CMI Entrance Exam 2019 Question Paper for B.Sc Maths & Computer

CHENNAI MATHEMATICAL INSTITUTE
Undergraduate Programme in Mathematics and Computer Science/Physics
Common Entrance Examination
15th May 2019

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Enter your Admit Card Number : U

IMPORTANT INSTRUCTIONS!

• Ensure that this booklet has all 18 printed sides: this cover page and 17 numbered pages.
Pages numbered 1 to 3 contain ten questions in part A plus the space to answer part A on page
3. Page numbered 4 contains six questions in part B. Pages 6 to 17 contain answer sheets for each
question in part B. For rough work use the colored blank pages at the end.

• Time allowed is 3 hours. Total points: 100 = 40 for part A + 60 for part B.

• Part A will be used for screening. Part B will be graded only if you score a certain minimum
in part A. However your scores in both parts will be used while making the final decision. Specific
instructions for each part are given below.

• Advice: Please ensure that you have about 2 hours left for part B.

For office use only

× Points Remarks Points Remarks

Part A B1

Part B B2

Total B3

B4

B5

B6

Total

2019 Entrance Examination for the BSc Programmes at CMI

Read the instructions on the front of the booklet carefully!

Part A. Write your final answers on page 3.

Part A is worth a total of (4 × 10 = 40) points. Points will be given based only on clearly
legible final answers filled 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 “infinite”/“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. For a natural number m, define Φ1 (m) to be the number of divisors of m and for k ≥ 2
define Φk (m) := Φ1 (Φk−1 (m)). For example, Φ2 (12) = Φ1 (6) = 4. Find the minimum k
such that
Φk (20192019 ) = 2.
2. Let f be a real valued continuous function defined on R satisfying
f 0 (tan2 θ) = cos 2θ + tan θ sin 2θ, for all real numbers θ.
If f 0 (0) = − cos 12
π
then find f (1).
3. You have a piece of land close to a river, running straight. You are required to cut off a
rectangular portion of the land, with the river forming one of the sides of the rectangle
so, your fence will have three sides to it. You only have 60 meters of fencing. The
maximum area that you can enclose is . . . . . . .
4. The sum
S = 1 + 111 + 11111 + · · · + 11 · · · 1}
| {z
2k + 1
is equal to . . . . . . .
5. You are given an 8×8 chessboard. If two distinct squares are chosen uniformly at random
find the probability that two rooks placed on these squares attack each other. Recall
that a rook can move either horizontally or vertically, in a straight line.
6. For how many natural numbers n is n6 + n4 + 1 a square of a natural number?
7. A broken calculator has all its 10 digit keys and two operation keys intact. Let us
call these operation keys A and B. When the calculator displays a number n pressing
A changes the display to n + 1. When the calculator displays a number n pressing B
changes the display to 2n. For example, if the number 3 is displayed then the key strokes
ABBA changes the display in the following steps 3 → 4 → 8 → 16 → 17.
If 1 is on the display what is the least number of key strokes needed to get 260 on the
display?

1

8. Let π = π1 π2 . . . . . . πn be a permutation of the numbers 1, 2, 3, . . . , n. We say π has its
first ascent at position k < n if π1 > π2 . . . > πk and πk < πk+1 . If π1 > π2 > . . . >
πn−1 > πn we say π has its first ascent in position n. For example when n = 4 the
permutation 2134 of has its first ascent at position 2.
The number of permutations which have their first ascent at position k is . . . . . . .
For questions 9 and 10 below, some statements are given. For each statement,
state if it is true or false. Write your answer to each question as a sequence
of three/ four letters (T for True and F for False) in correct order.

9. Consider f : R × R → R defined as follows:
1
f (a, b) := lim loge [ena + enb ].
n→∞ n

(a) f is not onto i.e. the range of f is not all of R.
(b) For every a the function x 7→ f (a, x) is continuous everywhere.
(c) For every b the function x 7→ f (x, b) is differentiable everywhere.
(d) We have f (0, x) = x for all x ≥ 0.

10. Let f : R → R.
R1
(a) There is no continuous function f for which f (x)(1 − f (x))dx < 41 .
0
R1
(b) There is only one continuous function f for which 0 f (x)(1 − f (x))dx = 41 .
R1
(c) There are infinitely many continuous functions f for which 0 f (x)(1−f (x))dx = 41 .

