1
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1 Let $$y=y(x)$$ be the solution of the differential equation

$$\frac{d y}{d x}=\frac{4 y^{3}+2 y x^{2}}{3 x y^{2}+x^{3}}, y(1)=1$$.

If for some $$n \in \mathbb{N}, y(2) \in[n-1, n)$$, then $$n$$ is equal to _____________.

2
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1 Let $$f$$ be a twice differentiable function on $$\mathbb{R}$$. If $$f^{\prime}(0)=4$$ and $$f(x) + \int\limits_0^x {(x - t)f'(t)dt = \left( {{e^{2x}} + {e^{ - 2x}}} \right)\cos 2x + {2 \over a}x}$$, then $$(2 a+1)^{5}\, a^{2}$$ is equal to _______________.

3
JEE Main 2022 (Online) 29th June Evening Shift
Numerical
+4
-1 Let y = y(x), x > 1, be the solution of the differential equation $$(x - 1){{dy} \over {dx}} + 2xy = {1 \over {x - 1}}$$, with $$y(2) = {{1 + {e^4}} \over {2{e^4}}}$$. If $$y(3) = {{{e^\alpha } + 1} \over {\beta {e^\alpha }}}$$, then the value of $$\alpha + \beta$$ is equal to _________.

4
JEE Main 2022 (Online) 29th June Morning Shift
Numerical
+4
-1 Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 < x < {\pi \over 2}$$ with $$y\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {32}}$$. If $$y\left( {{\pi \over 3}} \right) = {{{\pi ^2}} \over {18}}{e^{ - {{\tan }^{ - 1}}(\alpha )}}$$, then the value of 3$$\alpha$$2 is equal to ___________.

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