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Let a and b be the roots of equation Ax^2+Bx+C=0.Evaluate the value of determinant | 31+a+b1+a^2 +b^2|| 1+a+b.1+a^2+b^21+a^3+b^3|| 1+a^2+b^2.1+a^3+b^3.1+a^4+b^4.|

bhumika gupta , 7 Years ago
Grade 12
anser 1 Answers
mycroft holmes

Last Activity: 7 Years ago

The given determinant can be written as a product:
\begin{vmatrix} 1 & 1 &1 \\ 1 & a& b\\ 1& a^2& b^2 \end{vmatrix} \ \times \begin{vmatrix} 1 & 1 &1 \\ 1 & a& b\\ 1& a^2 & b^2 \end{vmatrix}
 
= \begin{vmatrix} 1 & 1 &1 \\ 1 & a& b\\ 1& a^2& b^2 \end{vmatrix} ^2
 
Using the well known expansion for the Vandermonde Determinant, we can evaluate this as (a-b)^2 (a-1)^2(b-1)^2
 
Since a,b are roots of the quadratic Ax2+Bx+C, we have
 
(a-b)^2 = \frac{B^2-4AC}{A^2}
 
and (a-1)^2 (b-1)^2 = [(a-1)(b-1)]^2 = P(1)^2 = (A+B+C)^2
 
Hence given expressions equals
 
\frac{(B^2-4AC)(A+B+C)^2}{A^2}

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