GATE CS 2022 Q.3: Quantitative Aptitude

Let $r$ be a root of the equation $x^2 + 2x + 6 = 0$ .

Then the value of the expression $(r + 2)(r + 3)(r + 4)(r + 5)$ is

$-51$

126

$-126$

Official answer and solution

Solution

Pair the factors symmetrically and substitute the quadratic relation $r^2 + 2r = -6$ .

Since $r$ is a root of $x^2 + 2x + 6 = 0$ , we have $r^2 + 2r + 6 = 0$ , which gives $r^2 + 2r = -6$ .

Consider the product $(r + 2)(r + 3)(r + 4)(r + 5)$ . Group the factors with equal sum of constants:

(r + 2)(r + 5) = r^2 + 7r + 10 = (r^2 + 2r) + 5r + 10 = -6 + 5r + 10 = 5r + 4

(r + 3)(r + 4) = r^2 + 7r + 12 = (r^2 + 2r) + 5r + 12 = -6 + 5r + 12 = 5r + 6

Multiplying these two grouped expressions:

(5r + 4)(5r + 6) = 25r^2 + 50r + 24 = 25(r^2 + 2r) + 24

Substituting $r^2 + 2r = -6$ :

25(-6) + 24 = -150 + 24 = -126

Final answer

Option (D)

Common trap

Attempting to find the complex roots $r = -1 \pm i\sqrt{5}$ explicitly and expanding powers of complex numbers.

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