Organisation : MSc Data Science
Exam : MSc Data Science Entrance Exam
Document Type : Question Paper
Year : 2022
Website : https://www.cmi.ac.in/
CMI MSc Data Science Question Paper
The entrance examination is a test of aptitude in mathematics, statistics and computer science. The questions will test familiarity with school level mathematics, discrete mathematics, probability theory, and programming.
Related / Similar Question Paper : CMI MSc Data Science Entrance 2021 Question Paper
MSc Data Science Question Paper
4. Let E and F be events such that P (E ∩ F c) = 0.2, P (F ∩ Ec) = 0.3, P ((E ∩ F )c) = 0.7, where, for an event E, the notation Ec denotes the complement of the event. Then we can conclude that
(a) P (E ∪ F ) = 0.8.
(b) P (Ec ∩ F c) = 0.3.
(c) P (F ) = 0.6.
(d) P (E) = 0.6.
5. Let A, B be n × n invertible matrices of real numbers. Let C = I + AAT , D = I + BAAT BT . We can conclude that
(a) (AB)−1 = (I + B−1A−1)
(b) (AB)−1 = A−1B−1
(c) C = CT
(d) D = DT
6. A matrix C is said to be symmetric if CT = C. Which of the following is/are true? Let A, B be n × n matrices.
(a) If A is symmetric and invertible, then A−1 is also symmetric and invertible.
(b) If A and B are symmetric, then C = AB is also symmetric.
(c) If A and B are invertible, then C = AB is also invertible.
(d) If A and B are symmetric, then D = A + B is also symmetric.
7. Which of the following statements is/are true?
(a) Let A be a n × m matrix with n < m. Then there is a nonzero solution y with Ay = 0 only if A has full row rank.
(b) Let A be a n × m matrix with n > m. There is a nonzero solution y with Ay = 0.
(c) The row rank of an n × m matrix is equal to its column rank only when n = m.
(d) Let A be an n × n matrix. Suppose A = BC, where B has size n × r and C has size r × n. The rank of A is less than or equal to r.
8. Let n ≥ 5 be a natural number, let X = {x1, x2, …, xn}, and let Y = {y1, y2}. Let F be the set of functions from X to Y and G be the set of bijective functions from X to Y . Then
(a) The number of functions in F equals n2.
(b) The number of functions in F equals 2n.
(c) The number of functions in G equals n2 − 2.
(d) The number of functions in G equals 0.
13. We say that a subset S of a finite set U is large if |S| > |U \ S|. Here U \ S denotes the elements of U which are not in S and the notation |T | denotes the number of elements in a set T . Let x be the number of large subsets of the set X = {1, 2, . . . , 10}, and let y be the number of large subsets of Y = {1, 2, . . . , 9}. Which of the following is true?
(a) x = 260, y = 256
(b) x = 386, y = 256
(c) x = 386, y = 130
(d) x = 512, y = 256.
16. At a conference attended by 1235 people, some attendees shake hands with other attendees. As a part of the local COVID-19 tracing protocol the organizers ask each attendee to note down the number of other attendees with whom they shook hands. Let N be the sum of all the numbers noted down by the attendees. That is, N is the sum, taken over all attendees, of the number of other attendees with whom they shook hands. Which of the following is guaranteed to be true about N ?
(a) N is always a multiple of 1235
(b) N is always a multiple of 2
(c) N is always a multiple of 3
(d) N is always a multiple of 5
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