GATE AG 2025 • Q11 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Matrices and determinants

Question Statement:

If $a, b, c$ are the roots of $2x^3 - 3x^2 + px - 1 = 0$ and the sum of two of the roots is 1, find $p$:

Options:

  • (A) 2
  • (B) 1
  • (C) 3
  • (D) 1/3

Official Answer Key:

C

Step-by-Step Solution & Explanation

Sum of all roots a+b+c = 3/2 (= −(−3)/2). If a+b = 1, then c = 3/2 − 1 = 1/2. Product abc = −(−1)/2 = 1/2 ⇒ ab = (1/2)/(1/2) = 1. Sum of pairwise products ab+bc+ca = p/2. bc+ca = c(a+b) = (1/2)(1) = 1/2. So ab+bc+ca = 1 + 1/2 = 3/2 = p/2 ⇒ p = 3.

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