Indian Statistical Institute (ISI) 2007 B.Sc Mathematics b math admission test - Question Paper
B.MATH ADMISSION 2007
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B.Math.(Hons.) Admission Test:2007
Short-Answer Type Test Time: 2 hours
1. For any positive integer k, prove that
2(y/k+H \fk) < p < 2(sfk \/k 1).
Also, compute the integral part of A- + H-----b 000
2. Let a\ 1 and an = n (an_i + 1) for all n > 2. Define
CL\ CLji
Compute lim Pn.
n~->oo
3. Let ABCD be a quadrilateral such that the sum of a pair of opposite sides equals the sum of the other pair of opposite sides (i.e., AB + CD = AD + BC). Prove that the circles inscribed in triangles ABC and ACD are tangent to each other.
4. For a set S we denote its cardinality by |S|. Let ei,...,ek be non-negative
integers. Let Ak (respectively Bk) be the set of all fc-tuples (/i, .., fk) of
k
integers such that 0 < / < e* for all i and J2 fi is even (respectively odd).
il
Show that \Ak\ \Bk\ = 0 or 1.
5. Find the point in the closed unit disc D = {(x,y) | x2 + y2 < 1} at which the function f (x,y) = x + y attains its maximum.
6. Let ao = 0 < ai < a2 < < an be real numbers. Suppose p(t) is a real valued polynomial of degree n such that
aJ +1
p(t)dt = 0 for all 0 < j < n 1.
aj
Show that, for 0 < j < n 1, the polynomial p (t) has exactly one root in the interval (cij, aJ+i).
7. Let M be a point in the triangle ABC such that
Area {ABM) = 2.Area (ACM).
Show that the locus of all such points is a straight line.
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8. In how many ways can one fill an n x n matrix with 1 so that the product of the entries in each row and each column equals 1?
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