MCA Entrance Previous Year Question Bank

The absolute minimum value of the function $f(x)=|x^2-x+1|+[x^2-x+1]$, where $[t]$ denotes the greatest integer function, in the interval $[-1,2]$, is:

$y=f(x)={\sin }^3\left(\frac{\pi }{3}\cos \left(\frac{\pi }{3\sqrt{2}}(-4x^3+5x^2+1{)}^{\frac{3}{2}}\right)\right)$. Then, at $x=1$:

Let $[x]$ denote the greatest integer function and $f(x)=\max \{1+x+[x],2+x,x+2[x]\}$, $0\le x\le 2$. Let $m$ be the number of points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in $(0,2)$ where $f$ is not differentiable. Then, $(m+n{)}^2$ is equal to:

Let $f(x)={\sum }_{k=1}^{10}k{x}^k,x\in \mathbb{R}$. If $2f(2)-f^{\prime }(2)=119(2{)}^n+1$, then $n$ is equal to ________.

If the local maximum value of the function $f(x)={\left(\frac{\sqrt{3}e}{2\sin x}\right)}^{{\sin }^{2}x},x\in \left(0,\frac{\pi }{2}\right)$ is $\frac{k}{e}$, then ${\left(\frac{k}{e}\right)}^8+\frac{k^8}{e^5}+k^8$ is equal to

Let $f$ be a continuous function satisfying: \int_{0}^{t^{2}} (f(x) + x^{2})dx = \frac{4}{3}t^{3}, \quad \forall t > 0 Then $f\left(\frac{{\pi }^2}{4}\right)$ is equal to:

Let $f:(-2,2)\to \mathbb{R}$ be defined by: $$f(x)=\{\begin{array}{llcc}x[x],& -2& lt;x& lt;0\\ (x-1)[x],& 0\le x& lt;2& \end{array}$$ where $[x]$ is the greatest integer function. Find $m+n$ where $m$ is points of non-continuity and $n$ is points of non-differentiability.

If $2x^y+3y^x=20$, then $\frac{dy}{dx}$ at $(2,2)$ is equal to :

Let $y={\log }_e\left(\frac{1-x^2}{1+x^2}\right)$, -1<x<1. Then at $x=\frac{1}{2}$, the value of $225(y^{\prime }-y^{\prime \prime })$ is equal to

Let $f:\mathbb{R}\to \mathbb{R}$ be a twice differentiable function such that $(\sin x\cos y)-(f(2x+2y)-f(2x-2y))=(\cos x\sin y)+(f(2x+2y)+f(2x-2y))$, for all $x,y\in \mathbb{R}$ and $f^{\prime }(0)=\frac{1}{2}$. If $f(x)=\frac{1}{2}$, then the value of $24f^{\prime \prime }\left(\frac{5\pi }{3}\right)$ is