1
JEE Advanced 2016 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2

Let $$f:\left[ { - {1 \over 2},2} \right] \to R$$ and $$g:\left[ { - {1 \over 2},2} \right] \to R$$ be function defined by $$f(x) = [{x^2} - 3]$$ and $$g(x) = |x|f(x) + |4x - 7|f(x)$$, where [y] denotes the greatest integer less than or equal to y for $$y \in R$$. Then

A
f is discontinuous exactly at three points in $$\left[ { - {1 \over 2},2} \right]$$.
B
f is discontinuous exactly at four points in $$\left[ { - {1 \over 2},2} \right]$$.
C
g is NOT differentiable exactly at four points in $$\left( { - {1 \over 2},2} \right)$$.
D
g is NOT differentiable exactly at five points in $$\left( { - {1 \over 2},2} \right)$$.
2
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2

Let $$g:R \to R$$ be a differentiable function with $$g(0) = 0$$, $$g'(0) = 0$$ and $$g'(1) \ne 0$$. Let

$$f(x) = \left\{ {\matrix{ {{x \over {|x|}}g(x),} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$

and $$h(x) = {e^{|x|}}$$ for all $$x \in R$$. Let $$(f\, \circ \,h)(x)$$ denote $$f(h(x))$$ and $$(h\, \circ \,f)(x)$$ denote $$f(f(x))$$. Then which of the following is (are) true?

A
f is differentiable at x = 0.
B
h is differentiable at x = 0.
C
$$f\, \circ \,h$$ is differentiable at x = 0.
D
$$h\, \circ \,f$$ is differentiable at x = 0.
3
JEE Advanced 2014 Paper 1 Offline
MCQ (More than One Correct Answer)
+3
-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,
A
g(x) is continuous but not differentiable at a
B
g(x) is differentiable on R
C
g(x) is continuous but not differentiable at b
D
g(x) is continuous and differentiable at either a or b but not both
4
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
$${{ - 17} \over 2}$$
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