Let P be a clamped B-spline curve of degree two defined by the control points,
The control points are : $\binom{-2}{-2},\binom{-2}{0},\binom{0}{2},\binom{2}{2},\binom{2}{0},\binom{0}{-2} $
and over the knot vector of $\tau$ := $ (0, 0, 0, \frac{1}{4},\frac{1}{2} ,\frac{3}{4}, 1, 1, 1)$.
Does P pass through the point $q := \Biggl(\begin{smallmatrix} \frac{-1}{2} \\ \frac{1}{2} \end{smallmatrix} \Biggr)$ ?
Could we get different results for other clamped (possibly non-uniform) knot vectors instead of $\tau $?
The question doesn't really make sense. We have 6 control points and 9 knot values, and these completely determine a quadratic b-spline curve of degree 2.
The word "clamped" refers to an end condition that you use when constructing a spline curve. It serves to add two more constraints so that you end up with a linear system that has the same number of equations as unknowns. Since our b-spline curve is completely determined by the given data, it doesn't matter whether it was constructed with "clamped" ends or not.
I'd say that your instructor is trying to make you think by asking you a nonsensical question, or you copied the question incorrectly.