FAC1004 Mechanical Skills Drill — 90-Min Sprint
Course: FAC1004 — Advanced Mathematics II (Computing) Focus: Pure mechanical computation, based on leaks & past UAS papers Time: 90 minutes Rules: Show all working. No calculator for exact trig/ln forms.
Part A — Part A Style (Mixed Topics, Short Answer)
Leak pattern: True/False, fill-in-blanks, state equations, correct equations, Pascal triangle.
A1 | True or False — Justify if False
(a) "$\pi$ is a complex number." — T/F? Justify.
(b) $\text{sech}(x) = \dfrac{e^{-x}}{e^{x} + e^{-x}}$ — T/F? Justify.
(c) $\sinh^{-1}(x) = \ln\left(x + \sqrt{x^{2} + 1}\right)$ — T/F? Justify.
(d) $\cosh^2 x - 1 = \sinh^2 x$ — T/F? Justify.
(e) All hyperbolic functions are invertible without domain restriction — T/F? Justify.
A2 | Fill in the Blanks (Bernoulli DE Theory)
Consider a Bernoulli DE: $\displaystyle\frac{dy}{dx} + P(x),y = Q(x),y^{n}$.
(a) The substitution to linearise is $v = _______$.
(b) After substitution, the linear DE in $v$ is $_______ = _______$.
(c) The integrating factor is $\mu(x) = _______$.
A3 | State the Equation (Locus from 3 Items)
Given: $|z| > 1$, $|z - 3| \leq 4$, and $\text{Re}(z) < \text{Im}(\bar{z})$.
(a) Write the Cartesian equation for each boundary.
(b) State what geometric shape each inequality represents.
A4 | Correct the Equation
Each of the following is wrong. Correct it.
(a) $\displaystyle\frac{d}{dx}[\cos^{-1} x] = \frac{1}{\sqrt{1-x^2}}$
(b) $\displaystyle\frac{d}{dx}[\cosh x] = -\sinh x$
(c) $\tan^{-1}(-x) = \tan^{-1} x$
(d) $\displaystyle\int \frac{dx}{\sqrt{x^2 - a^2}} = \sinh^{-1}\left(\frac{x}{a}\right) + C$
A5 | Pascal's Triangle & Expansion
(a) Write Pascal's triangle up to row 5.
(b) Use Pascal's triangle to expand $(\cos x + i\sin x)^5$ and hence express $\cos 5x$ in terms of powers of $\cos x$.
Part B — Complex Numbers (20 min)
B1 | Cartesian ↔ Polar ↔ Exponential
Convert to the required form:
(a) $z = -2\sqrt{3} - 2i$. Find $|z|$, $\arg(z)$, and express in polar form $r(\cos\theta + i\sin\theta)$ then exponential form $re^{i\theta}$.
(b) $z = 4e^{-\frac{5\pi}{6}i}$. Find $\text{Re}(z)$, $\text{Im}(z)$, and express in Cartesian form $a + bi$.
(c) $z = e^{3 - \frac{\pi}{4}i}$. Find $|z|$ and $\arg(z)$.
B2 | De Moivre — Powers & Roots
Given $z = -1 + i$:
(a) Express $z$ in exponential form.
(b) Use De Moivre to find $z^6$ in Cartesian form.
(c) Find all four 4th roots of $z$. Express each in exponential form $re^{i\theta}$.
B3 | Locus — Find Cartesian Equation
(a) Given $|2z + 4 - 6i| = |2z - 2 + 2i|$, find the Cartesian equation.
(b) Given the inequalities $|z| \leq 2$, $|z - 2| < |z + 2|$, and $\arg(z) \geq \frac{\pi}{4}$, find the Cartesian equations of all boundaries involved.
B4 | Impedance — Application of Complex Numbers
An AC circuit has a resistor $R = 8,\Omega$, inductor $L = 3,\text{H}$, capacitor $C = 0.02,\text{F}$, and voltage source $V = 100,\text{V}$ at angular frequency $\omega = 5,\text{rad/s}$.
