MCA Entrance Previous Year Question Bank

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 ________.

Let $n\ge 5$ be an integer. If $9^n-8n-1=64\alpha$ and $6^n-5n-1=25\beta$, then $\alpha -\beta$ is:

If the constant term in the expansion of ${\left(3x^3-2x^2+\frac{5}{x^5}\right)}^{10}$ is $2^k\cdot l$, where $l$ is an odd integer, then the value of $k$ is equal to:

Let the sixth term in the binomial expansion of ${\left(\sqrt{2^{{\log }_2(10-3^x)}}+\sqrt[5]{2^{(x-2){\log }_{2}3}}\right)}^m$, in the increasing powers of $2^{(x-2){\log }_{2}3}$, be $21$. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $x$ is:

If the term without $x$ in the expansion of ${\left(x^{\frac{2}{3}}+\frac{a}{x^3}\right)}^{22}$ is $7315$, then $|a|$ is equal to:

The remainder, when $19^{200}+23^{200}$ is divided by $49$, is:

The value of $\frac{1}{1!50!}+\frac{1}{3!48!}+\frac{1}{5!46!}+\cdots +\frac{1}{49!2!}+\frac{1}{51!0!}$ is:

The coefficient of $x^{-6}$, in the expansion of ${\left(\frac{4x}{5}+\frac{5}{2x^2}\right)}^9$, is ________.

If the constant term in the binomial expansion of ${\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^l}\right)}^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^{\alpha }\beta$, where \beta < 0 is an odd number, then $|\alpha l-\beta |$ is equal to

The remainder on dividing $5^{99}$ by $11$ is: