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.
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.