3. If px^4 + qx^3 + rx^2 + sx + t = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi></mtd><mtd><mi>x</mi><mo>-</mo><mn>1</mn></mtd><mtd><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mn>2</mn><mo>-</mo><mi>x</mi></mtd><mtd><mi>x</mi><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>-</mo><mn>3</mn></mtd><mtd><mi>x</mi><mo>+</mo><mn>4</mn></mtd><mtd><mn>3</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math> then t is equal to -
Step-by-Step Solution
Key Concept: The constant term t in the polynomial expansion of the determinant is obtained by evaluating the determinant at x = 0.
To find the constant term t, we put x = 0 in the given equation. The determinant becomes: <br> |(0+0, 0-1, 0+3), (0+1, 2-0, 0-3), (0-3, 0+4, 3*0)| = |(0, -1, 3), (1, 2, -3), (-3, 4, 0)| <br> Expanding along the first row: <br> 0(0 - (-12)) - (-1)(0 - 9) + 3(4 - (-6)) <br> = 0 + 1(-9) + 3(10) <br> = -9 + 30 = 21. <br> Thus, t = 21.
Correct Answer: 3