The sensitivity S S\,S of a precision transducer depends on the frequency ω \omega\,ω of pressure oscillations according to the model
S(ω)=(1+kω)−12,∣kω∣<1 S(\omega) = (1 + k\omega)^{-\frac{1}{2}}, \quad |k\omega| < 1 S(ω)=(1+kω)−21,∣kω∣<1where k k\,k is a constant. Given that the binomial expansion of S(ω)S(\omega)S(ω) in ascending powers of ω \omega\,ω up to the term in ω3 \omega^3\,ω3 is
1+0.6ω+Pω2+Qω3 1 + 0.6\omega + P\omega^2 + Q\omega^3 1+0.6ω+Pω2+Qω3(i) find the value of kkk,
(ii) find the value of the constant P P\,P and the constant QQQ.
Use the expansion to find an approximate value to 10.88\displaystyle \frac{1}{\sqrt{0.88}}0.881.
Show your working and give your answer to 6 decimal places.
308 exam-style questions on OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.