These Ten-Year-Series (TYS) worked solutions with video explanations for 2020 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
It is given that $\text{f}(x)=\ln \left( 1+\sin 3x \right)$.
(i)
Show that $\text{f}”(x)=\frac{k}{1+\sin 3x}$, where $k$ is a constant to be found.
[3]
(i) Show that $\text{f}”(x)=\frac{k}{1+\sin 3x}$, where $k$ is a constant to be found.
[3]
(ii)
Hence find the first three non-zero terms of the Maclaurin expansion of $\text{f}(x)$.
[4]
(ii) Hence find the first three non-zero terms of the Maclaurin expansion of $\text{f}(x)$.
[4]
Share with your friends!
Do not use calculator in answering this question.
Three complex numbers are ${{z}_{1}}=1+\sqrt{3}\mathbf{i}$, ${{z}_{2}}=1-\mathbf{i}$ and ${{z}_{3}}=2\left( \cos \frac{1}{6}\pi +\mathbf{i}\sin \frac{1}{6}\pi \right)$.
(i)
[4]
(i) Find $\frac{{{z}_{1}}}{{{z}_{2}}{{z}_{3}}}$ in the form $r\,(\cos \theta +\mathbf{i}\sin \theta )$, where $r>0$ and $-\pi <\theta \le \pi $.
[4]
A fourth complex number, ${{z}_{4}}$, is such that $\frac{{{z}_{1}}{{z}_{4}}}{{{z}_{2}}{{z}_{3}}}$ is purely imaginary and $\left| \frac{{{z}_{1}}{{z}_{4}}}{{{z}_{2}}{{z}_{3}}} \right|=1$ .
(ii)
[3]
(ii) Find the possible values of ${{z}_{4}}$ in the form $r\,(\cos \theta +\mathbf{i}\sin \theta )$, where $r>0$ and $-\pi <\theta \le \pi $.
[3]
Share with your friends!
(a)
[2]
(a) Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that $\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{a}$, find the relationship between $\mathbf{a}$ and $\mathbf{b}$.
[2]
(b)
[3]
(b) The points $P$,$Q$ and $R$ have position vectors $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively. The points $P$ and $Q$ are fixed and $R$ varies.
[3]
[3]
(ii) Given instead that $\mathbf{r}=\left( \begin{matrix}x \\y \\z \\\end{matrix} \right)$, $\mathbf{p}=\left( \begin{matrix}-1 \\2 \\4 \\\end{matrix} \right)$, $\mathbf{q}=\left( \begin{matrix}3 \\-5 \\2 \\\end{matrix} \right)$ and that $(\mathbf{r}-\mathbf{p})\cdot \mathbf{q}=0$, find the relationship between $x$, $y$ and $z$. Describe the set of all possible positions of the point $R$ in this case.
[4]
Share with your friends!
(i)
[1]
(i) By considering the gradients of two lines, explain why ${{\tan }^{-1}}(2)-{{\tan }^{-1}}\left( -\frac{1}{2} \right)=\frac{1}{2}\pi $.
[1]
The curves ${{C}_{1}}$ and ${{C}_{2}}$ have equations $y=\frac{1}{{{x}^{2}}+1}$ and $y=\frac{k}{3x+4}$ respectively, where $k$ is a constant and $k>0$.
(ii)
[3]
(ii) Find the set of values of $k$ such that ${{C}_{1}}$ and ${{C}_{2}}$ intersect.
[3]
It is now given that $k=2$.
(iii)
[3]
(iii) Sketch ${{C}_{1}}$ and ${{C}_{2}}$ on the same graph, giving the coordinates of any points where ${{C}_{1}}$ or ${{C}_{2}}$ cross the axes and the equations of any asymptotes.
[3]
(iv)
[5]
(iv) Find the exact area of the region bounded by ${{C}_{1}}$ and ${{C}_{2}}$, simplifying your answer.
[5]
Share with your friends!
Scientists are investigating the effect of a disease on the number of sheep on a small island. They discover that every year the death rate of the sheep is greater than the birth rate of the sheep. The difference every year between the death rate and the birth rate for the population of sheep on the island is $3%$. The number of sheep on the island is $P$ at a time $t$ years after the scientists begin observations.
(i)
Write down a differential equation relating $P$ and $t$.
[2]
(i) Write down a differential equation relating $P$ and $t$.
[2]
(ii)
[4]
[4]
The scientists import sheep at a constant uniform rate of $n$ sheep per year. (The difference every year between the death rate and the birth rate remains at $3%$.)
(iii)
Write down a differential equation to model the new situation.
[2]
[2]
(iv)
Solve the differential equation to find an expression for $P$ in terms of $t$ and $n$.
[4]
[4]
(v)
Given that the number of sheep settles down to $500$ after many years, find $n$.
[2]
[2]
Share with your friends!
In sport science, studies are made to optimise performance in all aspects of sport from fitness to technique.
