Specialist Maths questions and solutions

Browse original practice and official past-paper questions. Each question has its own link, with its source and worked solution.

524 questions · Page 6 of 22

Q8 · 2022 QCAA · Paper 1 · 1 mark
Use the substitution $u=\tan(x)$ to determine $\displaystyle \int \tan(x)\sec^2(x)\,dx$ .
Integration and applications of integration
Q9 · 2022 QCAA · Paper 1 · 1 mark
A random variable $X$ is normally distributed with a mean of $36$ and a standard deviation of $4$ .
The respective mean and standard deviation of the distribution of $\bar X$ from repeated random samples of size $9$ are
Statistical inference
Q10 · 2022 QCAA · Paper 1 · 1 mark
A plane is represented by the equation $x-2z=5$ . A vector normal to this plane is
Vectors and matrices
Q11 · 2022 QCAA · Paper 1 · 6 marks
The position vector of a particle, $\mathbf r_1$ (cm), over time, $t$ (s), is given by $\mathbf r_1(t)=(2t+1)\hat{\mathbf i}+(t+3)\hat{\mathbf j}-(2t-3)\hat{\mathbf k}.$
Vectors and matrices
Q12 · 2022 QCAA · Paper 1 · 6 marks
Given $z_1=a+bi$ , $z_2=c+di$ for all $a,b,c,d\in\mathbb R$ and $z_2\ne0$ , prove the identity $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}.$
Question and worked solution
Q13 · 2022 QCAA · Paper 1 · 6 marks
View question and marking material
Question and worked solution
Q14 · 2022 QCAA · Paper 1 · 4 marks
The slope field for the differential equation $\frac{dy}{dx}=\frac{-0.5(y-4)}{x},\qquad x\ne0,$ using $-6\le x\le6$ and $-6\le y\le6$ is shown.
Question and worked solution
Q15 · 2022 QCAA · Paper 1 · 4 marks
Consider the equation $z^3=1$ , where $z\in\mathbb C$ .
Complex numbers 2
Q16 · 2022 QCAA · Paper 1 · 7 marks
Consider this system of equations that corresponds to three planes: $x+5y=1+2z,$ $x+z=3y+3,$ $8y-\lambda=3z.$
Vectors and matrices
Q17 · 2022 QCAA · Paper 1 · 5 marks
The region between the $x$ -axis and the curve of the function $y=1+\sin(2x)$ for $0\le x\le\frac{\pi}{2}$ is rotated about the $x$ -axis to form a solid of revolution. Determine the volume of this solid. Express your answer in simplest form.
Integration and applications of integration
Q18 · 2022 QCAA · Paper 1 · 5 marks
It is proposed that the following expression is divisible by $1+\operatorname{cis}(\theta)$ for $n\in\mathbb Z^+$ , $1+\operatorname{cis}(\theta)\ne0$ . $\sum_{r=0}^{2n+1}\operatorname{cis}(r\theta)$ Evaluate the reasonableness of the proposition.
Proof by mathematical induction
Q19 · 2022 QCAA · Paper 1 · 7 marks
The function $f(x)$ passes through the origin. The gradient function of $f(x)$ is defined as $g(x)=e^x\sin^{-1}(e^x)$ . Determine $f(x)$ .
Question and worked solution
Q1 · 2022 QCAA · Paper 2 · 1 mark
A solution of the equation $z^2=ai$ , where $a\in\mathbb{R}$ , is $z=-2-2i$ . The other solution is
Complex numbers 2
Q2 · 2022 QCAA · Paper 2 · 1 mark
The win/draw/loss results after a netball competition involving five teams is represented in matrix $M$ .
$M=\begin{bmatrix}0&1&2&0&2\\1&0&0&1&1\\0&2&0&0&0\\2&1&2&0&2\\0&1&2&0&0\end{bmatrix}$
Key: Team P drew with Team Q, defeated Team R and Team T, and lost to Team S
The model $M+M^2+M^3$ is used to rank the teams. The final positions from first to fift…
Vectors and matrices
Q3 · 2022 QCAA · Paper 2 · 1 mark
Determine the solution of the differential equation $\dfrac{dy}{dx}=\dfrac{\sin(2x)}{\cos(2x)}$ given $y=0$ when $x=\dfrac{\pi}{5}$ .
Rates of change and differential equations
Q4 · 2022 QCAA · Paper 2 · 1 mark
The time taken for students to answer questions in a class is assumed to be a random variable $X$ with an exponential distribution that has the probability density function $f(x)=\begin{cases}\lambda e^{-\lambda x},&x\ge0\\0,&\text{otherwise}\end{cases}$ . The mean of $X$ is $\dfrac1\lambda$ . The mean length of time taken for students to answer questions in t…
Statistical inference
Q5 · 2022 QCAA · Paper 2 · 1 mark
A random sample of the petrol price per litre at 50 petrol stations produced a sample mean of $\char36 1.52$ and a standard deviation of $\char36 0.14$ . Based on this sample and using a $z$ -value of 1.5, an approximate confidence interval for $\mu$ is
Statistical inference
Q6 · 2022 QCAA · Paper 2 · 1 mark
A 4 kg object moves in a straight line over time, $t$ (s), where $0\le t\le5$ with velocity $v=9+8t-t^2$ $(\mathrm{m\ s^{-1}})$ . Determine the momentum of the object when $t=3$ .
Vectors and matrices
Q7 · 2022 QCAA · Paper 2 · 1 mark
Given $\mathbf a=(3n+2)\hat{\mathbf i}+2\hat{\mathbf j}$ , $\mathbf b=(n-2)\hat{\mathbf j}$ and $\mathbf a\times\mathbf b=(1-2n)\hat{\mathbf k}$ , the possible values of $n$ are
Vectors and matrices
Q8 · 2022 QCAA · Paper 2 · 1 mark
Determine the gradient of the tangent to the curve $y^2-3x=5$ at the point $(1,2\sqrt2)$ .
Rates of change and differential equations
Q9 · 2022 QCAA · Paper 2 · 1 mark
Consider the matrix equation $X\begin{bmatrix}0&0&1\\0&1&1\\1&1&1\end{bmatrix}=\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}$ . Matrix $X$ is
Vectors and matrices
Q10 · 2022 QCAA · Paper 2 · 1 mark
In a town, the mean number of residents per household is 3.79 people with a standard deviation of 1.47 people. Using a random sample of 45 households from the town, determine the probability that the mean number of residents per household will be more than 4.
Statistical inference
Q11 · 2022 QCAA · Paper 2 · 6 marks
An aerial view of the surface of a dam, 6 km in length, is symmetrically positioned on a Cartesian plane as shown. A dam wall is located along the $y$ -axis.
The surrounding edge of the dam can be modelled by the ellipse $\frac{(x-2)^2}{16}+\frac{y^2}{9}=1,\qquad 0\le x\le6.$
Question and worked solution
Q12 · 2022 QCAA · Paper 2 · 5 marks
A scientist collects data for a species of tree frog in a protected area. Details for the female tree frog population are shown below.
Age (years) 0–1 1–2 2–3 3–4
Population in Year 1 150 101 84 62
Birth (breeding) rate 0.4 0.7 0.5 0.1
Survival rate 0.6 0.3 0.2 0
The scientist uses a…
Vectors and matrices

Age (years)	0–1	1–2	2–3	3–4
Population in Year 1	150	101	84	62
Birth (breeding) rate	0.4	0.7	0.5	0.1
Survival rate	0.6	0.3	0.2	0