Question
If all real values of $$x$$ obtained from the equation $${4^x} - \left( {a - 3} \right){2^x} + a - 4 = 0$$ are non-positive then
A.
$$a \in \left( {4,5} \right]$$
B.
$$a \in \left( {0,4} \right)$$
C.
$$a \in \left( {4, + \infty } \right)$$
D.
None of these
Answer :
$$a \in \left( {4,5} \right]$$
Solution :
$$\eqalign{
& {\left( {{2^x}} \right)^2} - \left( {a - 4} \right){2^x} - {2^x} + \left( {a - 4} \right) = 0 \cr
& \Rightarrow \,\,\left( {{2^x} - a + 4} \right)\left( {{2^x} - 1} \right) = 0 \cr
& \therefore \,\,{2^x} = 1,a - 4.\,\,{\text{As }}x \leqslant 0,0 < a - 4 \leqslant 1. \cr} $$