Question

The least possible integral value of a for which the equation $x^2-2(a-1)x+(2a+1)=0$ has both the roots positive, is:

• A. 1
• B. 3
• C. 4
• D. 6

Answer 1

The correct option is C 4

Let $f\left(x\right)=x^2-2(a-1)x+(2a+1)=0$
Then, f(x)=0 will have both roots positive, if
(i) $D\ge 0$ (ii) Sum of the roots > 0 and (iii) $f\left(0\right)>0$
Now, $D\ge 0\Rightarrow 4(a-1{)}^2-4(2a+1)\ge 0\Rightarrow a^2-4a\ge 0\Rightarrow a\le 0ora\ge 4\dots (i)$
Sum of the roots > 0
$\Rightarrow 2(a-1)>0\Rightarrow a>1\dots \left(ii\right)$
and $f\left(0\right)>0\Rightarrow 2(a-1)\Rightarrow a>-\frac{1}{2}\dots \left(iii\right)$
From (i), (ii) and (iii) we ge $a\ge 4$,
Hence, the least integral value of a is 4.