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 13 of 22

1. Q62 · Original practice · 6 marks
   The equation $z^5=10+10i$ has five roots.
   Complex numbers
2. Q63 · Original practice · 1 mark
   The region under $y=2-x$ from $x=0$ to $x=2$ is rotated about the $x$-axis. The volume is
   Integration
3. Q64 · Original practice · 6 marks
   A sphere has centre $C(2,-1,3)$ and radius 5. It is cut by the plane $$2x-y+2z=9.$$ The intersection is a circle.
   Vector calculus
4. Q65 · Original practice · 1 mark
   For $x=0,0.5,1.0,1.5,2.0$, a function has values $1.0,1.5,2.2,3.1,4.2$. Simpson’s rule with four intervals gives $\displaystyle\int_0^2 f(x)\,dx\approx$
   Integration
5. Q66 · Original practice · 6 marks
   Two autonomous vehicles move in a plane, with position in kilometres after $t$ hours given by $$\mathbf r_A(t)=(2t,1+t),\qquad\mathbf r_B(t)=(12-t,7+0.5t),\qquad t\ge0.$$ Determine when they are closest together and find their minimum separation, giving answers to three decimal places where appropriate.
   Vector calculus
6. Q67 · Original practice · 1 mark
   An exponential random variable has mean 5. The probability that it exceeds 8 is approximately
   Exponential distributions
7. Q68 · Original practice · 6 marks
   A projectile is launched from level ground with speed $30\ \mathrm{m\,s^{-1}}$. It must pass through the point $(40,15)$, where coordinates are in metres. Take $g=10\ \mathrm{m\,s^{-2}}$ and ignore air resistance. Determine the two possible acute launch angles, to the nearest $0.1^\circ$.
   Vector calculus
8. Q69 · Original practice · 1 mark
   If $T\sim\operatorname{Exp}(0.3)$, the 90th percentile of $T$ is approximately
   Exponential distributions
9. Q70 · Original practice · 6 marks
   A population has Leslie matrix $$L=\begin{pmatrix}0&1.4&2.0\\0.5&0&0\\0&0.6&0\end{pmatrix},\qquad\mathbf p_0=\begin{pmatrix}200\\120\\80\end{pmatrix}.$$
   Matrices
10. Q71 · Original practice · 1 mark
    The curve $x^2+xy+2y^2=8$ passes through $(2,1)$. The gradient there is
    Differential equations
11. Q72 · Original practice · 6 marks
    Five competitors have dominance matrix $$D=\begin{pmatrix}0&1&1&0&1\\0&0&1&1&0\\0&0&0&1&1\\1&0&0&0&1\\0&1&0&0&0\end{pmatrix}.$$ A dominance score is defined as the row sum of $D+D^2$.
    Matrices
12. Q73 · Original practice · 1 mark
    The volume of a sphere is increasing at $36\pi\ \mathrm{cm^3\,s^{-1}}$. When the radius is $3\ \mathrm{cm}$, $dr/dt$ is
    Differential equations
13. Q74 · Original practice · 5 marks
    The curve $x^2y+y^3=10$ passes through $(1,2)$.
    Differential equations
14. Q75 · Original practice · 1 mark
    The solution of $dy/dx=2xy$ with $y(0)=3$ has $y(1)$ equal to
    Differential equations
15. Q76 · Original practice · 5 marks
    Water in a hemispherical bowl of radius $R=3\ \mathrm m$ has depth $h\ \mathrm m$. The volume of water is $$V=\pi h^2\left(R-\frac h3\right).$$ Water is entering at $0.5\ \mathrm{m^3\,min^{-1}}$. Determine $dh/dt$ when $h=1\ \mathrm m$.
    Differential equations
16. Q77 · Original practice · 1 mark
    For $dy/dx=(y-1)(3-y)$ with $1<y(0)<3$, the solution approaches
    Differential equations
17. Q78 · Original practice · 7 marks
    A population $P$ satisfies $$\frac{dP}{dt}=0.3P\left(1-\frac P{500}\right),\qquad P(0)=50.$$
    Differential equations
18. Q79 · Original practice · 1 mark
    A cooling model is $T=20+60e^{-0.2t}$, with $t$ in minutes. The time when $T=30$ is approximately
    Differential equations
19. Q80 · Original practice · 6 marks
    An object cools in a room maintained at $22^\circ\mathrm C$. Its temperature $T$ satisfies Newton’s law of cooling. Initially $T=90^\circ\mathrm C$, and after 10 minutes $T=60^\circ\mathrm C$.
    Differential equations
20. Q81 · Original practice · 1 mark
    A substance has a half-life of 12 hours. The proportion remaining after 30 hours is approximately
    Differential equations
21. Q82 · Original practice · 6 marks
    A radioactive sample initially has mass $120\ \mathrm{mg}$. After 8 days its mass is $90\ \mathrm{mg}$. Assume exponential decay.
    Differential equations
22. Q83 · Original practice · 1 mark
    A $5\ \mathrm{kg}$ block is on a smooth plane inclined at $20^\circ$. A $30\ \mathrm N$ force acts up the plane. Taking $g=9.8\ \mathrm{m\,s^{-2}}$, the acceleration up the plane is approximately
    Mechanics
23. Q84 · Original practice · 5 marks
    The differential equation is $$\frac{dy}{dx}=x(1+y),\qquad y(0)=0.$$
    Differential equations
24. Q85 · Original practice · 1 mark
    A particle satisfies $v\,dv/dx=3x$ and $v=2$ when $x=0$. When $x=2$, its speed is
    Mechanics