1
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Morning Slot

Let  $$f\left( x \right) = \left\{ {\matrix{ {\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left| x \right| \le 2} \cr {8 - 2\left| x \right|} & {2 < \left| x \right| \le 4} \cr } } \right.$$

Let S be the set of points in the interval (– 4, 4) at which f is not differentiable. Then S
A
equals $$\left\{ { - 2, - 1,1,2} \right\}$$
B
equals $$\left\{ { - 2, - 1,0,1,2} \right\}$$
C
equals $$\left\{ { - 2,2} \right\}$$
D
is an empty set

Explanation

$$f\left( x \right) = \left\{ {\matrix{ {8 + 2x,} & { - 4 \le x \le - 2} \cr {{x^2},} & { - 2 \le x \le - 1} \cr {\left| x \right|,} & { - 1 < x < 1} \cr {{x^2},} & {1 \le x \le 2} \cr {8 - 2x,} & {2 < x \le 4} \cr } } \right.$$


f(x) is not differentiable at

x = $$\left\{ { - 2, - 1,0,1,2} \right\}$$

$$ \Rightarrow $$  S = {$$-$$2, $$-$$ 1, 0, 1, 2}

2
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Evening Slot

Let f : ($$-$$1, 1) $$ \to $$ R be a function defined by f(x) = max $$\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}.$$ If K be the set of all points at which f is not differentiable, then K has exactly -
A
one element
B
three elements
C
five elements
D
two elements

Explanation

f : ($$-$$ 1, 1) $$ \to $$ R

f(x) = max {$$-$$ $$\left| x \right|, - \sqrt {1 - {x^2}} $$}


Non-derivable at 3 points in ($$-$$1, 1)
3
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 11th January Morning Slot

Let $$f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.$$ and

$$g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).$$

Then, in the interval (–2, 2), g is :
A
non continuous
B
differentiable at all points
C
not differentiable at two points
D
not differentiable at one point

Explanation

$$\left| {f\left( x \right)} \right| = \left\{ {\matrix{ 1 & , & { - 2 \le x < 0} \cr {1 - {x^2}} & , & {0 \le x < 1} \cr {{x^2} - 1} & , & {1 \le x \le 2} \cr } } \right.$$

and  $$f\left( {\left| x \right|} \right) = {x^2} - 1,x \in \left[ { - 2,2} \right]$$

Hence  $$g(x) = \left\{ {\matrix{ {{x^2}} & , & {x \in \left[ { - 2,0} \right]} \cr 0 & , & {x \in \left[ {0,1} \right)} \cr {2\left( {{x^2} - 1} \right)} & , & {x \in \left[ {1,2} \right]} \cr } } \right.$$

It is not differentiable at x = 1
4
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 11th January Morning Slot

Let [x] denote the greatest integer less than or equal to x. Then $$\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( {\left| x \right| - \sin \left( {x\left[ x \right]} \right)} \right)}^2}} \over {{x^2}}}$$
A
equals $$\pi $$ + 1
B
equals 0
C
does not exist
D
equals $$\pi $$

Explanation

R.H.L. $$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( {\left| x \right| - \sin \left( {x\left[ x \right]} \right)} \right)}^2}} \over {{x^2}}}$$

(as x $$ \to $$ 0+ $$ \Rightarrow $$  [x] $$=$$ 0)

$$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {x^2}} \over {{x^2}}}$$

$$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right)} \over {\left( {\pi {{\sin }^2}x} \right)}} + 1 = \pi + 1$$

L.H.L. $$=$$ $$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( { - x + \sin x} \right)}^2}} \over {{x^2}}}$$

(as x $$ \to $$ 0$$-$$ $$ \Rightarrow $$  [x] $$=$$ $$-$$1)

$$\mathop {\lim }\limits_{x \to {0^ + }} {{\tan \left( {\pi {{\sin }^2}x} \right)} \over {\pi {{\sin }^2}x}}\,.\,{{\pi {{\sin }^2}x} \over {{x^2}}} + {\left( { - 1 + {{\sin x} \over x}} \right)^2} \Rightarrow \pi $$

R.H.L.  $$ \ne $$  L.H.L.

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