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

Posted by Amit Goyal on Apr 21, 2010; 6:26am
URL: http://discussion-forum.276.s1.nabble.com/Sample-Questions-of-ISI-ME-I-Mathematics-2010-Discussion-tp4850019p4935346.html

Next four:
If f(1) = 0, f'(x) > f(x) for all x > 1, then f(x) is
(a) positive valued for all x > 1,
(b) negative valued for all x > 1,
(c) positive valued on (1, 2) but negative valued on [2,∞).
(d) None of these.

Consider the constrained optimization problem
max (ax + by) subject to (cx + dy) ≤ 100, x ≥ 0, y ≥ 0
where a, b, c, d are positive real numbers such that d/b > (c + d)/(a + b)
The unique solution (x*, y*) to this constrained optimization problem is
(a) (x*=100/a, y*=0),
(b) (x*=100/c, y*=0),
(c) (x*=0, y*=100/b),
(d) (x*=0, y*=100/d).

For any real number x, let [x] be the largest integer not exceeding x. The
domain of definition of the function f(x) = 1/ √(|[|x|-2]|-3) is
(a) [−6, 6],
(b) (−∞,−6) ∪ (+6,∞),
(c) (−∞,−6] ∪ [+6,∞).
(d) None of these.

Let f: R → R and g: R → R be defined as
f(x) = −1, x < −0.5
      = −0.5, −0.5 ≤ x < 0
      = 0, x = 0
      = 1, x > 0
and g(x) = 1 + x − [x], where [x] is the largest integer not exceeding x.
Then f(g(x)) equals
(a) −1,
(b) −0.5,
(c) 0,
(d) 1.