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 16 of 22

1. Q134 · Original practice · 1 mark
   The polar point $4\operatorname{cis}(\frac{\pi}{2})$ is
   Complex numbers
2. Q135 · Original practice · 1 mark
   If sample size is multiplied by $16$, confidence-interval width is approximately
   Statistical inference
3. Q136 · Original practice · 1 mark
   For $\frac{\mathrm{d}P}{\mathrm{d}t}=0.3P(1-\frac{P}{50})$, the positive equilibrium is
   Differential equations
4. Q137 · Original practice · 1 mark
   Under $w=(2-i)z, z=1+i$ maps to
   Complex numbers
5. Q138 · Original practice · 1 mark
   For $a=(2,1,0), b=(1,-1,2), |a \times b|$ is
   Vectors in three dimensions
6. Q139 · Original practice · 1 mark
   For $z=1.96, s=8$ and $n=64$, the approximate margin of error for a mean is
   Statistical inference
7. Q140 · Original practice · 5 marks
   Two groups of sizes $40$ and $60$ have sample means $84$ and $78$ respectively. Treat the combined data as one sample with standard deviation $10$ and use $z=1.96$.
   Statistical inference
8. Q141 · Original practice · 6 marks
   A particle has position $r(t)=(t^{2}, 2t, 3-t^{2})$ for $t\ge 0$.
   Vector calculus
9. Q142 · Original practice · 6 marks
   Values of $f(x)$ at $x=0,0.5,1.0,1.5,2.0$ are $1.000,1.649,2.718,4.482,7.389$.
   Integration
10. Q143 · Original practice · 6 marks
    A population satisfies $\frac{\mathrm{d}P}{\mathrm{d}t}=0.4P(1-\frac{P}{80})$, with $P(0)=20$.
    Differential equations
11. Q144 · Original practice · 6 marks
    Let $z$ satisfy $z^{3}=-8i$.
    Complex numbers
12. Q145 · Original practice · 7 marks
    A particle executes SHM with $x(t)=0.12 \cos (4t-\frac{\pi}{3})$ metres.
    Mechanics
13. Q146 · Original practice · 7 marks
    A $2.0\,\mathrm{kg}$ particle moves in the $xy$-plane under force $F=(6-2t)i+(4t)j\,\mathrm{N}$. At $t=0$ its velocity is $3i-j\,\mathrm{m}\,\mathrm{s}^{-1}$ and position is the origin.
    Mechanics
14. Q147 · Original practice · 7 marks
    A sample of $n=144$ measurements has mean $25.6$ and standard deviation $3.6$. Use $z=2.576$ for a $99\%$ interval.
    Statistical inference
15. Q148 · Original practice · 1 mark
    The four roots of $z^4=-16$ are shown on the Argand plane. For each root, a new complex number is defined by

    $$w=z+\frac{4}{z}.$$

    Which statement describes the set of distinct values of $w$?
    Further complex numbers
16. Q149 · Original practice · 6 marks
    Two particles move in the plane for $0\leq t\leq4$, where $t$ is measured in seconds. Their position vectors, in metres, are

    $$\mathbf r_A(t)=t\mathbf i+(t-2)^2\mathbf j,\qquad \mathbf r_B(t)=(t+1)\mathbf i+\mathbf j.$$

    Their paths are shown. The points $A_0$ and $B_0$ indicate their positions at $t=0$.
    Vector calculus
17. Q150 · Original practice · 1 mark
    An instrument records $Y=1.2X+0.6$, where $X$ is the true measurement and $Y$ is the recorded measurement, both in millimetres. This calibration relationship is exact.

    A random sample of 64 independent measurements has recorded sample mean $\overline y=18.6$ and recorded sample standard deviation $s_Y=2.4$. Assume the conditions for an approximate normal co…
    Statistical inference
18. Q151 · Original practice · 8 marks
    The region $R$ is bounded by $y=\ln x$, the $x$-axis and $x=e$, as shown. A vertical line $x=c$, where $1<c<e$, divides $R$ into two regions of equal area.
    Integration techniques; applications of integral calculus
19. Q152 · Original practice · 8 marks
    A population is modelled as a continuous quantity $N(t)$, where $t$ is measured in days. It satisfies

    $$\frac{dN}{dt}=kN(K-N),\qquad k>0,\quad K>0.$$

    The graph shows the growth rate as a function of population. The axis intercepts and the labelled point P are exact. Initially, $N(0)=50$.
    Rates of change and differential equations
20. Q153 · Original practice · 1 mark
    The points $A$, $B$ and $C$ represent the three roots of $z^3=-8i$ on the Argand plane. The point $P$ represents $z=2$. What is the exact value of $PA\times PB\times PC$?
    Further complex numbers
21. Q154 · Original practice · 7 marks
    A sequence of partial sums is defined by

    $$S_n=\sum_{k=1}^{n}\frac{k}{(k+1)!},\qquad n\in\mathbb Z^+.$$

    Here $m!$ denotes the factorial of the positive integer $m$.
    Mathematical induction
22. Q155 · Original practice · 6 marks
    A point $P$ has coordinates $(4,1,3)$. A plane $\Pi$ has equation

    $$x+2y+2z=3.$$

    The point $H$ is the perpendicular projection of $P$ onto $\Pi$. The point $P'$ is the reflection of $P$ in $\Pi$, so $H$ is the midpoint of $PP'$. The diagram is schematic.
    Vectors in two and three dimensions
23. Q156 · Original practice · 7 marks
    An insect population has juvenile and adult stages. The model is

    $$\begin{pmatrix}J_{n+1}\\ A_{n+1}\end{pmatrix} =\begin{pmatrix}0&2\\0.5&0.5\end{pmatrix} \begin{pmatrix}J_n\\ A_n\end{pmatrix},\qquad \begin{pmatrix}J_0\\A_0\end{pmatrix}=\begin{pmatrix}200\\100\end{pmatrix}.$$

    The entries are expected population counts; fractional values are permitted. Th…
    Further matrices: Leslie models
24. Q157 · Original practice · 6 marks
    A particle moves for $0\le t<2\pi$ seconds with position vector, in metres,

    $$\mathbf r(t)=3\cos t\,\mathbf i+2\sin t\,\mathbf j.$$

    Its path is shown. The point $S$ is its position at $t=0$.
    Vector calculus: elliptical motion