# 2021 A Level H2 Math

These Ten-Year-Series (TYS) worked solutions with video explanations for 2021 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.

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Paper 1
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##### 2021 A Level H2 Math Paper 1 Question 1

A function $\text{f}$ is defined by $\text{f}\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d$. The graph of $y=\text{f}\left( x \right)$ passes through the points $\left( 1,5 \right)$ and $\left( -1,-3 \right)$. The graph has a turning point at $x=1$, and $\int_{0}^{1}{\text{f}\left( x \right)}\text{d}x=6$.
Find the values of $a$, $b$, $c$ and $d$.

[5]

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##### 2013 A Level H2 Math Paper 1 Question 2

(a)

Sketch, on the same axes, the graphs of $y=\left| {{x}^{2}}-5 \right|$ and $y=2x-1$.

[2]

(a) Sketch, on the same axes, the graphs of $y=\left| {{x}^{2}}-5 \right|$ and $y=2x-1$.

[2]

(b)

Find the exact solutions of $\left| {{x}^{2}}-5 \right|=2x-1$.

[4]

(b) Find the exact solutions of $\left| {{x}^{2}}-5 \right|=2x-1$.

[4]

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##### 2021 A Level H2 Math Paper 1 Question 6

A curve $C$ has equation $y=\frac{1}{\sqrt{4ax-{{x}^{2}}}}$, where $a>0$.

(a)

Sketch $C$ and give the equations of any asymptotes, in terms of $a$ where appropriate.

[4]

(a) Sketch $C$ and give the equations of any asymptotes, in terms of $a$ where appropriate.

[4]

(b)

Find the smallest possible value of $y$ in terms of $a$.

[1]

(b) Find the smallest possible value of $y$ in terms of $a$.

[1]

(c)

Describe the transformation that maps the graph of $C$ onto the graph of $y=\frac{1}{\sqrt{4{{a}^{2}}-{{x}^{2}}}}$.

[3]

(c) Describe the transformation that maps the graph of $C$ onto the graph of $y=\frac{1}{\sqrt{4{{a}^{2}}-{{x}^{2}}}}$.

[3]

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##### 2021 A Level H2 Math Paper 1 Question 7

It is given that $y={{\text{e}}^{{{\sin }^{-1}}x}}$, for $-1<x<1$.

(a)

Show that $\left( 1-{{x}^{2}} \right)\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=x\frac{\text{d}y}{\text{d}x}+y$.

[4]

(a) Show that $\left( 1-{{x}^{2}} \right)\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=x\frac{\text{d}y}{\text{d}x}+y$.

[4]

(b)

Find the first $4$ terms of the Maclaurin expansion of ${{\text{e}}^{{{\sin }^{-1}}x}}$.

[5]

(b) Find the first $4$ terms of the Maclaurin expansion of ${{\text{e}}^{{{\sin }^{-1}}x}}$.

[5]

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##### 2021 A Level H2 Math Paper 1 Question 9

A function $\text{f}$ is defined by $\text{f}\left( x \right)={{\text{e}}^{x}}\cos x$, for $0\le x\le \frac{1}{2}\pi$.

(a)

Using calculus, find the stationary point of $\text{f}\left( x \right)$ and determine its nature.

[5]

(a) Using calculus, find the stationary point of $\text{f}\left( x \right)$ and determine its nature.

[5]

(b)

Integrate by parts twice to show that

$\int{{{\text{e}}^{2x}}\cos 2x\text{d}x=\frac{1}{4}{{\text{e}}^{2x}}\left( \sin 2x+\cos 2x \right)+c}$.

[4]

(b) Integrate by parts twice to show that

$\int{{{\text{e}}^{2x}}\cos 2x\text{d}x=\frac{1}{4}{{\text{e}}^{2x}}\left( \sin 2x+\cos 2x \right)+c}$.

[4]

(c)

The graph of $y=\text{f}\left( x \right)$ is rotated completely about the $x$-axis. Find the exact volume generated.

[4]

(c) The graph of $y=\text{f}\left( x \right)$ is rotated completely about the $x$-axis. Find the exact volume generated.

[4]

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##### 2021 A Level H2 Math Paper 1 Question 11

Civil engineers design bridges to span over expressways. The diagram below represents a bridge over an expressway, $PS$.

