If $${\left( {\sqrt 2 } \right)^x} + {\left( {\sqrt 3 } \right)^x} = {\left( {\sqrt {13} } \right)^{\frac{x}{2}}}$$ then the number of values of $$x$$ is
A.
$$2$$
B.
$$4$$
C.
$$1$$
D.
none of these
Answer :
$$1$$
Solution :
$$\eqalign{
& {2^{\frac{x}{2}}} + {3^{\frac{x}{2}}} = {\left( {\sqrt {13} } \right)^{\frac{x}{2}}} \cr
& \Rightarrow \,\,{\left( {\frac{2}{{\sqrt {13} }}} \right)^{\frac{x}{2}}} + {\left( {\frac{3}{{\sqrt {13} }}} \right)^{\frac{x}{2}}} = 1 \cr} $$
which is of the form $${\cos ^{\frac{x}{2}}}\alpha + {\sin ^{\frac{x}{2}}}\alpha = 1.$$
$$\therefore \,\,\frac{x}{2} = 2.$$
Releted MCQ Question on Algebra >> Quadratic Equation
Releted Question 1
If $$\ell ,m,n$$ are real, $$\ell \ne m,$$ then the roots by the equation: $$\left( {\ell - m} \right){x^2} - 5\left( {\ell + m} \right)x - 2\left( {\ell - m} \right) = 0$$ are