A laboratory tracks the pressure variance VVV, in kilopascals, within a reaction vessel. The variance is modelled by the function V(t)=(t+2)(t−4)(2t−5)V(t) = (t+2)(t-4)(2t-5)V(t)=(t+2)(t−4)(2t−5), where t t\,t is the time in minutes relative to a baseline event.
A secondary model is defined as g(t)=12V(t−3)+k\displaystyle g(t) = \frac{1}{2}V(t - 3) + kg(t)=21V(t−3)+k. Given that the graph of y=g(t)y = g(t)y=g(t) passes through the point (3,14)(3, 14)(3,14), determine the value of the constant kkk.
The pressure variance model is shifted horizontally such that h(t)=V(t+m)h(t) = V(t + m)h(t)=V(t+m). Given that the graph of y=h(t)y = h(t)y=h(t) passes through the origin (0,0)(0,0)(0,0), find all possible values of the constant mmm.
Find V′(t)V'(t)V′(t).
Determine the set of values of t t\,t for which the rate of change of the pressure variance is less than -6.
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.