2

Answers to part A
This is the only place that will be seen for grading part A. So carefully and clearly write the
answers to each question on the designated line below. Write only the final answers, do not
show any intermediate work. Illegible/unclear answers will not be considered. ×

A1.

A2.

A3.

A4.

A5.

A6.

A7.

A8.

A9.

A10.

3

Part B. Write complete solutions for these questions from page 6 onwards.

Part B is worth a total of 60 marks. 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.

1. For a natural number n denote by Map(n) the set of all functions f : {1, 2, . . . , n} →
{1, 2, . . . , n}. For f, g ∈ Map(n), f ◦ g denotes the function in Map(n) that sends x to
f (g(x)). [10 marks]

(a) Let f ∈ Map(n). If for all x ∈ {1, . . . , n} f (x) 6= x, show that f ◦ f 6= f .
(b) Count the number of functions f ∈ Map(n) such that f ◦ f = f .
2019
2. (a) Count the number of roots
p w√of the equation z − 1 = 0 over complex numbers
that satisfy |w + 1| ≥ 2 + 2. [5 marks]
(b) Find all real numbers x that satisfy following equation: [5 marks]
8x + 27x 7
x x
= .
12 + 18 6
R∞
3. Evaluate 0
(1 + x2 )−(m+1) dx, where m is a natural number. [10 marks]

4. Let ABCD be a parallelogram. Let O be a point in its interior such that ∠AOB +
∠DOC = 180◦ . Show that ∠ODC = ∠OBC. [10 marks]

5. Three positive real numbers x, y, z satisfy

x2 + y 2 = 32
y 2 + yz + z 2 = 42
√
x2 + 3xz + z 2 = 52 .
√
Find the value of 2xy + xz + 3yz. [10 marks]
Z e x 
d 4
6. (a) Compute log(t) cos (t) dt . [4 marks]
dx 0
Rx
(b) For x > 0 define F (x) = 1 t log(t)dt. [6 marks]
i. Determine the open interval(s) (if any) where F (x) is decreasing and the open
interval(s) (if any) where F (x) is increasing.
ii. Determine all the local minima of F (x) (if any) and the local maxima of F (x)
(if any) .

4

Write answers to part B from the next page.
If you need extra space for any problem, continue on one of the colored blank
pages at the end and write a note to that effect.

5

Answers to part B

1. For a natural number n denote by Map(n) the set of all functions f : {1, 2, . . . , n} →
{1, 2, . . . , n}. For f, g ∈ Map(n), f ◦ g denotes the function in Map(n) that sends x to
f (g(x)). [10 marks]

(a) Let f ∈ Map(n). If for all x ∈ {1, . . . , n} f (x) 6= x, show that f ◦ f 6= f .
(b) Count the number of functions f ∈ Map(n) such that f ◦ f = f .

6

7

2019
2. (a) Count the number of roots
p w√of the equation z − 1 = 0 over complex numbers
that satisfy |w + 1| ≥ 2 + 2. [5 marks]

8

(b) Find all real numbers x that satisfy following equation: [5 marks]
8x + 27x 7
x x
= .
12 + 18 6

9

R∞
3. Evaluate 0
(1 + x2 )−(m+1) dx, where m is a natural number. [10 marks]

10

11

4. Let ABCD be a parallelogram. Let O be a point in its interior such that

∠AOB + ∠DOC = 180◦ .

Show that ∠ODC = ∠OBC. [10 marks]

12

13

5. Three positive real numbers x, y, z satisfy

x2 + y 2 = 32
y 2 + yz + z 2 = 42
√
x2 + 3xz + z 2 = 52 .
√
Find the value of 2xy + xz + 3yz. [10 marks]

14

15

Z e x 
d 4
6. (a) Compute log(t) cos (t) dt . [4 marks]
dx 0

16

Rx
(b) For x > 0 define F (x) = 1
t log(t)dt. [6 marks]
i. Determine the open interval(s) (if any) where F (x) is decreasing and the open
interval(s) (if any) where F (x) is increasing.
ii. Determine all the local minima of F (x) (if any) and the local maxima of F (x)
(if any) .

17