In the diagram, $PQ$ and $SR$ are parallel to the $y$-axis, and $PQ=SR$. The arch of the bridge, $QR$, forms part of the curve $OQRC$ with parametric equations

$x=a\left( \theta -\sin \theta \right)$, $y=a\left( 1-\cos \theta \right)$, for $0\le \theta \le 2\pi$,

where $a$ is a positive constant. The units of $x$ and $y$ are metres.

At the point $Q$, $\theta =\beta$ and at the point $R$, $\theta =2\pi -\beta$

(a)

Find, in terms of $a$ and $\beta$, the distance $PS$.

[2]

(a)Find, in terms of $a$ and $\beta$, the distance $PS$.Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

[2]

(b)

Show that the area of the shaded region on the diagram, representing the area under the bridge, is

$\frac{1}{2}{{a}^{2}}\left( 6\pi -6\beta +8\sin \beta -\sin 2\beta \right)$

[6]

(b) Show that the area of the shaded region on the diagram, representing the area under the bridge, isÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â Â

$\frac{1}{2}{{a}^{2}}\left( 6\pi -6\beta +8\sin \beta -\sin 2\beta \right)$

[6]

(c)

It is given that the area under the bridge, in square metres, is $7.8159{{a}^{2}}$. Find the value of $\beta$.

[1]

(c) It is given that the area under the bridge, in square metres, is $7.8159{{a}^{2}}$. Find the value of $\beta$.Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

[1]

(d)

The width of the expressway, $PS$ is $50$ metres. Find the greater and least heights of the arch, $QR$, above the expressway.

[4]

(d) The width of the expressway, $PS$ is $50$ metres. Find the greater and least heights of the arch, $QR$, above the expressway.Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â Â

[4]

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##### 2021 A Level H2 Math Paper 2 Question 2

The diagram shows a sketch of the curve $y=\text{f}\left( x \right)$. The region under the curve between $x=1$ and $x=5$, shown shaded in the diagram, is $A$. This region is split into $5$ vertical strips of equal width, $h$.

(a)

State the value of $h$ and show, using a sketch, that $\sum\limits_{n=0}^{4}{\left( \text{f}\left( 1+nh \right) \right)}\,h$ is less than the area of $A$.

[2]

(a) State the value of $h$ and show, using a sketch, that $\sum\limits_{n=0}^{4}{\left( \text{f}\left( 1+nh \right) \right)}\,h$ is less than the area of $A$.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

[2]

(b)

Find a similar expression that is greater than the area of $A$.

[1]

(b) Find a similar expression that is greater than the area of $A$.

[1]

You are now given that $\text{f}\left( x \right)=\frac{1}{20}{{x}^{2}}+1$.

(c)

Use the expression given in part (a) and your expression from part (b) to find lower and upper bounds for the area of $A$.

[2]

(c) Use the expression given in part (a) and your expression from part (b) to find lower and upper bounds for the area of $A$.

[2]

(d)

Sketch the graph of a function $y=\text{g}\left( x \right)$, between $x=1$ and $x=5$, for which the area between the curve, the $x$-axis and the lines $x=1$ and $x=5$ is less than $\sum\limits_{n=0}^{4}{\left( \text{g}\left( 1+nh \right) \right)}\,h$.

[1]

(d) Sketch the graph of a function $y=\text{g}\left( x \right)$, between $x=1$ and $x=5$, for which the area between the curve, the $x$-axis and the lines $x=1$ and $x=5$ is less than $\sum\limits_{n=0}^{4}{\left( \text{g}\left( 1+nh \right) \right)}\,h$.

[1]

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##### 2021 A Level H2 Math Paper 2 Question 3

(a)

The function $\text{h}$ is defined by $\text{h}:x\mapsto \frac{1}{2}{{x}^{2}}+3$, for $x\in \mathbb{R}$.
The function $\text{g}$ is defined by $\text{g}:x\mapsto \frac{x+1}{5x-1}$, for $x\in \mathbb{R}$, $x\ne 0.2$.

(a) The function $\text{h}$ is defined by $\text{h}:x\mapsto \frac{1}{2}{{x}^{2}}+3$, for $x\in \mathbb{R}$.
The function $\text{g}$ is defined by $\text{g}:x\mapsto \frac{x+1}{5x-1}$, for $x\in \mathbb{R}$, $x\ne 0.2$.

