These Ten-Year-Series (TYS) worked solutions with video explanations for 2018 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-243359@troppus.
(i)
Given that $y=\frac{\ln x}{x}$, find $\frac{\text{d}y}{\text{d}x}$ in terms of $x$.
[2]
(i) Given that $y=\frac{\ln x}{x}$, find $\frac{\text{d}y}{\text{d}x}$ in terms of $x$.
[2]
(ii)
Hence, or otherwise, find the exact value of $\int_{1}^{\text{e}}{\frac{\ln x}{{{x}^{2}}}\text{d}x}$, showing your working.
[4]
(ii) Hence, or otherwise, find the exact value of $\int_{1}^{\text{e}}{\frac{\ln x}{{{x}^{2}}}\text{d}x}$, showing your working.
[4]
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(i)
Find the exact roots of the equation $\left| 2{{x}^{2}}+3x-2 \right|=2-x$.
[4]
(i) Find the exact roots of the equation $\left| 2{{x}^{2}}+3x-2 \right|=2-x$.
[4]
(ii)
On the same axes, sketch the curves with equations $y=\left| 2{{x}^{2}}+3x-2 \right|$ and $y=2-x$.
Hence solve exactly the inequalityÂ
$\left| 2{{x}^{2}}+3x-2 \right|<2-x$.
[4]
(ii) On the same axes, sketch the curves with equations $y=\left| 2{{x}^{2}}+3x-2 \right|$ and $y=2-x$.
Hence solve exactly the inequalityÂ
$\left| 2{{x}^{2}}+3x-2 \right|<2-x$.
[4]
Â
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Functions $\text{f}$ and $\text{g}$ are defined by
$\text{f}:x\mapsto \frac{x+a}{x+b}$ for $x\in \mathbb{R},$ $x\ne -b,$ $a\ne -1,$
$g:x\mapsto x$ for $x\in \mathbb{R}$.
It is given that $\text{ff}=g$.
Find the value of $b$.
Find ${{\text{f}}^{-1}}(x)$ in terms of $x$ and $a$.
[5]
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Vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a}\ne \mathbf{0}$ and $\mathbf{a}\times 3\mathbf{b}=2\mathbf{a}\times \mathbf{c}$.
(i)
Show that $3\mathbf{b}-2\mathbf{c}=\lambda \mathbf{a}$, where $\lambda $ is a constant.
[2]
(i) Show that $3\mathbf{b}-2\mathbf{c}=\lambda \mathbf{a}$, where $\lambda $ is a constant.
[2]
(ii)
It is now given that $\mathbf{a}$ and $\mathbf{c}$ are unit vectors, that the modulus of $\mathbf{b}$ is $4$ and that the angle between $\mathbf{b}$ and $\mathbf{c}$ is $60{}^\circ $. Using a suitable scalar product, find exactly the two possible values of $\lambda $.
[5]
(ii) It is now given that $\mathbf{a}$ and $\mathbf{c}$ are unit vectors, that the modulus of $\mathbf{b}$ is $4$ and that the angle between $\mathbf{b}$ and $\mathbf{c}$ is $60{}^\circ $. Using a suitable scalar product, find exactly the two possible values of $\lambda $.
[5]
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A curve $C$ has equation $\frac{{{x}^{2}}-4{{y}^{2}}}{{{x}^{2}}+x{{y}^{2}}}=\frac{1}{2}$.
(i)
Show that $\frac{\text{d}y}{\text{d}x}=\frac{2x-{{y}^{2}}}{2xy+16y}$.
[3]
(i) Show that $\frac{\text{d}y}{\text{d}x}=\frac{2x-{{y}^{2}}}{2xy+16y}$.
[3]
The points $P$ and $Q$ on $C$ each have $x$-coordinate $1$. The tangents to $C$ at $P$ and $Q$ meet at the point $N$.
(ii)
Find the exact coordinates of $N$.
[6]
(ii) Find the exact coordinates of $N$.
[6]
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A curve $C$ has parametric equations
$x=2\theta -\sin 2\theta ,$ $y=2{{\sin }^{2}}\theta $,
for $0\le \theta \le \pi $.
(i)
Show that $\frac{\text{d}y}{\text{d}x}=\cot \theta .$
[4]
[4]
(ii)
The normal to the curve at the point where $\theta =\alpha $ meets the $x$-axis at the point $A$. Show that the $x$-coordinate of $A$ is $k\alpha $, where $k$ is a constant to be found.
[4]
(ii) The normal to the curve at the point where $\theta =\alpha $ meets the $x$-axis at the point $A$. Show that the $x$-coordinate of $A$ is $k\alpha $, where $k$ is a constant to be found.
[4]
Do not use a calculator in answering this part.
(iii)
The distance between two points along a curve is the arc-length. Scientists and engineers need to use arc-length in applications such as finding the work done in moving an object along the path described by a curve or the length of the cabling used on a suspension bridge.
The arc-length between two points on $C$, where $\theta =\beta $ and $\theta =\gamma $, is given by the formula
$\int_{\beta }^{\gamma }{\sqrt{{{\left( \frac{\text{d}x}{\text{d}\theta } \right)}^{2}}+{{\left( \frac{\text{d}y}{\text{d}\theta } \right)}^{2}}}\text{d}\theta }$
Find the total length of $C$.
[5]
(iii) The distance between two points along a curve is the arc-length. Scientists and engineers need to use arc-length in applications such as finding the work done in moving an object along the path described by a curve or the length of the cabling used on a suspension bridge.
The arc-length between two points on $C$, where $\theta =\beta $ and $\theta =\gamma $, is given by the formula
$\int_{\beta }^{\gamma }{\sqrt{{{\left( \frac{\text{d}x}{\text{d}\theta } \right)}^{2}}+{{\left( \frac{\text{d}y}{\text{d}\theta } \right)}^{2}}}\text{d}\theta }$
Find the total length of $C$.
