# 2018 A Level H2 Math

2021

2022

2023

2020

2019

2018

2017

2016

2015

2014

2013

2012

2011

2010

2009

2008

2007

2006

2005

2004

2003

2002

2001

2000

1995

1994

1993

1991

1990

1986

1984

1983

1982

### Â  Â 1971-1980

1978

1977

1973

Paper 1
Paper 2
##### 2018 A Level H2 Math Paper 1 Question 1

(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]

##### 2018 A Level H2 Math Paper 1 Question 4

(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

##### 2018 A Level H2 Math Paper 1 Question 5

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]

##### 2018 A Level H2 Math Paper 1 Question 6

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]

##### 2018 A Level H2 Math Paper 1 Question 7

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]

##### 2018 A Level H2 Math Paper 1 Question 9

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]

(i) Show that $\frac{\text{d}y}{\text{d}x}=\cot \theta .$

[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]

##### 2018 A Level H2 Math Paper 1 Question 10

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)

Show that, under certain conditions on $V$ which should be stated, $L\frac{{{\text{d}}^{2}}I}{\text{d}{{t}^{2}}}+R\frac{\text{d}I}{\text{d}t}+\frac{I}{C}=0$

[2]

(i) Show that, under certain conditions on $V$ which should be stated, $L\frac{{{\text{d}}^{2}}I}{\text{d}{{t}^{2}}}+R\frac{\text{d}I}{\text{d}t}+\frac{I}{C}=0$

[2]

It is now given that the differential equation in part (i) holds for the rest of the question.

(ii)

Given that $I=\text{A}t{{e}^{-\frac{Rt}{2L}}}$ is a solution of the differential equation, where $\text{A}$ is a positive constant, show that $C=\frac{4L}{{{R}^{2}}}$.

[5]

(ii) Given that $I=\text{A}t{{e}^{-\frac{Rt}{2L}}}$ is a solution of the differential equation, where $\text{A}$ is a positive constant, show that $C=\frac{4L}{{{R}^{2}}}$.

[5]

(iii)

In a particular circuit, $R=4,\,L=3$ and $C=0.75$. Find the maximum value of $I$ in terms of $A$, showing that this value is a maximum.

[4]

(iii) In a particular circuit, $R=4,\,L=3$ and $C=0.75$. Find the maximum value of $I$ in terms of $A$, showing that this value is a maximum.

[4]

(iv)

Sketch the graph of $I$ against $t$.

[2]

(iv) Sketch the graph of $I$ against $t$.

[2]

##### 2018 A Level H2 Math Paper 1 Question 11

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$\text{a}=0.2$(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]