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

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

$$f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}$$ where $$[t]$$ is the greatest integer less than or equal to $$t$$. If $$\mathop {\lim }\limits_{x \to -1 } f(x)$$ exists, then the value of $$\int\limits_{0}^{4} f(x) d x$$ is equal to

A
$$-$$1
B
$$-$$2
C
1
D
2
2
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1

Let $$I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x$$. Then

A
$${\pi \over 2} < I < {{3\pi } \over 4}$$
B
$${\pi \over 5} < I < {{5\pi } \over {12}}$$
C
$${{5\pi } \over {12}} < I < {{\sqrt 2 } \over 3}\pi$$
D
$${{3\pi } \over 4} < I < \pi$$
3
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 $$x=\frac{1}{8}$$
4
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
$$20(\pi+2)$$
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