# 2020 A Level H2 Math

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Paper 1
Paper 2
##### 2020 A Level H2 Math Paper 1 Question 3

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]

##### 2020 A Level H2 Math Paper 1 Question 4

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)

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]

(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)

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]

(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]

##### 2020 A Level H2 Math Paper 1 Question 5

(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]

(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)

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]

(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]

(i) Given that $\mathbf{q}$ is non-zero and $(\mathbf{r}-\mathbf{p})\times \mathbf{q}=\mathbf{0}$, describe geometrically the set of all possible positions of the point $R$.

[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]

##### 2020 A Level H2 Math Paper 1 Question 9Â

(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]

(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)

Find the set of values of $k$ such that ${{C}_{1}}$ and ${{C}_{2}}$ intersect.

[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)

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]

(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)

Find the exact area of the region bounded by ${{C}_{1}}$ and ${{C}_{2}}$, simplifying your answer.

[5]

(iv) Find the exact area of the region bounded by ${{C}_{1}}$ and ${{C}_{2}}$, simplifying your answer.

[5]

##### 2020 A Level H2 Math Paper 1 Question 10Â

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)

Solve this differential equation to find an expression for $P$ in terms of $t$. Explain what happens to the number of sheep if this situation continues over many years.

[4]

(ii) Solve this differential equation to find an expression for $P$ in terms of $t$. Explain what happens to the number of sheep if this situation continues over many years.

[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]

(iii) Write down a differential equation to model the new situation.

[2]

(iv)

Solve the differential equation to find an expression for $P$ in terms of $t$ and $n$.

[4]

(iv) Solve the differential equation to find an expression for $P$ in terms of $t$ and $n$.

[4]

(v)

Given that the number of sheep settles down to $500$ after many years, find $n$.

[2]

(v) Given that the number of sheep settles down to $500$ after many years, find $n$.

[2]

##### 2020 A Level H2 Math Paper 1 Question 11

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)

By expressing $\theta$ as the difference of two angles, or otherwise, show that

$\tan \theta =\frac{4x}{{{x}^{2}}+4a+{{a}^{2}}}$.

[3]

(i) By expressing $\theta$ as the difference of two angles, or otherwise, show that

$\tan \theta =\frac{4x}{{{x}^{2}}+4a+{{a}^{2}}}$.

[3]

(ii)

Find, in terms of $a$, the value of $x$ which maximises $\tan \theta$, simplifying your answer. Find also the corresponding value of $\tan \theta$. (You need not show that your answer gives a maximum.)

[3]

(ii) Find, in terms of $a$, the value of $x$ which maximises $\tan \theta$, simplifying your answer. Find also the corresponding value of $\tan \theta$. (You need not show that your answer gives a maximum.)

[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)

Explain why a player may decide not to take the kick from the optimal point.

[1]

(iii) Explain why a player may decide not to take the kick from the optimal point.

[1]

(iv)

Show that, when $\theta$ is the optimal angle, $\tan KDA=\sqrt{\frac{a}{4+a}}$. Find the approximate value of angle $KDA$ when $a$ is much greater than $4$.

[3]

(iv) Show that, when $\theta$ is the optimal angle, $\tan KDA=\sqrt{\frac{a}{4+a}}$. Find the approximate value of angle $KDA$ when $a$ is much greater than $4$.

[3]

(v)

It is given that the length of the scoring line $XY$ is $50$m. Find the range in which the optimal angle lies as the location of $A$ varies between $X$ and $C$.

[2]

(v) It is given that the length of the scoring line $XY$ is $50$m. Find the range in which the optimal angle lies as the location of $A$ varies between $X$ and $C$.

[2]

##### 2020 A Level H2 Math Paper 2 Question 2

(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]

##### 2020 A Level H2 Math Paper 2 Question 3

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]

##### 2020 A Level H2 Math Paper 2 Question 10

Carbon steel is made by adding carbon to iron; this makes the iron stronger, though less flexible. A steel manufacturing company makes carbon steel in the form of round bars. The process is designed to manufacture bars in which the amount of carbon in each bar, by weight, is $1.5\%$. It is known that the percentage of carbon in the steel bars is distributed normally, and that the standard deviation is $0.09\%$.

After comments from customers, the production manager wishes to test, at the $5\%$ level of significance, if the percentage of carbon in the steel bars is, in fact, $1.5\%$. He examines a random sample of $15$ bars to determine the percentage of carbon in each bar.

(i)

Find the critical region for this test.

[4]

(i) Find the critical region for this test.

[4]

The company recently launched a new line of flat bars made from mild steel. In mild steel the amount of carbon in each bar, by weight, is $0.25\%$.

Comments from customers suggests that these bars are not sufficiently flexible, which makes the production manager suspect that too much carbon has been added. He decides to perform a hypothesis test on a random sample of $40$ of the new flat bars to find out if this is the case.

(ii)

Explain why the production manager takes a sample of $40$ flat bars for this test when he only took a sample of $15$ round bars in his earlier test.

[2]

(ii) Explain why the production manager takes a sample of $40$ flat bars for this test when he only took a sample of $15$ round bars in his earlier test.

[2]

The amount of carbon, $x\%$, in a random sample of $40$ flat bars are summarised as follows.

$n=40$Â  Â  Â  Â $\sum{x=10.16}$Â  Â  Â  Â $\sum{{{x}^{2}}=2.586342}$

(iii)

Calculate unbiased estimates of the population mean and variance for the percentage amount of carbon in the flat bars.

[2]

(iii) Calculate unbiased estimates of the population mean and variance for the percentage amount of carbon in the flat bars.

[2]

(iv)

Test, at the $2\frac{1}{2}\%$ level of significance, whether the mean amount of carbon in the flat bars is more than $0.25\%$. You should state your hypotheses and define any symbols that you use.

[5]

(iv)

Test, at the $2\frac{1}{2}\%$ level of significance, whether the mean amount of carbon in the flat bars is more than $0.25\%$. You should state your hypotheses and define any symbols that you use.

[5]

##### Suggested Handwritten Solutions

In the earlier test, the percentage of carbon in the carbon steel is known to be normally distributed. Thus, the sample mean would also be normally distributed.
However, the percentage of carbon in the mild steel is not known to be normally distributed. A large sample (in this case $40$ bars) is required so that the sample mean would be approximately normal by Central Limit Theorem.

In the earlier test, the percentage of carbon in the carbon steel is known to be normally distributed. Thus, the sample mean would also be normally distributed.
However, the percentage of carbon in the mild steel is not known to be normally distributed. A large sample (in this case $40$ bars) is required so that the sample mean would be approximately normal by Central Limit Theorem.