Question

Consider $f ( x )= x ^2-( b +1) x +2( a -5)$ where $a , b \in N$. Identify which of the following statement(s) is(are) correct?

• A. If $f ( x )=0$ has two distinct real roots for all values of ' $b$ ', then sum of values of ' $a$ ' is 15 .
• B. If $f ( x )=0$ has two distinct real roots for all values of ' $b$ ', then sum of values of ' $a$ ' is 10 .
• C. If $f (-1)>0$ for all values of ' $a$ ' then least value of ' $b$ ' is 7 .
• D. If $f (-1)>0$ for all values of ' $a$ ' then least value of ' $b$ ' is 6 .

Answer 1

Answer: (A), (C)

$ f(x)=x^2-(b+1) x+2(a-5), a, b \in N$
$D=(b+1)^2-8(a-5) $
$D=(b+1)^2+8(5-a)>0 \forall b \in N$
$\Rightarrow a \leq 5 $
$\therefore \text { Sum of all values of } a=15 .$
$\Rightarrow f(-1)=1+b+1+2(a-5)>0 \forall a \in N $
$\Rightarrow 2 a+b-8>0 \forall a \in N$
$ b \geq 7 $