The number of integral values of m for which the equation ` ( 1 + m ^ 2 ) x ^ 2 - 2 ( 1 + 3m ) x + ( 1 + 8m ) = 0 ` has no real root is :

Options

A. 1
B. 2
C. 3
D. infinitely many

Answer

D

Explanation

Let D be the discriminant of the given equation. Then, `D = 4(1+3m)^(2)-4(1+m^(2))(1+8m)`
`rArr" "D=4{9m^(2)+6m+1-8m^(3)-m^(2)-8m-1}`
`rArr" "D=4(-8m^(3) + 8m^(2) - 2m)`
`rArr" "D=-8m(4m^(2)-4m+1)=-8m(2m-1)^(2)` ltBrgt If the given equation has no real roots, then
`D lt 0 rArr -8m(2m-1)^(2) lt 0 rArr m gt 0 and m ne (1)/(2)`
Hence, m can take infinitely many integral values.