Question

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

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Lines with direction ratios}(1,–c,–b),(–c,1,–a)\text{and}(–b,–a,1)& \text{(P)}& -1\\ & \text{are coplanar. Then}{a}^2+b^2+c^2+2abc\text{is}& & \\ \text{(B)}& \text{If the lines}x=ay+1,z=by+2\text{and}x=cy+3,z=dy+4& \text{(Q)}& 1\\ & \text{are perpendicular, then}ac+bd\text{is equal to}& & \\ \text{(C)}& \text{If}(a,b,c)\text{lies on a plane which forms}△ABC\text{with coordinate axes}& \text{(R)}& 3\\ & \text{whose centroid lies on}(\alpha ,\beta ,\gamma ),\text{then}\frac{a}{\alpha }+\frac{b}{\beta }+\frac{c}{\gamma }\text{is equal to}& & \\ \text{(D)}& \text{Let}\left[x\right]\text{denote the greatest integer less than or equal to}x.\text{Then}& \text{(S)}& 0\\ & f\left(x\right)=\left[x\sin \pi x\right]\text{is not differentiable when}x\text{is equal to}& & \\ & & \text{(T)}& 2\end{array}$

Which of the following is the only CORRECT combination?

• A. $\left(A\right)\to \left(R\right),\left(B\right)\to \left(P\right)$
• B. $\left(A\right)\to \left(Q\right),\left(B\right)\to \left(P\right)$
• C. $\left(A\right)\to \left(Q\right),\left(B\right)\to \left(S\right)$
• D. $\left(A\right)\to \left(T\right),\left(B\right)\to \left(S\right)$

Answer 1

The correct option is B $\left(A\right)\to \left(Q\right),\left(B\right)\to \left(P\right)$

$\left|\begin{array}{ccc}1& -c& -b\\ -c& 1& -a\\ -b& -a& 1\end{array}\right|=0\Rightarrow 1(1-a^2)+c(-c-ab)-b(ac+b)=0\Rightarrow a^2+b^2+c^2+2abc=1$

Lines $x=ay+1,z=by+2$ and$x=cy+3,z=dy+4$
$L_1:\frac{x-1}{a}=\frac{y}{1}=\frac{z-2}{b}{L}_2:\frac{x-3}{c}=\frac{y}{1}=\frac{z-2}{d}{L}_1\perp L_2\Rightarrow ac+1+bd=0\Rightarrow ac+bd=-1$