(i) Find $\text{gh}\left( 2 \right)$.

[2]

(i) Find $\text{gh}\left( 2 \right)$.

[2]

(ii) Find the value of $x$ for which $\text{g}\left( x \right)=1.4$.

[1]

(ii) Find the value of $x$ for which $\text{g}\left( x \right)=1.4$.

[1]

(b)

The function $\text{f}$ is defined by $\text{f}:x\mapsto \frac{x+a}{2x+b}$, for $x\in \mathbb{R}$, $x\ne k$.

(b) The function $\text{f}$ is defined by $\text{f}:x\mapsto \frac{x+a}{2x+b}$, for $x\in \mathbb{R}$, $x\ne k$.

(i) Give an expression for $k$ and explain why this value of $x$ has to be excluded from the domain of $\operatorname{f}$.

[2]

(i) Give an expression for $k$ and explain why this value of $x$ has to be excluded from the domain of $\operatorname{f}$.

[2]

The function $\text{f}$ is such that $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$ for all $x$ in the domain of $\text{f}$.

The function $\text{f}$ is such that $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$ for all $x$ in the domain of $\text{f}$.

(ii) Determine the possible values of $a$ and of $b$.

[3]

(ii) Determine the possible values of $a$ and of $b$.

[3]

(iii) Find an expression for ${{\text{f}}^{-1}}\left( -4 \right)$.

[1]

(iii) Find an expression for ${{\text{f}}^{-1}}\left( -4 \right)$.

[1]

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##### 2021 A Level H2 Math Paper 2 Question 10

In this question you should state the parameters of any normal distributions you use.

A company makes $3$-legged wooden stools from $4$ solid components â€“ a seat in the form of a disc, and $3$ legs each in the form of a long, thin, cylinder. The seats and legs are bought in bulk from another company. Over a period of time it is found that the masses of the seats are normally distributed; $80\%$ of the seats have mass less than $2.1$ kg, and $15\%$ of the seats have mass less than $1.95$ kg.

(a)

Find the mean mass of the seats and show that the standard deviation is $0.0799$ kg, correct to $3$ significant figures.

[3]

(a) Find the mean mass of the seats and show that the standard deviation is $0.0799$ kg, correct to $3$ significant figures.

[3]

The masses of the legs, in kg, follow the distribution $\text{N}\left( 1.2,\text{ }{{0.02}^{2}} \right)$.

(b)

Find the expected number of legs with mass more than $1.21$ kg in a randomly chosen batch of $500$ legs.

[2]

(b) Find the expected number of legs with mass more than $1.21$ kg in a randomly chosen batch of $500$ legs.

[2]

(c)

Find the probability that the total mass of a randomly chosen seat and $3$ randomly chosen legs is between $5.6$ kg and $5.7$ kg.

[3]

(c) Find the probability that the total mass of a randomly chosen seat and $3$ randomly chosen legs is between $5.6$ kg and $5.7$ kg.

[3]

In order to make the stools, circular holes are drilled in the seats and the legs are fitted into them. In this process, the mass of seats is modelled as being reduced by $9\%$ and the masses of the legs are unchanged.

(d)

Find the probability that the total mass of a randomly chosen drilled seat and $3$ randomly chosen legs is less than $5.6$ kg.

[3]

(d) Find the probability that the total mass of a randomly chosen drilled seat and $3$ randomly chosen legs is less than $5.6$ kg.

[3]

The holes made in the seats have diameters, in mm, that follow the distribution $\text{N}\left( 31,\text{ }{{0.4}^{2}} \right)$ and the diameters of the legs, in mm, follow the distribution $\text{N}\left( 30.7,\text{ }{{0.3}^{2}} \right)$. If the diameter of a leg is greater than the diameter of a hole, then the leg has to be sanded down to make it fit. If the diameter of a hole is more than $0.8$ mm greater than the diameter of a leg, then padding has to be added when the leg is glued to the seat.

(e)

A stool is made of a randomly chosen drilled seat and $3$ randomly chosen legs. The legs are paired up with the holes at random. Find the probability that the $3$ legs can be fitted without the need for any sanding or padding.

[4]

(e) A stool is made of a randomly chosen drilled seat and $3$ randomly chosen legs. The legs are paired up with the holes at random. Find the probability that the $3$ legs can be fitted without the need for any sanding or padding.

[4]

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