Let $f(x)=2^x-x^2, x \in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f^{\prime}(x)$ intersect the $x$-axis,then the value of $m+n$ is

Consider the polynomial $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.
$1.$ The real number $s$ lies in the interval
$(A)$ $\left(-\frac{1}{4}, 0\right)$ $(B)$ $\left(-1,-\frac{3}{4}\right)$
$(C)$ $\left(-\frac{3}{4},-\frac{1}{2}\right)$ $(D)$ $\left(0, \frac{1}{4}\right)$
$2.$ The area bounded by the curve $y=f(x)$ and the lines $x=0, y=0$ and $x=t$,lies in the interval
$(A)$ $\left(\frac{3}{4}, 3\right)$ $(B)$ $\left(\frac{21}{64}, \frac{11}{16}\right)$
$(C)$ $(9,10)$ $(D)$ $\left(0, \frac{21}{64}\right)$
$3.$ The function $f^{\prime}(x)$ is
$(A)$ increasing in $\left(-t,-\frac{1}{4}\right)$ and decreasing in $\left(-\frac{1}{4}, t\right)$
$(B)$ decreasing in $\left(-t,-\frac{1}{4}\right)$ and increasing in $\left(-\frac{1}{4}, t\right)$
$(C)$ increasing in $(-t, t)$ $(D)$ decreasing in $(-t, t)$
Give the answer for questions $1, 2$ and $3.$

Let $f(x) = \sin x + (x^3 - 3x^2 + 4x - 2) \cos x$ for $x \in (0, 1)$. Consider the following statements:
$I.$ $f$ has a zero in $(0, 1)$.
$II.$ $f$ is monotone in $(0, 1)$.
Then,

Match the functions of List $I$ with the items of List $II$.

List $I$	List $II$
$A. 3x^4 - 2x^3 - 6x^2 + 6x + 1$	$(I)$ has minimum value at $x = 4$
$B. x + \frac{1}{x}, \forall x < 0$	$(II)$ has maximum value at $x = -1$
$C. x^4(7 - x)^3$	$(III)$ has maximum value at $x = 4$
$D. x^4 + (8 - x)^4$	$(IV)$ is decreasing in $[2, \infty)$
	$(V)$ is increasing in $[2, \infty)$

Two differentiable functions $f(x)$ and $g(x)$ are such that $f''(x) > 0$ and $g''(x) < 0$ for all $x \in (a,b)$ and $\int_{a}^{b} f(x) dx = \int_{a}^{b} g(x) dx$. If $f(x) = g(x)$ for $x = \alpha, \beta \in (a,b)$ $(\alpha < \beta)$,then:

$f(x)=4 \log _{e}(x-1)-2 x^{2}+4 x+5, x>1$,which one of the following is $NOT$ correct?