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 5 of 22

1. Q3 · 2023 QCAA · Paper 2 · 1 mark
   Given that $2i$ is a root of $z^2-pz-q=0$, where $p,q\in\mathbb R$, determine the values of $p$ and $q$.
   Complex numbers 2
2. Q4 · 2023 QCAA · Paper 2 · 1 mark
   The position of a particle can be modelled using $\mathbf r=\cos(t)\hat{\mathbf i}-2\sin(t)\hat{\mathbf j}$, $t\ge0$. Which curve best represents the path of the particle?
   Vectors and matrices
3. Q5 · 2023 QCAA · Paper 2 · 1 mark
   A plane contains the origin and the points $(1,2,3)$ and $(3,2,1)$. A vector normal to the plane is
   Vectors and matrices
4. Q6 · 2023 QCAA · Paper 2 · 1 mark
   Two coplanar forces of magnitudes 12 N and 10 N act on an object in the directions shown.

   Determine the magnitude of the resultant force acting on the object.
   Vectors and matrices
5. Q7 · 2023 QCAA · Paper 2 · 1 mark
   Matrix $N$ represents the results for a competition involving four teams.

   $$N=\begin{bmatrix}0&0&1&1\\1&0&0&0\\0&1&0&0\\0&1&1&0\end{bmatrix}$$

   Key: Team P lost to team Q but won against teams R and S.

   Using the ranking model $N+0.5N^2$, the teams that placed first, second and third respectively are
   Vectors and matrices
6. Q8 · 2023 QCAA · Paper 2 · 1 mark
   Given $f(x)=\tan^{-1}(2x)$, determine $f'(3)$.
   Rates of change and differential equations
7. Q9 · 2023 QCAA · Paper 2 · 1 mark
   The time in minutes between the arrival of customers at a certain shop is assumed to be a random variable $X$ with an exponential distribution that has the probability density function $f(x)=\begin{cases}0.12e^{-0.12x},&x\ge0\\0,&\text{otherwise}\end{cases}$. A customer arrives at the shop. The probability that the next customer arrives within 30 to 60 secon…
   Statistical inference
8. Q10 · 2023 QCAA · Paper 2 · 1 mark
   The Argand diagram that represents the solutions to $z^4=16\operatorname{cis}\left(\frac{2\pi}{3}\right)$, $z\in\mathbb C$, is
   Complex numbers 2
9. Q11 · 2023 QCAA · Paper 2 · 4 marks
   The bounded region between the graphs of the functions $y=-1+\sec\left(\frac{x}{5}\right)$ and $y=0.1x^2$ over a certain domain is shaded as shown. The functions intersect at the origin and point A.
   Integration and applications of integration
10. Q12 · 2023 QCAA · Paper 2 · 7 marks
    Consider the complex number $z=-3+2i$.
    Complex numbers 2
11. Q13 · 2023 QCAA · Paper 2 · 4 marks
    The wait time for customers put on hold when calling complaint departments is assumed to be normally distributed. A company claims that the mean wait time for their customers is 7.6 minutes. The following data represents the wait time (minutes) from a random sample of 12 customers who called the complaint department of this company.

    | 8.3 | 12.7 | 9.1 | 7.3…
    Statistical inference
12. Q14 · 2023 QCAA · Paper 2 · 4 marks
    At a certain location, a biologist measures the width of a river to be 12 m. She also records the depth of the river at regular 2 m interval widths as shown.

    Width (m)	0	2	4	6	8	10	12
    Depth (m)	0.52	2.15	3.70	4.27	3.32	1.28	0.59

    The biologist estimates the cross-sectional area of…
    Question and worked solution
13. Q15 · 2023 QCAA · Paper 2 · 7 marks
    The travel time for students attending a certain university is assumed to be normally distributed, with a population mean of 25.2 minutes and standard deviation of 4.7 minutes. Travel times are collected from a random sample of 120 of these students and used to calculate a sample mean, $\bar X_1$, in minutes.
    Statistical inference
14. Q16 · 2023 QCAA · Paper 2 · 6 marks
    A curve modelled by the relation $xy^2-y+\cos^{-1}(2x)=1$, where $-0.35\le x\le0.27$ and $0\le y\le1$, intersects the $y$-axis at point A. Determine the equation of the tangent to the curve at point A.
    Question and worked solution
15. Q17 · 2023 QCAA · Paper 2 · 6 marks
    An object is projected upwards from ground level with an initial velocity of 15 m s$^{-1}$ at an angle of $54^\circ$ to the horizontal. The object just passes over a drone hovering in the air. An observer is positioned directly below the drone and at a horizontal distance of 20 m from where the object is projected. The observer commented that: • it took the…
    Rates of change and differential equations
16. Q18 · 2023 QCAA · Paper 2 · 5 marks
    Consider the complex solutions to the following equation, where $0<\arg(z)<\pi$. $$(z+1)(z^{14}-z^{13}+z^{12}-z^{11}+\cdots+z^4-z^3+z^2-z)=1-z.$$ Let $w_1$ be the solution with the maximum possible real part and $w_2$ be the solution with the maximum possible imaginary part. Show that $\frac{w_1^4}{w_2}\in\mathbb Z$.
    Question and worked solution
17. Q19 · 2023 QCAA · Paper 2 · 7 marks
    The height of Year 9 students at a school is assumed to be normally distributed with a population mean height of $\mu$ cm. A teacher at the school measured the height of all the students in her Year 9 class. This data was used to calculate an approximate 95% confidence interval for $\mu$ of $(163.7,166.9)$ cm. The teacher repeated the procedure using data fr…
    Statistical inference
18. Q1 · 2022 QCAA · Paper 1 · 1 mark
    Let $z=a+3i$ and $w=-3+bi$, where $a,b\in\mathbb R$.

    If $z=w$, then
    Complex numbers 2
19. Q2 · 2022 QCAA · Paper 1 · 1 mark
    Which statement regarding sample means is true?
    Statistical inference
20. Q3 · 2022 QCAA · Paper 1 · 1 mark
    A particle travels in a straight line over time, $t$, with a constant acceleration, $a(t)$.

    Which function could represent the particle’s displacement, $x(t)$?
    Vectors and matrices
21. Q4 · 2022 QCAA · Paper 1 · 1 mark
    When using proof by mathematical induction to prove De Moivre’s theorem expressed as $(r\operatorname{cis}(\theta))^n=r^n\operatorname{cis}(n\theta)$ $\forall n\in\mathbb Z^+$, which statement would be correct in the proof of the inductive step?
    Proof by mathematical induction
22. Q5 · 2022 QCAA · Paper 1 · 1 mark
    Four random samples of different sizes were taken to estimate a certain population mean, given a known population standard deviation. A $95\%$ confidence interval was calculated for each sample.

    Which sample used the largest sample size?
    Statistical inference
23. Q6 · 2022 QCAA · Paper 1 · 1 mark
    The Cartesian equation for a sphere with centre $(-2,3,-4)$ and radius $9$ is
    Vectors and matrices
24. Q7 · 2022 QCAA · Paper 1 · 1 mark
    Two forces act concurrently on a $2$ kg object placed at the origin.

    The magnitude of the acceleration of the object is
    Vectors and matrices