The impedance is: $Z = R + i\left(\omega L - \frac{1}{\omega C}\right)$
(a) Compute $\omega L$ and $\frac{1}{\omega C}$, hence find $Z$ in Cartesian form.
(b) Express $Z$ in polar form $re^{i\theta}$.
(c) If $V = IZ$ (Ohm's law in AC), find the current $I$ in polar form.
B5 | Complex Logarithm
Given $z = -5 + 5i$:
(a) Find $|z|$ and $\arg(z)$ (principal argument).
(b) Find the general complex logarithm $\ln(z)$.
(c) Find the principal complex logarithm $\text{Ln}(z)$.
Part C — Inverse Trigonometry (12 min)
C1 | Principal Values
Find the principal value:
(a) $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
(b) $\cos^{-1}\left(\frac{1}{2}\right)$
(c) $\tan^{-1}\left(-\sqrt{3}\right)$
C2 | Composition Simplification
Simplify:
(a) $\sin\left(\cos^{-1}\frac{3}{5}\right)$
(b) $\tan\left(\sin^{-1}\frac{5}{13}\right)$
(c) $\sec\left[\cos^{-1}\left(-\frac{1}{2}\right)\right]$
C3 | Tan Inverse Sum Formula
Prove that: $$\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) = \tan^{-1}\left(\frac{1}{2}\right)$$
C4 | Derivatives of Inverse Trig
Differentiate:
(a) $f(x) = \sin^{-1}(2x)$
(b) $f(x) = \tan^{-1}(e^{x})$
(c) $f(x) = x^2 \cos^{-1}(3x)$ — use product rule
Part D — Hyperbolic Functions (12 min)
D1 | Exponential Form & Identities
(a) Express $\sinh(2x)$ in exponential form and simplify.
(b) Using definitions of $\sinh x$ and $\cosh x$, prove $\cosh^2 x - \sinh^2 x = 1$.
(c) Find all possible values of $\cosh x$ satisfying $\sinh^2 x - 3\cosh x + 3 = 0$. [Hint: $\cosh^2 x - \sinh^2 x = 1$]
D2 | Derivatives of Hyperbolic Functions
Differentiate:
(a) $\frac{d}{dx}\left[\sinh(5x)\right]$
(b) $\frac{d}{dx}\left[\ln(\cosh x)\right]$
(c) $\frac{d}{dx}\left[\tanh^3 x\right]$
D3 | Integrals of Hyperbolic Functions
Evaluate:
(a) $\int \sinh(4x) , dx$
(b) $\int \cosh^3 x , \sinh x , dx$
(c) $\int \tanh x , dx$
Part E — Inverse Hyperbolic Functions (12 min)
E1 | Logarithmic Form
Write in logarithmic form:
(a) $\sinh^{-1}\left(\frac{3}{4}\right)$
(b) $\cosh^{-1}(5)$
(c) $\tanh^{-1}\left(\frac{1}{3}\right)$
E2 | Derivatives of Inverse Hyperbolic
Differentiate:
(a) $\frac{d}{dx}\left[\sinh^{-1}\left(\frac{x}{2}\right)\right]$
(b) $\frac{d}{dx}\left[\tanh^{-1}(3x)\right]$ — state the domain restriction
(c) $\frac{d}{dx}\left[\cosh^{-1}(e^x)\right]$ — state the domain restriction
E3 | Integrals Leading to Inverse Hyperbolic
Evaluate:
(a) $\int \frac{dx}{\sqrt{25 + x^2}}$
(b) $\int \frac{dx}{\sqrt{x^2 - 9}}$ for $x > 3$
(c) $\int \frac{dx}{16 - x^2}$ for $|x| < 4$
Part F — Differential Equations (22 min)
F1 | Bernoulli DE
Solve the Bernoulli differential equation: $$2x\frac{dy}{dx} + y = \frac{2x}{y}\tan^{-1}x$$
Steps: (a) Identify $n$ and state the substitution $v = y^{1-n}$. (b) Transform to a linear DE in $v$. (c) Find the integrating factor $\mu(x)$. (d) Solve for $v$, then back-substitute to show: $$y^2 = \frac{(x^2 + 1)\tan^{-1}x + C}{x} - 1$$
F2 | Non-Homogeneous DE — Linearly Independent
Solve: $$(y - x - 2),dx + (4y + x - 3),dy = 0$$
Steps: (a) Check linear independence: compute $a_1b_2 - a_2b_1$. (b) Solve the system $a_1h + b_1k + c_1 = 0$ and $a_2h + b_2k + c_2 = 0$ to find $(h, k)$. (c) Substitute $x = u + h$, $y = v + k$ to get a homogeneous DE. (d) Solve using $v = mu$, then back-substitute.
