Question

$\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The possible value(s) of}a\text{for which the largest}& \text{(P)}& 9\\ & \text{value of}{\sin }^{2}x-2a\sin x+a+3\text{is}7\text{is/are}& & \\ \text{(B)}& \text{The possible value(s) of}a\text{for which the smallest}& \text{(Q)}& 16\\ & \text{value of}{x}^4-a{x}^2+2a-1\text{for}x\in [-1,2]\text{is}& & \\ & -7\text{, is/are}& & \\ \text{(C)}& \text{If a relation}R\text{is defined on set of integers as}& \text{(R)}& -3\\ & R=\left\{\right(x,y):4x^2+9y^2\le 36\}\text{, then possible}& & \\ & \text{element(s) in the domain is/are}& & \\ \text{(D)}& \text{If}\sin x+\cos x=\frac{1}{5}\text{, then}|12\tan x|\text{is equal to}& \text{(S)}& 1\\ & & \text{(T)}& 11\end{array}$

Which of the following is the only CORRECT combination?

• A. $\text{(A)}\to \text{(Q)},\text{(S)}$
• B. $\text{(A)}\to \text{(Q)},\text{(R)}$
• C. $\text{(B)}\to \text{(Q)},\text{(S)}$
• D. $\text{(B)}\to \text{(R)},\text{(T)}$

Answer 1

The correct option is D $\text{(B)}\to \text{(R)},\text{(T)}$

$\left(A\right)$
Let $f\left(x\right)={\sin }^{2}x-2a\sin x+a+3$
Let $\sin x=t$, where $t\in [-1,1]$
Given equation is $t^2-2at+(a+3)$
In $t\in [-1,1]$
Largest value can occur at $t=1$ or $t=-1$
At $t=1,-a+4=7$
$\Rightarrow a=-3$
At $t=-1$, $1+3a+3=7$
$\Rightarrow a=1$
$\text{(A)}\to \text{(R)},\text{(S)}$

$\left(B\right)$
$f\left(x\right)=x^4-a{x}^2+2a-1,x\in [-1,2]$
Let $t=x^2,t\in [0,4]$
$g\left(t\right)=t^2-at+(2a-1)$ in $t\in [0,4]$
Smallest value can occur at
$g\left(0\right)=2a-1=-7$
$\Rightarrow a=-3g\left(4\right)=15-2a=-7$
$\Rightarrow a=11g\left(\frac{a}{2}\right)=\frac{a^2}{4}-\frac{a^2}{2}+2a-1=-7\Rightarrow a^2-8a-24=0\Rightarrow a=4\pm 2\sqrt{10}$
$\text{(B)}\to \text{(R)},\text{(T)}$