If $$\alpha $$ be a repeated root of a quadratic equation $$f(x)=0$$ and $$A(x), B(x)$$ and $$C(x)$$ be polynomials of degree $$3$$, $$4$$ and $$5$$ respectively,
then show that $$\left| {\matrix{
{A\left( x \right)} & {B\left( x \right)} & {C\left( x \right)} \cr
{A\left( \alpha \right)} & {B\left( \alpha \right)} & {C\left( \alpha \right)} \cr
{A'\left( \alpha \right)} & {B'\left( \alpha \right)} & {C'\left( \alpha \right)} \cr
} } \right|$$ is
divisible by $$f(x)$$, where prime denotes the derivatives.
Answer
Solve it.
2
IIT-JEE 1982
Subjective
Let $$f$$ be a twice differentiable function such that
$$f''\left( x \right) = - f\left( x \right),$$ and $$f'\left( x \right) = g\left( x \right),h\left( x \right) = {\left[ {f\left( x \right)} \right]^2} + {\left[ {g\left( x \right)} \right]^2}$$