Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

Posted by Amit Goyal on Apr 10, 2010; 3:22am
URL: http://discussion-forum.276.s1.nabble.com/Sample-Questions-of-ISI-ME-I-Mathematics-2010-Discussion-tp4850019p4880689.html

Good work Nidhi and Anisha

If a < b < c < d, then the equation (x − a)(x − b) + 2(x − c)(x − d) = 0 has
(a) both the roots in the interval [a, b],
(b) both the roots in the interval [c, d],
(c) one root in the interval (a, b) and the other root in the interval (c, d),
(d) one root in the interval [a, b] and the other root in the interval [c, d].
About this problem i am skeptical that there is some missing information because if we pick a = 0, b = 1, c = 100, d = 101, then there is no x for which (x)(x − 1) + 2(x − 100)(x − 101) = 0. The reason is if i pick x in the interval [0, 1] then x(x -1) will lie between -1/4 and 0 and 2(x − 100)(x − 101) will take the value larger than 2(99)(100) so the sum (x)(x − 1) + 2(x − 100)(x − 101) cannot be 0. And if i pick x in the interval [100, 101] then 2(x − 100)(x − 101)  will lie between -1/2 and 0 and x(x -1) will take value larger than (99)(100) so the sum (x)(x − 1) + 2(x − 100)(x − 101) cannot be 0. Thus none of the options.

Let f and g be two differentiable functions on (0, 1) such that f(0) = 2, f(1) = 6, g(0) = 0 and g(1) = 2. Then there exists θ ∈ (0, 1) such that f'(θ) equals
(b) 2g'(θ)

The minimum value of log(x)a + log(a)x, for 1 < a < x, is
(b) greater than 2

The value of ∫(1/(2x(1+√x)))dx over [4, 9] equals
(c) 2log(e)3 − 3log(e)2

Read log(x)a as log value of a when x is the base, And likewise others.
Read ∫(1/(2x(1+√x)))dx over [4, 9] as the value of definite integral of the said function over the range [4, 9]