cmi.ac.in Entrance Examination for BSc Maths & Computer Question Paper : Chennai Mathematical Institute

Name of the Organisation : Chennai Mathematical Institute
Exam : Entrance Exam
Subject : BSc (Hons) Mathematics and Computer Science
Year : 2016
Document Type : Previous Years’ Question Papers

Website : http://www.cmi.ac.in/admissions/syllabus.php
Download Model/Sample Question Paper :
Maths & Computer 2010 : https://www.pdfquestion.in/uploads/13290-ugmath2010.pdf
Maths & Computer 2011 : https://www.pdfquestion.in/uploads/13290-ugmath2011.pdf
Maths & Computer 2012 : https://www.pdfquestion.in/uploads/13290-ugmath2012.pdf
Maths & Computer 2013 : https://www.pdfquestion.in/uploads/13290-ugmath2013.pdf
Maths & Computer 2014 : https://www.pdfquestion.in/uploads/13290-ugmath2014.pdf
Maths & Computer 2015 : https://www.pdfquestion.in/uploads/13290-ugmath2015.pdf
Maths & Computer 2016 : https://www.pdfquestion.in/uploads/13290-ugmath2016.pdf

Entrance Examination for BSc Maths & Computer Question Paper :

Part A. Write your final answers on page 3. :
** Part A is worth a total of (4 * 10 = 40) points.

** Points will be given based only on clearly legible final answers filled in the correct place on page 3.
** Write all answers for a single question on the designated line and in the order in which they are asked, separated by commas.

** Unless specified otherwise, each answer is either a rational number or, where appropriate, one of the phrases “infinite”/“does not exist”/“not possible to decide”. Write integer answers in the usual decimal form. Write non-integer rationals as ratios of two coprime integers.

1. Four children K, L, M and R are about to run a race. They make some predictions as follows.
K says: M will win. Myself will come second.
R says: M will come second. L will be third.
M says: L will be last. R will be second.
After the race, it turns out that each person has made exactly one correct and one incorrect prediction. Write the result of the race in the order from first to the last.

2. A country’s GDP grew by 7.8% within a period. During the same period the country’s per-capita-GDP (= ratio of GDP to the total population) increased by 10%. During this period, the total population of the country increased/decreased by %. (Choose the correct option and fill in the blank if possible.)

3. You are told that n = 110179 is the product of two primes p and q. The number of positive integers less than n that are relatively prime to n (i.e. those m such that gcd(m, n) = 1) is 109480. Write the value of p + q. Then write the values of p and q.

4. A step starting at a point P in the XY -plane consists of moving by one unit from P in one of three directions: directly to the right or in the direction of one of the two rays that make the angle of ±120

Part B. Write complete solutions for these questions from page 6 onwards :
** Part B is worth a total of (6 ? 14 = 84) points. Solve these questions in the space provided for each question from page 6. You may solve only part of a question and get partial credit. Clearly explain your entire reasoning. No credit will be given without reasoning.

1. Out of the 14 students taking a test, 5 are well prepared, 6 are adequately prepared and 3 are poorly prepared. There are 10 questions on the test paper. A well prepared student can answer 9 questions correctly, an adequately prepared student can answer 6 questionscorrectly and a poorly prepared student can answer only 3 questions correctly. For each probability below, write your final answer as a rational number in lowest form.

(a) If a randomly chosen student is asked two distinct randomly chosen questions from the test, what is the probability that the student will answer both questions correctly?

Note :
The student and the questions are chosen independently of each other. “Random” means that each individual student/each pair of questions is equally likely to be chosen.

(b) Now suppose that a student was chosen at random and asked two randomly chosen questions from the exam, and moreover did answer both questions correctly. Find the probability that the chosen student was well prepared.

2. By definition the region inside the parabola y = x2 is the set of points (a, b) such that b & a2. We are interested in those circles all of whose points are in this region. A bubble at a point P on the graph of y = x2 is the largest such circle that contains P. (You may assume the fact that there is a unique such circle at any given point on the parabola.)
(a) A bubble at some point on the parabola has radius 1. Find the center of this bubble.
(b) Find the radius of the smallest possible bubble at some point on the parabola. Justify.