CAT 2024 Slot 2 QA Q16 Equations

Question 16 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.

The roots \(\alpha, \beta\) of the equation \(3 x^2+\lambda x-1=0\), satisfy \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}=15\). The value of \(\left(\alpha^3+\beta^3\right)^2\), is
  1. A
    9
  2. B
    16
  3. C
    4
  4. D
    1

Answer: (C) 4 Equations MCQ

Solution

  1. \(\alpha + \beta = -\frac{\lambda}{3} = s\), \(\alpha\beta = -\frac13\).
  2. \(\frac{1}{\alpha^2}+\frac{1}{\beta^2} = \frac{s^2 - 2\alpha\beta}{(\alpha\beta)^2} = 9\left(s^2 + \frac23\right) = 9s^2 + 6 = 15\), so \(s^2 = 1\).
  3. \(\alpha^3+\beta^3 = s^3 - 3\alpha\beta s = s^3 + s = s(s^2+1)\).
  4. \((\alpha^3+\beta^3)^2 = s^2(s^2+1)^2 = 1 \times 4 = 4\).

Same concept in other years

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

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