Exam Questions & Step-by-Step Markschemes (67 Problems)
Question 1 • 2024 Exam Specimen
6 Marks
Question 2 • 2024 Exam Specimen
6 Marks
Question 3 • Exam Practice
6 Marks
Consider the function f(x) = \frac{\ln x}{x^2} for x > 0 .\n\n(a) Find f'(x) . [3 marks]\n\n(b) Hence, find the coordinates of the local maximum point on the graph of f . [3 marks]
Answer: (a) \ f'(x) = \frac{1 - 2\ln x}{x^3}, \quad (b) \ \left(\sqrt{e}, \frac{1}{2e}\right)
Question 4 • Exam Practice
8 Marks
(a) Use the standard expansion for (1 + t)^{-1} to show that the Maclaurin series for \arctan x is given by \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots [4 marks]\n\n(b) Hence, approximate the value of \int_{0}^{1/2} \frac{\arctan x}{x} \, dx using the first three non-zero terms. [4 marks]
Answer: (a) \ \text{Proved by term-by-term integration}, \quad (b) \ \frac{3509}{7200} \approx 0.487
Question 5 • Exam Practice
8 Marks
A closed chemical process involves two reactants with concentrations x(t) and y(t) governed by:\n\n \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} -4 & 2 \\ 1 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}, \quad x(0) = 4, \, y(0) = 1 \n\n(a) Find the eigenvalues and eigenvectors of the coefficient matrix. [4 marks]\n\n(b) Write down the particular solution for x(t) and y(t) . [3 marks]\n\n(c) Classify the nature of the equilibrium at (0, 0) . [1 mark]
Answer: (a) \ \lambda_1 = -2, \lambda_2 = -5; \ \mathbf{v}_1 = (1, 1)^T, \mathbf{v}_2 = (-2, 1)^T, \quad (b) \ x(t) = 2e^{-2t} + 2e^{-5t}, y(t) = 2e^{-2t} - e^{-5t}, \quad (c) \ \text{Stable Node (Sink)}
Question 6 • Exam Practice
7 Marks
The curve C is defined by y = \ln(2x - 1) for x > 0.5 . The region R is bounded by C , the lines y = 0 , y = 2 , and the y -axis. Find the exact volume of the solid generated when R is rotated through 2\pi radians about the y -axis.
Answer: V = \frac{\pi(e^4 + 4e^2 - 1)}{8}