The amplitude of a signal in a circuit, S(t)S(t)S(t), can be modeled by the function
S(t)=(2−t4)8 S(t) = \left(2 - \frac{t}{4}\right)^8 S(t)=(2−4t)8where t t\,t represents time in milliseconds.
Find the first 4 terms, in ascending powers of ttt, of the binomial expansion of S(t)S(t)S(t), giving each term in its simplest form.
A noise-cancelling filter F(t)F(t)F(t) is applied such that the resulting signal is given by the product P(t)=F(t)S(t)P(t) = F(t)S(t)P(t)=F(t)S(t). Given that
F(t)=(3+2t)2 F(t) = \left(3 + \frac{2}{t}\right)^2 F(t)=(3+t2)2determine the constant term in the series expansion of P(t)P(t)P(t).
308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.