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 10 of 22

1. Q9 · 2020 QCAA · Paper 2 · 1 mark
   Two objects, P and Q, move in three-dimensional space such that their positions, $\mathbf r$, over time, $t$, are described by the following vectors until they collide.

   $\mathbf r_P=(t^2-4t)\hat{\mathbf i}+(2t^2-t+3)\hat{\mathbf j}-(6-5t)\hat{\mathbf k}$

   $\mathbf r_Q=(-t^2+2t)\hat{\mathbf i}+(3t+t^2)\hat{\mathbf j}+t^2\hat{\mathbf k}$

   The objects will col…
   Vectors and matrices
2. Q10 · 2020 QCAA · Paper 2 · 1 mark
   The time taken by the Year 7 students at a particular school to complete a standardised test is known to be normally distributed. A researcher claims that the population mean is $8.2$ minutes.

   The mean time taken to complete this test by a sample of $10$ of these students is $8.1$ minutes with a standard deviation of $1.2$ minutes.

   The $95\%$ confidence in…
   Statistical inference
3. Q11 · 2020 QCAA · Paper 2 · 5 marks
   Teams A, B, C, D and E participated in a competition with the following results:

   • A defeated D.
   • B defeated A, C and E.
   • C defeated A and E.
   • D defeated B, C and E.
   • E defeated A.

   To rank the teams at the end of the competition, the organisers constructed a dominance matrix, $N$, that is partially completed.
   Vectors and matrices
4. Q12 · 2020 QCAA · Paper 2 · 9 marks
   For a certain experiment, the number of yeast cells, $N$, after $t$ hours in a test tube can be modelled by the differential equation $$\frac{dN}{dt}=\frac1{1000}N(1000-N),\qquad t\ge0.$$
   Rates of change and differential equations
5. Q13 · 2020 QCAA · Paper 2 · 6 marks
   Data records show that the speeds of cars at a particular location on a highway are normally distributed with a mean of 98.7 km h$^{-1}$ and a standard deviation of 4.1 km h$^{-1}$. The speed limit at this location is 100 km h$^{-1}$. A police officer plans to randomly select a sample of 20 cars.
   Statistical inference
6. Q14 · 2020 QCAA · Paper 2 · 5 marks
   The time, $t$ (months), that it takes before a phone owner cracks the screen on their phone can be modelled by an exponentially distributed random variable with probability density function $$f(t)=\begin{cases}0.16e^{-0.16t},&t\ge0,\\0,&\text{otherwise.}\end{cases}$$
   Statistical inference
7. Q15 · 2020 QCAA · Paper 2 · 4 marks
   The position vectors of points P and Q are $2\hat{\mathbf i}-3\hat{\mathbf j}+\hat{\mathbf k}$ and $2\hat{\mathbf i}+2\hat{\mathbf j}-4\hat{\mathbf k}$ respectively. Let O be the origin.
   Vectors and matrices
8. Q16 · 2020 QCAA · Paper 2 · 6 marks
   Consider the identity $$\cos(4\theta)=A\cos^4(\theta)+B\sin^2(\theta)+C,$$ where $A,B,C\in\mathbb Z$.
   Question and worked solution
9. Q17 · 2020 QCAA · Paper 2 · 7 marks
   An object is released from rest at a height of 100 m above the ground. The motion of the vertical descent of the object is modelled by $$v\frac{dv}{dx}=9.8-0.004v^2,\qquad v\ge0,$$ where $v$ is the velocity (m s$^{-1}$) and $x$ is the displacement from the ground (m). Determine the velocity of the object when it strikes the ground.
   Rates of change and differential equations
10. Q18 · 2020 QCAA · Paper 2 · 6 marks
    The mass of a certain species of kangaroo is known to be normally distributed with a mean mass of $\mu$ kg and standard deviation of $\sigma$ kg. When one kangaroo is randomly selected, $P(X>83.2)=0.145$. When a sample of 12 kangaroos is randomly selected, $P(\bar X<74.1)=0.079$. A 90% approximate confidence interval for $\mu$ is calculated using a random sa…
    Statistical inference
11. Q19 · 2020 QCAA · Paper 2 · 7 marks
    An object is swinging at the end of a 0.5 m length of string in a vertical circular path with a constant angular speed, completing each revolution in 0.24 seconds. The object is projected from a height of 0.3 m above the ground in a vertical plane and just passes over a narrow pole as shown in the diagram. The pole is 2.05 m high and its base is 14 m horizon…
    Vectors and matrices
12. Q1 · Original practice · 1 mark
    Original Specialist Mathematics practice question 1: integration
    Integration
13. Q2 · Original practice · 1 mark
    Original Specialist Mathematics practice question 2: roots of complex numbers
    Complex numbers
14. Q3 · Original practice · 1 mark
    Original Specialist Mathematics practice question 3: line and plane
    Vectors in three dimensions
15. Q4 · Original practice · 1 mark
    Original Specialist Mathematics practice question 4: induction and divisibility
    Proof and induction
16. Q5 · Original practice · 1 mark
    Original Specialist Mathematics practice question 5: confidence interval sample size
    Statistical inference
17. Q6 · Original practice · 4 marks
    Original Specialist Mathematics practice question 6: partial fractions
    Integration
18. Q7 · Original practice · 5 marks
    Original Specialist Mathematics practice question 7: complex roots and factorisation
    Complex numbers
19. Q8 · Original practice · 5 marks
    Original Specialist Mathematics practice question 8: line-plane geometry
    Vectors in three dimensions
20. Q9 · Original practice · 6 marks
    Original Specialist Mathematics practice question 9: motion in a plane
    Vector calculus
21. Q10 · Original practice · 6 marks
    Original Specialist Mathematics practice question 10: transformed confidence interval
    Statistical inference
22. Q11 · Original practice · 1 mark
    Let $z_1=2\operatorname{cis}(\pi/3)$ and $z_2=3\operatorname{cis}(-\pi/6)$. The product $z_1z_2$ is
    Complex numbers
23. Q12 · Original practice · 6 marks
    Let $z=-3+3\sqrt3\,i$.
    Complex numbers
24. Q13 · Original practice · 1 mark
    The complex number $z=\dfrac{1-i\sqrt3}{1+i}$ can be written in polar form as
    Complex numbers