Question

Let $f(x)=x^{2010}+x^{1010}-x^{510}+x^{210}+x^2$. If $f(x)$ is divided by $x^2\left(x^2-1\right)$, then we get remainder as $g(x)$, function of $x$.
If roots of g(x) = 0 lies between the roots of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ then
[ Note : [k] denotes greatest integer function less than or equal to k.]

• A. number of integral values of a will be 0.
• B. number of values of a will be infinite.
• C. number of different values of [a] will be 1.
• D. number of different values of [a] will be 2.

Answer 1

Answer: (A), (B), (C)

Let remainder $g ( x )$ be $ax ^3+ bx ^2+ cx + d$
$\Theta f ( x )= x ^2\left( x ^2-1\right) Q ( x )+ ax x ^3+ bx ^2+ cx + d$, where $Q ( x )$ is quotient
$\therefore $ RHS should have common factor $x^2$
$\therefore c = d =0$
$f (1)= a + b =3$
and $f (-1)=- a + b =3$
___________________
$\therefore b =3 \text { and } a =0$
$\therefore g ( x )=3 x ^2$ which is many one into function.
$\therefore g ( x )=0 \Rightarrow x =0$ lies between roots of
$x^2-2(a+1) x+a(a-1)=0$
$\therefore a(a-1)< 0 \Rightarrow 0