If $x + y = 16$ and $x^2 + y^2$ is minimum,then the values of $x$ and $y$ are

If a continuous function $f$ defined on the real line $R$ assumes positive and negative values in $R$,then the equation $f(x)=0$ has a root in $R$. For example,if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum value is negative,then the equation $f(x)=0$ has a root in $R$.
Consider $f(x)=k e^x-x$ for all real $x$,where $k$ is a real constant.
$1.$ The line $y=x$ meets $y=k e^x$ for $k \leq 0$ at
$(A)$ no point $(B)$ one point $(C)$ two points $(D)$ more than two points
$2.$ The positive value of $k$ for which $k e^x-x=0$ has only one root is
$(A)$ $1/e$ $(B)$ $1$ $(C)$ $e$ $(D)$ $\log_e 2$
$3.$ For $k>0$,the set of all values of $k$ for which $k e^x-x=0$ has two distinct roots is
$(A)$ $(0, 1/e)$ $(B)$ $(1/e, 1)$ $(C)$ $(1/e, \infty)$ $(D)$ $(0, 1)$
Give the answer for questions $1, 2$ and $3$.

The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is . . . . . . .

Find the semi-vertical angle of a right circular cone of a given slant height,if the volume of the cone is maximum.

For the function $f(x) = x^4(12\ln x - 7)$,match the following columns:

Column-$I$	Column-$II$
$(A)$ If $(a, b)$ is a point of inflection,then $a - b$ equals	$(P)$ $3$
$(B)$ If $e^t$ is the point of local minimum,then $12t$ equals	$(Q)$ $1$
$(C)$ If the graph is concave down on $(d, e)$,then $d + 3e$ equals	$(R)$ $4$
$(D)$ If the graph is concave up on $(p, \infty)$,then $p$ equals	$(S)$ $8$

The smaller side of the rectangle with the largest area, that can be inscribed inside a semi-circle of radius $2 \ units$ is of length