Matrices & Determinants
Determinants
Grade Class 12
Question:
Let f(x) = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>4</mn><mi>sin</mi><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>4</mn><mi>sin</mi><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>1</mn><mo>+</mo><mn>4</mn><mi>sin</mi><mn>2</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math>, then the maximum value of f(x), is-
Step-by-Step Solution
Key Concept: Apply row operations to simplify the determinant. Perform R1 -> R1 - R3 and R2 -> R2 - R3 to get a simpler form, then expand.
Let the determinant be f(x). Applying R1 -> R1 - R3 and R2 -> R2 - R3: <br> f(x) = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>1</mn><mo>+</mo><mn>4</mn><mi>sin</mi><mn>2</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math> <br> Expanding along R1: <br> f(x) = 1(1 + 4sin 2x + cos^2 x) - 0 + (-1)(0 - sin^2 x) <br> f(x) = 1 + 4sin 2x + cos^2 x + sin^2 x <br> f(x) = 1 + 4sin 2x + 1 = 2 + 4sin 2x <br> Since the maximum value of sin 2x is 1, the maximum value of f(x) = 2 + 4(1) = 6.
Correct Answer: C