Madhava Mathematics Competition/Exam Model Question Paper 2017 : madhavacompetition.com

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Madhava Mathematics Competition/Exam Model Question Paper 2017 : madhavacompetition.com

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1 Madhava Mathematics Competition Question Paper

Madhava Mathematics Competition Question Paper

** The competition is named after Madhava, who introduced in the fourteenth centurry, profound mathematical ideas that are now part of Calculus.

Related / Similar Question Paper : Madhava Mathematics Competition MMC 2021 Question Paper

Topics

** Calculus of one variable (sequences, series, continuity, differentiation etc.), elementary algebra (polynomials, complex numbers and integers), elementary combinatorics, general puzzle type/ mixed type problems.

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Instruction

Max. Marks :100
Time: 12.00 noon to 3.00 p.m.
** Part I carries 20 marks, Part II carries 30 marks and Part III carries 50 marks.

Part I

** N.B. Each question in Part I carries 2 marks.
1. Let A(t) denote the area bounded by the curve y = e??jxj; the X?? axis and the straight lines x = ??t; x = t; then lim t!1 A(t) is
A) 2
B) 1
C) 1/2
D) e.

2. How many triples of real numbers (x; y; z) are common solutions of the equations
x + y = 2; xy ?? z2 = 1?
A) 0
B) 1
C) 2
D) infinitely many.

3. For non-negative integers x; y the function f(x; y) satises the relations f(x; 0) = x and f(x; y + 1) = f(f(x; y); y): Then which of the following is the largest?
A) f(10; 15)
B) f(12; 13)
C) f(13; 12)
D) f(14; 11):

4. Suppose p; q; r; s are 1; 2; 3; 4 in some order. Let x = 1 p + 1 q + 1 r + 1 s We choose p; q; r; s so that x is as large as possible, then s is
A) 1
B) 2
C) 3
D) 4.

6. There are 8 teams in pro-kabaddi league. Each team plays against every other exactly once. Suppose every game results in a win, that is, there is no draw. Let w1;w2; ;w8 be number of wins and l1; l2; ; l8 be number of loses by teams T1; T2; ; T8; then
A) w2 1 + + w2 8 = 49 + (l2 1 + + l2 8)
B) w2 1 + + w2 8 = l2 1 + + l2 8:
C) w2 1 + + w2 8 = 49 ?? (l21 + + l2 8)
D) None of these.

7. The remainder when m + n is divided by 12 is 8, and the remainder when m ?? n is divided by 12 is 6. If m > n, then the remainder when mn divided by 6 is
A) 1
B) 2
C) 3
D) 4 .

9. The maximum of the areas of the rectangles inscribed in the region bounded by the curve y = 3 ?? x2 and X??axis is
A) 4
B) 1
C) 3
D) 2.

10. How many factors of 253652 are perfect squares?
A) 24
B) 20
C) 30
D) 36.

Part II

N.B. Each question in Part II carries 6 marks.
1. How many 15??digit palindromes are there in each of which the product of the non-zero digits is 36 and the sum of the digits is equal to 15? (A string of digits is called a palindrome if it reads the same forwards and backwards. For example 04340, 6411146.)
2. Let H be a nite set of distinct positive integers none of which has a prime factor greater than 3: Show that the sum of the reciprocals of the elements of H is smaller than 3: Find two dierent such sets with sum of the reciprocals equal to 2:5:
3. Let A be an n n matrix with real entries such that each row sum is equal to one. Find the sum of all entries of A2015:
4. Let f : R ! R be a dierentiable function such that f(0) = 0; f0(x) > f(x) for all x 2 R: Prove that f(x) > 0 for all x > 0:
5. Give an example of a function which is continuous on [0; 1]; dierentiable on (0; 1) and not dierentiable at the end points. Justify.

Part III

1. There are some marbles in a bowl. A, B and C take turns removing one or two marbles from the bowl, with A going rst, then B, then C, then A again and so on. The player who takes the last marble from the bowl is the loser and the other two players are the winners. If the game starts with N marbles in the bowl, for what values of N can B and C work together and force A to lose?
2. Let p(x) be a polynomial with positive integer coecients. You can ask the question: What is p(n) for any positive integer n? What is the minimum number of questions to be asked to determine p(x) completely? Justify. [13]
2. Let f : R ! R be a function such that f0(0) exists. Suppose n 6= n; 8n 2 N and both sequences fng and fng converge to zero. Prove that lim n!1 Dn = f0(0) under EACH of the following conditions:
(a) n < 0 < n; 8n 2 N:
(b) 0 < n < n and
M; 8n 2 N; for some M > 0:
(c) f0(x) exists and is continuous for all x 2 (??1; 1):

Rules For Madhava Competition
1) The Examination will commence on Sunday, January 8, 2017 from 12 noon to 3 p.m.
2) The Seat numbers of the students registered for the competition have been sent to their colleges. Students are suggested to carefully note the same.

3) Seating arrangement will be displayed at the respective centres.
4) Students should take their seats before 11.45 a.m.
5) Students should bring the college identity card at the time of Examination.
6) Use of calculators is not allowed.
7) Cell phones are strictly prohibited inside the examination hall.

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