1. Let ibe a root of the equation x2+ 1 = 0 and let ωbe a root of the
equation x2+x+ 1 = 0. Construct a polynomial
f(x) = a0+a1x+. . . +anxn
where a0, a1, . . . , anare all integers such that f(i+ω)=0.
2. Let abe a fixed real number. Consider the equation
(x+ 2)2(x+ 7)2+a= 0, x ∈R,
where Ris the set of real numbers. For what values of a, will the equation
have exactly one double-root?
3. Let Aand Bbe variable points on x-axis and y-axis respectively such
that the line segment AB is in the first quadrant and of a fixed length
2d. Let Cbe the mid-point of AB and Pbe a point such that
(a) Pand the origin are on the opposite sides of AB and,
(b) P C is a line segment of length dwhich is perpendicular to AB.
Find the locus of P.
4. Let a real-valued sequence {xn}n≥1be such that
lim
n→∞ nxn= 0.
Find all possible real values of tsuch that limn→∞ xn(log n)t= 0.
5. Prove that the largest pentagon (in terms of area) that can be inscribed
in a circle of radius 1 is regular (i.e., has equal sides).
6. Prove that the family of curves
x2
a2+λ+y2
b2+λ= 1
satisfies dy
dx(a2−b2) = (x+ydy
dx) (xdy
dx −y).
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