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.
303 exam-style questions on OCR (MEI) A Level Maths 1.2 Algebra, covering 1.2.1 Algebraic vocabulary and notation, 1.2.2 Solve linear equations, 1.2.3 Change the subject of a formula, 1.2.4 Solve quadratic equations, 1.2.5 Discriminant of a quadratic, 1.2.6 Linear simultaneous equations, 1.2.7 One linear one quadratic simultaneous equations, 1.2.8 Points of intersection and solutions, 1.2.9 Linear inequalities, 1.2.10 Quadratic inequalities, 1.2.11 Expressing solutions of inequalities, 1.2.12 Use and manipulate surds, 1.2.13 Rationalise the denominator, 1.2.14 Laws of indices, 1.2.15 Negative, fractional and zero indices, 1.2.16 Proportional relationships, 1.2.17 Partial fractions (A-level only), and 1.2.18 Simplify rational expressions (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.