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.
368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.