A physical experiment measures the spectral intensity III of a pulsed laser relative to its frequency shift sss. The measurements are modeled by the parametric equations
s=5+4sint s = 5 + 4 \sin t s=5+4sint I=187+cos2t I = \frac{18}{7 + \cos 2t} I=7+cos2t18where −π2≤t≤π2-\frac{\pi}{2} \le t \le \frac{\pi}{2}−2π≤t≤2π.
Show that the relationship between intensity and frequency shift can be expressed as
I=144(13−s)(s+3)p≤s≤q I = \frac{144}{(13 - s)(s + 3)} \quad p \le s \le q I=(13−s)(s+3)144p≤s≤qwhere ppp and qqq are constants to be found.
Hence, determine a Cartesian equation for this relationship in the form
I=as+b+cs+dp≤s≤q I = \frac{a}{s + b} + \frac{c}{s + d} \quad p \le s \le q I=s+ba+s+dcp≤s≤qwhere a,b,ca, b, ca,b,c and ddd are constants.
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.