Organisation : Chennai Mathematical Institute
Exam : B.Sc Mathematics & Computer Science Entrance Exam
Document Type : Question Paper
Year : 2022
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 MSc/PhD Computer Science Entrance 2021 Question Paper
B.Sc Maths And Computer Science Question Paper
A1. Suppose a0, a1, a2, a3, . . . is an arithmetic progression with a0 and a1 positive integers.
Let g0, g1, g2, g3, . . . be the geometric progression such that g0 = a0 and g1 = a1.
Statements
(1) We must have (a5)2 ≥ a0a10.
(2) The sum a0 + a1 + · · · + a10 must be a multiple of the integer a5.
(3) If ∑∞ i=0 ai is +∞ then ∑∞ i=0 gi is also +∞.
(4) If ∑∞ i=0 gi is finite then ∑∞ i=0 ai is −∞. A2. Any two events X and Y are called mutually exclusive when the probability P (X and Y ) = 0 and they are called exhaustive when P (X or Y ) = 1. Suppose A and B are events and the probability of each of these two events is strictly between 0 and 1 (i.e., 0 < P (A) < 1 and 0 < P (B) < 1). Statements
(5) A and B are mutually exclusive if and only if not A and not B are exhaustive.
(6) A and B are independent if and only if not A and not B are independent.
(7) A and B cannot be simultaneously independent and exhaustive.
(9) Regardless of the value of k, the matrix A is not invertible, i.e., there is no 3 × 3 matrix B such that BA = the 3 × 3 identity matrix.
(10) There is a unique k such that determinant of A is 0.
(13) If I is true, then II is true.
(14) If II is true, then III is true.
(15) If III is false, then I is false.
(16) No two of the three given conditions are equivalent to each other. (Two statements being equivalent means each implies the other.) A5. Statements
(17) Let a = 1 ln 3 . Then 3a = e.
(18) sin(0.02) < 2 sin(0.01).
(19) arctan(0.01) > 0.01.
(20) 4 ∫ 1 0 arctan(x)dx = π − ln 4 . A6. Let f (x) = 1| ln x|(1x + cos x) . Statements
(21) As x → ∞, the sign of f (x) changes infinitely many times.
(22) As x → ∞, the limit of f (x) does not exist.
(23) As x → 1, f (x) → ∞.
(24) As x → 0+, f (x) → 1. A7. Let f0(x) = x. For x > 0, define functions inductively by fn+1(x) = xfn(x). So f1(x) = xx, f2(x) = xxx , etc. Note that f0(1) = f ′ 0(1) = 1. Statements
(25) As x → 0+, f1(x) → 1.
(26) As x → 0+, f2(x) → 1. (27) ∫ 10 f3(x)dx = 1. (28) The derivative of f123 at x = 1 is 1. A8. Let N = {1, 2, 3, 4, 5, 6, 7, 8, 9} and L = {a, b, c}. Statements
(29) Suppose we arrange the 12 elements of L ∪ N in a line such that all three letters appear consecutively (in any order). The number such arrangements is less than 10! × 5.
(30) More than half of the functions from N to L have the element b in their range.
(31) The number of one-to-one functions from L to N is less than 512.
(32) The number of functions from N to L that do not map consecutive numbers to consecutive letters is greater than 512. (e.g., f (1) = b and f (2) = a or c is not allowed. f (1) = a and f (2) = c is allowed. So is f (1) = f (2).) A9. In this question z denotes a non-real complex number, i.e., a number of the form a + ib (with a, b real) whose imaginary part b is nonzero. Let f (z) = z222 + 1 z222 . Statements
(33) If |z| = 1, then f (z) must be real.
(34) If z + 1 z = 1, then f (z) = 2. (35) If z + 1 z is real, then |f (z)| ≤ 2.
(36) If f (z) is a real number, then f (z) must be positive. A10. Suppose that cards numbered 1, 2, . . . , n are placed on a line in some sequence (with each integer i ∈ [1, n] appearing exactly once). A move consists of interchanging the card labeled 1 with any other card. If it is possible to rearrange the cards in increasing order by doing a series of moves, we say that the given sequence can be rectified. Statements
(37) The sequence 9 1 2 3 4 5 6 7 8 can be rectified in 8 moves and no fewer moves.
(38) The sequence 1 3 4 5 6 7 8 9 2 can be rectified in 8 moves and no fewer moves.
(39) The sequence 1 3 4 2 9 5 6 7 8 cannot be rectified.
(40) There exists a sequence of 99 cards that cannot be rectified.
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