A biologist models the concentration ratio, RRR, of two specific enzymes in a cell culture over time ttt (in hours) using the function
R(t)=8t2+6t−54t2+5t+6+16t+2,t∈R, t>1 R(t) = \frac{8t^2 + 6t - 54}{t^2 + 5t + 6} + \frac{16}{t + 2}, \quad t \in \mathbb{R}, \ t > 1 R(t)=t2+5t+68t2+6t−54+t+216,t∈R, t>1Show that R(t)=8t−2t+2R(t) = \frac{8t - 2}{t + 2}R(t)=t+28t−2.
Show, using calculus, that RRR is an increasing function. You must make your reasoning clear.
The function hhh is defined by
h(x)=1+lnx,x≥1 h(x) = 1 + \ln x, \quad x \ge 1 h(x)=1+lnx,x≥1Find h−1(x)h^{-1}(x)h−1(x), stating its domain.
Determine the exact value of aaa for which hR(a)=2hR(a) = 2hR(a)=2.
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.
