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

Let $$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{2 \times 2}$$, where $$\mathrm{a}_{\mathrm{ij}} \neq 0$$ for all $$\mathrm{i}, \mathrm{j}$$ and $$\mathrm{A}^{2}=\mathrm{I}$$. Let a be the sum of all diagonal elements of $$\mathrm{A}$$ and $$\mathrm{b}=|\mathrm{A}|$$. Then $$3 a^{2}+4 b^{2}$$ is equal to :

A
4
B
3
C
14
D
7
2
JEE Main 2023 (Online) 6th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x > 0$$. Then $$18 \int_\limits{1}^{2} f(x) d x$$ is equal to :

A
$$10 \log _{\mathrm{e}} 2+6$$
B
$$5 \log _{e} 2-3$$
C
$$10 \log _{\mathrm{e}} 2-6$$
D
$$5 \log _{\mathrm{e}} 2+3$$
3
JEE Main 2023 (Online) 6th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$I(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d x$$. If $$I(0)=0$$, then $$I\left(\frac{\pi}{4}\right)$$ is equal to :

A
$$\log _{e} \frac{(\pi+4)^{2}}{32}-\frac{\pi^{2}}{4(\pi+4)}$$
B
$$\log _{e} \frac{(\pi+4)^{2}}{16}-\frac{\pi^{2}}{4(\pi+4)}$$
C
$$\log _{e} \frac{(\pi+4)^{2}}{16}+\frac{\pi^{2}}{4(\pi+4)}$$
D
$$\log _{e} \frac{(\pi+4)^{2}}{32}+\frac{\pi^{2}}{4(\pi+4)}$$
4
JEE Main 2023 (Online) 6th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$$ and $$\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$$. If $$\vec{d}$$ is a vector perpendicular to both $$\vec{b}$$ and $$\vec{c}$$, and $$\vec{a} \cdot \vec{d}=18$$, then $$|\vec{a} \times \vec{d}|^{2}$$ is equal to :

A
680
B
720
C
760
D
640
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