Organisation : Chennai Mathematical Institute
Exam : B.Sc Mathematics & Computer Science Entrance Exam
Document Type : Question Paper
Year : 2021
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 : IITB UCEED 2021 Question Paper
B.Sc Maths & CS Question Paper
1. Consider the two equations numbered [1] and [2]:log2021 a = 2022 a[1] 2021b = 2022 b[2]
(a) Equation [1] has a unique solution.
(b) Equation [2] has a unique solution.
(c) There exists a solution a for [1] and a solution b for [2] such that a = b.
(d) There exists a solution a for [1] and a solution b for [2] such that a+b is an integer.
2. A prime p is an integer 2 whose only positive integer factors are 1 and p.
(a) For any prime p the number p2 p is always divisible by 3.
(b) For any prime p > 3 exactly one of the numbers p 1 and p + 1 is divisible by 6.
(c) For any prime p > 3 the number p2 1 is divisible by 24.
(d) For any prime p > 3 one of the three numbers p + 1,p + 3 and p + 5 is divisible by 8.
3. We want to construct a triangle ABC such that angle A is 20.21 , side AB has length 1 and side BC has length x where x is a positive real number. Let N(x) = the number of pairwise noncongruent triangles with the required properties.
(a) There exists a value of x such that N(x) = 0.
(b) There exists a value of x such that N(x) = 1.
(c) There exists a value of x such that N(x) = 2.
(d) There exists a value of x such that N(x) = 3.
4. Consider polynomials of the form f(x) = x3 + ax2 + bx + c where a, b, c are integers. Name the three (possibly non-real) roots of f(x) to be p, q, r.
(a) If f(1) = 2021, then f(x)=(x1)(x2 +sx+t) + 2021 where s, t must be integers.
(b) There is such a polynomial f(x) with c = 2021 and p = 2.
(c) There is such a polynomial f(x) with r = 12
.(d) The value of p2 + q2 + r2 does not depend on the value of c.
5. For any complex number z define P(z) = the cardinality of {zk|k is a positive integer}, i.e., the number of distinct positive integer powers of z. It may be useful to remember that π is an irrational number.
(a) For each positive integer n there is a complex number z such that P(z) = n.
(b) There is a unique complex number z such that P(z) = 3.
(c) If |z| 6= 1, then P(z) is infinite.
(d) P(ei) is infinite.
6. A stationary point of a function f is a real number r such that f0(r) = 0. A polynomial need not have a stationary point (e.g. x3 + x has none). Consider a polynomial p(x).
(a) If p(x) is of degree 2022, then p(x) must have at least one stationary point.
(b) If the number of distinct real roots of p(x) is 2021, then p(x) must have at least 2020 stationary points.
(c) If the number of distinct real roots of p(x) is 2021, then p(x) can have at most 2020 stationary points.
(d) If r is a stationary point of p(x) AND p00(r) = 0, then the point (r, p(r)) is neither a local maximum nor a local minimum point on the graph of p(x).
7. Given three distinct positive constants a, b, c we want to solve the simultaneous equations ax + by = p2 bx + cy = p3
(a) There exists a combination of values for a, b, c such that the above system has infinitely many solutions (x, y).
(b) There exists a combination of values for a, b, c such that the above system has exactly one solution (x, y).
(c) Suppose that for a combination of values for a, b, c, the above system has NO solution. Then 2b<a + c.
(d) Suppose 2b<a + c. Then the above system has NO solution.
8. Given two distinct nonzero vectors v1 and v2 in 3 dimensions, define a sequence of vectors by vn+2 = vn ⇥ vn+1 (so v3 = v1 ⇥ v2, v4 = v2 ⇥ v3 and so on). Let S = {vn|n = 1, 2,…} and U = { vn |vn||n = 1, 2,…}. (Note: Here ⇥ denotes the cross product of vectors and |v| denotes the magnitude of the vector v. The vector 0 with 0 magnitude, if it occurs in S, is counted. But in that case of course the 0 vector is not considered while listing elements of U.)
(a) There exist vectors v1 and v2 for which the cardinality of S is 2.
(b) There exist vectors v1 and v2 for which the cardinality of S is 3.
(c) There exist vectors v1 and v2 for which the cardinality of S is 4.
(d) Suppose that for some v1 and v2, the set S is infinite. Then the set U is also infinite.
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