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 4 of 22

1. Q17 · 2024 QCAA · Paper 2 · 6 marks
   An object moves with a constant speed of $v$ in a circular path. The position vector of the object is given by $$\mathbf r=r\cos(\omega t)\hat{\mathbf i}+r\sin(\omega t)\hat{\mathbf j}$$ where - $r$ is the radius (metres) of the circle - $\omega$ is the angular velocity (radians per second) - $t$ is the time (seconds) of motion for $t\ge0$.

   Use vector calcu…
   Vectors and matrices
2. Q18 · 2024 QCAA · Paper 2 · 6 marks
   A random variable $X$ is normally distributed, with a known mean $\mu$ and standard deviation $\sigma$. In figure 1, the shaded region between 4 and $\mu$ represents 30% of the distribution of $X$.

   Consider the distribution of $\bar X$ based on repeated random sampling of $X$ using a certain sample size. In figure 2, the shaded region between $\mu$ and 6 re…
   Statistical inference
3. Q19 · 2024 QCAA · Paper 2 · 6 marks
   An experiment researching the population changes of a certain species of insect was conducted over a four-week period. The insect has two distinct stages in its two-week lifespan. Each stage is approximately one week in length. A constant proportion of females survive from stage 1 into stage 2. The ratio of the reproduction rate for females in stage 2 to fem…
   Vectors and matrices
4. Q1 · 2023 QCAA · Paper 1 · 1 mark
   The position of a particle is given by $\mathbf r=(t+2)\hat{\mathbf i}+t^2\hat{\mathbf j}$ for $t\ge0$. Determine the corresponding Cartesian equation.
   Vectors and matrices
5. Q2 · 2023 QCAA · Paper 1 · 1 mark
   Consider the proof of the following proposition using mathematical induction. $$\sum_{r=1}^{n} r(r+1)=\frac13n(n+1)(n+2)\quad\forall n\in\mathbb Z^+.$$ An appropriate assumption statement within the proof is
   Proof by mathematical induction
6. Q3 · 2023 QCAA · Paper 1 · 1 mark
   One solution of $z^3-z^2-7z-2=0$ is $z=-2$. Which equation could be used to determine the remaining solutions?
   Complex numbers 2
7. Q4 · 2023 QCAA · Paper 1 · 1 mark
   The age-specific population distribution of a particular species of animal is shown.

   Age (years)	0–1	1–2	2–3	3–4
   Female population	94	82	37	6
   Breeding rate	0	1.3	0.9	0.2
   Survival rate	0.6	0.8	0.4	0

   The Leslie matrix based on this data is
   Vectors and matrices
8. Q5 · 2023 QCAA · Paper 1 · 1 mark
   A confidence interval for a parameter is a range of values within which the
   Statistical inference
9. Q6 · 2023 QCAA · Paper 1 · 1 mark
   The shaded region defined as $\{z:|z+2-i|\le5\}\cap\{z:\operatorname{Re}(z)<1\},\ z\in\mathbb C$ is best represented by
   Complex numbers 2
10. Q7 · 2023 QCAA · Paper 1 · 1 mark
    The differential equation for which the solution is a logistic equation of the form $y=\dfrac{a}{b+Ce^{-at}}$ where $a$, $b$ and $C$ are constants is
    Rates of change and differential equations
11. Q8 · 2023 QCAA · Paper 1 · 1 mark
    Point $A$ is the centre of a sphere and point $B$ lies on its surface as shown.

