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.
303 exam-style questions on OCR (MEI) A Level Maths 1.2 Algebra, covering 1.2.1 Algebraic vocabulary and notation, 1.2.2 Solve linear equations, 1.2.3 Change the subject of a formula, 1.2.4 Solve quadratic equations, 1.2.5 Discriminant of a quadratic, 1.2.6 Linear simultaneous equations, 1.2.7 One linear one quadratic simultaneous equations, 1.2.8 Points of intersection and solutions, 1.2.9 Linear inequalities, 1.2.10 Quadratic inequalities, 1.2.11 Expressing solutions of inequalities, 1.2.12 Use and manipulate surds, 1.2.13 Rationalise the denominator, 1.2.14 Laws of indices, 1.2.15 Negative, fractional and zero indices, 1.2.16 Proportional relationships, 1.2.17 Partial fractions (A-level only), and 1.2.18 Simplify rational expressions (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.