Question

Let $f(x)=x^2+ax+b\cos x$, a being an integer and $b$ is a real number. Find the number of ordered pairs $(a,b)$ for which the equations $f(x)=0$ and $f(f(x))=0$ have the same (nonempty) set of real roots

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

Let $\alpha$ be a root of $f(x)=0$
$\therefore f(\alpha )=0$ and $f(f(\alpha ))=0$
$\Rightarrow f(0)=0$
$\Rightarrow b=0$
$\therefore f(x)=x(x+a)=0$
$\Rightarrow x=0$ or $x=-a$
$f(f(x))=x(x+a)\left(x^2+ax+a\right)=0$
$\therefore x^2+ax+a=0$
should have no real roots besides $0$ and
$-aD=a^2-4a<0$
$\Rightarrow 0<a<4$
If the roots of $x^2+ax+a=0$ is either $x=0$
or $x=-a$ then $a=0$
$\therefore \alpha \in [0,4)$
$\Rightarrow a=0,1,2,3$
Number of ordered pairs $=4$.