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.
566 exam-style questions on OCR A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Indices, 1.2.2 Surds, 1.2.3 Simultaneous equations, 1.2.4 Quadratic functions, 1.2.5 Completing the square, 1.2.6 Solving quadratic equations, 1.2.7 Inequalities, 1.2.8 Expressing solutions of inequalities, 1.2.9 Representing inequalities graphically, 1.2.10 Manipulating polynomials, 1.2.11 Simplifying rational expressions, 1.2.12 The modulus function (A-level only), 1.2.13 Graphs of functions, 1.2.14 Sketching curves from equations, 1.2.15 Sketching reciprocal curves, 1.2.16 Interpreting solutions graphically, 1.2.17 Intersection points of graphs, 1.2.18 Proportional relationships, 1.2.19 Graph of the modulus of a linear function (A-level only), 1.2.20 Solving modulus equations graphically (A-level only), 1.2.21 Definition of a function (A-level only), 1.2.22 Inverse and composite functions (A-level only), 1.2.23 Simple graph transformations, 1.2.24 Combinations of graph transformations (A-level only), 1.2.25 Partial fractions (A-level only), and 1.2.26 Models in context. Each one has a worked solution and a mark scheme showing where the marks go.