NinetynineCAT 2025 Slot 2 QA Q16 Equations
Question 16 of the Quantitative Ability (QA) section, CAT 2025 Slot 2, with CAT's official answer and a worked solution.
The equations \( 3 x ^ { 2 } - 5 x + p = 0 \) and \( 2 x ^ { 2 } - 2 x + q = 0 \) have one common root. The sum of the other roots of these two equations is
- A\( \frac { 2 } { 3 } - p + \frac { 3 } { 2 } q \)
- B\( \frac { 8 } { 3 } + p + \frac { 1 } { 3 } q \)
- C\( \frac { 2 } { 3 } - 2 p + \frac { 2 } { 3 } q \)
- D\( \frac { 8 } { 3 } - p + \frac { 3 } { 2 } q \)
Answer: (D) ( 8 )/( 3 ) - p + ( 3 )/( 2 ) q Equations MCQ
Solution
- Let the common root be \(\alpha\). Other roots: \(\frac53 - \alpha\) and \(1 - \alpha\); their sum is \(\frac83 - 2\alpha\).
- Eliminate \(\alpha^2\): \(2(3\alpha^2 - 5\alpha + p) - 3(2\alpha^2 - 2\alpha + q) = 0 \Rightarrow -4\alpha + 2p - 3q = 0\).
- So \(\alpha = \frac{2p - 3q}{4}\), and sum \(= \frac83 - \frac{2p - 3q}{2} = \frac83 - p + \frac32 q\).