1
IIT-JEE 2012 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1

For every integer n, let an and bn be real numbers. Let function f : R $$\to$$ R be given by

$$f(x) = \left\{ {\matrix{ {{a_n} + \sin \pi x,} & {for\,x \in [2n,2n + 1]} \cr {{b_n} + \cos \pi x,} & {for\,x \in (2n - 1,2n)} \cr } } \right.$$, for all integers n. If f is continuous, then which of the following hold(s) for all n ?

A
an $$-$$ 1 $$-$$ bn $$-$$ 1 = 0
B
an $$-$$ bn = 1
C
an $$-$$ bn $$+$$ 1 = 1
D
an $$-$$ 1 $$-$$ bn = $$-$$1
2
IIT-JEE 2011 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1

Let f : R $$\to$$ R be a function such that $$f(x + y) = f(x) + f(y),\,\forall x,y \in R$$. If f(x) is differentiable at x = 0, then

A
f(x) is differentiable only in a finite interval containing zero.
B
f(x) is continuous $$\forall x \in R$$.
C
f'(x) is constant $$\forall x \in R$$.
D
f(x) is differentiable except at finitely many points.
3
IIT-JEE 2011 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1

If $$f(x) = \left\{ {\matrix{ { - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr { - \cos x} & { - {\pi \over 2} < x \le 0} \cr {x - 1} & {0 < x \le 1} \cr {\ln x} & {x > 1} \cr } } \right.$$, then

A
f(x) is continuous at x = $$-$$ $$\pi$$/2.
B
f(x) is not differentiable at x = 0.
C
f(x) is differentiable at x = 1.
D
f(x) is differentiable at x = $$-$$3/2.
4
IIT-JEE 2009 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2

Let $$L = \mathop {\lim }\limits_{x \to 0} {{a - \sqrt {{a^2} - {x^2}} - {{{x^2}} \over 4}} \over {{x^4}}},a > 0$$. If L is finite, then

A
$$a = 2$$
B
$$a = 1$$
C
$$L = {1 \over {64}}$$
D
$$L = {1 \over {32}}$$
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