A

Step 1: Identify the components of the current.
The current consists of a direct current (d.c.) component: I_d = 3 ext{ A} and an alternating current (a.c.) component: I_a = 4 ext{ sin } heta = 4 ext{ sin } ( ext{ } ext{ } ext{ } ext{ u } t).

Step 2: Write the expression for the resultant current.
The total current (I) at any time (t) can be expressed as the sum of the d.c. and a.c. components:
I(t) = I_d + I_a = 3 + 4 ext{ sin } ( u t).

Step 3: Calculate the effective value (RMS value) of the resultant current.
The effective value (or RMS value) of a d.c. component is simply its value.
The RMS value of a sinusoidal a.c. component is given by: \frac{I_a}{\sqrt{2}}. Here, \( I_a = 4 \), so:
\text{RMS value of } I_a = \frac{4}{\sqrt{2}} = 2\sqrt{2} ext{ A}.

Step 4: Combine the d.c. component with the effective value of the a.c. component. Since the d.c. component does not vary with time, the effective value of the resultant current is given by the square root of the sum of squares of the d.c. and a.c. RMS values:
I_{rms} = \sqrt{I_d^2 + I_{a_{rms}}^2} = \sqrt{3^2 + (2\sqrt{2})^2} = \sqrt{9 + 8} = \sqrt{17} ext{ A}.

Thus, the effective value of the resultant current is \sqrt{17} ext{ A}.