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 14 of 22

1. Q86 · Original practice · 5 marks
   A $3\ \mathrm{kg}$ particle moves along the $x$-axis under a force $F=6x\ \mathrm N$ in the positive $x$ direction. When $x=1\ \mathrm m$, its speed is $2\ \mathrm{m\,s^{-1}}$.
   Mechanics
2. Q87 · Original practice · 1 mark
   A particle in SHM satisfies $v^2=25(A^2-x^2)$. If $x=3$ when $|v|=20$, the amplitude $A$ is
   Mechanics
3. Q88 · Original practice · 5 marks
   A $4\ \mathrm{kg}$ block is on a smooth plane inclined at $25^\circ$ to the horizontal. A $30\ \mathrm N$ force acts at $15^\circ$ above the plane in the uphill direction. Take $g=9.8\ \mathrm{m\,s^{-2}}$.
   Mechanics
4. Q89 · Original practice · 1 mark
   A vertical-motion model uses $dv/dt=-9.8-0.4v$, where upward velocity is positive. The terminal speed is
   Mechanics
5. Q90 · Original practice · 7 marks
   A particle satisfies $$\frac{d^2x}{dt^2}=-4x,\qquad x(0)=3,\qquad v(0)=-4.$$
   Mechanics
6. Q91 · Original practice · 1 mark
   A population has mean 80 and standard deviation 15. For samples of size 25, the sample mean has mean and standard deviation
   Statistical inference
7. Q92 · Original practice · 7 marks
   A ball is projected vertically upward with speed $20\ \mathrm{m\,s^{-1}}$. Its velocity satisfies $$\frac{dv}{dt}=-10-0.5v,$$ where upward is positive and $t$ is in seconds.
   Mechanics
8. Q93 · Original practice · 1 mark
   A population has $\mu=50$, $\sigma=10$, and $n=100$. Using the normal model, $P(\bar X>52)$ is approximately
   Statistical inference
9. Q94 · Original practice · 6 marks
   The region bounded by $y=\ln x$, $y=0$ and $x=e$ is rotated about the $y$-axis.
   Integration
10. Q95 · Original practice · 1 mark
    For $n=100$, $\bar x=20.4$, $s=5$, and $z=1.96$, the approximate 95% confidence interval for $\mu$ is
    Statistical inference
11. Q96 · Original practice · 6 marks
    A waiting time $T$ (minutes) is exponentially distributed with parameter $\lambda=0.12$.
    Exponential distributions
12. Q97 · Original practice · 1 mark
    Using $z=1.96$ and planning with $s=12$, the smallest sample size that gives margin of error at most $1.5$ is
    Statistical inference
13. Q98 · Original practice · 5 marks
    A population has mean 18 and standard deviation 4.5. Random samples of size 81 are taken.
    Statistical inference
14. Q99 · Original practice · 1 mark
    An approximate 95% confidence interval for a population mean is $(48.2,51.8)$. Which statement is best supported?
    Statistical inference
15. Q100 · Original practice · 6 marks
    A sample of size 64 has $\bar x=42.6$ and $s=5.2$.
    Statistical inference
16. Q101 · Original practice · 1 mark
    For $$L=\begin{pmatrix}0&1.2&1.8\\0.6&0&0\\0&0.7&0\end{pmatrix},\qquad \mathbf p=\begin{pmatrix}100\\80\\40\end{pmatrix},$$ the youngest class after one step is
    Matrices
17. Q102 · Original practice · 6 marks
    Two independent samples from the same type of population give the following summaries. $$\begin{array}{c|ccc}\text{Sample}&n&\bar x&s\\\hline A&100&75.2&10\\B&400&74.8&10\end{array}$$ Using $z=1.96$:
    Statistical inference
18. Q103 · Original practice · 1 mark
    For the dominance matrix $$D=\begin{pmatrix}0&1&1&0\\0&0&1&1\\0&0&0&1\\1&0&0&0\end{pmatrix},$$ the entry $(D^2)_{1,4}=2$ means that competitor 1 has
    Matrices
19. Q104 · Original practice · 6 marks
    A machine-filling process is studied using a sample of 144 packets. The sample has $\bar x=502\ \mathrm g$ and $s=12\ \mathrm g$.
    Statistical inference
20. Q105 · Original practice · 1 mark
    A projectile is launched from level ground at $25\ \mathrm{m\,s^{-1}}$ at $40^\circ$ above the horizontal. Taking $g=9.8\ \mathrm{m\,s^{-2}}$, its horizontal range is approximately
    Vector calculus
21. Q106 · Original practice · 6 marks
    A curve satisfies $$\frac{dy}{dx}=\frac{x}{1+y^2},\qquad y(0)=0.$$
    Differential equations
22. Q107 · Original practice · 1 mark
    A particle has $\mathbf r(t)=(4\cos2t,4\sin2t)$. The magnitude of its acceleration is
    Vector calculus
23. Q108 · Original practice · 7 marks
    A drone and a moving beacon have positions $$\begin{aligned}\mathbf r_D(t)&=(2t,t),\\\mathbf r_B(t)&=(8\cos(0.4t),8\sin(0.4t)),\end{aligned}\qquad0\le t\le6.$$ Coordinates are in kilometres and $t$ is in hours.
    Vectors in three dimensions
24. Q109 · Original practice · 1 mark
    For $\mathbf r(t)=(3\cos t,2\sin t)$, the angle between velocity and acceleration at $t=\pi/4$ is approximately
    Vector calculus