| Re: ISI B.Math. Sample Paper
Here is the list of few questions of ISI B.Math. exam which you are looking for .
ISI B.Math. Sample Papers
1. Let x; y; z be non-zero real numbers. Suppose ®; ¯; ° are complex
numbers such that j®j = j¯j = j°j = 1. If x+y+z = 0 = ®x+¯y+°z,
then prove that ® = ¯ = °.
2. Let c be a ¯xed real number. Show that a root of the equation
x(x + 1)(x + 2) ¢ ¢ ¢ (x + 2009) = c
can have multiplicity at most 2. Determine the number of values of c
for which the equation has a root of multiplicity 2.
3. Let 1; 2; 3; 4; 5; 6; 7; 8; 9; 11; 12; ¢ ¢ ¢ be the sequence of all the positive
integers which do not contain the digit zero. Write fang for this se-
quence. By comparing with a geometric series, show thatPn
1
an
< 90.
4. Find the values of x; y for which x2 + y2 takes the minimum value
where (x + 5)2 + (y ¡ 12)2 = 142.
5. Let p be a prime number bigger than 5. Suppose, the decimal expan-
sion of 1=p looks like 0:a1a2 ¢ ¢ ¢ ar where the line denotes a recurring
decimal. Prove that 10r leaves a remainder of 1 on dividing by p.
6. Let a; b; c; d be integers such that ad ¡ bc is non-zero. Suppose b1; b2
are integers both of which are multiples of ad ¡ bc. Prove that there
exist integers simultaneously satisfying both the equalities ax + by =
b1; cx + dy = b2.
7. Compute the maximum area of a rectangle which can be inscribed in
a triangle of area M.
8. Suppose you are given six colours and, are asked to colour each face
of a cube by a di®erent colour. Determine the di®erent number of
colourings possible.
9. Let f(x) = ax2 + bx + c where a; b; c are real numbers. Suppose
f(¡1); f(0); f(1) 2 [¡1; 1]. Prove that jf(x)j • 3=2 for all x 2 [¡1; 1].
10. Given odd integers a; b; c, prove that the equation ax2 + bx + c = 0
cannot have a solution x which is a rational number.
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