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.
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.