An interference is observed due to two coherent sources, A and B, separated by a distance $4\lambda$ along the Y-axis, where $\lambda$ is the wavelength of the source. A detector D is moved along the positive X-axis. The number of points on the X-axis, excluding $x = 0$ and $x \to \infty$, at which maxima will be observed, is

Let source B be at $(0,0)$ and source A be at $(0, 4\lambda)$.

The path difference $\Delta p$ at a point $D(x, 0)$ on the positive X-axis is given by:

$\Delta p = AD - BD = \sqrt{x^2 + (4\lambda)^2} - x$

Let us check the boundary values of the path difference:
At $x = 0$, $\Delta p = \sqrt{0 + 16\lambda^2} - 0 = 4\lambda$.
As $x \to \infty$, $\Delta p = \lim_{x \to \infty} x\left(1 + \dfrac{16\lambda^2}{x^2}\right)^{1/2} - x = 0$.

For constructive interference (maxima), the path difference must be an integer multiple of the wavelength:

$\Delta p = n\lambda$

Since the points $x = 0$ and $x \to \infty$ are explicitly excluded, the path difference must strictly lie in the range:

$0 < \Delta p < 4\lambda$

Therefore, the possible values for $\Delta p$ are $1\lambda, 2\lambda,$ and $3\lambda$.

This corresponds to $n = 1, 2, 3$.

Thus, there are exactly $3$ points on the positive X-axis where maxima will be observed.

Answer: $3$