1
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1
Change Language

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 _____________.

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2
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1
Change Language

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 _______________.

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3
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1
Change Language

Let $${a_n} = \int\limits_{ - 1}^n {\left( {1 + {x \over 2} + {{{x^2}} \over 3} + \,\,.....\,\, + \,\,{{{x^{n - 1}}} \over n}} \right)dx} $$ for every n $$\in$$ N. Then the sum of all the elements of the set {n $$\in$$ N : an $$\in$$ (2, 30)} is ____________.

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4
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

If the circles $${x^2} + {y^2} + 6x + 8y + 16 = 0$$ and $${x^2} + {y^2} + 2\left( {3 - \sqrt 3 } \right)x + 2\left( {4 - \sqrt 6 } \right)y = k + 6\sqrt 3 + 8\sqrt 6 $$, $$k > 0$$, touch internally at the point $$P(\alpha ,\beta )$$, then $${\left( {\alpha + \sqrt 3 } \right)^2} + {\left( {\beta + \sqrt 6 } \right)^2}$$ is equal to ________________.

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