NinetynineCAT 2024 Slot 1 QA Q20 Equations
Question 20 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.
If the equations \(x^2+m x+9=0, x^2+n x+17=0\) and \(x^2+(m+n) x+35=0\) have a common negative root, then the value of \((2 m+3 n)\) is
TITA: type in the answer. There are no options and no negative marking.
Answer: 38 Equations TITA
Solution
- Let \(r\) be the common root: \(r^2 + mr + 9 = 0\), \(r^2 + nr + 17 = 0\), \(r^2 + (m+n)r + 35 = 0\).
- Add the first two: \(2r^2 + (m+n)r + 26 = 0\). Subtract the third: \(r^2 - 9 = 0\), so \(r = -3\) (negative).
- From the first: \(9 - 3m + 9 = 0 \Rightarrow m = 6\). From the second: \(9 - 3n + 17 = 0 \Rightarrow n = \frac{26}{3}\).
- \(2m + 3n = 12 + 26 = 38\).
Shortcut. Notice \(9 + 17 = 26\) while the third constant is 35: the difference \(9 = r^2\) gives the root immediately.