Let f ( x ) = 1 x - 1 x 2 ( x - 1 ) ( x - 1 ) ( x - 2 ) x ( x - 1 ) 3 ( x - 1 ) ( x - 2 ) ( x - 1 ) ( x - 2 ) ( x - 3 ) x ( x - 1 ) ( x - 2 ) & D r = 1 10 2 r 70 17 3 r + 1 1 1 1 . The value of ∑ r = 1 10 D r is

The determinant D r can be expanded along the third column or simplified using row/column operations. Since D r is linear in r , \sum r = 1 10 D r = 10 2 ( D 1 + D 10 ) . Calculating D 1 and D 10 and summing them gives 575.

Matrices & Determinants

Properties of determinants

Grade Class 12

Question:

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><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><mi>x</mi><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> & <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>r</mi></msub><mo>=</mo><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 <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>D</mi><mi>r</mi></msub></math> is

575

Step-by-Step Solution

Key Concept: The determinant D_r is a linear function of r. The sum of a linear function over an arithmetic progression can be calculated using the sum of the first and last terms multiplied by the number of terms divided by 2.

The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>r</mi></msub></math> can be expanded along the third column or simplified using row/column operations. Since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>r</mi></msub></math> is linear in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>\sum</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>D</mi><mi>r</mi></msub><mo>=</mo><mfrac><mn>10</mn><mn>2</mn></mfrac><mo>(</mo><msub><mi>D</mi><mn>1</mn></msub><mo>+</mo><msub><mi>D</mi><mn>10</mn></msub><mo>)</mo></math>. Calculating <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>10</mn></msub></math> and summing them gives 575.

Correct Answer: 575

Question:

Step-by-Step Solution

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