The concentration of a specialized catalyst in a chemical reaction, CCC, is modeled by the function C(t)=(3+αt)4C(t) = (3 + \alpha t)^4C(t)=(3+αt)4 for small values of time t>0t > 0t>0, where α\alphaα is a non-zero constant.
Given that the coefficient of the t2t^2t2 term in the binomial expansion of (3+αt)4(3 + \alpha t)^4(3+αt)4 is 135013501350, find the possible values of α\alphaα.
A related adjusted concentration function, G(t)G(t)G(t), is defined by the product
G(t)=(127+2t3)(3+αt)4 G(t) = \left( \frac{1}{27} + \frac{2}{t^3} \right) (3 + \alpha t)^4 G(t)=(271+t32)(3+αt)4Using your values from part (a), determine the possible values for the term independent of ttt in the binomial expansion of G(t)G(t)G(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.