F3 | Non-Homogeneous DE — Linearly Dependent
Solve: $$(x + y),dx + (3x + 3y - 4),dy = 0$$
Steps: (a) Show linear dependence ($a_1b_2 - a_2b_1 = 0$). (b) Substitute $u = x + y$. (c) Reduce to separable equation and solve.
F4 | Exact DE
Show that the following DE is exact and find its general solution: $$(4x^3 - 8xy^2 - y\sin xy),dx + (16y^3 - 8x^2y - x\sin xy),dy = 0$$
Steps: (a) Compute $\frac{\partial M}{\partial y}$ and $\frac{\partial N}{\partial x}$ — verify they are equal. (b) Integrate $M$ w.r.t. $x$ to get $F(x,y)$. (c) Integrate $N$ w.r.t. $y$ and compare with $F(x,y)$. (d) Write final solution $F(x,y) = C$.
F5 | Write 2 Ways to Solve a DE
Consider the Bernoulli DE: $\displaystyle x\frac{dy}{dx} - y = \frac{y^2}{x}$.
(a) Way 1: Identify $n$, state the substitution $v = y^{1-n}$, transform to linear form, find the integrating factor, and write the general solution.
(b) Way 2: Without stating the substitution explicitly, explain how else you could solve this DE (hint: what form can it be rearranged into?).
F6 | Application — Mixing Problem
A tank with capacity $222,$m$^3$ contains $22,$m$^3$ of distilled water with $20,$kg of sugar dissolved. Sugar solution at concentration $2,$kg/m$^3$ is pumped in at $5,$m$^3$/min. The mixture is pumped out at $3,$m$^3$/min. Assume uniform mixing.
(a) Write the differential equation for $A(t)$, the mass of sugar (kg) at time $t$ (min).
(b) Find the volume $V(t)$ as a function of time.
(c) Write the net rate equation $\frac{dA}{dt} = \text{rate in} - \text{rate out}$.
(d) Solve the DE to show: $$A = 4t + 44 - \frac{24\sqrt{22^3}}{(2t + 22)^{3/2}}$$
Quick Reference — Derivatives to Use
| Function | Derivative |
|---|---|
| $\sin^{-1} x$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\cos^{-1} x$ | $\frac{-1}{\sqrt{1-x^2}}$ |
| $\tan^{-1} x$ | $\frac{1}{1+x^2}$ |
| $\sinh x$ | $\cosh x$ |
| $\cosh x$ | $\sinh x$ |
| $\tanh x$ | $\text{sech}^2 x$ |
| $\sinh^{-1} x$ | $\frac{1}{\sqrt{1+x^2}}$ |
| $\cosh^{-1} x$ | $\frac{1}{\sqrt{x^2-1}}$ |
| $\tanh^{-1} x$ | $\frac{1}{1-x^2}$ |
| $\text{sech}^{-1} x$ | $\frac{-1}{x\sqrt{1-x^2}}$ |
Built from FAC1004 lecture notes, UAS 22-23, 23-24, 24-25 papers, and Exam Leaks 2025-2026. Pure mechanical computation — no theory questions. 🚀