MCA Entrance Previous Year Question Bank

Let $f:R\to R$ be defined by: $f(x)=\{\begin{array}{llc}{\max }_{t\le x}\{t^3-3t\}& ;x\le 2& \\ x^2+2x-6& ;2& lt;x\le 3\\ [x-3]+9& ;3& lt;x\le 5\\ 2x+1& ;x& gt;5\end{array}$ Where $[t]$ is the greatest integer function. Let $m$ be the number of points where $f$ is not differentiable and $I={\int }_{-2}^{2}f(x)dx$. Then $(m,I)$ is:

If y(x) = x^x, x > 0, then $y^{\prime \prime }(2)-2y^{\prime }(2)$ is equal to:

The number of integral values of $k$, for which one root of the equation $2x^2-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$, is:

If $f(x)=x^2+g^{\prime }(1)x+g^{\prime \prime }(2)$ and $g(x)=f(1)x^2+xf(x)+f^{\prime }(x)$, then the value of $f(4)-g(4)$ is equal to:

Let a differentiable function $f$ satisfy $f(x)+{\int }_3^x\frac{f(t)}{t}dt=\sqrt{x+1},x\ge 3$. Then $12f(8)$ is equal to:

$\text{Let }{f}^1(x)=\frac{3x+2}{2x+3},x\in \mathbb{R}-\left\{\frac{-3}{2}\right\}$ $\text{For }n\ge 2,\text{ define }{f}^n(x)=f^1(f^{n-1}(x))$ $\text{If }{f}^5(x)=\frac{ax+b}{bx+a},\gcd (a,b)=1,\text{ then }a+b\text{ is equal to}$

Suppose $f:R\to (0,\infty )$ be a differentiable function such that $5f(x+y)=f(x)\cdot f(y),\forall x,y\in R$. If $f(3)=320$, then ${\sum }_{n=0}^{5}f(n)$ is equal to:

If the function $f(x)=\{\begin{array}{llcc}(1+|\cos x|{)}^{\frac{\lambda }{|\cos x|}},& 0& lt;x& lt;\frac{\pi }{2}\\ \mu ,& x=\frac{\pi }{2}& & \\ e^{\frac{\cot 6x}{\cot 4x}},& \frac{\pi }{2}& lt;x& lt;\pi \end{array}$ is continuous at $x=\frac{\pi }{2}$, then $9\lambda +6{\log }_e\mu +{\mu }^6-e^{6\lambda }$ is equal to

Let $f(x)=\{\begin{array}{ll}{x}^2\sin \left(\frac{1}{x}\right)& ,x\ne 0\\ 0& ,x=0\end{array}$ Then at $x=0$

Let $[x]$ be the greatest integer $\le x$. Then the number of points in the interval $(-2,1)$, where the function $f(x)=|[x]|+\sqrt{x-[x]}$ is discontinuous, is ___.