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# CMI B.Sc Mathematics & Computer Science Entrance 2020 Question Paper

Organisation : Chennai Mathematical Institute
Exam : B.Sc Mathematics & Computer Science Entrance Exam
Document Type : Question Paper
Year : 2020

Contents :

## 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 2018 Question Paper ## B.Sc Maths & CS Question Paper

1. Each student in a small school has to be a member of at least one of THREE school clubs. It is known that each club has 35 members. It is not known how many students are members of two of the three clubs, but it is known that exactly 10 students are members of all three clubs. What is the largest possible total number of students in the school? What is the smallest possible total number of students in the school?

2. Let P be the plane containing the vectors (6; 6; 9) and (7; 8; 10). Find a unit vector that is perpendicular to (2;??3; 4) and that lies in the plane P. (Note: all vectors are considered as line segments starting at the origin (0; 0; 0). In particular the origin lies in the plane P.)

3. A fair die is thrown 100 times in succession. Find probabilities of the following events.
(i) 4 is the outcome of one or more of the rst three throws.
(ii) Exactly 2 of the last 4 throws give an outcome divisible by 3 (i.e., outcome 3 or 6).

4. Write your answers to each question below as a series of three letters Y (for Yes) or N (for No). Leave space between the group of three letters answering (i), the answers to (ii) and the answers to (iii). Consider the graphs of functions
(i) Does f have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(ii) Does g have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(ii) Does h have a horizontal asymptote? A vertical asymptote? A removable discontinuity?

5. Recall the function arctan(x), also denoted as tan-1(x). Complete the sentence: arctan(20202019) + arctan(20202021) 2 arctan(20202020); because in the relevant region, the graph of y = arctan(x) .

Fill in the rst blank with one of the following: is less than / is equal to / is greater than. Fill in the second blank with a single correct reason consisting of one of the following phrases: is bounded / is continuous / has positive rst derivative / has negative rst derivative / has positive second derivative / has negative second derivative / has an in ection point.

6. The polynomial p(x) = 10×400 + ax399 + bx398 + 3x + 15, where a; b are real constants, is given to be divisible by x2 – 1.
(i) If you can, nd the values of a and b. Write your answers as a = ; b = . If it is not possible to decide, state so.
(ii) If you can, nd the sum of reciprocals of all 400 (complex) roots of p(x). Write your answer as sum = . If it is not possible to decide, state so.

Note that 25 x 16- 19 x 21 = 1. Using this or otherwise, nd positive integers a; b and c, all <= 475 = 25 x 19, such that
** a is 1 mod 19 and 0 mod 25,
** b is 0 mod 19 and 1 mod 25, and
** c is 4 mod 19 and 10 mod 25.
(Recall the mod notation: since 13 divided by 5 gives remainder 3, we say 13 is 3 mod 5.) 