The cross-sectional profile of a custom-designed skate ramp is modeled by the curve C1C_1C1 with equation y=f(x)y = f(x)y=f(x), where f(x)=(6−x)(2x+5)2f(x) = (6 - x)(2x + 5)^2f(x)=(6−x)(2x+5)2 and xxx represents horizontal distance from a sensor in meters.
Sketch C1C_1C1 showing the coordinates of any point where the curve touches or crosses the coordinate axes.
Hence or otherwise, (i) find the values of xxx for which f(3x)=0f(3x) = 0f(3x)=0. (ii) find the value of the constant kkk such that the curve with equation y=f(x)+ky = f(x) + ky=f(x)+k passes through the origin.
A second curve C2C_2C2 has equation y=g(x)y = g(x)y=g(x), where g(x)=f(x−2)g(x) = f(x - 2)g(x)=f(x−2).
(c) (i) Find, in simplest form, g(x)g(x)g(x). You may leave your answer in a factorised form. (ii) Hence, or otherwise, find the yyy-intercept of curve C2C_2C2.
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.