The temperature, V V\,V degrees Celsius, of a chemical reaction chamber over a period of time t t\,t minutes is modelled by the function
V(t)=∣kt−30∣−12,t≥0 V(t) = |kt - 30| - 12, \quad t \ge 0 V(t)=∣kt−30∣−12,t≥0where k k\,k is a positive constant.
The graph of y=V(t)y = V(t)y=V(t) intersects the vertical axis at the point C C\,C and has a minimum point at MMM.
(i) Write down the V V\,V coordinate of CCC. (ii) Find, in terms of kkk, the t t\,t coordinate of MMM.
Find, in terms of kkk, the range of values of t t\,t for which V(t)<0V(t) < 0V(t)<0.
A cooling phase is defined by the linear function L(t)=6−2tL(t) = 6 - 2tL(t)=6−2t.
Given that the line y=L(t)y = L(t)y=L(t) intersects the graph y=V(t)y = V(t)y=V(t) at two distinct points, find the range of possible values for kkk.
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.