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 Edexcel A Level Maths Algebraic Methods with exam-style questions for A Level Maths. 226 questions covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division, matched to the Edexcel A Level Maths (9MA0) 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.
