Question

List Match Sets
$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{I}& \text{The sides of a rectangle of greatest}& & \\ & \text{perimeter which is inscribed in a semicircle}& \text{(P)}& 108\\ & \text{of radius}\sqrt{5}\text{are}{\lambda }_1\text{and}{\lambda }_2\text{then}4({\lambda }_1^2+{\lambda }_2^2)\text{equals}& & \\ \text{II}& \text{If the number of points of minima of}f\left(x\right)=& \text{(Q)}& 2\\ & \left|\frac{x^2-2}{x^2-1}\right|\text{is}\lambda \text{then}18\lambda \text{is}& & \\ \text{III}& \text{If}f\left(x\right)=e^{2x}-2(a^2-21)e^x+8x+5& \text{(R)}& 68\\ & \text{is monotonically increasing for all}xϵR,& & \\ & \text{then the number of integers in range of 'a' are}& & \\ \text{IV}& \text{The volume of a rectangular closed box is 72}& & \\ & \text{and the base sides are in the ratio 1 : 2.}& \text{(S)}& 54\\ & \text{The least total surface area is}& & \end{array}$

Which of the following is the only CORRECT combination?

• A. (III), (P)
• B. (IV), (P)
• C. (III), (Q)
• D. (II), (P)

Answer 1

The correct option is B (IV), (P)

I
$P=2(\sqrt{5}sin\theta +2\sqrt{5}cos\theta )$
$\frac{dP}{dQ}=2\sqrt{5}(cos\theta -2sin\theta )=0$
$tan\theta =\frac{1}{2}$
So ${\lambda }_1=\sqrt{5}sin\theta =1$
${\lambda }_2=2\sqrt{5}cos\theta =4$
${\lambda }_1^2+{\lambda }_2^2=17$

II $f\left(x\right)=\left|\frac{x^2-2}{x^2-1}\right|$

Points of minima are $-\sqrt{2},0,\sqrt{2}$

III $f^{\prime }\left(x\right)=2e^{2x}-2(a^2-21)e^x+8⩾0$
coefficient of the quadratic equation in $e^x$ is 1 so for $e^{2x}-(a^2-21)e^x+4⩾0$ to be true
D $\le 0$
$(a^2-21{)}^2-16\le 0$
$(a^2-25)(a^2-17)\le 0$
$(a-5)(a+5)(a-\sqrt{1}7)(a+\sqrt{1}7)\le 0$

$aϵ[-5,-\sqrt{1}7]\cup [\sqrt{1}7,5]$
So, number of integral values of $a=2$

IV
Let sides are $x,2xandh$
$x^{2}h=36$
$S=2(2x^2+hx+2hx)$
$S=4x^2+6hx$
$S=4x^2+\frac{216}{x}$
$\frac{ds}{dx}=8x-\frac{216}{x^2}=0$
$\Rightarrow$ x = 3 and h = 4
So S = 108.