Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mi>cos</mi><mn>2</mn><mi>x</mi></mtd><mtd><mn>4</mn><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi>x</mi><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><msup><mi>tan</mi><mn>2</mn></msup><mfrac><mi>x</mi><mn>2</mn></mfrac></mtd><mtd><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>cos</mi><mn>4</mn><mi>x</mi></mtd><mtd><msup><mi>tan</mi><mn>2</mn></msup><mfrac><mi>x</mi><mn>2</mn></mfrac></mtd><mtd><mi>sin</mi><mn>4</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math> = a_0 + a_1(cos x) + a_2(cos^2 x) + ........ + a_n(cos^n x), then a_0 is -
Step-by-Step Solution
Key Concept: The value of a_0 is the value of the determinant when cos x = 0. Since the expression is a polynomial in cos x, setting cos x = 0 eliminates all terms except a_0.
To find a_0, we set cos x = 0. This implies x = \Theta/2. Substituting this into the determinant and evaluating it yields 0.
Correct Answer: 1