IPMAT Indore 2023 · Quantitative Ability · Question 18

Step 1 — exploit the GP. Since {a}_{1},{a}_{2},{a}_{3} are in GP, {a}_{2}^{2}={a}_{1}{a}_{3}. Look at the discriminant of the first equation:

\Delta ={\left( 2{a}_{2}\right)}^{2}-4{a}_{1}{a}_{3}=4\left({a}_{2}^{2}-{a}_{1}{a}_{3}\right)=0

So the first equation has a repeated root — and therefore the “common root” is forced to be that one:

x=-\frac{2{a}_{2}}{2{a}_{1}}=-\frac{{a}_{2}}{{a}_{1}}

Step 2 — write everything with the common ratio. Let {a}_{2}={a}_{1}r and {a}_{3}={a}_{1}{r}^{2}. The common root is x=-r.

Step 3 — substitute into the second equation.

{b}_{1}{r}^{2}-2{b}_{2}r+{b}_{3}=0

Step 4 — divide through by {a}_{3}={a}_{1}{r}^{2}. This is the step that produces the answer:

\frac{{b}_{1}{r}^{2}}{{a}_{1}{r}^{2}}-\frac{2{b}_{2}r}{{a}_{1}{r}^{2}}+\frac{{b}_{3}}{{a}_{1}{r}^{2}}=0

\frac{{b}_{1}}{{a}_{1}}-\frac{2{b}_{2}}{{a}_{1}r}+\frac{{b}_{3}}{{a}_{1}{r}^{2}}=0

Since {a}_{1}r={a}_{2} and {a}_{1}{r}^{2}={a}_{3}:

\frac{{b}_{1}}{{a}_{1}}-2\,\frac{{b}_{2}}{{a}_{2}}+\frac{{b}_{3}}{{a}_{3}}=0

Step 5 — read off the conclusion. Rearranging,

2\,\frac{{b}_{2}}{{a}_{2}}=\frac{{b}_{1}}{{a}_{1}}+\frac{{b}_{3}}{{a}_{3}}

which is exactly the condition for \frac{{b}_{1}}{{a}_{1}},\frac{{b}_{2}}{{a}_{2}},\frac{{b}_{3}}{{a}_{3}} to be in arithmetic progression.

Answer: Option 2

Shortcut. The GP condition is a disguised “discriminant is zero”. The moment you see {a}_{1},{a}_{2},{a}_{3} in GP attached to {a}_{1}{x}^{2}+2{a}_{2}x+{a}_{3}, note that \Delta =4\left({a}_{2}^{2}-{a}_{1}{a}_{3}\right)=0 — the first quadratic is a perfect square {a}_{1}{\left( x+r \right)}^{2}, with the double root x=-r.

That removes all choice: the common root must be -r. Plug it into the second equation and divide by {a}_{3}, and the AP falls out.

Why “2b₂” is in the question: the coefficient 2 exists precisely so that the middle term becomes 2{b}_{2}/{a}_{2} and the AP condition reads cleanly. A gratuitous-looking 2 in a quadratic is almost always a signal that an AP or a discriminant is coming.