1
JEE Main 2023 (Online) 6th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Among the statements :

(S1) : $$2023^{2022}-1999^{2022}$$ is divisible by 8

(S2) : $$13(13)^{n}-12 n-13$$ is divisible by 144 for infinitely many $$n \in \mathbb{N}$$

A
both (S1) and (S2) are incorrect
B
only (S1) is correct
C
only (S2) is correct
D
both (S1) and (S2) are correct
2
JEE Main 2023 (Online) 6th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If the solution curve $$f(x, y)=0$$ of the differential equation

$$\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0$$,

passes through the points $$(1,0)$$ and $$(\alpha, 2)$$, then $$\alpha^{\alpha}$$ is equal to :

A
$$e^{\sqrt{2} e^{2}}$$
B
$$e^{2 e^{\sqrt{2}}}$$
C
$$e^{e^{2}}$$
D
$$e^{2 e^{2}}$$
3
JEE Main 2023 (Online) 6th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f(x)$$ be a function satisfying $$f(x)+f(\pi-x)=\pi^{2}, \forall x \in \mathbb{R}$$. Then $$\int_\limits{0}^{\pi} f(x) \sin x d x$$ is equal to :

A
$$\pi^{2}$$
B
$$\frac{\pi^{2}}{2}$$
C
$$2 \pi^{2}$$
D
$$\frac{\pi^{2}}{4}$$
4
JEE Main 2023 (Online) 6th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

$$\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}$$ is equal to :

A
$$\sqrt{2}$$
B
1
C
$$\frac{1}{\sqrt{2}}$$
D
0
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