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.
532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, 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). Each one has a worked solution and a mark scheme showing where the marks go.