The rate of accumulation of a specific enzyme in a bioreactor, R(t)R(t)R(t) in mmol/L per hour, is modeled by the function:
R(t)=7t−2(t+3)2,t≥0 R(t) = \frac{7t - 2}{(t + 3)^2}, \quad t \ge 0 R(t)=(t+3)27t−2,t≥0where t t\,t is the time in hours since the start of the reaction.
Express R(t)R(t)R(t) in the form At+3+B(t+3)2\displaystyle \frac{A}{t + 3} + \frac{B}{(t + 3)^2}t+3A+(t+3)2B, where A A\,A and B B\,B are constants to be found.
The total accumulation of the enzyme, KKK, between t=1t = 1t=1 and t=5t = 5t=5 hours is given by K=∫15R(t) dtK = \int_{1}^{5} R(t) \, dtK=∫15R(t)dt.
Show that K=p+lnqK = p + \ln qK=p+lnq, where p p\,p and q q\,q are rational numbers to be determined.
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.