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 $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The vertices of a triangle are}(1,a),& \text{(P)}& a+b=13\\ & (2,b)\text{and}(c^2,-3).\text{If the centroid of}& & \\ & \text{the triangle lies on the}x\text{-axis, then}& & \\ \text{(B)}& \text{If}a,b,c(a<b<c)\text{are the roots of}& \text{(Q)}& ab=12\\ & x^3-6x^2+11x-6=0\text{, then}& & \\ \text{(C)}& \text{If}a,b,c\text{lie between 2 and 18 such that}& \text{(R)}& (a,b)=(-5,8)\\ & \text{(i)}a+b+c=25& & \\ & \text{(ii)}2,a,b\text{are in A.P.}& & \\ & \text{(iii)}b,c,18\text{are in G.P., then}& & \\ \text{(D)}& \text{If}a,b,c\text{are in G.P. with common ratio}& \text{(S)}& a+b=8\\ & r(r>1)\text{such that}abc=216\text{and}& & \\ & ab+bc+ca=156\text{, then}& & \\ & & \text{(T)}& a+b=3\end{array}$

Which of the following is the only CORRECT combination?

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

Answer 1

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

$\left(A\right)$
Given vertices : $(1,a),(2,b),(c^2,-3)$
Centroid lies on $x$-axis.
$\Rightarrow \frac{a+b-3}{3}=0$
$\Rightarrow a+b=3$
$\text{(A)}\to \text{(R)},\text{(T)}$

$\left(B\right)$
Given $a,b,c$ are the roots of $x^3-6x^2+11x-6-0(a<b<c)$
By verification, $x=1$ is a root.
$\therefore (x-1)(x^2-5x+6)=0\Rightarrow (x-1)(x-2)(x-3)=0\Rightarrow x=1,x=2,x=3\Rightarrow a=1,b=2,c=3$
$\text{(B)}\to \text{(T)}$