Question

Let $f: R \rightarrow\{a, b\}$ be a function defined by $f(x)=x^2(p-4)-x\left(q^2-3 q+2\right)+$ $\operatorname{sgn}\left(x^2-2 r x+(r+2)\right)$, where $p, q, r \in I$. Identify which of the following statement(s) is/are correct?
[Note: $\operatorname{sgn}( y )$ 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 \rightarrow\{ a , b \}$
$f(x)=x^2(p-4)-x\left(q^2-3 q+2\right)+\operatorname{sgn}\left(x^2-2 r n+(r+2)\right)$
(A) For $f ( x )$ to be surjective, $p =4, q =1,2$
and
$x ^2-2 rn + r +2=0 $
$\downarrow $
$D =0$
$4 r ^2-4( r +2)=0 \Rightarrow r ^2- r -2=0 \Rightarrow r =-1,2$
$\therefore$ 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 $x ^2-2 rn + r +2$
$\Rightarrow 4 r ^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.