In a materials science laboratory, the expansion factor of a synthetic polymer PPP relative to temperature change ttt is modeled by the function:
P(t)=(2+kt4)5 P(t) = \left(2 + \frac{kt}{4}\right)^5 P(t)=(2+4kt)5where kkk is a non-zero constant.
Find the first 4 terms, in ascending powers of ttt, of the binomial expansion of P(t)P(t)P(t). Give each term in its simplest form.
To account for sensor drift, a corrected expansion model R(t)R(t)R(t) is defined as:
R(t)=(5+3t)P(t) R(t) = (5 + 3t)P(t) R(t)=(5+3t)P(t)Given that in the expansion of R(t)R(t)R(t), the coefficient of the ttt term is equal to the constant term, determine the value of kkk.
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.