Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mi>α</mi></mtd><mtd><mi>β</mi></mtd><mtd><mi>γ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>t</mi></math>, where t is an even prime number & α, β, γ are the integral roots of the equation x<sup>3</sup> - 14x<sup>2</sup> + Px - 36 = 0. The value of P is -
Step-by-Step Solution
Key Concept: The determinant evaluates to (γ - β) - (γ - α) + (β - α) = γ - 2β + α = t. Since t is an even prime, t = 2. Given α, β, γ are roots of x^3 - 14x^2 + Px - 36 = 0, we have α + β + γ = 14, αβ + βγ + γα = P, and αβγ = 36. Solving for integers, we find roots 2, 6, 6 (not distinct) or others. Checking the condition α + γ - 2β = 2 and α + β + γ = 14, we get 3β = 12, so β = 4. Then α + γ = 10 and αγ = 36/4 = 9. Roots are 1 and 9. Thus P = αβ + βγ + γα = 4(1+9) + 9 = 49.
The determinant is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>(</mo><mn>2</mn><mi>γ</mi><mo>-</mo><mn>3</mn><mi>β</mi><mo>)</mo><mo>-</mo><mn>1</mn><mo>(</mo><mi>γ</mi><mo>-</mo><mn>3</mn><mi>α</mi><mo>)</mo><mo>+</mo><mn>1</mn><mo>(</mo><mi>β</mi><mo>-</mo><mn>2</mn><mi>α</mi><mo>)</mo><mo>=</mo><mi>γ</mi><mo>-</mo><mn>2</mn><mi>β</mi><mo>+</mo><mi>α</mi><mo>=</mo><mi>t</mi></math>. Since t is an even prime, t = 2. Given roots α, β, γ of x<sup>3</sup> - 14x<sup>2</sup> + Px - 36 = 0, we have α + β + γ = 14 and αβγ = 36. From α + γ - 2β = 2 and α + β + γ = 14, we get 3β = 12, so β = 4. Then α + γ = 10 and αγ = 9. The roots are 1, 4, 9. P = αβ + βγ + γα = 4(1+9) + 9 = 49. 49 is an odd natural number.
Correct Answer: 3