QCEVault

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 8 of 22

  1. Q18 · 2021 QCAA · Paper 1 · 6 marks
    This differential equation can be used to determine the current II (amperes) at time tt (seconds) with voltage VV (volts) in an electric circuit containing a resistance RR (ohms): kdIdt+RI=V,k\frac{dI}{dt}+RI=V, where kk, RR and VV are positive constants and t≥0t\ge0. Assuming that there is no current in the electric circuit initially, show that the size of t…
    Rates of change and differential equations
  2. Q19 · 2021 QCAA · Paper 1 · 7 marks
    The velocity vectors of two objects A and B (in m s−1^{-1}) at time tt (in s) are given respectively by vA=6sin⁡(3t)i^+6cos⁡(3t)j^,\mathbf v_A=6\sin(3t)\hat{\mathbf i}+6\cos(3t)\hat{\mathbf j}, vB=cos⁡(t)i^−sin⁡(t)j^.\mathbf v_B=\cos(t)\hat{\mathbf i}-\sin(t)\hat{\mathbf j}. Objects A and B are initially at (−2,0,2)(-2,0,2) and (0,1,−1)(0,1,-1) respectively. Determine the position of Object A when it is 4 metr…
    Rates of change and differential equations
  3. Q1 · 2021 QCAA · Paper 2 · 1 mark
    The time taken to complete orders at a pizza store is normally distributed with a mean time (μ\mu) of 1010 minutes.
    The owner of the pizza store records the time taken to complete orders for a random sample of 2020 pizzas each day over a 3030-day period. From this data, an approximate 90%90\% confidence interval for μ\mu is calculated at the end of each da…
    Statistical inference
  4. Q2 · 2021 QCAA · Paper 2 · 1 mark
    Determine the area of the shaded region between the graphs of the functions y=13sec⁡(x3)y=\frac13\sec\left(\frac{x}{3}\right) and y=2cos⁡(x2)y=2\cos\left(\frac{x}{2}\right), as shown.
    Integration and applications of integration
  5. Q3 · 2021 QCAA · Paper 2 · 1 mark
    Given n∈Z+n\in\mathbb Z^+, for which proposition can the initial statement for mathematical induction be proven?
    Proof by mathematical induction
  6. Q4 · 2021 QCAA · Paper 2 · 1 mark
    The mean time that visitors spend at an art exhibition is 3939 minutes and the standard deviation is 66 minutes.
    Determine the approximate probability that the mean time spent at the exhibition by a random sample of 3535 visitors is between 3838 and 4040 minutes.
    Statistical inference
  7. Q5 · 2021 QCAA · Paper 2 · 1 mark
    A vector normal to the plane that contains the vectors (130)\begin{pmatrix}1\\3\\0\end{pmatrix} and (102)\begin{pmatrix}1\\0\\2\end{pmatrix} is
    Vectors and matrices
  8. Q6 · 2021 QCAA · Paper 2 · 1 mark
    The Cartesian equation of a sphere is given by x2+y2+z2+2x−2y=7x^2+y^2+z^2+2x-2y=7.
    The centre and radius of the sphere are
    Vectors and matrices
  9. Q7 · 2021 QCAA · Paper 2 · 1 mark
    The altitude angle of OA→\overrightarrow{OA} is represented as φ\varphi.
    Given the coordinates of A are (3,4,6)(3,4,6), the altitude angle of OA→\overrightarrow{OA} in radians is
    Vectors and matrices
  10. Q8 · 2021 QCAA · Paper 2 · 1 mark
    The imaginary part of (cis⁡(π8))−2\left(\operatorname{cis}\left(\frac{\pi}{8}\right)\right)^{-2} is
    Complex numbers 2
  11. Q9 · 2021 QCAA · Paper 2 · 1 mark
    Two vertical forces act on a skydiver with a mass of 8585 kg, as shown.
    When the magnitude of the air resistance is 6262 N, the magnitude of the acceleration of the skydiver is
    Vectors and matrices
  12. Q10 · 2021 QCAA · Paper 2 · 1 mark
    A random variable is normally distributed with a mean, μ\mu. An approximate 95%95\% confidence interval for μ\mu from a sample from this distribution is (209.7,221.9)(209.7,221.9).
    An approximate confidence interval for μ\mu based on the same sample, using a confidence level greater than 95%95\%, could be
    Statistical inference
  13. Q11 · 2021 QCAA · Paper 2 · 4 marks
    OABC is a triangular-based pyramid, as shown. The vertices are A(1,2,5)(1,2,5), B(−1,2,−2)(-1,2,-2) and C(0,5,2)(0,5,2). Use a vector method to determine the area of the shaded face ABC.
