MADHAVA MATHEMATICS COMPETITION (MMC)-2014

Organized by:

Department of Mathematics, S. P. College, Pune &

Homi Bhabha Centre for Science Education (TIFR), Mumbai

Under the aegis of: National Board for Higher Mathematics

The competition is named after Madhava, who introduced in the fourteenth century, profound mathematical ideas that are now part of Calculus. His most famous achievements include the Madhava-Leibnitz series for π, the Madhava-Newton power series for Sine and Cosine functions and a numerical value of π that is accurate to eleven decimal places. The “Madhava School” consisting of a long chain of teacher-student continuity flourished for well over two centuries, making significant contributions to mathematics and astronomy.

Eligibility:

Second Year B. Sc. Students.

Interested students of First year B.Sc. and Third Year B.Sc. are also eligible to participate in this competition. A separate merit list will be prepared for Third Year B. Sc. students.

Prizes:

Attractive Cash Prizes and Medals for Rank Holders.

Several Cheer prizes. Participation Certificate to all students. Nurture Camp for select students.

The competition is now in its fourth year. It is being organized in 13 regions in India viz., Pune, Ahmednagar, Nasik, Mumbai, Ahmedabad, Baroda, Hyderabad, Ernakulum, Kolkata, Bhubaneswar, Allahabad and Goa.

Registration Fee: Rs. 50/- Last Date for Registration: November, 30, 2013. Day and Date of Examination: Sunday, January 05, 2014.

Registration for the test will be through the Departments of Mathematics, of the respective colleges. The head of the department of Mathematics of each college is requested to collect the registration fees & then submit it along with a registration form to Centre Coordinator of Bhubaneswar Region (i.e. towards Sk. Sarif Hassan, Centre Coordinator, Bhubaneswar).

Centre Coordinator (for Bhubaneswar Region): Sk. Sarif Hassan Institute of Mathematics & Applications Andharua, Bhubaneswar 751003 Odisha, India Email: sarimif@gmail.com Mobile: 09692389457

For previous years question papers write an e-mail to: sarimif@gmail.com

For more details about the competition: Please visit: www.hbcse.tifr.res.in

MADHAVA MATHEMATICS COMPETITION

(A Mathematics Competition for Undergraduate Students) 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

Registration Form for Participating Colleges To, Sk. Sarif Hassan Centre Coordinator Madhava Mathematics Competition, Bhubaneswar Region. Faculty in Mathematics, Institute of Mathematics & Applications Andharua, Bhubaneswar 751003 Odisha, India Email: sarimif@gmail.com Mobile no: 09692389457 Name of the College

Address

District Office Phone No. (STD)

Name of the contact person with Email-id and Mob. No.

Sr. No.

Name of the Student

Sex (M/F)

Class

Head of Dept. of Mathematics Seal of College/Institute Principal

MADHAVA MATHEMATICS COMPETITION, 8 January 2012

Part IN.B. Each question in Part I carries 2 marks.

1. Let n be a fixed positive integer. The value of k for which∫ k1 xn−1dx = 1

n is

(a) 0 (b) 2n (c) (2n

)1n (d) 2

1n

2. Let S = {a, b, c}, T = {1, 2}. If m denotes the number of one-one functions andn denotes the number of onto functions from S to T, then the values of m and n

respectively are(a) 6,0 (b) 0,6 (c) 5,6 (d) 0,8.

3. In the binary system,12

can be written as(a) 0.01111 · · · (b) 0.01000 · · · (c) 0.0101 · · · (d) None of these

4. For the equation |x|2 + |x| − 6 = 0(a) there is only one root. (b) the sum of the roots is -1.(c) the sum of the roots is 0. (d) the product of the roots is -6.

5. The value of limn→∞

1 · 1! + 2 · 2! + · · ·+ n · n!(n + 1)!

(a) 1 (b) 2 (c)12

(d) does not exist.

