CAT Quantitative Aptitude Questions | CAT Algebra - Progressions questions

CAT Questions | Algebra | Progressions - AP GP HP

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, 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

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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, 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.


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