1
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$w_{1}$$ be the point obtained by the rotation of $$z_{1}=5+4 i$$ about the origin through a right angle in the anticlockwise direction, and $$w_{2}$$ be the point obtained by the rotation of $$z_{2}=3+5 i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_{1}-w_{2}$$ is equal to :

A
$$-\pi+\tan ^{-1} \frac{8}{9}$$
B
$$-\pi+\tan ^{-1} \frac{33}{5}$$
C
$$\pi-\tan ^{-1} \frac{8}{9}$$
D
$$\pi-\tan ^{-1} \frac{33}{5}$$
2
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$y=y(x)$$ be a solution curve of the differential equation.

$$\left(1-x^{2} y^{2}\right) d x=y d x+x d y$$.

If the line $$x=1$$ intersects the curve $$y=y(x)$$ at $$y=2$$ and the line $$x=2$$ intersects the curve $$y=y(x)$$ at $$y=\alpha$$, then a value of $$\alpha$$ is :

A
$$\frac{1+3 e^{2}}{2\left(3 e^{2}-1\right)}$$
B
$$\frac{3 e^{2}}{2\left(3 e^{2}-1\right)}$$
C
$$\frac{1-3 e^{2}}{2\left(3 e^{2}+1\right)}$$
D
$$\frac{3 e^{2}}{2\left(3 e^{2}+1\right)}$$
3
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

Let $$(\alpha, \beta, \gamma)$$ be the image of the point $$\mathrm{P}(2,3,5)$$ in the plane $$2 x+y-3 z=6$$. Then $$\alpha+\beta+\gamma$$ is equal to :

A
10
B
9
C
5
D
12
4
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of the integral $$\int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x$$ is equal to :

A
$$\log _{e}\left(\frac{(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$$
B
$$\log _{e}\left(\frac{\sqrt{2}(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$$
C
$$\log _{e}\left(\frac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$$
D
$$\log _{e}\left(\frac{\sqrt{2}(3-\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$$
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