Determine the exact values of k for which the curves y = x^2 − kx and y = 3 (k + 1) + kx − x^2 touch. 

To ascertain the specific values of k for which the curves defined by the equations y = x^2 − kx and y = 3(k + 1) + kx − x^2 exhibit tangential contact, one must analyse the conditions under which these two curves intersect at a single point. This scenario occurs when the two equations yield a common solution that satisfies both the position and the slope of the curves at that point. Therefore, it is essential to equate the two expressions and subsequently derive the conditions that lead to a double root in the resulting polynomial equation.

By setting the two equations equal to each other, we can rearrange the terms to form a standard quadratic equation. The resulting equation will take the form of a quadratic in x, which can be expressed as a function of k. To ensure that the curves touch, the discriminant of this quadratic must equal zero, indicating that there is exactly one solution for x. This condition will yield a specific relationship involving k, which can then be solved to find the exact values of k that satisfy the tangential condition.

In the context of A Level Maths May Revision, this problem exemplifies the application of quadratic equations and the concept of tangency in coordinate geometry. By carefully analysing the derived conditions and solving for k, one can gain a deeper understanding of the interplay between algebraic expressions and their graphical representations. This exploration not only reinforces fundamental mathematical principles but also enhances problem-solving skills essential for advanced studies in mathematics.