Step-by-step Solution:

We are asked to find the number of solutions for the pair of equations in the interval $[0, 2 au]$.

The given equations are:

1. $2{\sin ^2}\theta - \cos 2\theta = 0$
2. $2{\cos ^2}\theta - 3\sin \theta = 0$

Step 1: Simplify the first equation.

We use the double-angle identity for cosine: $\cos 2\theta = 1 - 2{\sin ^2}\theta$. Substituting this into the first equation: $2{\sin ^2}\theta - (1 - 2{\sin ^2}\theta) = 0$ $2{\sin ^2}\theta - 1 + 2{\sin ^2}\theta = 0$ $4{\sin ^2}\theta - 1 = 0$ $4{\sin ^2}\theta = 1$ ${\sin ^2}\theta = \frac{1}{4}$ (Equation 3)

This implies $\sin \theta = \pm \frac{1}{2}$.

Step 2: Simplify the second equation.

We use the Pythagorean identity: ${\cos ^2}\theta = 1 - {\sin ^2}\theta$. Substituting this into the second equation: $2(1 - {\sin ^2}\theta) - 3\sin \theta = 0$ $2 - 2{\sin ^2}\theta - 3\sin \theta = 0$ $2{\sin ^2}\theta + 3\sin \theta - 2 = 0$ (Equation 4)

Step 3: Solve the system of equations.

We need to find the values of $\theta$ that satisfy both simplified equations simultaneously. We can substitute the result from Equation 3, ${\sin ^2}\theta = \frac{1}{4}$, into Equation 4.

$2\left(\frac{1}{4}\right) + 3\sin \theta - 2 = 0$ $\frac{1}{2} + 3\sin \theta - 2 = 0$ $3\sin \theta - \frac{3}{2} = 0$ $3\sin \theta = \frac{3}{2}$ $\sin \theta = \frac{1}{2}$

Step 4: Find the solutions in the given interval.

We need to find the number of values of $\theta$ in the interval $[0, 2\pi]$ that satisfy $\sin \theta = \frac{1}{2}$.

The sine function is positive in the first and second quadrants.

• In the first quadrant, the solution is $\theta = \frac{\pi}{6}$.
• In the second quadrant, the solution is $\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.

Both $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ lie within the interval $[0, 2\pi]$.

Let's verify if these solutions are consistent with our initial simplification. If $\sin \theta = \frac{1}{2}$, then ${\sin ^2}\theta = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$, which is consistent with Equation 3.

Therefore, the common solutions are the angles $\theta$ for which $\sin\theta = 1/2$. There are two such solutions in the interval $[0, 2\pi]$.

Conclusion:

The number of solutions for the given pair of equations in the interval $[0, 2\pi]$ is 2. This corresponds to option C.