    The equation of the sphere is
    Vectors and matrices
12. Q9 · 2023 QCAA · Paper 1 · 1 mark
    The geometric interpretation of a certain system of three equations with no solution is shown. Given two of the equations are $x+y-z=0.5$ and $x-y-z=0.5$, the third equation could be
    Vectors and matrices
13. Q10 · 2023 QCAA · Paper 1 · 1 mark
    A random variable is drawn from a population with the distribution shown in the histogram. A number of samples of size 10 were randomly selected from this distribution and the sample means, $\bar x$, were recorded. The histogram that most likely represents the distribution of the sample means is
    Statistical inference
14. Q11 · 2023 QCAA · Paper 1 · 5 marks
    Determine the following definite integrals.
    Integration and applications of integration
15. Q12 · 2023 QCAA · Paper 1 · 5 marks
    Given $A=\begin{pmatrix}1&-2\\1&2\end{pmatrix}$, $B=\begin{pmatrix}0&2\\1&3\end{pmatrix}$ and $C=\begin{pmatrix}-1&-1\\0&3\end{pmatrix}$, determine $X$ in the matrix equation $XA-XC=B$.
    Vectors and matrices
16. Q13 · 2023 QCAA · Paper 1 · 5 marks
    Given $z\in\mathbb C$, where $z\ne0$, prove $$\left|\frac{|z|}{z\bar z}\right|=|z^{-1}|.$$
    Question and worked solution
17. Q14 · 2023 QCAA · Paper 1 · 6 marks
    Consider a cube with three edges positioned along the $x$-, $y$- and $z$-axes on the Cartesian plane as shown. Points O, A and B are vertices of the cube.
    Question and worked solution
18. Q15 · 2023 QCAA · Paper 1 · 5 marks
    The sum of a geometric progression with $n$ terms, where the first term is 1 and the common ratio is $r$, is given by $$1+r+r^2+r^3+\cdots+r^{n-1}=\frac{r^n-1}{r-1}\quad(r\ne1).$$ Prove that this rule is true $\forall n\in\mathbb Z^+$ using mathematical induction by completing the steps of the proof as indicated.
    Proof by mathematical induction
19. Q16 · 2023 QCAA · Paper 1 · 5 marks
    A curve is defined by the parametric equations $x=2\tan(\theta)$ and $y=3\sin(2\theta)$, where $0\le\theta<\frac{\pi}{2}$. Given that $\frac{dy}{dx}$ can be expressed in the form $a\cos^4(\theta)+b\cos^2(\theta)$, where $a,b\in\mathbb R$, determine the values of $a$ and $b$.
    Question and worked solution
20. Q17 · 2023 QCAA · Paper 1 · 7 marks
    An object of mass 2 kg is moving with a constant velocity (m s$^{-1}$) of $\mathbf v=3\hat{\mathbf i}+\hat{\mathbf k}$. At an instant, two forces (N), $\mathbf F_1=5t\hat{\mathbf j}-3\hat{\mathbf k}$ and $\mathbf F_2=-t\hat{\mathbf j}+\hat{\mathbf k}$, act simultaneously on the object for $t$ seconds, where $0\le t\le2$. Determine the magnitude of the moment…
    Rates of change and differential equations
21. Q18 · 2023 QCAA · Paper 1 · 6 marks
    A particular solution to the differential equation $$\frac{dy}{dx}=\frac{x}{(x^2+1)\tan y},$$ where $x\ge0$ and $-\frac{\pi}{2}<y\le0$, passes through the origin. Determine this solution in the form $x=f(y)$. Leave your answer in simplified form.
    Rates of change and differential equations
22. Q19 · 2023 QCAA · Paper 1 · 6 marks
    Object A is released from the origin with constant velocity, $\mathbf v_A$, such that its position after $t$ seconds is given by $$\mathbf r_A(t)=2\sqrt3t\hat{\mathbf i}+3t\hat{\mathbf j}+2t\hat{\mathbf k},\quad t\ge0.$$ At a later time, object B is released from point $P(3\sqrt3,6,0)$ and travels towards point $Q(5\sqrt3,8,4)$ with constant velocity,…
    Rates of change and differential equations
23. Q1 · 2023 QCAA · Paper 2 · 1 mark
    The acceleration $(\mathrm{m\,s^{-2}})$ of an object moving with simple harmonic motion is modelled by $a=-2.95x$, where $x$ is its displacement (m) from the origin. Determine the period of the motion in seconds.
    Rates of change and differential equations
24. Q2 · 2023 QCAA · Paper 2 · 1 mark
    The standard deviation for the scores of 1000 students completing an entry test at a certain university is 13. A researcher takes repeated random samples of the test results, with each sample comprising 40 scores, and calculates the mean score for each sample. Determine the standard deviation of the distribution of the sample mean scores.
    Statistical inference