Matrices & Determinants
Properties of 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></mtd><mtd><mi>x</mi><mo>-</mo><mn>1</mn></mtd><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mn>2</mn><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mtd><mtd><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo></mtd><mtd><mi>x</mi><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mtd></mtr><mtr><mtd><mn>3</mn><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo></mtd><mtd><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mtd><mtd><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math> & D<sub>r</sub> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd><mtd><mn>2</mn><mi>r</mi></mtd></mtr><mtr><mtd><mn>70</mn></mtd><mtd><mn>17</mn></mtd><mtd><mn>3</mn><mi>r</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math>. The value of f(50) - D<sub>5</sub> is -
Step-by-Step Solution
Key Concept: Evaluate the determinant f(x) by factoring out common terms from rows and columns, and evaluate D_r by expanding or using properties of determinants.
For f(x), notice that (x-1) is a common factor in row 2 and (x-1)(x-2) is a common factor in row 3. After factoring, the determinant simplifies significantly. For D_r, substitute r=5 and evaluate the 3x3 determinant.
Correct Answer: 4