The signal response L L\,L of a specialized optical sensor is modeled by the function
L(θ)=(2−12θ)6 L(\theta) = \left(2 - \frac{1}{2}\theta\right)^6 L(θ)=(2−21θ)6where θ \theta\,θ represents the incident angle in radians.
Determine the first four terms, in ascending powers of θ\thetaθ, of the binomial expansion of L(θ)L(\theta)L(θ).
In a dual-sensor array, the combined response R(θ)R(\theta)R(θ) is given by
R(θ)=(2−12θ)6+(2+12θ)6 R(\theta) = \left(2 - \frac{1}{2}\theta\right)^6 + \left(2 + \frac{1}{2}\theta\right)^6 R(θ)=(2−21θ)6+(2+21θ)6Given that θ \theta\,θ is small enough that terms in θ4 \theta^4\,θ4 and higher powers of θ \theta\,θ may be neglected, show that
R(θ)≈A+Bθ2 R(\theta) \approx A + B\theta^2 R(θ)≈A+Bθ2where A A\,A and B B\,B are constants to be found.
Practise AQA A Level Maths 1.7 D: Sequences and series with exam-style questions for A Level Maths. 267 questions covering 1.7.1 Binomial expansion, 1.7.2 Types of sequence (A-level only), 1.7.3 Sigma notation (A-level only), 1.7.4 Arithmetic sequences and series (A-level only), 1.7.5 Geometric sequences and series (A-level only), and 1.7.6 Sequences and series 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.
