JEE Main 2019 (Online) 9th January Evening Slot | Limits, Continuity and Differentiability Question 149 | Mathematics

JEE Main 2019 (Online) 9th January Evening Slot

MCQ (Single Correct Answer)

-1

For each x$$ \in $$R, let [x] be the greatest integer less than or equal to x.

Then $$\mathop {\lim }\limits_{x \to {0^ - }} \,\,{{x\left( {\left[ x \right] + \left| x \right|} \right)\sin \left[ x \right]} \over {\left| x \right|}}$$ is equal to :

$$-$$ sin 1

sin 1

JEE Main 2019 (Online) 9th January Evening Slot

MCQ (Single Correct Answer)

-1

Let f be a differentiable function from

R to R such that $$\left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \over 2}}},$$

for all  $$x,y \in $$ R.

If   $$f\left( 0 \right) = 1$$

then   $$\int\limits_0^1 {{f^2}} \left( x \right)dx$$  is equal to :

$${1 \over 2}$$

JEE Main 2019 (Online) 9th January Morning Slot

MCQ (Single Correct Answer)

-1

Let f : R $$ \to $$ R be a function defined as
$$f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x < 3} \cr {b + 5x} & ; & {3 \le x < 5} \cr {30} & ; & {x \ge 5} \cr } } \right.$$

Then, f is

continuous if a = 0 and b = 5

continuous if a = –5 and b = 10

continuous if a = 5 and b = 5

not continuous for any values of a and b

JEE Main 2019 (Online) 9th January Morning Slot

MCQ (Single Correct Answer)

-1

$$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$

exists and equals $${1 \over {2\sqrt 2 }}$$

exists and equals $${1 \over {4\sqrt 2 }}$$

exists and equals $${1 \over {2\sqrt 2 (1 + \sqrt {2)} }}$$

does not exists