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 9 of 22

1. Q4 · 2020 QCAA · Paper 1 · 1 mark
   Consider points A and B as shown.

   The position vector representing the midpoint of $AB$ is
   Vectors and matrices
2. Q5 · 2020 QCAA · Paper 1 · 1 mark
   Determine $\displaystyle \int 4x(3x^2+5)^3\,dx$.
   Integration and applications of integration
3. Q6 · 2020 QCAA · Paper 1 · 1 mark
   Given $z=2-2i$ and $w=-3+i$, calculate $z^2-\overline{w}$.
   Complex numbers 2
4. Q7 · 2020 QCAA · Paper 1 · 1 mark
   The diagram shows a slope field.

   The differential equation represented by the slope field is
   Rates of change and differential equations
5. Q8 · 2020 QCAA · Paper 1 · 1 mark
   An equation of the line passing through the points $A(2,4,5)$ and $B(3,-2,1)$ is
   Vectors and matrices
6. Q9 · 2020 QCAA · Paper 1 · 1 mark
   The scores on a test are assumed to be normally distributed.

   Researchers use the results from a random sample of scores to calculate a confidence interval for the population mean. However, a shorter confidence interval width is required so the researchers decide to use a second sample for their calculations.

   Assuming that the standard deviations for both s…
   Statistical inference
7. Q10 · 2020 QCAA · Paper 1 · 1 mark
   The subset of the complex plane that represents $\arg(z+i-1)+\frac{\pi}{4}=0$ for $z\in\mathbb C$ is
   Complex numbers 2
8. Q11 · 2020 QCAA · Paper 1 · 7 marks
   The vertices of a regular hexagon are positioned on the circumference of a unit circle as shown on the Argand plane.

   Consider the complex number $w$, as shown on the plane.
   Complex numbers 2
9. Q12 · 2020 QCAA · Paper 1 · 8 marks
   Consider the vertices A, B and C of the rectangular prism as shown.
   Vectors and matrices
10. Q13 · 2020 QCAA · Paper 1 · 4 marks
    For an exponentially distributed random variable $X$ with parameter $\lambda>0$, $$E(X)=\int_0^\infty x\lambda e^{-\lambda x}\,dx.$$ Use integration by parts to determine $E(X)$. Express your answer in simplest form.
    Integration and applications of integration
11. Q14 · 2020 QCAA · Paper 1 · 4 marks
    The motion of an object that moves in a straight line is given by $v(x)=\cos^{-1}(2x)$, where $v$ is the velocity (m s$^{-1}$) and $x$ is the displacement (m) from the origin.
    Rates of change and differential equations
12. Q15 · 2020 QCAA · Paper 1 · 6 marks
    The points O $(0,0,0)$, A $(-6,2,-2)$ and C $(3,1,2)$ are represented in three-dimensional space in the diagram.

    OABC forms a parallelogram in three-dimensional space.
    Question and worked solution
13. Q16 · 2020 QCAA · Paper 1 · 6 marks
    Given $\cos(\theta)\ne0$ for all $n\in\mathbb{Z}^+$, use mathematical induction to prove $$\cos\theta-\cos3\theta+\cos5\theta-\cdots+(-1)^{n+1}\cos((2n-1)\theta)=\frac{1-(-1)^n\cos(2n\theta)}{2\cos\theta}.$$
    Proof by mathematical induction
14. Q17 · 2020 QCAA · Paper 1 · 7 marks
    Determine the smallest positive value of $a$ given $$\int_{-a}^{a}\left[1+\left(\frac{\sec(2x)+\tan(2x)}{\csc(2x)+1}\right)^2\right]dx=1.$$
    Question and worked solution
15. Q18 · 2020 QCAA · Paper 1 · 6 marks
    Consider the function $P(z)=2z^4+az^3+6z^2+az+b$, where $a,b\in\mathbb{Z}^+$. One of the roots of $P(z)$ is $z=-i$. Determine the possible value/s for $a$ and $b$ such that all remaining roots of $P(z)$ have an imaginary component.
    Question and worked solution
16. Q19 · 2020 QCAA · Paper 1 · 7 marks
    A circular-based bowl has been positioned symmetrically on a Cartesian plane as shown in the diagram. The bowl has a shape that can be generated by rotating the curve $y=\dfrac{4}{8-x}-1$ about the $y$-axis for $4\le x\le7.6$ cm. The bowl is being filled with a liquid at the rate of $7\pi$ cm$^3$ s$^{-1}$. Determine the rate at which the depth of liquid is i…
    Vectors and matrices
17. Q1 · 2020 QCAA · Paper 2 · 1 mark
    The position $x$ (m) at time $t$ (s) of a $7$ kg particle moving in a straight line is given by

    $x=3t^3-5t^2+2t-4$ for $0\le t\le10$.

    Determine the time when the particle has a momentum of $620$ kg m s$^{-1}$.
    Vectors and matrices
18. Q2 · 2020 QCAA · Paper 2 · 1 mark
    The Leslie matrix for a certain endangered species is given.

    $\mathbf L=\begin{bmatrix}0.8&2.4&0.3\\0.4&0&0\\0&0.55&0\end{bmatrix}$.

    A group of the species was moved into a secure property at the start of 2018. The initial female population is given.

    $\mathbf N_0=\begin{bmatrix}150\\80\\40\end{bmatrix}$.

    The best estimate of the total female population a…
    Vectors and matrices
19. Q3 · 2020 QCAA · Paper 2 · 1 mark
    The masses of packages of cheese produced by a company are assumed to be normally distributed with a known mean of $\mu$ grams and a standard deviation of $7.37$ grams.

    The packages of cheese are labelled to contain $500$ grams.

    Given there is a $25\%$ probability that the mean mass of $20$ randomly selected packages will be less than the labelled amount,…
    Statistical inference
20. Q4 · 2020 QCAA · Paper 2 · 1 mark
    A particle is moving with simple harmonic motion described by the equation $x=1.32\cos\left(\frac{\pi t}{2}\right)$ where $x$ (m) is the displacement of the particle from a central position over time $t$ (s), $t\ge0$.

    The maximum speed of the particle is
    Rates of change and differential equations
21. Q5 · 2020 QCAA · Paper 2 · 1 mark
    The gradient of the tangent at point A on the curve $y^2=4x$ is $1.36$.

    The $x$-coordinate of A is
    Rates of change and differential equations
22. Q6 · 2020 QCAA · Paper 2 · 1 mark
    Solve the matrix equation for $\mathbf X$.

    $\begin{bmatrix}0&1\\2&3\end{bmatrix}\mathbf X\begin{bmatrix}4&5\\6&7\end{bmatrix}=\begin{bmatrix}8&9\\0&1\end{bmatrix}$.
    Vectors and matrices
23. Q7 · 2020 QCAA · Paper 2 · 1 mark
    The heights of all students at a school were measured. A mean height of $157.0$ cm was calculated from this data.

    A random sample of $35$ students from this school was selected. The mean height of this sample was $159.7$ cm with a standard deviation of $8.7$ cm.

    The smallest confidence level that could be used to produce a confidence interval that contains…
    Statistical inference
24. Q8 · 2020 QCAA · Paper 2 · 1 mark
    Let $u=1+i$ and $v=-12+5i$.

    $\operatorname{Re}(u^5-|v|)$ is
    Complex numbers 2