GATE AG 2023 • Q9 MCQ • 2 Marks

Section 8: General Aptitude

Quantitative Aptitude

Subtopic: Numerical Computation

Question Statement:

The coefficient of x⁴ in the polynomial (x − 1)³(x − 2)³ is equal to _______.

Options:

  • (A) 33
  • (B) −3
  • (C) 30
  • (D) 21

Official Answer Key:

A

Step-by-Step Solution & Explanation

Expand (x−1)³(x−2)³ = [(x−1)(x−2)]³ = (x²−3x+2)³. Using the multinomial expansion of (x²−3x+2)³, the x⁴ term arises from choosing exponents (a,b,c) on (x²)^a(−3x)^b(2)^c with a+b+c=3 and 2a+b=4. Valid combinations: (a,b,c)=(2,0,1) and (1,2,0). • (2,0,1): coefficient = 3!/(2!0!1!) × 1² × 1 × 2 = 3×2 = 6 • (1,2,0): coefficient = 3!/(1!2!0!) × 1 × (−3)² × 1 = 3×9 = 27 Total coefficient of x⁴ = 6+27 = 33. Answer: (A) 33. (Verified independently by symbolic expansion.)

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