In a high-frequency signal filter design, the gain GGG relative to a frequency shift ω\omegaω is modelled by the function G(ω)=(25+2ω)−12G(\omega) = (25 + 2\omega)^{-\frac{1}{2}}G(ω)=(25+2ω)−21. The first three terms, in ascending powers of ω\omegaω, of the binomial expansion of G(ω)G(\omega)G(ω) are given by
(25+2ω)−12≈a−1125ω+36250ω2(25 + 2\omega)^{-\frac{1}{2}} \approx a - \frac{1}{125}\omega + \frac{3}{6250}\omega^2(25+2ω)−21≈a−1251ω+62503ω2where aaa is a constant.
State the range of values of ω\omegaω for which this expansion is valid.
Choose from the options below:
∣ω∣<225∣ω∣<252∣ω∣<5∣ω∣<25|\omega| < \frac{2}{25} \quad |\omega| < \frac{25}{2} \quad |\omega| < 5 \quad |\omega| < 25∣ω∣<252∣ω∣<225∣ω∣<5∣ω∣<25Find the value of aaa.
Choose from the options below:
12515525\frac{1}{25} \quad \frac{1}{5} \quad 5 \quad 2525151525308 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.