The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>a</mi><mi>a</mi><mo>+</mo><mi>b</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>b</mi><mi>a</mi><mo>+</mo><mi>c</mi></mtd></mtr><mtr><mtd><mi>a</mi><mi>a</mi><mo>+</mo><mi>b</mi></mtd><mtd><mi>b</mi><mi>a</mi><mo>+</mo><mi>c</mi></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> is equal to zero, if -
(A) a, b, c are in AP
(B) a, b, c are in GP
(C) a is a root of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math>
(D) (x - a) is a factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math>
Step-by-Step Solution
Key Concept: The determinant represents the condition for a, b, c to be in GP. Expanding the determinant or using row operations R3 -> R3 - a*R1 - R2 shows that the determinant is -(ax^2 + 2bx + c)(ac - b^2). Thus, it is zero if ac - b^2 = 0 (GP) or if ax^2 + 2bx + c = 0.
Let the determinant be D. Performing R3 -> R3 - a*R1 - R2, we get D = |a b ax+b; b c bx+c; 0 0 -(ax^2+2bx+c)|. Expanding along the third row, D = -(ax^2+2bx+c)(ac-b^2). For D=0, either ac-b^2=0 (a,b,c in GP) or ax^2+2bx+c=0. If a,b,c are in GP, b^2=ac, so D=0. If a is a root of ax^2+bx+c=0, then a(a^2)+b(a)+c=0, which is not directly related to the factor ax^2+2bx+c. However, the options suggest multiple correct answers.
Correct Answer: 1, 2, 3