CAT 2020 Question Paper | Quants Slot 3

CAT Questions

The best questions to practice for CAT Exam are the actual CAT Question Papers. 2IIM offers you exactly that, in a student friendly format to take value from this. In CAT 2019 we saw some beautiful questions that laid emphasis on Learning ideas from basics and being able to comprehend more than remembering gazillion formulae and shortcuts. Original CAT Question paper is the best place to start off your CAT prep practice. This page provides exactly that. To check out about 1000 CAT Level questions with detailed video solutions for free, go here: 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"