Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
If a, b, c are in AP, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd><mi>x</mi><mo>+</mo><mi>a</mi></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd><mi>x</mi><mo>+</mo><mn>3</mn></mtd><mtd><mi>x</mi><mo>+</mo><mi>b</mi></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mn>3</mn></mtd><mtd><mi>x</mi><mo>+</mo><mn>4</mn></mtd><mtd><mi>x</mi><mo>+</mo><mi>c</mi></mtd></mtr></mtable></mfenced></math> equals -
(A) a + b + c
(B) x + a + b + c
(C) 0
(D) none of these
Step-by-Step Solution
Key Concept: Since a, b, c are in AP, 2b = a + c. Apply row operations R1 -> R1 + R3 - 2R2 to show the first row becomes zero.
Given a, b, c are in AP, we have 2b = a + c. Let the determinant be \Delta. Applying R1 -> R1 + R3 - 2R2: The elements of the first row become: (x+1) + (x+3) - 2(x+2) = 0, (x+2) + (x+4) - 2(x+3) = 0, (x+a) + (x+c) - 2(x+b) = 2x + a + c - 2x - 2b = (a+c) - 2b = 0. Since all elements of the first row are zero, the value of the determinant is 0.
Correct Answer: 3