An engineering student, Mia, is analyzing a system transfer function H(s)=5s2−2(s+2)(s−1)2H(s) = \frac{5s^2 - 2}{(s + 2)(s - 1)^2}H(s)=(s+2)(s−1)25s2−2. She attempts to decompose it into partial fractions for an inverse Laplace transform calculation.
Her incorrect attempt is shown below.
Step 1: 5s2−2(s+2)(s−1)2≡Ps+2+Q(s−1)2\frac{5s^2 - 2}{(s + 2)(s - 1)^2} \equiv \frac{P}{s + 2} + \frac{Q}{(s - 1)^2}(s+2)(s−1)25s2−2≡s+2P+(s−1)2Q
Step 2: 5s2−2≡P(s−1)2+Q(s+2)5s^2 - 2 \equiv P(s - 1)^2 + Q(s + 2)5s2−2≡P(s−1)2+Q(s+2)
Step 3: Let s=−2⇒18=9P⇒P=2s = -2 \Rightarrow 18 = 9P \Rightarrow P = 2s=−2⇒18=9P⇒P=2 Let s=1⇒3=3Q⇒Q=1s = 1 \Rightarrow 3 = 3Q \Rightarrow Q = 1s=1⇒3=3Q⇒Q=1
Mia's Result: 5s2−2(s+2)(s−1)2≡2s+2+1(s−1)2\frac{5s^2 - 2}{(s + 2)(s - 1)^2} \equiv \frac{2}{s + 2} + \frac{1}{(s - 1)^2}(s+2)(s−1)25s2−2≡s+22+(s−1)21
(i) By using a counter-example, show that the result obtained by Mia cannot be correct.
(ii) Explain the conceptual mistake Mia made in Step 1.
Determine the correct partial fraction decomposition for H(s)H(s)H(s), with constant numerators.
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.