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 1 of 22

1. Q1 · 2025 QCAA · Paper 1 · 1 mark
   Use the substitution $u=x^2$ to express $\displaystyle\int 2xe^{x^2}\,dx$ in terms of $u$.
   Integration and applications of integration
2. Q2 · 2025 QCAA · Paper 1 · 1 mark
   The position vector of a particle at time $t$ is given by $\mathbf r=\sin(t)\hat{\mathbf i}+\cos(t)\hat{\mathbf j}$. The path of the particle is
   Vectors and matrices
3. Q3 · 2025 QCAA · Paper 1 · 1 mark
   The polynomial $P(z)=z^3+pz^2+qz+1$ has complex conjugate roots when
   Complex numbers 2
4. Q4 · 2025 QCAA · Paper 1 · 1 mark
   $X$ is a random variable with mean $\mu$ and standard deviation $\sigma$. From random samples of $X$ values, each of size $n$, the sample mean is calculated. This sampling and calculation is repeated a large number of times. The mean of the distribution of the sample means would be approximately
   Statistical inference
5. Q5 · 2025 QCAA · Paper 1 · 1 mark
   Within the method of proof using mathematical induction, for which sum is the initial statement true?
   Proof by mathematical induction
6. Q6 · 2025 QCAA · Paper 1 · 1 mark
   At time $t$, a particle travels with a velocity of $\mathbf v=\left(\dfrac{2}{1+t^2}\right)\hat{\mathbf i}-2t\hat{\mathbf j}$. Determine a general expression for the position vector, $\mathbf r$, of the particle during this motion.
   Vectors and matrices
7. Q7 · 2025 QCAA · Paper 1 · 1 mark
   For $z\in\mathbb C$, a subset of the complex plane of the form $\operatorname{Arg}(z-z_1)=\theta$ is shown.

   The values of $z_1$ and $\theta$ respectively are
   Complex numbers 2
8. Q8 · 2025 QCAA · Paper 1 · 1 mark
   The position vectors of two objects over time, $t$, where $t\ge0$, are given by $\mathbf r_1(t)=-2\hat{\mathbf i}+t^2\hat{\mathbf j}$ and $\mathbf r_2(t)=at\hat{\mathbf i}+4\hat{\mathbf j}$ (where $a\in\mathbb R$). Given that the two objects collide, the value of $a$ is
   Vectors and matrices
9. Q9 · 2025 QCAA · Paper 1 · 1 mark
   An approximate confidence interval for a population mean is calculated based on a sample size of 16 and is found to have width, $w$. A second confidence interval is calculated from another sample with the same sample standard deviation. Given that the same $z$-value was used for both intervals and the width of the second interval is $2w$, what is the size of…
   Statistical inference
10. Q10 · 2025 QCAA · Paper 1 · 1 mark
    Given $z_1$ and $z_2$ are complex numbers, which statement is always true?
    Complex numbers 2
11. Q11 · 2025 QCAA · Paper 1 · 5 marks
    Consider the system of linear equations represented using the augmented matrix shown. $$\left[\begin{array}{ccc|c}1&-1&-1&-6\\-2&1&1&1\\0&0&4&4\end{array}\right]$$ Key: $R_1$ represents the row 1 values.
    Vectors and matrices
12. Q12 · 2025 QCAA · Paper 1 · 6 marks
    A line, $l$, is given by the equation $$\frac{x+1}{-2}=\frac{y-3}{2}=z-2.$$
    Question and worked solution
13. Q13 · 2025 QCAA · Paper 1 · 5 marks
    The velocity (m s$^{-1}$) of a 3 kg object moving in a straight line is given by $$v=2\cos^{-1}\left(\frac{x}{3}\right),\qquad0\le x<3,$$ where $x$ is its position (m) from the origin.
    Rates of change and differential equations
14. Q14 · 2025 QCAA · Paper 1 · 5 marks
    A factory produces cans of juice. Each can has a labelled volume of 500 mL.

    The factory manager conducted a random sample of 100 cans to assess whether the labelled volume was being met satisfactorily in production. The mean volume of the sample was 498.9 mL with a sample standard deviation of 4.0 mL.
    Statistical inference
15. Q15 · 2025 QCAA · Paper 1 · 5 marks
    Integration by parts and area.
    Integration and applications of integration
16. Q16 · 2025 QCAA · Paper 1 · 5 marks
    A parallelepiped is a three-dimensional figure where all six faces are parallelograms. It can be defined by vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$, as shown. The origin $O$ and points $P$, $Q$, $R$ and $S$ are vertices of the parallelepiped.

    Use vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$ to prove that the diagonal from $P$ to $R$ and the dia…
    Vectors and matrices
17. Q17 · 2025 QCAA · Paper 1 · 7 marks
    The radius of a cylinder decreases at a constant rate of $0.5$ m s$^{-1}$, while maintaining a constant height of four metres. Given that the cylinder has an initial volume of $100\pi$ m$^3$, determine the rate of change of the volume (m$^3$ s$^{-1}$) of the cylinder after four seconds.
    Rates of change and differential equations
18. Q18 · 2025 QCAA · Paper 1 · 6 marks
    An object is projected at an acute angle of $\theta$ below the horizontal, with an initial speed of 30 m s$^{-1}$ from a position 90 m above ground level. The object hits the ground 90 m horizontally from its projection point. Use vector calculus to determine $\theta$ in its simplest form. Assume that the magnitude of mean acceleration due to gravity on Eart…
    Rates of change and differential equations
19. Q19 · 2025 QCAA · Paper 1 · 6 marks
    Let $w\in\mathbb C$ be any fifth root of unity, where $w\notin\mathbb R$. Show that $$w^3(1+w)(1+w^3)\in\mathbb Z^-.$$
    Complex numbers 2
20. Q1 · 2025 QCAA · Paper 2 · 1 mark
    Rounded to the nearest degree, an angle equivalent to the arctangent of $2$ is
    Question and worked solution
21. Q2 · 2025 QCAA · Paper 2 · 1 mark
    Which unit vector product is correct?
    Vectors and matrices
22. Q3 · 2025 QCAA · Paper 2 · 1 mark
    If all roots of the equation $z^3=1$ are plotted on an Argand plane and then joined by straight lines, the shape formed is
    Complex numbers 2
23. Q4 · 2025 QCAA · Paper 2 · 1 mark
    The expression $9\times 2^{n+1}+2^n$, where $n\in\mathbb Z^+$, is divisible by
    Proof by mathematical induction
24. Q5 · 2025 QCAA · Paper 2 · 1 mark
    The height (cm) of people in a certain population is normally distributed with a standard deviation of $7.42$ cm.

    A researcher takes repeated random samples of $15$ people and calculates the mean height for each sample.

    The expected standard deviation (cm) of the distribution of these sample mean heights would be approximately
    Statistical inference