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