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 gs.ude.htamnagmitobfsctd-811acc@troppus.
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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(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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Do not use a calculator in answering this question
The complex number $z$ is given by
$z=\frac{{{\left( \cos \left( \frac{1}{16}\pi \right)+\text{i}\sin \left( \frac{1}{16}\pi \right) \right)}^{2}}}{\cos \left( \frac{1}{8}\pi \right)-\text{i}\sin \left( \frac{1}{8}\pi \right)}$.
(a)
Find $\left| z \right|$ and $\arg \left( z \right)$. Hence find the value of ${{z}^{2}}$.
[3]
(a) Find $\left| z \right|$ and $\arg \left( z \right)$. Hence find the value of ${{z}^{2}}$.
[3]
(b)
(i) Show that $\left( \cos \theta +\text{i}\sin \theta \right)\left( 1+\cos \theta -\text{i}\sin \theta \right)=1+\cos \theta +\text{i}\sin \theta $.
[2]
(b)(i) Show that $\left( \cos \theta +\text{i}\sin \theta \right)\left( 1+\cos \theta -\text{i}\sin \theta \right)=1+\cos \theta +\text{i}\sin \theta $.
[2]
(ii) Hence, or otherwise, find the value of ${{\left( 1+z \right)}^{4}}+{{\left( 1+{{z}^{*}} \right)}^{4}}$.
[2]
(ii) Hence, or otherwise, find the value of ${{\left( 1+z \right)}^{4}}+{{\left( 1+{{z}^{*}} \right)}^{4}}$.
[2]
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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.
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(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 $a$.
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(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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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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The lines ${{l}_{1}}$ and ${{l}_{2}}$ have equations
${{\mathbf{r}}_{1}}=\left( \begin{matrix}
3 \\
2 \\
0 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
2 \\
-3 \\
1 \\
\end{matrix} \right)$ and ${{\mathbf{r}}_{2}}=\left( \begin{matrix}
-4 \\
1 \\
-3 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
1 \\
2 \\
4 \\
\end{matrix} \right)$
respectively, where $\lambda $ and $\mu $ are parameters.
(a)
Find a cartesian equation of the plane containing ${{l}_{1}}$ and the point $\left( 1,-3,-1 \right)$.
[4]
(a) Find a cartesian equation of the plane containing ${{l}_{1}}$ and the point $\left( 1,-3,-1 \right)$.
[4]
(b)
Show that ${{l}_{1}}$ is perpendicular to ${{l}_{2}}$.
[2]
(b) Show that ${{l}_{1}}$ is perpendicular to ${{l}_{2}}$.
[2]
(c)
(i) Find values of $\lambda $ and $\mu $ such that ${{\mathbf{r}}_{1}}-{{\mathbf{r}}_{2}}$ is perpendicular to both ${{l}_{1}}$ and ${{l}_{2}}$. State the position vectors of the points where the common perpendicular meets ${{l}_{1}}$ and ${{l}_{2}}$.
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(c)(i) Find values of $\lambda $ and $\mu $ such that ${{\mathbf{r}}_{1}}-{{\mathbf{r}}_{2}}$ is perpendicular to both ${{l}_{1}}$ and ${{l}_{2}}$. State the position vectors of the points where the common perpendicular meets ${{l}_{1}}$ and ${{l}_{2}}$.
[6]
(ii) Find the length of this common perpendicular.
[2]
(ii) Find the length of this common perpendicular.
[2]
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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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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
[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, is                                                        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
[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
[4]
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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
[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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(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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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 – 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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A company manufactures a wide variety of components for use in domestic appliances.
(a)
The company introduces a new type of light fitting for refrigerators. Each day a supervisor selects a sample of $100$ of the light fittings for testing.
(a) The company introduces a new type of light fitting for refrigerators. Each day a supervisor selects a sample of $100$ of the light fittings for testing.
(i) How should the light fittings be selected? Give a reason for this method of selection.
[2]
(i) How should the light fittings be selected? Give a reason for this method of selection.
[2]
The supervisor records the number of faulty light fittings found on each of the $150$ working days.
Her results are shown in the table.
The supervisor records the number of faulty light fittings found on each of the $150$ working days.
Her results are shown in the table.
(ii) Use the information in the table to estimate $p$, the probability that a light fitting is faulty.
[1]
(ii) Use the information in the table to estimate $p$, the probability that a light fitting is faulty.
[1]
(iii) Assuming that the number of faulty fittings found each day follows the binomial distribution $\text{B}\left( 100,\text{ }p \right)$, find the expected number of days on which $3$ faulty fittings are found in a period of $150$ days.
[2]
(iii) Assuming that the number of faulty fittings found each day follows the binomial distribution $\text{B}\left( 100,\text{ }p \right)$, find the expected number of days on which $3$ faulty fittings are found in a period of $150$ days.
[2]
(b)
The company also makes heating elements for electric ovens. A fixed number of randomly chosen heating elements are tested each day and the number found to be faulty is denoted by $X$.
(b) The company also makes heating elements for electric ovens. A fixed number of randomly chosen heating elements are tested each day and the number found to be faulty is denoted by $X$.
(i) State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.
[2]
(i) State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.
[2]
Assume now that $X$ has the distribution $\text{B}\left( 80,\text{ }0.02 \right)$.
Assume now that $X$ has the distribution $\text{B}\left( 80,\text{ }0.02 \right)$.
(ii) Find the probability that, on a randomly chosen day, the number of elements found to be faulty is between $1$ and $4$ inclusive.
[2]
(ii) Find the probability that, on a randomly chosen day, the number of elements found to be faulty is between $1$ and $4$ inclusive.
[2]
(iii) Find the probability that, in a randomly chosen $5$-day working week, more than $3$ elements are found to be faulty on at least $2$ days.
[3]
(iii) Find the probability that, in a randomly chosen $5$-day working week, more than $3$ elements are found to be faulty on at least $2$ days.
[3]
(iv) Find the probability that, in a randomly chosen $5$-day working week, no more than $8$ faulty elements are found in total.
[2]
(iv) Find the probability that, in a randomly chosen $5$-day working week, no more than $8$ faulty elements are found in total.
[2]
The light fittings should be selected in a random, i.e. each light fitting has equal chance of getting selected. This ensures that the sample is unbiased and gives a fair representation of the population of light fittings manufactured by the company.
The light fittings should be selected in a random, i.e. each light fitting has equal chance of getting selected. This ensures that the sample is unbiased and gives a fair representation of the population of light fittings manufactured by the company.
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