CAT 2022 Slot 3 QA Q1 Equations

Question 1 of the Quantitative Ability (QA) section, CAT 2022 Slot 3, with CAT's official answer and a worked solution.

Suppose \(k\) is any integer such that the equation \(2 x^2+k x+5=0\) has no real roots and the equation \(x^2+(k-5) x+1=0\) has two distinct real roots for \(x\). Then, the number of possible values of \(k\) is
  1. A
    7
  2. B
    8
  3. C
    9
  4. D
    13

Answer: (C) 9 Equations MCQ

Solution

  1. No real roots of \(2x^2+kx+5=0\): \(k^2-40\lt0\), so \(k\in\{-6,\dots,6\}\).
  2. Distinct real roots of \(x^2+(k-5)x+1=0\): \((k-5)^2\gt4\), so \(k\gt7\) or \(k\lt3\).
  3. Common integers: \(-6\le k\le2\), which is 9 values.

Same concept in other years

See every question on Discriminant conditions for real, distinct or no real roots.

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