In a high-precision mechanical linkage system, the tolerance level T T\,T is modeled as a function of the misalignment factor δ \delta\,δ by the formula T(δ)=(4+cδ)5T(\delta) = (4 + c\delta)^5T(δ)=(4+cδ)5, where c c\,c is a constant. In a specific quality test, it is found that the coefficient of the δ2 \delta^2\,δ2 term in the binomial expansion of T(δ)T(\delta)T(δ) is 5760.
Determine the possible values of ccc.
The sensitivity S S\,S of the system is defined by the expression
S(δ)=(5512−14δ3)T(δ) S(\delta) = \left( \frac{5}{512} - \frac{1}{4\delta^3} \right) T(\delta) S(δ)=(5125−4δ31)T(δ)Find the possible values of the term independent of δ \delta\,δ in the expansion of S(δ)S(\delta)S(δ).
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.