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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 3 of 22

  1. Q12 · 2024 QCAA · Paper 1 · 7 marks
    Point A lies on a section of the ellipse 3x2+y2=103x^2+y^2=10 as shown. The coordinates of A are (2,y1)(\sqrt2,y_1).
    Integration and applications of integration
  2. Q13 · 2024 QCAA · Paper 1 · 4 marks
    P(z)=az2−iz+1−3iP(z)=az^2-iz+1-3i and Q(z)=z2+3iz+2aQ(z)=z^2+3iz+2a, where a∈Ca\in\mathbb C, have the same remainder when divided by z−iz-i. Use the remainder theorem to determine the value of aa.
    Question and worked solution
  3. Q14 · 2024 QCAA · Paper 1 · 5 marks
    The displacement (cm) of a particle from the origin as it travels in two-dimensional space at time tt for 0≤t<π20\le t<\frac{\pi}{2} seconds is given by r=(2sec⁡(t)−1)i^+tan⁡(t)j^.\mathbf r=(2\sec(t)-1)\hat{\mathbf i}+\tan(t)\hat{\mathbf j}.
    Question and worked solution
  4. Q15 · 2024 QCAA · Paper 1 · 6 marks
    A sketch of a partially completed slope field for the differential equation dydx=1−y\frac{dy}{dx}=1-y is shown.
    Rates of change and differential equations
  5. Q16 · 2024 QCAA · Paper 1 · 6 marks
    Use mathematical induction to prove that 12n+2(5n−1)12^n+2(5^{n-1}) is a multiple of 7 for n∈Z+n\in\mathbb Z^+.
    Proof by mathematical induction
  6. Q17 · 2024 QCAA · Paper 1 · 6 marks
    The acceleration (m s−2^{-2}) of an object that moves in a straight line in an easterly direction over time tt for 0≤t≤π60\le t\le\frac{\pi}{6} seconds is given by a=2(1+v2)a=2(1+v^2), where vv is its velocity (m s−1^{-1}). The object is initially at rest at a position that is ln⁡(2)\ln(\sqrt2) metres west of the origin. A student uses this information to calculate that…
    Rates of change and differential equations
  7. Q18 · 2024 QCAA · Paper 1 · 6 marks
    A random variable XX has a probability density function given by f(x)={ksin⁡−1(x),0≤x≤1,0,otherwise,f(x)=\begin{cases}k\sin^{-1}(x),&0\le x\le1,\\0,&\text{otherwise},\end{cases} where kk is a positive constant. Determine the value of kk.
    Statistical inference
  8. Q19 · 2024 QCAA · Paper 1 · 6 marks
    Consider complex numbers of the form w=x+iw=x+i, where xx is a positive real number. If Re⁡(w7)=0\operatorname{Re}(w^7)=0, determine all possible values of xx.
    Complex numbers 2
  9. Q1 · 2024 QCAA · Paper 2 · 1 mark
    Given that z=−2+3iz=-2+3i is a root of z3+az+b=0z^3+az+b=0, where a,b∈Ra,b\in\mathbb R, another root is
    Complex numbers 2
  10. Q2 · 2024 QCAA · Paper 2 · 1 mark
    Rounded to two decimal places, the zz-value used in the calculation of an approximate 95% confidence interval for μ\mu is
    Statistical inference
  11. Q3 · 2024 QCAA · Paper 2 · 1 mark
    Given a=j^+k^\mathbf a=\hat{\mathbf j}+\hat{\mathbf k} and b=2i^+k^\mathbf b=2\hat{\mathbf i}+\hat{\mathbf k}, determine a×b\mathbf a\times\mathbf b.
    Vectors and matrices
  12. Q4 · 2024 QCAA · Paper 2 · 1 mark
    The mass of biscuit packets produced by a company is normally distributed with a mean of 250 g and a standard deviation of 1.5 g. The distribution of the sample mean mass of these biscuit packets is formed using repeated random sampling of size 5. The mean and standard deviation of this distribution of sample means are
    Statistical inference
  13. Q5 · 2024 QCAA · Paper 2 · 1 mark
    The equation of a plane is 2x−4z−8=02x-4z-8=0. Determine the point where the plane intersects the zz-axis.