    Question and worked solution
  14. Q12 · 2021 QCAA · Paper 2 · 6 marks
    The life span of batteries manufactured by a company is assumed to be normally distributed with an unknown mean and standard deviation.
    A supervisor at the company randomly selects nn batteries and uses the life spans from this sample to calculate an approximate 95% confidence interval for the population mean of (2321.4, 2423.6)(2321.4,\,2423.6) hours.
    Question and worked solution
  15. Q13 · 2021 QCAA · Paper 2 · 6 marks
    The area under the graph of the function f(x)=0.2e−0.2xf(x)=0.2e^{-0.2x} for 1≤x≤91\le x\le9 is shaded.
    Question and worked solution
  16. Q14 · 2021 QCAA · Paper 2 · 5 marks
    The Tasmanian thornbill is a species of bird that has an average life span of three years. Female thornbills do not reproduce in their first year, but produce an average of four female offspring in each of their second and third years. The survival rate of each age group is estimated as 25% in their first year and 30% in their second year.
    A Leslie matrix,…
    Vectors and matrices
  17. Q15 · 2021 QCAA · Paper 2 · 8 marks
    Water is poured into a cone-shaped cup at a rate of 22 cm3^3 s−1^{-1}. The cup has a height of 12 cm and a radius of 6 cm, as shown. As the cup fills, the ratio of the height of the water hh to the surface radius of the water rr remains constant.
    Question and worked solution
  18. Q16 · 2021 QCAA · Paper 2 · 6 marks
    Let A=[122211221].A=\begin{bmatrix}1&2&2\\2&1&1\\2&2&1\end{bmatrix}. Given A4=pA3+qA2+rA+3IA^4=pA^3+qA^2+rA+3I, use matrix algebra to determine the value of the scalars pp, qq and rr.
    Vectors and matrices
  19. Q17 · 2021 QCAA · Paper 2 · 7 marks
    An object with a mass of 2 kg is released from rest at the top of a 1 metre long frictionless plane inclined at 30∘30^\circ to the horizontal. A force of PP newtons acting parallel to the plane opposes the motion of the object as it travels down the plane. When the object is xx metres from the top of the plane, its velocity is vv m s−1^{-1}. Given…
    Vectors and matrices
  20. Q18 · 2021 QCAA · Paper 2 · 6 marks
    Consider the polynomial P(z)=z3+az2+bz+cP(z)=z^3+az^2+bz+c, where a,b,c∈Ra,b,c\in\mathbb R and z∈Cz\in\mathbb C. Two of the roots of P(z)P(z) are also roots of z4+z3+z2+z+1z^4+z^3+z^2+z+1. The remaining root of P(z)P(z) is z=2z=2. Given z5−1=(z−1)(z4+z3+z2+z+1)z^5-1=(z-1)(z^4+z^3+z^2+z+1), determine a possible expression for P(z)P(z). Leave your answer in expanded form.
    Question and worked solution
  21. Q19 · 2021 QCAA · Paper 2 · 7 marks
    Consider the following information for a continuous random variable XX: E(X)=μ=∫−∞∞xp(x) dxE(X)=\mu=\int_{-\infty}^{\infty}xp(x)\,dx and Var⁡(X)=∫−∞∞(x−μ)2p(x) dx\operatorname{Var}(X)=\int_{-\infty}^{\infty}(x-\mu)^2p(x)\,dx. The waiting time (minutes) until workers at a certain call centre receive their nnth phone call, where n∈Z+n\in\mathbb Z^+, is a random variable TT with probability density…
    Statistical inference
  22. Q1 · 2020 QCAA · Paper 1 · 1 mark
    The indefinite integral ∫3x−A1−x2 dx\displaystyle \int \frac{3x-A}{1-x^2}\,dx can be determined using the partial fractions −11+x+21−x\displaystyle -\frac{1}{1+x}+\frac{2}{1-x}.
    The value of AA is
    Integration and applications of integration
  23. Q2 · 2020 QCAA · Paper 1 · 1 mark
    When using proof by mathematical induction to show that n(2n−1)(2n+1)n(2n-1)(2n+1) is divisible by 33 ∀ n∈Z+\forall\,n\in\mathbb{Z}^+, the inductive step requires proving
    Proof by mathematical induction
  24. Q3 · 2020 QCAA · Paper 1 · 1 mark
    According to a recent census, the mean hours worked per week by all Australian workers is 35.635.6 hours.
    A mean of 36.136.1 hours worked per week is calculated from a random selection of 500500 Australian workers.
    Based on this data, which of the following is correct?
    Statistical inference