CAT 2023 Slot 1 QA Q6 Equations

Question 6 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

Let \(\alpha\) and \(\beta\) be the two distinct roots of the equation \(2 x^2-6 x+k=0\), such that \((\alpha+\beta)\) and \(\alpha \beta\) are the distinct roots of the equation \(x^2+p x+p=0\). Then, the value of \(8(k-p)\) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 6 Equations TITA

Solution

  1. From \(2x^2-6x+k=0\): \(\alpha+\beta = 3\), \(\alpha\beta = \frac{k}{2}\).
  2. These are roots of \(x^2+px+p=0\): sum \(3+\frac k2 = -p\), product \(\frac{3k}{2} = p\).
  3. So \(3 + \frac k2 = -\frac{3k}{2}\), giving \(k = -\frac32\) and \(p = -\frac94\).
  4. \(8(k-p) = 8\left(-\frac32+\frac94\right) = 8\cdot\frac34 = 6\).

Same concept in other years

See every question on Symmetric functions of roots via sum and product (Vieta).

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