1
JEE Advanced 2013 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2

$$a \in R$$ (the set of all real numbers), a $$\ne$$ $$-$$1,

$$\mathop {\lim }\limits_{n \to \infty } {{({1^a} + {2^a} + ... + {n^a})} \over {{{(n + 1)}^{a - 1}}[(na + 1) + (na + 2) + ... + (na + n)]}} = {1 \over {60}}$$, Then a = ?

A
5
B
7
C
$${{ - 15} \over 2}$$
D
$${{ - 17} \over 2}$$
2
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
3
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.
4
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.
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