A model for the concentration of a catalyst, CCC, at a distance x x\,x millimetres from the center of a reaction chamber is given by the function
C(x)=(x−4)(3x2+16x+k)+340 C(x) = (x - 4)(3x^2 + 16x + k) + 340 C(x)=(x−4)(3x2+16x+k)+340where k k\,k is a constant.
State the value of the concentration at x=4x = 4x=4.
Given that (3x−2)(3x - 2)(3x−2) is a factor of C(x)C(x)C(x),
show that k=90k = 90k=90.
Hence
(i) fully factorise the expression for C(x)C(x)C(x),
(ii) determine the number of real values of x x\,x for which the concentration is zero, giving a reason for 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.