Madhava Mathematics Competition MMC 2021 Question Paper : madhavacompetition.in
Organisation : Homi Bhabha Centre for Science Education
Exam : Madhava Mathematics Competition (MMC)
Document Type : Previous Question Paper
Year : Jan 2021
Website : https://madhavacompetition.in/OldQuestionPapers.aspx
What is Madhava Mathematics Competition?
A Mathematics Competition for Undergraduate Students is Organized by Department of Mathematics, S.P. College, Pune and Homi Bhabha Centre for Science Education, T.I.F.R., Mumbai Funded by National Board for Higher Mathematics. A registration fee of Rs. 100/- will be charged. All the students appearing for the competition will receive a participation certificate
Question Paper of Madhava Mathematics Competition
Madhava Mathematics Competition January 2021 Question Paper
N.B.: Part I carries 20 marks, Part II carries 20 marks and Part III carries 10marks.Part I: MCQ with single correct answer. Each question in Part I carries 2 marks for correct answer and -1 mark for wrong answer.
1. For matrices X,Ydefine [X,Y] =XY−Y X. For X=(0 10 0),Y=(0 01 0),rank [X,Y]
(a) equals rank X
(b) is strictly less that rank (XY)
(c) is strictly greater than rank (XY)
(d) equals 0.
Ans:(c)
2. If a+b+c= 0,then the quadratic equation 3ax2+ 2bx+c= 0 has
(a) at least one root in (0,1).
(b) one root in (1,2) and other in (−1,0).
(c) both imaginary roots.
(d) a repeated root
Ans:(a)
3. Let f(x) be the function defined on R such that f(x) =x,for all x≤1 and f(x) =ax2+bx+c,for x >1.The triples (a,b,c) such that f(x) is differentiable at all real x are of the form:
(a) (a,1−2a,a)
(b) (1,0,0) only
(c) (a,−2a,a)
(d) (0,1,−1) only
Ans:(a)
4. The system of equations2x+py+ 6z= 8x+ 2y+qz= 5x+y+ 3z= 4has no solution for
(a)p6= 2,q6= 3
(b)p6= 2,q= 3
(c)p= 2,q= 3
(d)p= 2,q6= 3
Ans:(b)
5. Let f be any differentiable function on R with f(−2) = 16,f(4) = 4 and f(8) = 24. For which of the following three values of λ the equation f′(x) =λmust have a solution?
(a) 0,5,10
(b)−5,0.8,3
(c)−2,0.8,5
(d)−2,0,14
Ans:(c)
6. We have two glasses. One having milk and other having water in it, in exactly the same quantities. One spoon of milk is transferred to water and then one spoon of mixture is replaced in milk. Then which of the following is true?
(a) The amount of milk in water is bigger than the amount of water in milk.
(b) The amount of milk in water is smaller than the amount of water in milk.
(c) The amount of milk in water is equal to the amount of water in milk.
(d) The information is insufficient.
Ans:(c)
Part II: Numerical Questions N.B. The answer to each question in Part II is an integer. Each question in Part II carries 2 marks. No marks will be deducted for wrong answer.
1. For a complex number z,if|z+ 4|≤3,then the maximum value of|z+ 1|is …..
Ans: 6
2. The number of continuous functions f:R→R satisfying (f(x))2=xf(x) is …..
Ans: 4
Part III: Multiple Select Questions N.B. Each question in Part III carries 2 marks. No marks will be deducted for wrong answer. Each question may have more than one correct alternatives.A candidate gets 2 marks if he/she selects all the correct answers only and no wrong answers.
1. A four letter word is converted into a matrix form by writing its letters, say ABCD as the matrix(A BC D). Each such matrix is then replaced by the corresponding letter in the alphabet via the ruleA7→1, B7→2,…,Z7→26. The resulting matrix is further reduced modulo 5.If the matrix(2 00 4)has been obtained by the above procedure,the following words are among the possible original words
(a) GOOD
(b) QEOS
(c) GEOD
(d) QEED
Ans: (a),(b),(c),(d).
2. If 1,w1,w2,w3,w4,w5are distinct roots ofx6−1,then
(a) 1 +wi+w2i+w3i+w4i+w5i= 0 for i= 1,2,3,4,5.
(b) 1 +w2i+w4i+w6i+w8i+w10i= 0 for i= 1,2,3,4,5.
(c) 1 +w3i+w6i+w9i+w12i+w15i= 0 for i= 1,2,3,4,5.
(d) 1 +w5i+w10i+w15i+w20i+w25i= 0 for i= 1,2,3,4,5.
Ans: (a),(d).
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