CAT 2020 Question Paper | Quants Slot 3

This question is from Linear and Quadratic Equations. CAT Exam is known to test the fundamentals instead of high funda stuff. This question revolves around the idea of the discriminant. Questions like this have appeared many times in CAT Exam. Make sure you master this topic during your CAT Preparation. To explore 1000+ CAT Level Questions with detailed video and text solutions check out 2IIM CAT Question Bank.

Question 23 : Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m + n is

Video Explanation

Explanatory Answer

The roots of a Quadratic Equation of the form, ax² + bx + c = 0
are given by x = \\frac{-b ± √[b^{2} - 4ac]}{2a})

In order for these roots to be real, the portion under the square root should not be negative.
In other words 'b² - 4ac', determines wether the roots are real or imaginary.
Hence, rightly, it is called the Determinat(D).

If the Determinant, D is greater than or equal to 0, then the equation has real roots.
For D to be greater than or equal to 0, b² ≥ 4ac.

We are told that x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots,
That means, m² ≥ 4(2n) and (2n)² ≥ 4m.

Instead of trying to solve through equations, the two inequalities m² ≥ 8n and n² ≥ m.
Let's try to solve inputting specific numbers.

If n = 1; m² ≥ 8; m ≥ 3
If m ≥ 3 at n = 1, n² ≥ m stands invalid.

If n = 2; m² ≥ 16; m ≥ 4
If m ≥ 4 at n = 2, n² ≥ m stands valid, if m = 4 and n = 2

Hence 4,2 is the smallest(and the only) pair that m,n can take.
So, the minimum sum of m and n is 4+2 = 6.

The question is "Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m + n is"

Hence, the answer is, "6"