CAT Quantitative Aptitude Questions | CAT Algebra - Progressions questions

The question is about type of a progression. Roots of two quadratic progression is in a progression, what type it is? With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. CAT Exam tests the idea of progression often in the CAT Quantitative Aptitude section and this could also be tested in DI LR section of the CAT Exam as a part of a puzzle.

Question 19: If the equation px² + 2qx + r = 0 and dx² + 2ex + f = 0 have a common root, and p,q,r are in G.P., then in which type of progression is \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\)

1. \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in G.P
2. \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in A.P
3. \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in H.P
4. Insufficient Data

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Progressions: How you would find out the type of a progression?

Since p,q,r are in G.P. we have q² = pr. On solving px² +2qx +r = 0 we get
x = \\frac{-2q ± √(4q^2 - 4pr)}{2p}\\)
=> x = \\frac{-2q}{2p}\\) (q² = pr)
=> x = \\frac{-q}{p}\\)
Thus x = \\frac{-q}{p}\\) is the repeated root of px²+2qx+r = 0
∴ x = \\frac{-q}{p}\\) is also a root of dx²+2ex+f = 0
=> d(\\frac{-q}{p}\\))² + 2e(\\frac{-q}{p}\\)) + f = 0
=> \\frac{dq^2 - 2eqp + fp^2}{p^2}\\) = 0
=> dq² - 2eqp + fp² = 0
=> \\frac{d}{p}\\) - 2 * \\frac{e}{q}\\) + \\frac{fp}{q^2}\\) = 0 [On dividing by pq²]
=> \\frac{d}{p}\\) - 2 * \\frac{e}{q}\\) + \\frac{f}{r}\\) = 0
=> \\frac{d}{p}\\) + \\frac{f}{r}\\) = 2 * \\frac{e}{q}\\)
Hence it can be clearly seen that \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in A.P

The question is "If the equation px² + 2qx + r = 0 and dx² + 2ex + f = 0 have a common root, and p,q,r are in G.P., then in which type of progression is \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\)"

Hence the answer is "\\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in A.P."

Choice B is the correct answer.