6. Let f : R→ R and g : R→ R be differentiable functions such that f ′(x) > g′(x) forevery x. Then the graphs y = f(x) and y = g(x)(a) intersect exactly once. (b) intersect at most once.(c) do not intersect. (d) could intersect more than once.

7. The function |x|3 is(a) differentiable twice but not thrice at 0. (b) not differentiable at 0.(c) three times differentiable at 0. (d) differentiable only once at 0.

8. Let A be the n× n matrix (n ≥ 2), whose (i, j)th entry is i + j for all i, j = 1, 2, · · · , n.

The rank of A is(a) 2 (b) 1 (c) n (d) n− 1.

9. Let A = −1 + 0i be the point in the complex plane. Let PQR be an arc with centre atA and radius 2. If P = −1 +

√2 +√

2i, Q = 1 + 0i and R = −1 +√

2−√

2i

then the shaded region is given by(a) |z + 1| < 2, | arg(z + 1)| < π

2

(b) |z − 1| < 2, | arg(z − 1)| < π2

(c) |z + 1| > 2, | arg(z + 1)| < π4

(d) |z − 1| > 2, | arg(z + 1)| < π4

10. The solution ofdy

dx= ax+y is

(a) ax − a−y = c (b) a−x + a−y = c (c) a−x − ay = c (d) ax + a−y = c

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Part IIN.B. Each question in Part II carries 5 marks.

1. Let f : [0, 4] → [3, 9] be a continuous function. Show that there exists x0 such that

f(x0) =3x0 + 6

2.

2. Suppose a, b, c are all real numbers such that a+b+c > 0, abc > 0 and ab+bc+ac > 0.

Show that a, b, c are all positive.

3. Let f be a continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) =0, f(1) = 1 and f(2) = 2, then show that there exists x0 such that f ′′(x0) = 0.

4. Integers 1, 2, · · · , n are placed in such a way that each value is either bigger than allpreceding values or smaller than all preceding values. In how many ways this can bedone? (For example in case of n = 5, 3 2 4 1 5 is valid and 3 2 5 1 4 is not valid.)

Part IIIN.B. Each question in Part III carries 12 marks.

1. Consider an isosceles right triangle with legs of fixed length a.

Inscribe a rectangle and a circle inside the triangle as indicatedin the figure. Find the dimensions of the rectangle and theradius of the circle which make the total area of the rectangleand circle maximum.

2. Assume that f : R → R is a continuous, one-one function. If there exists a positiveinteger n such that fn(x) = x, for every x ∈ R, then prove that either f(x) = x orf2(x) = x. (Note that fn(x) = f(fn−1(x)).)

3. Consider f(x) =x3

6+

x2

2+

x

3+ 1. Prove that f(x) is an integer whenever x is an

integer. Determine with justification, conditions on real numbers a, b, c and d so thatax3 + bx2 + cx + d is an integer for all integers x.

4. Suppose A =

2 1 01 b d

1 b d + 1

, X =

x

y

z

, U =

f

g

h

. Find

conditions on A and U such that the system AX = U has no solution.

5. Let A and B be finite subsets of the set of integers. Show that|A + B| ≥ |A|+ |B| − 1. When does equality hold?(Here A + B = {x + y : x ∈ A, y ∈ B}. Also, |S| denotes the number of elements in theset S.)

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Part IN.B. Each question in Part I carries 2 marks.

1. If N = 1! + 2! + 3! + · · ·+ 2011!, then the digit in the units place of thenumber N is(a) 1 (b) 3 (c) 0 (d) 9.

2. The set of all points z in the complex plane satisfying z2 = |z|2 is a(a) pair of points (b) circle (c) union of lines (d) line.

3. If the arithmetic mean of two numbers is 26 and their geometric meanis 10, then the equation with these two numbers as roots is(a) x2 + 52x+ 100 = 0 (b) x2 − 52x− 100 = 0(c) x2 − 52x+ 100 = 0 (d) x2 + 52x− 10 = 0.

