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 15 of 22

1. Q110 · Original practice · 6 marks
   For a real parameter $t$, define $$A=(t,0,1),\qquad B=(1,t,0),\qquad C=(0,1,t).$$
   Vectors in three dimensions
2. Q111 · Original practice · 1 mark
   A non-real root of $z^{4}=16$ with positive imaginary part is
   Complex numbers
3. Q112 · Original practice · 1 mark
   The matrix $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ maps $(2,3)$ to
   Matrices
4. Q113 · Original practice · 1 mark
   A direction vector from $A=(2,-1,0)$ to $B=(5,1,-4)$ is
   Vectors in three dimensions
5. Q114 · Original practice · 1 mark
   An antiderivative of $\frac{5x+7}{(x+1)(x+2)}$ is
   Integration
6. Q115 · Original practice · 1 mark
   For $\frac{\mathrm{d}y}{\mathrm{d}x}=(x+2)(y-3)$, zero slope occurs on
   Differential equations
7. Q116 · Original practice · 1 mark
   The area of the parallelogram generated by $a=(1,2,0)$ and $b=(0,1,2)$ is
   Vectors in three dimensions
8. Q117 · Original practice · 1 mark
   If all else is unchanged, increasing sample size by a factor of $9$ changes an approximate confidence-interval margin of error by a factor of
   Statistical inference
9. Q118 · Original practice · 1 mark
   If $z=3\operatorname{cis}(\frac{\pi}{4})$ and $w=2\operatorname{cis}(\frac{\pi}{6})$, then $zw$ is
   Complex numbers
10. Q119 · Original practice · 1 mark
    For $z=x+iy, |z-2|=|z+2|$ describes
    Complex numbers
11. Q120 · Original practice · 1 mark
    Two non-parallel lines in $3D$ intersect if
    Vectors in three dimensions
12. Q121 · Original practice · 5 marks
    Let $z=1-\sqrt{3}i$ and $w=2+2i$.
    Complex numbers
13. Q122 · Original practice · 6 marks
    The lines $l_{1}$: $r=(1,2,0)+s(1,-1,2)$ and $l_{2}$: $r=(3,0,1)+t(2,1,-1)$ are given.
    Vectors in three dimensions
14. Q123 · Original practice · 5 marks
    A particle has acceleration $a(t)=4t-6$, with $v(0)=2$ and $x(0)=1$.
    Mechanics
15. Q124 · Original practice · 6 marks
    A $99\%$ confidence interval for a mean is centred at $40$ and has total width $5.152$. It uses $z=2.576$ and sample standard deviation $s=10$.
    Statistical inference
16. Q125 · Original practice · 5 marks
    Evaluate $\int x e^{2x} \mathrm{d}x$ and then evaluate it from $0$ to $1$.
    Integration
17. Q126 · Original practice · 6 marks
    A particle at $P=(2,-1,1)$ moves with direction $d=(1,2,-2)$ and reflects from the plane $2x-y+2z=3$. Reflection reverses the normal component of $d$.
    Mechanics
18. Q127 · Original practice · 5 marks
    Let $A=\begin{pmatrix}2&1\\1&1\end{pmatrix}$.
    Matrices
19. Q128 · Original practice · 6 marks
    A curve satisfies $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2x+1}{y-1}$ and passes through $(0,3)$.
    Differential equations
20. Q129 · Original practice · 6 marks
    Points $z=x+iy$ satisfy $|z|=5$ and are equidistant from $1+2i$ and $5+2i$.
    Complex numbers
21. Q130 · Original practice · 1 mark
    A confidence interval $(54.2,59.8)$ has centre and margin of error
    Statistical inference
22. Q131 · Original practice · 1 mark
    Forces $30\,\mathrm{N}$ east, $20\,\mathrm{N}$ north and $10\,\mathrm{N}$ west have resultant
    Mechanics
23. Q132 · Original practice · 1 mark
    Simpson’s rule with $h=1$ and ordinates $2,5,4$ gives
    Integration
24. Q133 · Original practice · 1 mark
    For $a=(1,2,2), b=(2,0,1)$, the scalar projection of $a$ onto $b$ is
    Vectors in three dimensions