Question

if the roots of the equation x² -2mx +m² - 1 =0 lie in the interval (-2,4) then,

a) m ~ (-1,3) b) m~(1,5)

c) m~(-3,1) d) m~(-1.5)

where ~ stand for "belongs to".

Answer 1

If roots ax2+bx+c=0 lie between k1 and k2 then of (k1) > 0, of (k2) > 0 and k1 < −b2a < k2
f(-2)>0, f(4) > 0 and -2 <m<4
⇒ 4+4m+m2−1>0, 16−8m+m2−1>0 and -2<m<4
⇒ m2+4m+3>0, m2−8m+15>0 and -2<m<4
⇒ (m+1)(m+3) > 0, (m-3)(m-5) > 0 and -2<m<4
⇒ -1<m<3 [combining all]