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.
11. (a) Let a, b, c be real numbers with a ¹ 0 and let a, b be the roots of the equation ax2 + bx + c = 0. Express the roots of a2x2 + abcx + c3 = 0 in terms of a,b.
(b) Let a, b, c be real numbers with a2 + b2 + c2 = 1. Show that the equation
12. (a) Let P be a point on the ellipse x2/a2 + y2/b2 =1, 0 < b < a. Let the line parallel to y-axis passing through P meet the circle x2 + y2 = a2 at the point Q such that P and Q are on the same side of x-axis. For two positive real numbers r and s, find the locus of the point R on PQ such that PR : RQ = r : s as P varies over the ellipse.
(b) If D is the area of a triangle with side lengths a, b, c then
show that D < 1/4 √((a+b+c)abc).
Also show that the equality occurs in the above inequality if and only if a = b = c.
13. A hemispherical tank of radius 2 metres is initially full of water and has an outlet of 12 cm2 cross-sectional are at the bottom. The outlet is opened at some instant.
The flow through the outlet is according to the law v(t) = 0.6 √(2gh(t)), where v(t) and h(t) are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank.
(Hint : Form a differential equation by relating the decrease of water level to the outflow).
14. (a) An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white?
(b) An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6 is thrown n times and the list of n numbers showing up is noted. What is the probability that, among the numbers 1, 2, 3, 4, 5, 6 only three numbers appear in this list?