A researcher models the depth of a sediment layer, D(p)D(p)D(p), in millimetres as a function of the pressure ppp in kilopascals using the modulus function:
D(p)=∣cp−30∣−6,p≥0 D(p) = |cp - 30| - 6, \quad p \ge 0 D(p)=∣cp−30∣−6,p≥0where ccc is a positive constant representing the compression coefficient.
The graph of y=D(p)y = D(p)y=D(p) has a minimum point MMM and intersects the yyy-axis at the point SSS.
(i) Find the yyy-coordinate of SSS. (ii) Find, in terms of ccc, the ppp-coordinate of MMM.
Find, in terms of ccc, the set of values of ppp for which the sediment depth is negative.
A second model suggests the depth follows the linear relationship y=18−4py = 18 - 4py=18−4p. Given that this linear model intersects the modulus model y=D(p)y = D(p)y=D(p) at two distinct points,
find the range of possible values for ccc.
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.