The thermal resilience index, RRR, of a specific polymer during a stress test is modeled by the function
R(t)=5t2−202t2+9t+10−22t+5,t∈R, t>1 R(t) = \frac{5t^2 - 20}{2t^2 + 9t + 10} - \frac{2}{2t + 5}, \quad t \in \mathbb{R}, \ t > 1 R(t)=2t2+9t+105t2−20−2t+52,t∈R, t>1where ttt is the duration of the test in hours.
Show that R(t)=5t−122t+5R(t) = \frac{5t - 12}{2t + 5}R(t)=2t+55t−12.
Show, using calculus, that RRR is an increasing function for all t>1t > 1t>1. You must make your reasoning clear.
The monitoring function HHH is defined by
H(t)=8+2lnt,t≥1 H(t) = 8 + 2 \ln t, \quad t \ge 1 H(t)=8+2lnt,t≥1Find H−1(x)H^{-1}(x)H−1(x).
Find the exact value of aaa for which HR(a)=9HR(a) = 9HR(a)=9.
Practise AQA A Level Maths 1.5 B: Algebra and functions with exam-style questions for A Level Maths. 358 questions covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
