Explanation:

In order to determine whether a set of functions is linearly dependent or independent, we need to check if there exist non-zero constants such that a linear combination of the functions equals zero.

The set of functions given is:
a) sinx, sin2x, and cos2x

We need to check if there exist constants A, B, and C, not all zero, such that:
A*sinx + B*sin2x + C*cos2x = 0

To check for linear dependence, we can simplify the equation using trigonometric identities.

Simplifying the equation:
Using the double angle formula for sine and cosine, we can rewrite the equation as follows:
A*sinx + B*sin2x + C*cos2x = 0
A*sinx + B*2*sinx*cosx + C*(1 - 2*sin^2x) = 0
(A + C)sinx + 2B*sinx*cosx + C - 2C*sin^2x = 0
(A + C)sinx + 2B*sinx*cosx + C - 2C(1 - cos^2x) = 0
(A + C)sinx + 2B*sinx*cosx + C - 2C + 2C*cos^2x = 0
(A + C)sinx + 2B*sinx*cosx + C(2cos^2x - 1) = 0

Now, for this equation to hold for all values of x, the coefficients of sinx, cosx, and the constant term must all be zero.

Setting up the system of equations:
We can equate the coefficients to zero and set up a system of equations:
A + C = 0
2B = 0
2C - 1 = 0

Solving the system of equations:
From the second equation, we have B = 0. This means that the coefficient of sin2x is zero.

Substituting B = 0 into the first equation, we get A + C = 0. This means that the coefficients of sinx and cos2x are linearly dependent on each other.

From the third equation, we have C = 1/2. This means that the coefficient of cos2x is not zero.

Therefore, the set of functions sinx, sin2x, and cos2x is linearly dependent because there exist non-zero constants A, B, and C such that A*sinx + B*sin2x + C*cos2x = 0.