Question

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

Let $[.]$ denote the greatest integer function and $\text{sgn}(.)$ denote the signum function.
$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}I=\underset{0}{\overset{\pi /4}{\int }}{\sin }^{3}x{\cos }^{3}xdx,\text{then}{I}^{-1}\text{is divisible by}& \text{(P)}& 1\\ \text{(B)}& \text{If}I=\underset{0}{\overset{2\pi }{\int }}\left[2\sin x\right]dx,\text{then}\left|\left[I\right]\right|\text{is divisible by}& \text{(Q)}& 2\\ \text{(C)}& \text{If}I=\underset{-2}{\overset{2}{\int }}\text{sgn}(x-1)dx,\text{then}|I|\text{is equal to}& \text{(R)}& 4\\ \text{(D)}& \text{If}I=\underset{0}{\overset{1}{\int }}\frac{x^3-1}{x+1}dx,\text{then}[3I+\ln 64]\text{is divisible by}& \text{(S)}& 8\\ & & \text{(T)}& 16\end{array}$

Which of the following is the only CORRECT combination ?

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

Answer 1

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

$\text{(C)}$
$I=\underset{-2}{\overset{2}{\int }}\text{sgn}(x-1)dx$
Put $x-1=t\Rightarrow dx=dt$
$I=\underset{-3}{\overset{1}{\int }}\text{sgn}\left(t\right)dt$
$=\underset{-3}{\overset{0}{\int }}(-1)dt+{\int }_0^{1}dt$
$=-3+1=-2$
$\text{(C)}\to \text{(Q)}$

$\text{(D)}$
$I=\underset{0}{\overset{1}{\int }}\frac{x^3+1-2}{x+1}dx=\underset{0}{\overset{1}{\int }}\left(\right(x^2-x+1)-\frac{2}{x+1})dx$
$={[\frac{x^3}{3}-\frac{x^2}{2}+x-2\ln (x+1\left)\right]}_0^1$
$=(\frac{1}{3}-\frac{1}{2}+1-2\ln 2)=\frac{5}{6}-\ln 4$

$3I=3(\frac{5}{6}-\ln 4)=\frac{5}{2}-3\ln 4=\frac{5}{2}-\ln 64$
Now, $[3I+\ln 64]=[\frac{5}{2}-\ln 64+\ln 64]$
$=\left[\frac{5}{2}\right]=2$
$\text{(D)}\to \text{(P)},\text{(Q)}$