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

Select one answerSingle correct

Pick an option to check

Solution

Step-by-step verified solution

Official answer: D) -126

Quantitative Aptitude

Since $r$ is a root of the quadratic equation $x^2 + 2x + 6 = 0$ , it satisfies:

r^2 + 2r + 6 = 0 \implies r^2 + 2r = -6

We want to evaluate the product:

E = (r + 2)(r + 3)(r + 4)(r + 5)

Group the first and last factors, and the two middle factors together:

E = \big[(r + 2)(r + 5)\big] \cdot \big[(r + 3)(r + 4)\big]

E = (r^2 + 7r + 10)(r^2 + 7r + 12)

Using $r^2 = -2r - 6$ , rewrite $r^2 + 7r$ :

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

Now substitute this into the expression for $E$ :

E = (5r - 6 + 10)(5r - 6 + 12) = (5r + 4)(5r + 6)

E = 25r^2 + 50r + 24 = 25(r^2 + 2r) + 24

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

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

Hence, the correct option is D.

Try the question first, then check your work.

Think the official key or this write-up is wrong?

Report error or challenge

Examiner trap

The misconception this question tests

This Quantitative Aptitude previous year question is Q3 of the GATE CSE 2022 paper. Find more GATE CSE PYQs on Quantitative Aptitude in General Aptitude, or browse all GATE CSE PYQs.

Free diagnostic

Where do you stand for GATE?

Take the 20-minute diagnostic: the 3 topics that cost you the most marks, and a rank range. Or sit 5 related Quantitative Aptitude PYQs now with no account.

One GATE CSE PYQ every morning

An official question by email at 7 AM IST, with a link to its verified solution. Unsubscribe in one click.

Quantitative Aptitude in GATE CSE

Quantitative Aptitude has 53 GATE CSE PYQs across 13 papers from 2011 to 2026. 26 of them are 1-mark MCQs like this one, and GATE CSE 2022 has 3.

How GATE tests Quantitative Aptitude

More from GATE CSE 2022

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.