Find two numbers whose sum is $24$ and whose product is as large as possible.

The sixth term of an $A.P.$ is equal to $2$. The value of the common difference $x$ of the $A.P.$ that makes the product $a_1 a_4 a_5$ least is given by:

Find the maximum and minimum values of the function given by $f(x) = |x + 2| - 1$.

Let $AD$ and $BC$ be two vertical poles at $A$ and $B$ respectively on a horizontal ground. If $AD = 8 \ m$,$BC = 11 \ m$ and $AB = 10 \ m$,then the distance (in meters) of point $M$ on $AB$ from the point $A$ such that $MD^2 + MC^2$ is minimum,is

Of all the closed cylindrical cans (right circular) with a given volume of $100 \text{ cm}^3$,find the dimensions of the can that has the minimum surface area.

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 $3$.