CAT 2023 Slot 3 QA Q3 Equations

Question 3 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.

For some real numbers \(a\) and \(b\), the system of equations \(x+y=4\) and \((a+5) x+\left(b^2-15\right) y=8 b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(a b\) is
  1. A
    15
  2. B
    33
  3. C
    55
  4. D
    25

Answer: (B) 33 Equations MCQ

Solution

  1. Infinitely many solutions means the second equation is a multiple of \(x+y=4\): \(\frac{a+5}{1} = \frac{b^2-15}{1} = \frac{8b}{4} = 2b\).
  2. \(b^2-15 = 2b \Rightarrow b^2-2b-15=0 \Rightarrow b=5\) or \(b=-3\).
  3. \(a = 2b-5\): \(b=5\) gives \(a=5, ab=25\); \(b=-3\) gives \(a=-11, ab=33\).
  4. Maximum \(ab = 33\).

Same concept in other years

See every question on Unique, none or infinitely many solutions of a 2×2 linear system.

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