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

Let $$y=y(x)$$ be the solution of the differential equation $$\frac{d y}{d x}+\frac{5}{x\left(x^{5}+1\right)} y=\frac{\left(x^{5}+1\right)^{2}}{x^{7}}, x > 0$$. If $$y(1)=2$$, then $$y(2)$$ is equal to :

A
$$\frac{693}{128}$$
B
$$\frac{697}{128}$$
C
$$\frac{637}{128}$$
D
$$\frac{679}{128}$$
2
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1

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) 10th April Morning Shift
MCQ (Single Correct Answer)
+4
-1

Let $$f$$ be a differentiable function such that $${x^2}f(x) - x = 4\int\limits_0^x {tf(t)dt}$$, $$f(1) = {2 \over 3}$$. Then $$18f(3)$$ is equal to :

A
160
B
210
C
150
D
180
4
JEE Main 2023 (Online) 6th April Evening Shift
MCQ (Single Correct Answer)
+4
-1

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
$$e^{2 e^{2}}$$
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