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 2 of 22

1. Q6 · 2025 QCAA · Paper 2 · 1 mark
   Determine the gradient of the tangent to $y^2=4x$ when $y=1$.
   Rates of change and differential equations
2. Q7 · 2025 QCAA · Paper 2 · 1 mark
   Given the complex number $z=\operatorname{cis}\!\left(-\frac{\pi}{2}\right)$, determine $\operatorname{Arg}(z^6)$.
   Complex numbers 2
3. Q8 · 2025 QCAA · Paper 2 · 1 mark
   Consider the plane that contains both the $x$-axis and the $y$-axis. A sphere centred at $(3,4,6)$ touches this plane.

   The length of the radius of the sphere is
   Vectors and matrices
4. Q9 · 2025 QCAA · Paper 2 · 1 mark
   The masses (grams) of a random sample of $40$ chocolate muffins produced by a local bakery are recorded.

   Using the sample standard deviation of $0.845$ grams, an approximate confidence interval for the population mean mass of chocolate muffins produced by this bakery is $(149.68,150.12)$ grams.

   The $z$-value used in this calculation is
   Statistical inference
5. Q10 · 2025 QCAA · Paper 2 · 1 mark
   Consider the solid of revolution formed by rotating a section of the curve $y=\sin(x)$ around the $x$-axis.

   Determine the volume of the solid of revolution.
   Integration and applications of integration
6. Q11 · 2025 QCAA · Paper 2 · 4 marks
   The mass of checked bags that passengers take on a Brisbane–Sydney flight is normally distributed with a mean of 21.3 kg and a standard deviation of 4.2 kg.

   A random sample of 16 checked bags was conducted.
   Statistical inference
7. Q12 · 2025 QCAA · Paper 2 · 7 marks
   A 10 kg object is travelling at ground level with a constant velocity.

   At an instant, two forces of 60 N and 42 N act simultaneously on the object parallel to the ground in the directions shown.

   Let unit vectors in the east and north directions be $\hat{\mathbf i}$ and $\hat{\mathbf j}$ respectively.
   Rates of change and differential equations
8. Q13 · 2025 QCAA · Paper 2 · 7 marks
   The sketch shows sections of the functions $$f(x)=-0.5x^2+7.5x-18$$ and $$g(x)=4\operatorname{cosec}\left(\frac{\pi x}{24}\right)-4.$$ Two points of intersection at $(12,0)$ and point $A$ are shown.
   Question and worked solution
9. Q14 · 2025 QCAA · Paper 2 · 8 marks
   The origin, $O$, is joined to points $A(1,2,5)$ and $B(-3,4,0)$ to form triangle $OAB$. Point $C$ is the point on $OB$ such that $AC$ is perpendicular to $OB$, as shown.
   Vectors and matrices
10. Q15 · 2025 QCAA · Paper 2 · 6 marks
    De Moivre’s theorem can be expressed as $$\bigl(r(\cos\theta+i\sin\theta)\bigr)^n=r^n\bigl(\cos(n\theta)+i\sin(n\theta)\bigr)\qquad\forall n\in\mathbb Z^+.$$ Prove De Moivre’s theorem using mathematical induction.
    Proof by mathematical induction
11. Q16 · 2025 QCAA · Paper 2 · 6 marks
    A certain population can be approximately modelled by the differential equation $$\frac{dP}{dt}=0.5P(1-0.2P)$$ where $P$ is the population in millions and $t$ is the number of years since 1 January 2025. Given that the population on 1 January 2025 was estimated at 0.3 million, use a calculus approach to estimate the population on 1 January 2030.
    Rates of change and differential equations
12. Q17 · 2025 QCAA · Paper 2 · 5 marks
    A variable, $X$, is assumed to be normally distributed with $\mu=24.311$ and $\sigma=5.102$. Two 90% confidence intervals for $\mu$ were calculated from two different random samples from $X$, with the second sample being smaller than the first sample by 60. Both confidence intervals were calculated using the population standard deviation rather than their re…
    Statistical inference
13. Q18 · 2025 QCAA · Paper 2 · 7 marks
    Polar curves are defined by points that are a variable distance of $r$ units from the origin and dependent on the angle $\theta$ (in radians) measured from the positive $x$-axis. Consider the polar curve $r=1+\cos(\theta)$. A table of four polar coordinates on this curve is shown.

    | $\theta$ | $r$ | | --- | --- | | $0$ | $2$ | | $\pi/6$ | $1+\sqrt3/2$ | |…
    Integration and applications of integration
14. Q1 · 2024 QCAA · Paper 1 · 1 mark
    Repeated random samples will be used to calculate a large number of 90% confidence intervals for a population mean $\mu$. Which statement best describes the possible outcomes?
    Statistical inference
15. Q2 · 2024 QCAA · Paper 1 · 1 mark
    Given that $\dfrac{A}{x-2}+\dfrac3x=\dfrac{x-6}{x(x-2)}$, determine the value of $A$.
    Integration and applications of integration
16. Q3 · 2024 QCAA · Paper 1 · 1 mark
    Consider a proof of the proposition $\displaystyle\sum_{j=1}^{n}(2j-1)=n^2\ \forall n\in\mathbb Z^+$ using mathematical induction. Within the proof of the inductive step, the proposition for $n=k+1$ could be expressed as
    Proof by mathematical induction
17. Q4 · 2024 QCAA · Paper 1 · 1 mark
    A plane contains the point $(1,3,1)$ and is normal to the vector $\hat{\mathbf i}+\hat{\mathbf j}+2\hat{\mathbf k}$. The vector equation of the plane is
    Vectors and matrices
18. Q5 · 2024 QCAA · Paper 1 · 1 mark
    The augmented matrix shown is produced when a Gaussian elimination technique is used to solve a certain system of equations with three variables: $\left[\begin{array}{ccc|c}1&4&2&-10\\0&2&0&5\\0&0&3&2\end{array}\right]$. Given that row 1 values of the matrix represent $x+4y+2z=-10$, the unique solution for $y$ is
    Vectors and matrices
19. Q6 · 2024 QCAA · Paper 1 · 1 mark
    Players P, Q, R and S played each other once in a competition where there were no draws.

    Only the following results are known. • Player P defeated players Q and R. • Player Q defeated two players. • Players R and S each defeated one player.

    Based on these results, a dominance matrix $N$ was partially constructed as shown.

    | | P | Q | R | S | | --- | --…
    Vectors and matrices
20. Q7 · 2024 QCAA · Paper 1 · 1 mark
    $A$, $B$ and $C$ are points in three-dimensional space. If $2\overrightarrow{AB}=\overrightarrow{BC}$, then
    Vectors and matrices
21. Q8 · 2024 QCAA · Paper 1 · 1 mark
    Given $z=2\operatorname{cis}\left(\dfrac{\pi}{3}\right)$, determine $z^3$.
    Complex numbers 2
22. Q9 · 2024 QCAA · Paper 1 · 1 mark
    Use a suitable double-angle identity to determine $\displaystyle\int 2\sin^2(x)\,dx$.
    Integration and applications of integration
23. Q10 · 2024 QCAA · Paper 1 · 1 mark
    The polynomial $P(z)=z^3-2iz^2+z-2i$ can be expressed in factorised form as $P(z)=(z-i)(z^2+biz+2)$, where $b\in\mathbb Z$. Determine the value of $b$.
    Complex numbers 2
24. Q11 · 2024 QCAA · Paper 1 · 4 marks
    The vector equation of a straight line is given by $$\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}2\\0\end{pmatrix}+k\begin{pmatrix}-1\\2\end{pmatrix},$$ where $k$ is a scalar.
    Vectors and matrices