1
JEE Main 2023 (Online) 6th April Morning Shift
Numerical
+4
-1
Change Language

Let $$y=y(x)$$ be a solution of the differential equation $$(x \cos x) d y+(x y \sin x+y \cos x-1) d x=0,0 < x < \frac{\pi}{2}$$. If $$\frac{\pi}{3} y\left(\frac{\pi}{3}\right)=\sqrt{3}$$, then $$\left|\frac{\pi}{6} y^{\prime \prime}\left(\frac{\pi}{6}\right)+2 y^{\prime}\left(\frac{\pi}{6}\right)\right|$$ is equal to ____________.

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2
JEE Main 2022 (Online) 27th July Morning Shift
Numerical
+4
-1
Change Language

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

$$\sin \left( {2{x^2}} \right){\log _e}\left( {\tan {x^2}} \right)dy + \left( {4xy - 4\sqrt 2 x\sin \left( {{x^2} - {\pi \over 4}} \right)} \right)dx = 0$$, $$0 < x < \sqrt {{\pi \over 2}} $$, which passes through the point $$\left(\sqrt{\frac{\pi}{6}}, 1\right)$$. Then $$\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|$$ is equal to ______________.

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3
JEE Main 2022 (Online) 26th July Evening Shift
Numerical
+4
-1
Change Language

Suppose $$y=y(x)$$ be the solution curve to the differential equation $$\frac{d y}{d x}-y=2-e^{-x}$$ such that $$\lim\limits_{x \rightarrow \infty} y(x)$$ is finite. If $$a$$ and $$b$$ are respectively the $$x$$ - and $$y$$-intercepts of the tangent to the curve at $$x=0$$, then the value of $$a-4 b$$ is equal to _____________.

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4
JEE Main 2022 (Online) 26th July Morning Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let a curve $$y=y(x)$$ pass through the point $$(3,3)$$ and the area of the region under this curve, above the $$x$$-axis and between the abscissae 3 and $$x(>3)$$ be $$\left(\frac{y}{x}\right)^{3}$$. If this curve also passes through the point $$(\alpha, 6 \sqrt{10})$$ in the first quadrant, then $$\alpha$$ is equal to ___________.

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