In a game, a player scores $3$ points by carrying the ball over the scoring line, shown in the diagram as $XY$. When a player has scored these $3$ points, an extra point is scored if the ball is kicked between two fixed vertical posts at $C$ and $D$. The kick can be taken from any point on the line $AB$, where $A$ is the point at which the player crossed the scoring line and $AB$ is perpendicular to $XY$.
The distance $CD$ is $4$m; $XC$ is equal to $DY$; the point $A$ is a distance $a$ m from $C$ and $A$ lies between $X$ and $C$. The kick is taken from the point $K$, where $AK$ is $x$m. The angle $CKD$ is $\theta $ (see diagram).
(i)
$\tan \theta =\frac{4x}{{{x}^{2}}+4a+{{a}^{2}}}$.
[3]
$\tan \theta =\frac{4x}{{{x}^{2}}+4a+{{a}^{2}}}$.
[3]
(ii)
[3]
[3]
The point corresponding to the value of $x$ found in part (ii) is called the optimal point. The corresponding value of $\theta $ is called the optimal angle.
(iii)
[1]
[1]
(iv)
[3]
[3]
(v)
[2]
[2]
Share with your friends!
(a)
A sequence is such that ${{u}_{1}}=p$ where $p$ is a constant and ${{u}_{n+1}}=2{{u}_{n}}-5$, for $n>0$.
(a) A sequence is such that ${{u}_{1}}=p$ where $p$ is a constant and ${{u}_{n+1}}=2{{u}_{n}}-5$, for $n>0$.
(i) Describe how the sequence behaves when
(A) $p=7$,
[1]
(B) $p=5$.
[1]
(i) Describe how the sequence behaves when
(A) $p=7$,
[1]
(B) $p=5$.
[1]
(ii) Find the value of $p$ for which ${{u}_{5}}=101$.
[2]
(ii) Find the value of $p$ for which ${{u}_{5}}=101$.
[2]
(b)
Another sequence is defined by ${{v}_{1}}=a$, ${{v}_{2}}=b$, where $a$ and $b$ are constants, and ${{v}_{n+2}}={{v}_{n}}+2{{v}_{n+1}}-7$, for $n>0$.
(b) Another sequence is defined by ${{v}_{1}}=a$, ${{v}_{2}}=b$, where $a$ and $b$ are constants, and ${{v}_{n+2}}={{v}_{n}}+2{{v}_{n+1}}-7$, for $n>0$.
For this sequence, ${{v}_{4}}=2{{v}_{3}}$.
(i) Find the value of $b$.
[3]
For this sequence, ${{v}_{4}}=2{{v}_{3}}$.
(i) Find the value of $b$.
[3]
(ii) Find an expression in terms of $a$ for ${{v}_{5}}$.
[1]
(ii) Find an expression in terms of $a$ for ${{v}_{5}}$.
[1]
(c)
The sum of the first $n$ terms of a series is ${{n}^{3}}-11{{n}^{2}}+4n$, where $n$ is a positive integer.
(c) The sum of the first $n$ terms of a series is ${{n}^{3}}-11{{n}^{2}}+4n$, where $n$ is a positive integer.
(i) Find an expression for the $n$th term of this series, giving your answer in its simplest form.
[2]
(i) Find an expression for the $n$th term of this series, giving your answer in its simplest form.
[2]
(ii) The sum of the first $m$ terms of this series, where $m>3$, is equal to the sum of the first three terms of this series. Find the value of $m$.
[2]
(ii) The sum of the first $m$ terms of this series, where $m>3$, is equal to the sum of the first three terms of this series. Find the value of $m$.
[2]
Share with your friends!
The curve $C$ is defined by the parametric equations
$x=3{{t}^{2}}+2$, $y=6t-1$ where $t\ge \frac{1}{6}$.
The line $N$ is the normal to $C$ at the point $\left( 14,\,\,11 \right)$.
(i)
Find the cartesian equation of $N$. Give your answer in the form $ax+by=c$, where $a$, $b$ and $c$ are integers to be determined.
[5]
(i) Find the cartesian equation of $N$. Give your answer in the form $ax+by=c$, where $a$, $b$ and $c$ are integers to be determined.
[5]
(ii)
Find the area enclosed by $C$, $N$ and the $x$-axis.
[4]
(ii) Find the area enclosed by $C$, $N$ and the $x$-axis.
[4]
(iii)
The curve $C$ and the line $N$ are both transformed by a $2$-way stretch, scale factor $2$ in the $x$-direction and scare factor $3$ in the $y$-direction, to form the curve $D$ and the line $M$.
(iii) The curve $C$ and the line $N$ are both transformed by a $2$-way stretch, scale factor $2$ in the $x$-direction and scare factor $3$ in the $y$-direction, to form the curve $D$ and the line $M$.
(a) Find the area enclosed by $D$,$M$ and the $x$-axis.
[1]
(a) Find the area enclosed by $D$,$M$ and the $x$-axis.
[1]
(b) Find the cartesian equation of $D$.
[2]
(b) Find the cartesian equation of $D$.
[2]
Share with your friends!
