The roots of the equation \((5-x)^{4}+(4-x)^{4}=(q-2 x)^{4}\) are (a) all imaginary (b) all real (c) Two real and two imaginary (d) None of these

If all the roots are either imaginary or real, it means that the equation must be a valid algebraic equation without any restrictions on the terms. Thus, let's explore these options by substituting each of the given options into the equation and see if it satisfies the requirements of an algebraic equation.

Let's assume that all the roots are imaginary, i.e., x is complex and of the form x = a + bi (where a and b are real numbers, i^2 = -1, and bi is the imaginary part). Now, substitute this in the given equation: \[((5-(a+bi))^4 + (4-(a+bi))^4 = (q-2(a+bi))^4\] Since all the terms are raised to the power of 4, their imaginary parts will cancel out (as \(i^4 = 1\)). This leaves us with the equation: \[((5-a)^4 + (4-a)^4 = (q-2a)^4\] This equation looks algebraic, but it doesn't guarantee that all the roots must be imaginary. So, this option doesn't make the given equation suitable for identifying the nature of the roots.

Let's assume that all the roots are real, i.e., x is a real number. In this case, the given equation can be written as: \[((5-x)^4 + (4-x)^4 = (q-2x)^4\] This is an algebraic equation with no restrictions on the terms. However, the given equation still doesn't guarantee that all the roots will be real.

Let's now examine the more general condition of two real and two imaginary roots. Consider x1 and x2 as the real roots and x3 and x4 as the imaginary roots (of the form x3 = a3 + bi, x4 = a4 + ci). For real roots (x1 and x2), the equation looks like: \[((5-x1)^4 + (4-x1)^4 = (q-2x1)^4\) and \[((5-x2)^4 + (4-x2)^4 = (q-2x2)^4\] For imaginary roots (x3 and x4), \[((5-(a3+bi))^4 + (4-(a3+bi))^4 = (q-2(a3+bi))^4\] and \[((5-(a4+ci))^4 + (4-(a4+ci))^4 = (q-2(a4+ci))^4\] When we look at these equations, it becomes clear that it is possible to have two distinct and independent pairs of equations (real and imaginary) that can satisfy the given equation. This gives rise to the possibility of having two real roots and two imaginary roots.

Therefore, after analyzing all options, we can conclude that the roots of the given equation are of the nature mentioned in option (c): Two real roots and two imaginary roots.