The potential energy V(x)V(x)V(x) of a mechanical system as a function of displacement x x\,x is modeled by the cubic polynomial
V(x)=x3+(k+5)x2+5(k+5)x+125 V(x) = x^3 + (k + 5)x^2 + 5(k + 5)x + 125 V(x)=x3+(k+5)x2+5(k+5)x+125where k k\,k is a constant.
Use the factor theorem to prove that (x+5)(x + 5)(x+5) is a factor of V(x)V(x)V(x) for all values of kkk.
The graph of y=V(x)y = V(x)y=V(x) meets the xxx-axis at exactly two distinct points.
(i) Sketch a possible graph of y=V(x)y = V(x)y=V(x), showing the coordinates of the intercepts with both axes.
(ii) Given that V(x)V(x)V(x) can be written in the form V(x)=(x+5)(x2+kx+25)V(x) = (x + 5)(x^2 + kx + 25)V(x)=(x+5)(x2+kx+25), determine the value of kkk. Fully justify your answer.
306 exam-style questions on CCEA A Level Maths 1.1 Algebra and functions, covering 1.1.1 Algebra and functions, 1.1.2 Algebra and functions, 1.1.3 Algebra and functions, 1.1.4 Algebra and functions, 1.1.5 Algebra and functions, 1.1.6 Algebra and functions, 1.1.7 Algebra and functions, 1.1.8 Algebra and functions, 1.1.9 Algebra and functions, 1.1.10 Algebra and functions, 1.1.11 Algebra and functions, 1.1.12 Algebra and functions, 1.1.13 Algebra and functions, and 1.1.14 Algebra and functions. Each one has a worked solution and a mark scheme showing where the marks go.