Question

Let $f:R\to \{a,b\}$ be a function defined by $f(x)=x^2(p-4)-x\left(q^2-3q+2\right)+$ $\mathrm{sgn}\left(x^2-2rx+(r+2)\right)$, where $p,q,r\in I$. Identify which of the following statement(s) is/are correct?
[Note: $\mathrm{sgn}(y)$ denotes the signum function of y.]

• A. If $f(x)$ is surjective then number of ordered triplets $(p,q,r)$ is 4 .
• B. If $f(x)$ is not surjective the number of ordered triplets $(p,q,r)$ is 4 .
• C. If $f(x)$ is surjective then $a+b=1$.
• D. If $f(x)$ is not surjective then one of the values of a and b must be 0 .

Answer 1

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

$f:R\to \{a,b\}$
$f(x)=x^2(p-4)-x\left(q^2-3q+2\right)+\mathrm{sgn}\left(x^2-2rn+(r+2)\right)$
(A) For $f(x)$ to be surjective, $p=4,q=1,2$
and
$x^2-2rn+r+2=0$
$\downarrow$
$D=0$
$4r^2-4(r+2)=0\Rightarrow r^2-r-2=0\Rightarrow r=-1,2$
$\therefore$ Number of ordered triplets $(p,q,r)=4$
(B) For $f(x)$ not to be surjective, $p=4,q=1,2$
and $D<0$ for $x^2-2rn+r+2$
$\Rightarrow 4r^2-4(r+2)<0$
$\Rightarrow r\in (-1,2)\Rightarrow r=0,1$
$\therefore$ Number of ordered triplets $(p,q,r)=4$
(C) When $f(x)$ is surjective, range of $f(x)$ is $\{0,1\}$
$\Rightarrow a+b=1$
(D) When $f(x)$ is not surjective, range of $f(x)$ is $\{1\}$
$\therefore$ One of the values of $a$ and $b$ must be 1.