Organisation : Chennai Mathematical Institute
Exam : B.Sc Mathematics & Computer Science Entrance Exam
Document Type : Question Paper
Year : 2018
Website : https://www.cmi.ac.in/admissions/syllabus.php
CMI B.Sc Question Paper
The entrance examination is a test of aptitude for Mathematics at the 12th standard level, featuring both objective questions and problems drawn mostly from the following topics: arithmetic, algebra, geometry, trigonometry, and calculus.
Related / Similar Question Paper : CMI B.Sc Mathematics & Computer Science Entrance 2020 Question Paper
B.Sc Maths & CS Question Paper
Part A:
Part A is worth a total of (4 x10 = 40) points. Points will be given based only on clearly legible nal answers lled 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 number (rational/ real/ complex) or, where appropriate, one of the phrases \innite”/\does not exist”/\not possible to decide”. Write in-teger answers in the usual decimal form. Write non-integer rationals as ratios of two integers.
1. Consider an equilateral triangle ABC with altitude 3 centimeters. A circle is inscribed in this triangle, then another circle is drawn such that it is tangent to the inscribed circle and the sides AB;AC. Infinitely many such circles are drawn; each tangent to the previous circle and the sides AB;AC. The gure shows the construction after 2 steps. Find the sum of the areas of all these circles.
2. List in increasing order all positive integers n <= 40 such that n cannot be written in the form a2 – b2, where a and b are positive integers.
3. How many non-congruent triangles are there with integer lengths a <= b <= c such that a + b + c = 20?
4. Consider a sequence of polynomials with real coefficients defined by p0(x) = (x2 + 1)(x2 + 2) (x2 + 1009) with subsequent polynomials dened by pk+1(x) := pk(x+1))-pk(x) for k >= 0. Find the least n such that pn(1) = pn(2) = = pn(5000):
Part B:
Part B is worth a total of 85 points (Question 1 is worth 10 points and the remaining questions are worth 15 points each). 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.
Answer the following questions
(a) A natural number k is called stable if there exist k distinct natural numbers a1, ……. ;,ak, each ai > 1, such that 1/a1+ ……… +1/ak= 1. Show that if k is stable then k + 1 is also stable. Using this or otherwise, nd all stable numbers.
(b) Let f be a dierentiable function defined on a subset A of the real numbers. Define whenever the above maximum is nite. For the function f(x) = -ln(x), determine the set of points for which f* is defined and find an expression for f*(y) involving only y and constants.
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