4. All points lying inside the triangle with vertices at the points (1, 3), (5, 0)and (−1, 2) satisfy(a) 3x+ 2y ≥ 0 (b) 2x+ y − 13 ≥ 0(c) 2x− 3y − 12 ≥ 0 (d) −2x+ y ≥ 0.

5. For n ≥ 3, let A be an n × n matrix. If rank of A is n − 2, then rankof adjoint of A is(a) n− 2 (b) 2 (c) 1 (d) 0.

6. Suppose f : R → R is an odd and differentiable function. Then forevery x0 ∈ R, f ′(−x0) is equal to(a) f ′(x0) (b) −f ′(x0) (c) 0 (d) None of these.

7. If S = {a, b, c} and the relation R on the set S is given byR = {(a, b), (c, c)}, then R is(a) reflexive and transitive (b) reflexive but not transitive(c) not reflexive but transitive (d) neither reflexive nor transitive.

8. Let a1 = 1, an+1 =

(1 + n

n

)an for n ≥ 1. Then the sequence {an} is

(a) divergent (b) decreasing (c) convergent (d) bounded.

9. The coefficient of x2n−2 in (x−1)(x+ 1)(x−2)(x+ 2) · · · (x−n)(x+n)is

(a) 0 (b)−n(n+ 1)(2n+ 1)

6(c)

n(n+ 1)(2n+ 1)

6(d)−n(n+ 1)

2.

10. The number of roots of 5x4 − 4x+ 1 = 0 in [0, 1] is(a) 0 (b) 1 (c) 2 (d) 3.

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Part IIN.B. Each question in Part II carries 5 marks.

1. If n ≥ 3 is an integer and k is a real number, prove that n is equal tothe sum of nth powers of the roots of the equation xn − kx− 1 = 0.

2. Find all positive integers n such that (n2n − 1) is divisible by 3.

3. Start with the set S = {3, 4, 12}. At any stage you may perform thefollowing operation: Choose any two elements a, b ∈ S and replace

them by

(3a− 4b

5

)and

(4a+ 3b

5

). Is it possible to transform the

set S into the set {4, 6, 12} by performing the above operation a finitenumber of times?

4. Let a < b. Let f be a continuous function on [a, b] and differentiableon (a, b). Let α be a real number. If f(a) = f(b) = 0, show that thereexists x0 ∈ (a, b) such that αf(x0) + f ′(x0) = 0.

Part IIIN.B. Each question in Part III carries 12 marks.

1. Let Mn be the n × n matrix with all 1’s along the main diagonal, di-rectly above the main diagonal and directly below the main diagonaland 0’s everywhere else. For example,

M3 =

1 1 01 1 10 1 1

, M4 =

1 1 0 01 1 1 00 1 1 10 0 1 1

. Let dn = detMn.

(a) Find d1, d2, d3, d4.(b) Find a formula expressing dn in terms of dn−1 and dn−2, for all

n ≥ 3.(c) Find d100.

2. Let p(x) = x2n − 2x2n−1 + 3x2n−2 − 4x2n−3 + · · · − 2nx+ (2n+ 1).Show that the polynomial p(x) has no real root.

3. Let f(x) = x10 + a1x9 + a2x

8 + · · · + a10 where ai’s are integers. If allthe roots of f(x) are from the set {1, 2, 3}, determine the number ofsuch polynomials. Further, if g(x) is the sum of all such polynomials

f(x), then show that the constant term of g(x) is1

2(312 + 1)− 212.

4. Let f : R→ R be a differentiable function such that

f(x+ h)− f(x) = hf ′(x+1

2h),

for all real x and h. Prove that f is a polynomial of degree atmost 2.

5. (a) Let n = 9. Express n as a sum of positive integers such that theirproduct is maximum. Find the value of the maximum product.(b) Repeat part (a) for n = 10 and n = 11.(c) Given a positive integer n ≥ 6, express n as a sum of positiveintegers such that their product is maximum. Find the value of themaximum product.

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