    Vectors and matrices
  14. Q6 · 2024 QCAA · Paper 2 · 1 mark
    Two concurrent forces represented in the polar form of F1=(1.21 N,120∘)F_1=(1.21\text{ N},120^\circ) and F2=(1.30 N,−160∘)F_2=(1.30\text{ N},-160^\circ) act on an object. Determine the magnitude of the resultant force.
    Vectors and matrices
  15. Q7 · 2024 QCAA · Paper 2 · 1 mark
    Determine the number of roots of w8=1w^8=1 that can be expressed in the form a+bia+bi, where a,b∈R+a,b\in\mathbb R^+.
    Complex numbers 2
  16. Q8 · 2024 QCAA · Paper 2 · 1 mark
    TT is a random variable. A random sample of four values of TT is collected and used to produce an approximate confidence interval for the population mean of (3.3,4.1)(3.3,4.1). Given that three of the sample values are 3.4, 3.6 and 3.9, the remaining sample value is
    Statistical inference
  17. Q9 · 2024 QCAA · Paper 2 · 1 mark
    Random variable XX has an exponential distribution with the probability density function f(x)={15e−x/5,x>00,otherwisef(x)=\begin{cases}\dfrac15e^{-x/5},&x>0\\0,&\text{otherwise}\end{cases}. Given that P(0≤X≤k)=0.5P(0\le X\le k)=0.5, determine kk.
    Statistical inference
  18. Q10 · 2024 QCAA · Paper 2 · 1 mark
    The acceleration (m s−2^{-2}) of an object at time, tt, for 0≤t≤20\le t\le2 seconds is given by a=2t+1a=\dfrac{2}{t+1}.
    Given that the object is initially at rest, its velocity–time graph is
    Rates of change and differential equations
  19. Q11 · 2024 QCAA · Paper 2 · 4 marks
    A company claims that the mean battery life of their latest model of smartphone is 9.5 hours.
    To test this claim, the battery lives of a random sample of 40 of the smartphones were measured. A sample mean of 9.31 hours and a standard deviation of 0.52 hours were calculated from this data.
    Statistical inference
  20. Q12 · 2024 QCAA · Paper 2 · 4 marks
    A system of linear equations is given by x−2y−2z=−6,−3x−y+z=2,2x+3y−5z=10.\begin{aligned} x-2y-2z&=-6,\\ -3x-y+z&=2,\\ 2x+3y-5z&=10. \end{aligned}
    Vectors and matrices
  21. Q13 · 2024 QCAA · Paper 2 · 5 marks
    A drone travels vertically from point AA at a constant speed of 8 m s−18\text{ m s}^{-1} over time tt for t≥0t\ge0 seconds. Observation of the drone is made from point BB, which is 50 m horizontally from point AA.
    When the drone is yy metres above point AA, it is zz metres from point BB as shown.
    Question and worked solution
  22. Q14 · 2024 QCAA · Paper 2 · 5 marks
    The height of Year 12 students at a school is normally distributed, with a mean height of 168.6 cm and standard deviation of 12.7 cm.
    The heights of a random sample of 20 of these students are recorded.
    Statistical inference
  23. Q15 · 2024 QCAA · Paper 2 · 8 marks
    The vectors representing the position (m) of particles A and B are given by rA=(4t−9)i^−2(5−t)j^−8k^\mathbf r_A=(4t-9)\hat{\mathbf i}-2(5-t)\hat{\mathbf j}-8\hat{\mathbf k} and rB=(t2+1)i^−3j^+(4−at2)k^\mathbf r_B=(t^2+1)\hat{\mathbf i}-3\hat{\mathbf j}+(4-at^2)\hat{\mathbf k} respectively, where tt is the time of motion for 0≤t≤100\le t\le10 seconds.
    Vectors and matrices
  24. Q16 · 2024 QCAA · Paper 2 · 6 marks
    Two subsets of the complex plane are S={z:∣z−1∣=4}S=\{z:|z-1|=4\} and T={z:arg⁡(z+i)=π3},T=\left\{z:\arg(z+i)=\frac{\pi}{3}\right\}, where z∈Cz\in\mathbb C. Determine the complex number/s where SS and TT intersect. Leave your answer/s in Cartesian form. Provide an Argand diagram with a sketch of subsets SS and TT as part of your solution.
    Complex numbers 2