JEE Advanced 2014 Paper 1 Offline | Limits, Continuity and Differentiability Question 33 | Mathematics

JEE Advanced 2014 Paper 1 Offline

MCQ (More than One Correct Answer)

-0

Let $$f:(a,b) \to [1,\infty )$$ be a continuous function and g : R $$\to$$ R be defined as $$g(x) = \left\{ {\matrix{ 0 & , & {x < a} \cr {\int_a^x {f(t)dt} } & , & {a \le x \le b} \cr {\int_a^b {f(t)dt} } & , & {x > b} \cr } } \right.$$ Then,

g(x) is continuous but not differentiable at a

g(x) is differentiable on R

g(x) is continuous but not differentiable at b

g(x) is continuous and differentiable at either a or b but not both

JEE Advanced 2013 Paper 2 Offline

MCQ (More than One Correct Answer)

-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 = ?

$${{ - 15} \over 2}$$

$${{ - 17} \over 2}$$

IIT-JEE 2012 Paper 2 Offline

MCQ (More than One Correct Answer)

-1

For every integer n, let a_n and b_n 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_{n $$-$$ 1} $$-$$ b_{n $$-$$ 1} = 0

a_n $$-$$ b_n = 1

a_n $$-$$ b_{n $$+$$ 1} = 1

a_{n $$-$$ 1} $$-$$ b_n = $$-$$1

IIT-JEE 2011 Paper 1 Offline

MCQ (More than One Correct Answer)

-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

f(x) is differentiable only in a finite interval containing zero.

f(x) is continuous $$\forall x \in R$$.

f'(x) is constant $$\forall x \in R$$.

f(x) is differentiable except at finitely many points.

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