MCA Entrance Previous Year Question Bank

Let $f(x)$ and $g(x)$ be two real polynomials of degree 2 and 1 respectively. If $f(g(x))=8x^2-2x$, and $g(f(x))=4x^2+6x+1$, then the value of $f(2)+g(2)$ is ________.

Let $M=\left[\begin{array}{cc}0& -\alpha \\ \alpha & 0\end{array}\right]$, where $\alpha$ is a non-zero real number and $N={\sum }_{k=1}^{49}{M}^{2k}$. If $(I-M^2)N=-2I$, then the positive integral value of $\alpha$ is ________.

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ________.

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 $f$ and $g$ be twice differentiable even functions on $(-2,2)$ such that $f(\frac{1}{4})=0,f(\frac{1}{2})=0,f(1)=1$, and $g(\frac{3}{4})=0,g(1)=2$. Then, the minimum number of solutions of $f(x)g^{\prime \prime }(x)+f^{\prime }(x)g^{\prime }(x)=0$ in $(-2,2)$ is equal to ________.

For real number a, b (a > b > 0), let: $\text{Area }\{(x,y):x^2+y^2\le a^2\text{ and }\frac{x^2}{a^2}+\frac{y^2}{b^2}\ge 1\}=30\pi$ $\text{Area }\{(x,y):x^2+y^2\ge b^2\text{ and }\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1\}=18\pi$ Then the value of $(a-b{)}^2$ is equal to ________.

The number of solutions of the equation $\sin x={\cos }^{2}x$ in the interval $(0,10)$ is ________.

Let $3,6,9,12,\dots$ up to $78$ terms and $5,9,13,17,\dots$ up to $59$ terms be two series. Then, the sum of the terms common to both the series 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 ________.

Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}+(\vec{b}\times \vec{c})=\vec{0}$ and $\vec{b}\cdot \vec{c}=5$. Then, the value of $3(\vec{c}\cdot \vec{a})$ is equal to ________.