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.
Practise AQA A Level Maths 1.5 B: Algebra and functions with exam-style questions for A Level Maths. 358 questions 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), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
