion; False Note that every differentiable function f: [0,1] --> (0, 1) is uniformly continuous by virtue of uniform continuity theorem which says every continuous map from closed bounded interval to R is uniformly continuous. However in this case the domain is […]
Problem: Let f: R --> R be defined by $latex f(x) = sin (x^3) $. Then f is continuous but not uniformly continuous. Discussion: True It is sufficient to show that there exists an $latex epsilon > 0 $ such that for all $latex \delta > 0 $ there exist $latex x_1 , x_2 \in […]
This post contains a problem from TIFR 2013 Math paper D based on Inequality of square root function. The inequality $ \sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n } $ is false for all in n such that $ 101 \le n \le 2000 $ False Discussion: $ \sqrt {n+1} - \sqrt n […]
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2009 based on Coordinate Geometry.
Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Exponents and Equations.
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2010 based on functional equation. You may use sequential hints.
Try this beautiful problem from the Pre-RMO, 2019 based on Trigonometry Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based on quadratic equation from PRMO 2016. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2009 based on Triangles and sides.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Rectangles and sides.
Try this beautiful problem from algebra, based on Arithmetic Progression from AMC-10B, 2004. You may use sequential hints to solve the problem
Try this beautiful problem from PRMO, 2016 based on Triangle You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Area of triangle from AMC-10A, 2009, Problem-10. You may use sequential hints to solve the problem.
This post discusses the solutions to the problems from IIT JAM Mathematical Statistics (MS) 2021 Question Paper - Set B. You can find solutions in video or written form. Note: This post is getting updated. Stay tuned for solutions, videos, and more. IIT JAM Mathematical Statistics (MS) 2021 Problems & Solutions (Set B) Problem 1 […]
Here, you will find all the questions of ISI Entrance Paper 2014 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. Problem 1: The system of inqualities $a-b^{2} \geq \frac{1}{4}$, $b-c^{2} \geq \frac{1}{4}$, $c-d^{2} \geq \frac{1}{4}$, $d-a^{2} \geq \frac{1}{4}$ has(A) no solutions(B) exactly one solution(C) […]
Here, you will find all the questions of ISI Entrance Paper 2013 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. Problem 1: Let $i=\sqrt{-1}$ and $S=\{i+i^{2}+\cdots+i^{n}: n \geq 1\} .$ The number of distinct real numbers in the set $S$ is (A) 1(B) 2(C) […]
Complex numbers and geometry are very closely related. We consider a problem from I.S.I. Entrance that uses this geometric character complex numbers.
Try a beautiful problem from complex numbers and geometry. It is from I.S.I. Entrance. We have created sequential hints to make this mathematical journey enjoyable!
This is a Test of Mathematics Solution Subjective 188 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Problem Consider the squares of an $ 8 X 8 $ chessboard filled with the […]
This is a Test of Mathematics Solution Subjective 181 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta Problem Suppose that one moves along […]
This is a Test of Mathematics Solution Subjective 177 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta Problem There are 1000 doors $ […]
This is a Test of Mathematics Solution Subjective 176 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta Problem Suppose that P(x) is a […]
This is a Test of Mathematics Solution Subjective 175 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta Problem Let \(\text{P(x)}=x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_{1}x+a_{0}\) be a polynomial […]