MCA Entrance Previous Year Question Bank

Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of \left( 2x^{\frac{1}{5}} - \frac{1}{x^{\frac{1}{5}}} \right)^{15}, x > 0, be $m$ and $n$ respectively. If $r$ is a positive integer such that $m{n}^2={}^{15}{C}_r\cdot 2^r$, then the value of $r$ is equal to ________.

Let y = y(x), x > 1, be the solution of the differential equation: $(x-1)\frac{dy}{dx}+2xy=\frac{1}{x-1},\text{ with }y(2)=\frac{1+e^4}{2e^4}$ If $y(3)=\frac{e^{\alpha }+1}{\beta e^{\alpha }}$, then the value of $\alpha +\beta$ is equal to ________.

If $y=y(x)$ is the solution of the differential equation $(1+e^{2x})\frac{dy}{dx}+2(1+y^2)e^x=0$ and $y(0)=0$, then $6{\left(y^{\prime }(0)+(y({\log }_e\sqrt{3}))\right)}^2$ is equal to:

Let $c,k\in \mathbb{R}$. If $f(x)=(c+1)x^2+(1-c^2)x+2k$ and $f(x+y)=f(x)+f(y)-xy$, for all $x,y\in \mathbb{R}$, then the value of $|2(f(1)+f(2)+f(3)+\cdots +f(20))|$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{\sqrt{2}y}{2{\cos }^{4}x-\cos 2x}=x{e}^{{\tan }^{-1}(\sqrt{2}\cos 2x)}$, 0 < x < \pi/2 with $y(\frac{\pi }{4})=\frac{{\pi }^2}{32}$. If $y(\frac{\pi }{3})=\frac{{\pi }^2}{18}{e}^{-{\tan }^{-1}(\alpha )}$, then the value of $3{\alpha }^2$ is equal to:

Let the solution curve of the differential equation $x\frac{dy}{dx}-y=\sqrt{y^2+16x^2}$, $y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal to:

Let $\alpha x=\exp (x^{\beta }{y}^{\gamma })$ be the solution of the differential equation $2x^{2}ydy-(1-x{y}^2)dx=0$, x > 0, $y(2)=\sqrt{{\log }_{e}2}$. Then $\alpha +\beta -\gamma$ equals:

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f^{\prime }(x)+f(x)={\int }_0^{2}f(t)dt$ If $f(0)=e^{-2}$, then $2f(0)-f(2)$ is equal to:

The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx}+\frac{x+a}{y-2}=0,y(1)=0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is:

If $y=y(x)$ is the solution curve of the differential equation $\frac{dy}{dx}+y\tan x=x\sec x$, $0\le x\le \frac{\pi }{3}$, $y(0)=1$, then $y\left(\frac{\pi }{6}\right)$ is equal to: