Question

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

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

Answer 1

Answer: (D)

$a > 0$ or $a < - 1$

Answer 2

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 ≥ 0 and $f(a) < 0$

Now, Discriminant ≥ 0

$4(2a + 1)^{2} - 8a(a + 1) \geq 0$ ⇒ $8(a^{2} + a + 1/2) \geq 0$ which is

always true.

Also $f(a) < 0$ ⇒ $2a^{2} - 2a(2a + 1) + a(a + 1) < 0$⇒ $- a^{2} - a < 0$

⇒ $a^{2} + a > 0$ ⇒ $a(1 + a) > 0$ ⇒ $a > 0$ or $a < - 1$