Question

Let p and q be the roots of the equation $x^2-2x+A=0$ and let r and s be the roots of the equation $x^2-18x+B=0$. If $p<q<r<s$ are in arithmetic progression, then A, B respectively equal to:

• A. 8, 17
• B. 3, 7
• C. -3, 11
• D. None of these

Answer 1

The correct option is D None of these

Let us consider $p=a-3d,q=a-d,r=a+d.ands=a+3d$
$\begin{array}{ccc}\therefore & 2a-4d=2& \dots \left(i\right)\\ and& 2a+4d=18& \dots \left(ii\right)\end{array}$
Solving equation (i) and (ii), we get
$\begin{array}{cc}& a=5,d=2\\ \therefore & p=-1,q=3,r=7,s=11\\ Hence& A=pq=-3\\ and& B=rs=77\end{array}$
Thus (d) is the right choice

Alternatively:
Just consider an option, and then substitute the values of A and B from the assumed option, if the roots p, q, r, s are in A.P., then the presumed option is correct, else not. Thus we get options a, b and c are incorrect, hence (d) is the right choice.