Question
The equation $${x^2} - 6x + 8 + \lambda \left( {{x^2} - 4x + 3} \right) = 0,\lambda \in R,$$ has
A.
real and unequal roots for all $$\lambda $$
B.
real roots for $$\lambda < 0$$ only
C.
real roots for $$\lambda > 0$$ only
D.
real and unequal roots for $$\lambda = 0$$ only
Answer :
real and unequal roots for all $$\lambda $$
Solution :
$$\eqalign{
& \left( {1 + \lambda } \right){x^2} - \left( {6 + 4\lambda } \right)x + 8 + 3\lambda = 0 \cr
& \therefore \,\,D = {\left( {6 + 4\lambda } \right)^2} - 4\left( {1 + \lambda } \right)\left( {8 + 3\lambda } \right) = {\lambda ^2} + \lambda + 1 > 0. \cr} $$