Question

If a be a repeated root of the quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degrees 3, 4 and 5 respectively, then $\left|\begin{array}{ccc}A\left(x\right)& B\left(x\right)& C\left(x\right)\\ A\left(a\right)& B\left(a\right)& C\left(a\right)\\ A^{\prime }\left(a\right)& B^{\prime }\left(a\right)& C^{\prime }\left(a\right)\end{array}\right|$

• A. is divisible by f(x) for all x
• B. is not divisible by f(x) for all x
• C. is equal to 0
• D. None of these

Answer 1

Answer: (A)

We must have $f(x)=k(x-a{)}^2$ where k is a constant. Now in order that the given determinant [Say, D(x)] be divisible by f(x) we must show that both D(x) and D'(x) vanish at x = a. Now
$D\left(a\right)=\left|\begin{array}{ccc}A\left(a\right)& B\left(a\right)& C\left(a\right)\\ A\left(a\right)& B\left(a\right)& C\left(a\right)\\ A^{\prime }\left(a\right)& B^{\prime }\left(a\right)& C^{\prime }\left(a\right)\end{array}\right|=0$
Also
$D^{\prime }\left(x\right)=\left|\begin{array}{ccc}{A}^{\prime }\left(x\right)& B^{\prime }\left(x\right)& C^{\prime }\left(x\right)\\ A\left(a\right)& B\left(a\right)& C\left(a\right)\\ A^{\prime }\left(a\right)& B^{\prime }\left(a\right)& C^{\prime }\left(a\right)\end{array}\right|+\left|\begin{array}{ccc}A\left(x\right)& B\left(x\right)& C\left(x\right)\\ 0& 0& 0\\ A^{\prime }\left(a\right)& B^{\prime }\left(a\right)& C^{\prime }\left(a\right)\end{array}\right|$
$+\left|\begin{array}{ccc}A\left(x\right)& B\left(x\right)& C\left(x\right)\\ A\left(a\right)& B\left(a\right)& C\left(a\right)\\ 0& 0& 0\end{array}\right|=\left|\begin{array}{ccc}{A}^{\prime }\left(x\right)& B^{\prime }\left(x\right)& C^{\prime }\left(x\right)\\ A\left(a\right)& B\left(a\right)& C\left(a\right)\\ A^{\prime }\left(a\right)& B^{\prime }\left(a\right)& C^{\prime }\left(a\right)\end{array}\right|$
which clearly gives D' (a) = 0, since first and third row become identical.