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 7 of 22

1. Q13 · 2022 QCAA · Paper 2 · 5 marks
   An article claims that the mean starting salary of graduates in Australia is currently $\char36 64\,800$ with a standard deviation of $\char36 4500$.

   To check the validity of this claim, an employment agent intends to collect data on the starting salaries of a random sample of 360 graduates.
   Statistical inference
2. Q14 · 2022 QCAA · Paper 2 · 5 marks
   An object is moving in a straight line with an acceleration represented by the differential equation $$\frac{dv}{dt}=-(4+v^2),$$ where $v$ is the object’s velocity (m s$^{-1}$) over time $t$ (s), where $t\ge0$, until it comes to rest.
   Rates of change and differential equations
3. Q15 · 2022 QCAA · Paper 2 · 5 marks
   Consider points A $(3,-1,3)$ and B $(1,1,6)$.
   Vectors and matrices
4. Q16 · 2022 QCAA · Paper 2 · 6 marks
   An object with a mass of 12 kg lies on a frictionless inclined plane. A rope is attached to the object at an angle of $25^\circ$ above the plane, as shown. The force of the rope, $T$ N, prevents the object from moving. When the rope is detached, the object moves down the plane with an acceleration of $5.6$ m s$^{-2}$. Determine the magnitude of $T$.
   Vectors and matrices
5. Q17 · 2022 QCAA · Paper 2 · 6 marks
   The mass of a population of elephants is known to be normally distributed. A biologist randomly selects a number of elephants from this population and measures their masses. The mean mass of the sample is 5206 kg with a standard deviation of 356 kg. The biologist uses the data to calculate a 90% confidence interval for the population mean mass of…
   Statistical inference
6. Q18 · 2022 QCAA · Paper 2 · 5 marks
   Consider the polynomials $P(z)=z^3+(i-a)z^2-2biz+3i$ and $Q(z)=z-2i$, where $a,b\in\mathbb R$. Given $P(z)/Q(z)$ has a remainder of $a-bi$, evaluate the reasonableness that $(z-(a-bi))$ is a factor of $P(z)$.
   Question and worked solution
7. Q19 · 2022 QCAA · Paper 2 · 7 marks
   A research organisation plans to use a drone to drop a scientific instrument vertically from a stationary position above the ocean surface. The acceleration (m s$^{-2}$) of the falling instrument can be modelled by $9.8-0.1v$, where $v$ is its velocity (m s$^{-1}$). In order for the instrument sensors to activate, its speed as it hits the ocean surface must…
   Rates of change and differential equations
8. Q1 · 2021 QCAA · Paper 1 · 1 mark
   Which of the following is a population parameter?
   Statistical inference
9. Q2 · 2021 QCAA · Paper 1 · 1 mark
   When the polynomial $P(z)=z^3-iz^2-z-i$ is divided by $z-i$, the remainder is
   Complex numbers 2
10. Q3 · 2021 QCAA · Paper 1 · 1 mark
    An object has a velocity $\mathbf v(t)=e^{-2t}\hat{\mathbf i}+\left(\frac1t\right)\hat{\mathbf k}$, where $t$ represents time $(t>0)$.

    The displacement $\mathbf r(t)$ of the object could be
    Vectors and matrices
11. Q4 · 2021 QCAA · Paper 1 · 1 mark
    The number of sunflower seeds in each packet produced by a company is known to be normally distributed with a standard deviation of $100$. A worker counts the number of seeds in a random sample of four packets and calculates the sample mean.

    Based on this sampling, the standard deviation of the distribution of the sample mean is
    Statistical inference
12. Q5 · 2021 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.

    $\left[\begin{array}{ccc|c}1&1&-3&4\\0&-1&5&-6\\0&0&1&0\end{array}\right]$

    The geometric interpretation of the solution to this system of equations is best represented by
    Vectors and matrices
13. Q6 · 2021 QCAA · Paper 1 · 1 mark
    The subset of the complex plane that represents $|z|=|z-2|$ for $z\in\mathbb C$ is
    Complex numbers 2
14. Q7 · 2021 QCAA · Paper 1 · 1 mark
    The mass of a particular variety of cake is claimed to be normally distributed with a mean of $660$ grams. A random sample of five of these cakes is found to have a mean mass of $600$ grams.

    Which option represents an approximate confidence interval for $\mu$ based on this sample?
    Statistical inference
15. Q8 · 2021 QCAA · Paper 1 · 1 mark
    Let $P(n)$ be the proposition that

    $\displaystyle \sum_{r=1}^{n}(r+1)3^{r-1}=n\times3^n\quad\forall n\in\mathbb Z^+$.

    Which option represents a correct formulation of the assumption that $P(k)$ is true $\forall k\in\mathbb Z^+$ in a proof using mathematical induction?
    Proof by mathematical induction
16. Q9 · 2021 QCAA · Paper 1 · 1 mark
    The slope field for the differential equation $\frac{dy}{dx}=y-x^2$ is shown.

    The solution curve to the differential equation that passes through the point $(3,-5)$ also passes through point
    Rates of change and differential equations
17. Q10 · 2021 QCAA · Paper 1 · 1 mark
    The 2016 Australian census recorded the number of bedrooms per household. The results are summarised in the histogram, as shown. Based on this data, the mean number of bedrooms per household was calculated to be $3.5$.

    Fifty samples of size $40$ were randomly selected from the census data and the sample means recorded.

    The histogram that most likely repres…
    Statistical inference
18. Q11 · 2021 QCAA · Paper 1 · 5 marks
    Let $f(x)=\tan^{-1}\left(\dfrac{x}{2}\right)$ for suitable values of $x$, where $f(x)\in(-\pi/2,\pi/2)$.
    Question and worked solution
19. Q12 · 2021 QCAA · Paper 1 · 8 marks
    Consider the plane $x-y-2z=15$.
    Vectors and matrices
20. Q13 · 2021 QCAA · Paper 1 · 6 marks
    Use $z=a+bi$ and $w=c+di$, where $a,b,c,d\in\mathbb R$, to prove $$|z-w|^2=|z|^2+|w|^2-2\operatorname{Re}(z\overline w).$$
    Question and worked solution
21. Q14 · 2021 QCAA · Paper 1 · 6 marks
    An object is projected vertically upwards from ground level. After the object has been in motion for $t$ seconds, its position vector through the air, in metres, is modelled by $$\mathbf r(t)=5t(8-t)\hat{\mathbf j}.$$
    Vectors and matrices
22. Q15 · 2021 QCAA · Paper 1 · 4 marks
    Use partial fractions to determine $$\int\frac{4x-17}{x^2-x-6}\,dx,$$ where $x\in\mathbb R$, $x\ne-2$, $x\ne3$. Express your answer in the form $\ln|f(x)|+c$.
    Question and worked solution
23. Q16 · 2021 QCAA · Paper 1 · 6 marks
    Use mathematical induction to prove that $2^{2n}+3n-1$ is divisible by 3 for all $n\in\mathbb Z^+$.
    Proof by mathematical induction
24. Q17 · 2021 QCAA · Paper 1 · 7 marks
    The area between the graphs of the functions $y=4x$ and $y=2x^2$ is rotated about the $y$-axis to form a solid of revolution with a volume of $V$ units$^3$. Determine the exact value of $V$.
    Integration and applications of integration