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 18 of 22

1. Q182 · Original practice · 1 mark
   A student checks that $n^2+n+41$ is prime for $n=0,\ldots,39$ and claims it is prime for every nonnegative integer. Which statement is correct?
   Mathematical induction and trigonometric proofs
2. Q183 · Original practice · 5 marks
   Four corners of a square game arena have complex coordinates satisfying $z^4=-81$. The dashed circle is centred at the origin.
   Further complex numbers
3. Q184 · Original practice · 10 marks
   Let $\omega=\operatorname{cis}(2\pi/3)$, so the cube roots of unity are $1,\omega,\omega^2$.
   Further complex numbers
4. Q185 · Original practice · 7 marks
   A game map has triangular zone with complex coordinates $0,2,2i$. A portal transforms coordinates by $w=(1+i)z+(2-i)$.
   Further complex numbers
5. Q186 · Original practice · 5 marks
   The polynomial $p(z)=z^3+az^2+bz+10$ has real coefficients a,b and zero $1+i$.
   Further complex numbers
6. Q187 · Original practice · 1 mark
   Which set contains all solutions of $z^2=(1+i)/(1-i)$?
   Further complex numbers
7. Q188 · Original practice · 5 marks
   A point $P=(2,1,2)$ and a line $L:\mathbf r=(1,0,0)+t(1,2,2)$ are given, with $t\in\mathbb R$.
   Vectors in two and three dimensions
8. Q189 · Original practice · 5 marks
   A triangular panel has vertices $A=(1,0,0)$, $B=(0,2,0)$ and $C=(0,0,3)$.
   Vectors in two and three dimensions
9. Q190 · Original practice · 8 marks
   A laser ray follows $\mathbf r=(1,0,2)+t(2,1,-1)$, $t\ge0$. A reflecting screen lies in $x+y+z=6$.
   Vectors in two and three dimensions
10. Q191 · Original practice · 6 marks
    Two straight rails in a sculpture are modelled by $L_1:(x,y,z)=(t,t,0)$ and $L_2:(x,y,z)=(s,1-s,1)$, with $s,t\in\mathbb R$.
    Vectors in two and three dimensions
11. Q192 · Original practice · 5 marks
    A river is 100 m wide, measured due north. Its current is $3\mathbf i$ m/s east. A boat has speed 5 m/s relative to the water. Let $\mathbf j$ point north. Ignore acceleration and current changes.
    Vectors in two and three dimensions
12. Q193 · Original practice · 6 marks
    Two animated drones have positions $\mathbf r_A=(t,2t)$ and $\mathbf r_B=(6-t,3)$ metres for $0\le t\le3$ seconds. Their paths are shown.
    Vector calculus
13. Q194 · Original practice · 5 marks
    For $0\le t\le3$, a particle has position $\mathbf r(t)=(t+1)\mathbf i+(t+1)^{-1}\mathbf j$ metres.
    Vector calculus
14. Q195 · Original practice · 1 mark
    Particles move on the radius-two circle with $\mathbf r_A=2(\cos t,\sin t)$ and $\mathbf r_B=2(\cos(2t+\pi),\sin(2t+\pi))$. What is their first meeting time after t=0?
    Vector calculus
15. Q196 · Original practice · 10 marks
    A camera rig has velocity $\mathbf v(t)=2t\mathbf i+3t^2\mathbf j$ m/s on $0\le t\le1$, and initial position (1,-1) m.
    Vector calculus
16. Q197 · Original practice · 5 marks
    In a tournament A beats B, B beats C and C beats A. Define $D_{ij}=1$ if i beats j and 0 otherwise, in order A,B,C. There are no draws.
    Matrices and applications
17. Q198 · Original practice · 5 marks
    A dominance matrix in order A,B,C,D is

    $$M=\begin{pmatrix}0&1&1&0\\0&0&1&1\\0&0&0&1\\1&0&0&0\end{pmatrix}.$$

    A score is the row sum of $M+M^2$.
    Matrices and applications
18. Q199 · Original practice · 5 marks
    A population follows $\binom{J_{n+1}}{A_{n+1}}=\begin{pmatrix}0&f\\0.4&0.6\end{pmatrix}\binom{J_n}{A_n}$ with f>0.
    Matrices and applications
19. Q200 · Original practice · 7 marks
    A three-stage insect model uses

    $$L=\begin{pmatrix}0&0&4\\0.5&0&0\\0&0.5&0\end{pmatrix}.$$

    Counts are expected values and may be fractional.
    Matrices and applications
20. Q201 · Original practice · 5 marks
    A population first undergoes transition $\begin{pmatrix}0&2\\0.5&0.5\end{pmatrix}$. After each transition a fraction q of the new adults is removed, $0\le q<1$. Juveniles are not removed.
    Matrices and applications
21. Q202 · Original practice · 1 mark
    Which plane contains all points equidistant from $A=(1,0,2)$ and $B=(3,2,0)$?
    Vectors in two and three dimensions
22. Q203 · Original practice · 5 marks
    A curved bookmark occupies the region below $y=x/(x^2+4)$ and above the x-axis for $0\le x\le2$.
    Integration techniques
23. Q204 · Original practice · 5 marks
    Let $f(x)=x\ln x$ on $1\le x\le e$.
    Integration techniques
24. Q205 · Original practice · 1 mark
    What is $\int_0^1\frac{3x+5}{(x+1)(x+2)}\,dx$ exactly?
    Integration techniques