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.
566 exam-style questions on OCR A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Indices, 1.2.2 Surds, 1.2.3 Simultaneous equations, 1.2.4 Quadratic functions, 1.2.5 Completing the square, 1.2.6 Solving quadratic equations, 1.2.7 Inequalities, 1.2.8 Expressing solutions of inequalities, 1.2.9 Representing inequalities graphically, 1.2.10 Manipulating polynomials, 1.2.11 Simplifying rational expressions, 1.2.12 The modulus function (A-level only), 1.2.13 Graphs of functions, 1.2.14 Sketching curves from equations, 1.2.15 Sketching reciprocal curves, 1.2.16 Interpreting solutions graphically, 1.2.17 Intersection points of graphs, 1.2.18 Proportional relationships, 1.2.19 Graph of the modulus of a linear function (A-level only), 1.2.20 Solving modulus equations graphically (A-level only), 1.2.21 Definition of a function (A-level only), 1.2.22 Inverse and composite functions (A-level only), 1.2.23 Simple graph transformations, 1.2.24 Combinations of graph transformations (A-level only), 1.2.25 Partial fractions (A-level only), and 1.2.26 Models in context. Each one has a worked solution and a mark scheme showing where the marks go.