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

Answer 1

The correct option is C $\text{(D)}\to \text{(P)},\text{(Q)}$

$\left(C\right)$
$x,y\in Z$
$4x^2+9y^2\le 36$

If $x=0,$ then $y^2\le \frac{36}{9}$
$\Rightarrow y^2\le 4\Rightarrow y\in \{0,\pm 1,\pm 2\}$

If $x=\pm 1,$ then $y^2\le \frac{32}{9}$
$\Rightarrow y\in \{0,\pm 1\}$

If $x=\pm 2,$ then $y^2\le \frac{20}{9}$
$\Rightarrow y\in \{0,\pm 1\}$

If $x=\pm 3,$ then $y^2\le 0$
$\Rightarrow y\in \left\{0\right\}$
$\therefore$ Domain $=\{0,\pm 1,\pm 2,\pm 3\}$
$\text{(C)}\to \text{(R)},\text{(S)}$

$\left(D\right)$
$\sin x+\cos x=\frac{1}{5}$
$\Rightarrow (\sin x+\cos x{)}^2=\frac{1}{25}\Rightarrow 1+\sin 2x==\frac{1}{25}\Rightarrow \frac{2\tan x}{1+{\tan }^{2}x}=\frac{-24}{25}\Rightarrow 12{\tan }^{2}x+25\tan x+12=0\Rightarrow (4\tan x+3\left)\right(3\tan x+4)=0$
$\Rightarrow \tan x=\frac{-3}{4}$ or $\tan x=\frac{-4}{3}$
$\Rightarrow |12\tan x|=9$ or $16$
$\text{(D)}\to \text{(P)},\text{(Q)}$