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>> Mathematics Mains-2001
Time : Three hours Max. Marks : 100
1. Let a1, a2, …………. be positive real numbers in geometric progression. For each n, let An, Gn, Hn be respectively, the arithmetic mean, geometric mean, and harmonic mean of a1, a2, ………, an. Find an expression for the geometric mean of G1, G2, ………., Gn in terms of A1, A2, ………., An, H1, H2, ……….., Hn.
2. Let a, b, c be positive real numbers such that b2 – 4ac > 0 and let a1 = c. Prove
by induction that
an+1 = (a αn2) / ( (b2 - 2a (α1- α2 +.…..+ αn ) )
is well-defined and an+1 < α_n/2 for all n = 1, 2, ………..
(Here, ‘well-defined’ means that the denominator in the expression for an+1 is not zero.)
3. Let – 1 < p < 1. Show that the equation 4x2 – 3x – p = 0 has a unique root in
the interval [1/2, 1] and identify it.
4. Let 2x2 + y2 – 3xy = 0 be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.
5. Evaluate ∫ sin-1 ( (2x+2) / √(4x2+8x+13) ) dx.
6. Let f(x), x > 0, be a non-negative continuous function, and
let F(x) = ∫0x f(t) dt, x > 0. If for some c > 0, f(x) < cF(x) for all x > 0,
then show that f(x) = 0 for all x > 0.
7. Let b ¹ 0 and for j = 0, 1, 2, …….., n, let Sj be the area of the region bounded
by the y-axis and the curve xeay = sin by, jπ/b ≤ y ≤ ((j+1)π)/b.
Show that S0, S1, S2, ………., Sn are in geometric progression. Also, find their sum for a = – 1 and b = π.
8. Let α Î R. Prove that a function f : R --> R is differentiable at α if and only if there is a function g : R --> R which is continuous at α and satisfies f(x) – f(α) = f(x) (x – α) for all x Î R.
9. Let C1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1touches C1 internally and C2 externally. Identify the locus of the centre of C.
10. Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.
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