Question

The values of a for which $2x^2$ - 2 (2a + 1)x + a (a + 1) = 0 may have one root less than a and other root greater than a are given by

• A. 1 > a > 0
• B. -1 < a < 0
• C.
• D. a > 0 or a < -1

Answer 1

The correct option is D

a > 0 or a < -1

The given condition suggest that a lies between the roots.

Let f(x) $=2x^2-2(2a+1)x+a(a+1)$

For ‘a’ to lie between the roots we must have Discriminant $\ge$ 0 and f(a) < 0

Now, Discriminant $\ge$ 0

$\Rightarrow 4(2a+1{)}^2$ – 8a(a + 1) $\ge$ 0

$\Rightarrow 8(a^2+a+\frac{1}{2})\ge 0$ which is always true

Also f(a) < 0 $\Rightarrow 2a^2$ – 2a(2a + 1) + a(a + 1) < 0

$\Rightarrow -a^2$ – a < 0 $\Rightarrow a^2$ + a > 0 $\Rightarrow$ a(1 + a) > 0

$\Rightarrow$ a > 0 or a < –1