A precision laser-cutting head moves along a path C C\,C in a vertical plane. Its horizontal displacement sss (cm) and vertical height hhh (cm) are modelled by the parametric equations
s=5+3sinθ s = 5 + 3 \sin \theta s=5+3sinθ h=167+cos2θ h = \frac{16}{7 + \cos 2\theta} h=7+cos2θ16for −π2≤θ≤π2-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}−2π≤θ≤2π.
Show that the path C C\,C has the Cartesian equation
h=72(11−s)(s+1)p≤s≤q h = \frac{72}{(11 - s)(s + 1)} \quad p \le s \le q h=(11−s)(s+1)72p≤s≤qwhere p p\,p and q q\,q are constants to be found.
Hence, find a Cartesian equation for C C\,C in the form
h=as+b+cs+dp≤s≤q h = \frac{a}{s + b} + \frac{c}{s + d} \quad p \le s \le q h=s+ba+s+dcp≤s≤qwhere a,b,c a, b, c\,a,b,c and d d\,d are constants.
532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.