CAT 2017 Question Paper | Quants Slot 2

The best way to gear up for the CAT Examis to have a crack at the CAT previous year paper so that you have a fair idea about the mental state the CAT Question Paper will put you in! Try your hands at this question from Quadratic Equation that appeared in CAT 2017 Question Paper. See for yourself, where you stand and where these previous year questions take you!

Question 24 : The minimum possible value of the sum of the squares of the roots of the equation x² + (a + 3)x - (a + 5) = 0 is

1. 1
2. 2
3. 3
4. 4

Video Explanation

Explanatory Answer

Method of solving this CAT Question from Quadratic Equations

Let p and q be two roots of the equation
We have to find the minimum value of (p² + q²)
p² + q² = (p + q)² – 2pq ----(i)
Sum of the roots = $\frac{−𝑏}{𝑎}$.
Therefore, p + q = -(a + 3)
Product of the roots = $\frac{c}{𝑎}$
Therefore, pq = -(a + 5)
Substituting these values back in (i)
p² + q² = [(-a + 3)² – 2(-a + 5)]
p² + q² = a² + 6a + 9 + 2a + 10
p² + q² = a² + 8a + 19
We have to find the minimum value of a² + 8a + 19.
By completion of squares, 19 can be split into 16 and 3
⟹ a² + 8a + 16 + 3
⟹ (a + 4)² + 3
The minimum possible value is 3 when a = -4.

The question is "The minimum possible value of the sum of the squares of the roots of the equation x² + (a + 3)x - (a + 5) = 0 is"

Hence, the answer is 3

Choice C is the correct answer

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