Question

Values of 'm' for which both the roots of equation $x^2$ - 2mx + $m^2$ - 1=0 are less than 4 are

• A. (-, -3]
• B. (-, 3)
• C. (3, )
• D. [-3, )

Answer 1

The correct option is B

(-, 3)

Conditions for both the roots to be less than a real value $x_0$, can be visualized through graph

Here, a = 1 (> 0), b = -2m, c = $m^2$ - 1

Observe that (i) f$x_0$ > 0

(ii) $x_0$ > -b/2a

Also for real roots to exist (iii) $b^2$ ≥ 4ac

i) f(4) > 0 ⇒ 16 - 8m + $m^2$ - 1 > 0

$m^2$ - 8m + 15 > 0

(m - 3)(m - 5) > 0

m < 3 (or) m > 5 ---------- (1)

ii) 4 > m ⇒ m < 4

------------ (2)

iii) D ≥ 0

$b^2$ - 4ac ≥ 0

4$m^2$ - 4(m² - 1) ≥ 0

4 ≥ 0 (Always true) ------- (3)

Intersection of i), ii) & iii) gives,

m $ϵ$ (-$\infty$, 3)