1
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1

Let a function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be defined as :

$$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$$

where $$\mathrm{b} \in \mathbb{R}$$. If $$f$$ is continuous at $$x=4$$, then which of the following statements is NOT true?

A
$$f$$ is not differentiable at $$x=4$$
B
$$f^{\prime}(3)+f^{\prime}(5)=\frac{35}{4}$$
C
$$f$$ is increasing in $$\left(-\infty, \frac{1}{8}\right) \cup(8, \infty)$$
D
$$f$$ has a local minima at $$x=\frac{1}{8}$$
2
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1

$$\int\limits_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x \text { is equal to }$$

A
$$10(\pi+4)$$
B
$$10(\pi+2)$$
C
$$20(\pi-2)$$
D
$$20(\pi+2)$$
3
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1
Out of Syllabus

If $$a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}}$$ and $$f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}}$$, $$x \in (0,1)$$, then :

A
$$2\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
B
$$f\left( {{a \over 2}} \right)f'\left( {{a \over 2}} \right) = \sqrt 2$$
C
$$\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
D
$$f\left( {{a \over 2}} \right) = \sqrt 2 f'\left( {{a \over 2}} \right)$$
4
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1
Out of Syllabus

$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {3 \over {{2^n}}}} }} + \,\,...\,\, + \,\,{1 \over {\sqrt {1 - {{{2^n} - 1} \over {{2^n}}}} }}} \right)$$ is equal to

A
$$\frac{1}{2}$$
B
1
C
2
D
$$-$$2
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