1
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1

$$\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$$

A
$$\frac{5 \pi+2+3 \sqrt{3}}{6}$$
B
0
C
$$\frac{\pi+2-3 \sqrt{3}}{6}$$
D
$$\pi$$
2
JEE Main 2023 (Online) 12th April Morning Shift
+4
-1

If the local maximum value of the function $$f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$$ , is $$\frac{k}{e}$$, then $$\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}$$ is equal to

A
$$e^{3}+e^{6}+e^{10}$$
B
$$e^{3}+e^{5}+e^{11}$$
C
$$e^{3}+e^{6}+e^{11}$$
D
$$e^{5}+e^{6}+e^{11}$$
3
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1
Out of Syllabus

Let $$f:[2,4] \rightarrow \mathbb{R}$$ be a differentiable function such that $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]$$ with $$f(2)=\frac{1}{2}$$ and $$f(4)=\frac{1}{4}$$.

Consider the following two statements :

(A) : $$f(x) \leq 1$$, for all $$x \in[2,4]$$

(B) : $$f(x) \geq \frac{1}{8}$$, for all $$x \in[2,4]$$

Then,

A
Neither statement (A) nor statement (B) is true
B
Only statement (A) is true
C
Only statement (B) is true
D
Both the statements $$(\mathrm{A})$$ and (B) are true
4
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1

Let $$\mathrm{g}(x)=f(x)+f(1-x)$$ and $$f^{\prime \prime}(x) > 0, x \in(0,1)$$. If $$\mathrm{g}$$ is decreasing in the interval $$(0, a)$$ and increasing in the interval $$(\alpha, 1)$$, then $$\tan ^{-1}(2 \alpha)+\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$ is equal to :

A
$$\frac{3 \pi}{4}$$
B
$$\pi$$
C
$$\frac{5 \pi}{4}$$
D
$$\frac{3 \pi}{2}$$
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