A chemical engineer models the rate of heat loss H(T)H(T)H(T), in kilowatts per degree Celsius, from a specific pressurized vessel using the formula
H(T)=2T2−3T−7T−4,T>4 H(T) = \frac{2T^2 - 3T - 7}{T - 4}, \quad T > 4 H(T)=T−42T2−3T−7,T>4where TTT is the temperature of the vessel in degrees Celsius.
Write H(T)H(T)H(T) in the form
AT+B+CT−4 AT + B + \frac{C}{T - 4} AT+B+T−4Cwhere AAA, BBB, and CCC are integers to be found.
Hence use algebraic integration to show that the total heat loss between temperature T=5T = 5T=5 and T=7T = 7T=7, given by ∫57H(T) dT\int_{5}^{7} H(T) \, dT∫57H(T)dT, is equal to
α+βln3 \alpha + \beta \ln 3 α+βln3where α\alphaα and β\betaβ are integers to be found.
532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, 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). Each one has a worked solution and a mark scheme showing where the marks go.