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 21 of 22

1. Q254 · Original practice · 12 marks
   A river is 100 m wide. Coordinates have $x$ east and $y$ north, with the near bank $y=0$ and far bank $y=100$. A rescue boat starts at $(0,0)$ and keeps a fixed velocity $(u_x,u_y)$ relative to the water, with $u_x^2+u_y^2=25$ and $u_y>0$. The current at northward coordinate $y$ is $0.12y$ m/s east. A raft moves east along the far bank from $(60,100)$ at…
   Vector calculus
2. Q255 · Original practice · 12 marks
   In a magnetic-launch simulation, a particle moves on a straight track with acceleration $a(x)=-20/(1+x)^2$ m/s$^2$ for $x\ge0$. It starts at $x=0$ with positive speed $v_0$. The acceleration depends only on position and applies on both outward and return motion, whenever a return occurs.
   Modelling motion
3. Q256 · Original practice · 12 marks
   In a robot simulation, energy $E$ joules obeys $E^{\prime}=-kE-v^2$, where $k=\ln2/5$ s$^{-1}$ and $v$ is the numerical speed in m/s. The coefficient of the $v^2$ term is included in the simulation units. Initially $E=420$. The robot travels for 5 s at constant speed $u$, then for 5 s at constant speed $w$, with $u,w\ge0$. It must travel exactly 30 m. A subs…
   Differential equations and related rates
4. Q257 · Original practice · 12 marks
   Three sensors measure an unknown mean $\mu$, but their population means are $\mu+\beta$, $\mu-\beta$ and $\mu+2\beta$, where $\beta$ is an unknown fixed calibration offset. Each sensor supplies $n$ independent normal readings; the three samples are independent. Population standard deviations are 3, 2 and 1 respectively. Let their sample means be…
   Statistical inference
5. Q258 · Original practice · 4 marks
   Let $z=a+bi$ and $w=c+di$, where $a,b,c,d\in\mathbb R$.
   Further complex numbers
6. Q259 · Original practice · 5 marks
   Let $z,w\in\mathbb C$, with $w\ne0$. You may use $u\overline u=|u|^2$ and the modulus product identity.
   Further complex numbers
7. Q260 · Original practice · 5 marks
   The complex coordinates of two adjacent sides of a parallelogram are $z$ and $w$. Its diagonals are represented by $z+w$ and $z-w$.
   Further complex numbers
8. Q261 · Original practice · 5 marks
   A graphics tool maps a complex number $z$ to $w=(z+2)/(2z+1)$. Assume $|z|=1$.
   Further complex numbers
9. Q262 · Original practice · 5 marks
   For this question, the principal argument is $\operatorname{Arg}z\in(-\pi,\pi]$. Let $z,w\ne0$.
   Further complex numbers
10. Q263 · Original practice · 6 marks
    Three lights have complex coordinates satisfying $z^3=8i$. Blank Argand axes are provided.
    Further complex numbers
11. Q264 · Original practice · 6 marks
    Consider $S=\{z:|z-1|=2\}$ and $T=\{z:\operatorname{Im}z=1\}$.
    Further complex numbers
12. Q265 · Original practice · 6 marks
    For $z\ne0$, use $\operatorname{Arg}z\in(-\pi,\pi]$. Define $R=\{z:1\le|z|\le2,\ 0<\operatorname{Arg}z\le\pi/2\}$.
    Further complex numbers
13. Q266 · Original practice · 6 marks
    Let $\theta\in\mathbb R$. The angle addition identities may be used.
    Mathematical induction and trigonometric proofs
14. Q267 · Original practice · 6 marks
    Consider $(\cos\theta+i\sin\theta)^4$.
    Mathematical induction and trigonometric proofs
15. Q268 · Original practice · 5 marks
    A nondegenerate parallelogram ABCD in three-dimensional space has position vectors $\mathbf a,\mathbf b,\mathbf c,\mathbf d$ in cyclic order. Let M and N be the midpoints of AB and DC.
    Vectors in two and three dimensions
16. Q269 · Original practice · 5 marks
    Let A, B and C be noncollinear points with position vectors $\mathbf a,\mathbf b,\mathbf c$. M is the midpoint of BC. Define G by $\mathbf g=(\mathbf a+\mathbf b+\mathbf c)/3$.
    Vectors in two and three dimensions
17. Q270 · Original practice · 6 marks
    A 4 kg block is initially at rest on a smooth plane inclined at $30^\circ$ above the horizontal. A rope pulls at $20^\circ$ above the plane with tension 18 N. Use $g=10$ m s$^{-2}$. The stimulus shows the geometry only; it contains no force arrows.
    Modelling motion
18. Q271 · Original practice · 6 marks
    The polynomial $P(z)=z^3+az^2+bz+2$ has $a,b\in\mathbb C$. Its remainder on division by $z-i$ is $2-2i$, and $P(1)=0$.
    Further complex numbers
19. Q272 · Original practice · 5 marks
    Let

    $$A=\begin{pmatrix}1&1&0\\0&1&1\\1&0&1\end{pmatrix},\quad B=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix},\quad C=\begin{pmatrix}2&2&5\\7&3&2\\2&1&0\end{pmatrix}.$$

    A $3\times3$ matrix X satisfies $XA+XB=C$.
    Further matrices
20. Q273 · Original practice · 7 marks
    A particle moves with $\mathbf r(t)=(1+2\sec t)\mathbf i+(2+\tan t)\mathbf j$, for $0\le t<\pi/2$. Blank coordinate axes are supplied.
    Vector calculus
21. Q274 · Original practice · 7 marks
    A breeding program models females in three one-year age classes. The table gives births per female during a step, the proportion surviving into the next age class, and initial counts. Offspring enter the youngest class; no females survive beyond the oldest class. Fractional expected counts are permitted.
    Further matrices
22. Q275 · Original practice · 6 marks
    A two-stage population follows

    $$\binom{J_{n+1}}{A_{n+1}}=\begin{pmatrix}f&2f\\s&0\end{pmatrix}\binom{J_n}{A_n},\quad f>0,\quad0<s\le1.$$

    The initial vector is $(40,20)^T$ and the vector after one step is $(32,20)^T$.
    Further matrices
23. Q276 · Original practice · 6 marks
    Three teams A, B and C play a mini-tournament. A defeats B, B defeats C, and C defeats A. Let $D_{ij}=1$ when team i defeats team j and 0 otherwise, with order A, B, C.
    Further matrices
24. Q277 · Original practice · 6 marks
    A partially completed slope field represents $dy/dx=2-y$. Slopes on $y=0$ are supplied; nodes on $y=2$ and $y=4$ are blank.
    Rates of change and differential equations