[5]
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An electrical circuit comprises a power source of $V$ volts in series with a resistance of $R$ ohms, a capacitance of $C$ farads and an inductance of $L$ henries. The current in the circuit, $t$ seconds after turning on the power, is $I$ amps and the charge on the capacitor is $q$ coulombs. The circuit can be used by scientists to investigate resonance, to model heavily damped motion and to tune into radio stations on a stereo tuner. It is given that $R,C$ and $L$ are constants, and that $I=0$ when $t=0$.
A differential equation for the circuit is $L\frac{\text{d}I}{\text{d}t}+RI+\frac{q}{C}=V$, where $I=\frac{\text{d}q}{\text{d}t}$.
(i)
[2]
[2]
It is now given that the differential equation in part (i) holds for the rest of the question.
(ii)
[5]
[5]
(iii)
[4]
[4]
(iv)
[2]
[2]
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Mr Wong is considering investing money in a savings plan. One plane, $\text{P}$, allows him to invest $\$100$ into the account on the first day of every month. At the end of each month, the total in the account is increased by $a\%$.
(i)
It is given that $\text{a}=0.2$
(a) Mr Wong invests $\$100$ on $1$ January $2016$. Write down how much this $\$100$ is worth at the end of $31$ December $2016$.
[1]
(b) Mr Wong invests $\$100$ on the first day of each of the $12$ months of $2016$. Find the total amount in the account at the end of $31$ December $2016$.
[3]
(c) Mr Wong continues to invest $\$100$ on the first day of each month. Find the month in which the total in the account will first exceed $\$3000$. Explain whether this occurs on the first or last day of the month.
[5]
An alternative plan, $\text{Q}$, also allows him to invest $\$100$ on the first day of every month. Each $\$100$ invested earns a fixed bonus of $\$\text{b}$ at the end of every month for which it has been in the account. This is added to the account. The accumulated bonuses themselves do not earn any further bonus.
(ii)
(a) Find, in terms of $\text{b}$, how much of $\$100$ invested on $1$ January $2016$ will be worth at the end of $31$ December $2016.$
[1]
[1]
(b) Mr Wong invests $\$100$ on the first day of each of the $24$ months in $2016$ and $2017$. Find the value of $\text{b}$ such that the total value of all the investments, including bonuses, is worth $\$2800$ at the end of $31$ December $2017$.
[3]
It is given instead that $\text{a}=1$ for plan $\text{P}$
(iii)
Find the value of $\text{b}$ for plan $\text{Q}$ such that both plans give the same total value in the account at the end of the ${{60}^{\text{th}}}$ month.
[3]
[3]
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In this question you may use expansions from the List of Formulae (MF26).
(i)
Find the Maclurin expansion of $\ln (\cos 2x)$ in ascending powers of $x$, up to and including the term ${{x}^{6}}$. State any value(s) of $x$ in the domain $0\le x\le \frac{1}{4}\pi $ for which the expansion is not valid.
[6]
[6]
(ii)
[3]
[3]
(iii)
Use your graphing calculator to find a second approximate value for $\int_{0}^{0.5}{\frac{\ln (\cos 2x)}{{{x}^{2}}}\text{d}x}$, giving your answer to $4$decimal places.
[1]
[1]
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In a computer game, a bug moves from left to right through a network of connected paths. The bug starts at $\text{S}$ and, at each junction, randomly takes the left fork with probability $p$ or the right fork with probability $q$, where $q=1-p$. The forks taken at each junction are independent. The bug finishes its journey at one of the $9$ endpoints labelled $\text{A}-\text{I}$ (see diagram).
(i)
[2]
[2]
(ii)
[4]
(ii) Given that the probability that the bug finishes its journey at $\text{D}$ is greater than the probability that the bug finishes its journey at any one of the other endpoints, find exactly the possible range of values of $p$.
[4]
In another version of the game, the probability that, at each junction, the bug takes the left fork is $0.9p$, the probability that the bug takes the right fork is $0.9q$ and the probability that the bug is swallowed up by a ‘black hole’ is $0.1$.
(iii)
[1]
[1]
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The events $A$, $B$ and $C$ are such that $P(A)=a$, $\text{P}\left( B \right)=b$ and $\text{P}\left( C \right)=c$. $A$ and $B$ are independent events. $A$ and $C$ are mutually exclusive events.
(i)
Find an expression for $\text{P}\left( A’\cap B’ \right)$ and hence prove that $A’$ and $B’$ are independent events.
[2]
(i) Find an expression for $\text{P}\left( A’\cap B’ \right)$ and hence prove that $A’$ and $B’$ are independent events.
[2]
(ii)
Find an expression for $\text{P}\left( A’\cap C’ \right)$. Draw a Venn diagram to illustrate to case when $A’$ and $C’$ are also mutually exclusive events. (You should not show event $B$on your diagram.)
[2]
(ii) Find an expression for $\text{P}\left( A’\cap C’ \right)$. Draw a Venn diagram to illustrate to case when $A’$ and $C’$ are also mutually exclusive events. (You should not show event $B$on your diagram.)
[4]
You are now given that $A’$ and $C’$ are not mutually exclusive, $P(A)=\frac{2}{5}$, $\text{P}\left( B\cap C \right)=\frac{1}{5}$ and $\text{P}\left( A’\cap B’\cap C’ \right)=\frac{1}{10}$.
(iii)
Find exactly the maximum and minimum possible values of $\text{P}\left( A\cap B \right)$.
[4]
(iii) Find exactly the maximum and minimum possible values of $\text{P}\left( A\cap B \right)$.
[1]
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