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)

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