These Ten-Year-Series (TYS) worked solutions with video explanations for 2014 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-9839b4@troppus.
The function $\text{f}$ is defined by
$\text{f}:x\mapsto \frac{1}{1-x},x\in \mathbb{R},x\ne 1,x\ne 0$ .
(i)
Show that ${{\text{f}}^{2}}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.
[4]
(i) Show that ${{\text{f}}^{2}}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.
[4]
(ii)
Find ${{\text{f}}^{3}}\left( x \right)$ and ${{\text{f}}^{2020}}\left( x \right)$ in simplified forms.
[3]
(ii) Find ${{\text{f}}^{3}}\left( x \right)$ and ${{\text{f}}^{2020}}\left( x \right)$ in simplified forms.
[3]
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It is given that $\text{f}\left( x \right)={{x}^{6}}-3{{x}^{4}}-7.$ The diagram shows the curve with equation $y=\text{f}\left( x \right)$and the line with equation$y=-7,$for $x\ge 0.$ The curve crosses the positive $x$-axis at $x=\alpha ,$and the curve and the line meet where $x=0$ and $x=\beta .$
(i)
Find the value of $\alpha ,$giving your answer correct to $3$ decimal places, and find the exact value of $\beta .$
[2]
(i) Find the value of $\alpha ,$giving your answer correct to $3$ decimal places, and find the exact value of $\beta .$
[2]
(ii)
Evaluate $\int_{\beta }^{\alpha }{\text{f}\left( x \right)\text{d}x},$giving your answer correct to $3$ decimal places.
[2]
(ii) Evaluate $\int_{\beta }^{\alpha }{\text{f}\left( x \right)\text{d}x},$giving your answer correct to $3$ decimal places.
[2]
(iii)
Find, in terms of $\sqrt{3},$the area of the finite region bounded by the curve and the line, for $x\ge 0.$
[3]
(iii) Find, in terms of $\sqrt{3},$the area of the finite region bounded by the curve and the line, for $x\ge 0.$
[3]
(iv)
Show that $\text{f}\left( x \right)=\text{f}\left( -x \right).$What can be said about the six roots of the equation $\text{f}\left( x \right)=0?$ [
4]
(iv) Show that $\text{f}\left( x \right)=\text{f}\left( -x \right).$What can be said about the six roots of the equation $\text{f}\left( x \right)=0?$
[4]
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The mass, $x$ grams, of a certain substance present in a chemical reaction at time $t$ minutes satisfies the differential equation $\frac{\text{d}x}{\text{d}t}=k\left( 1+x-{{x}^{2}} \right)$, where $0\le x\le \frac{1}{2}$ and $k$ is a constant. It is given that $x=\frac{1}{2}$ and $\frac{\text{d}x}{\text{d}t}=-\frac{1}{4}$ when $t=0$.
(i)
Show that $k=-\frac{1}{5}$.
[1]
(i) Show that $k=-\frac{1}{5}$.
[1]
(ii)
By first expressing $1+x-{{x}^{2}}$ in completed square from, find $t$ in terms of $x$.
[5]
(ii) By first expressing $1+x-{{x}^{2}}$ in completed square from, find $t$ in terms of $x$. [5]
(iii)
Hence find
(iii) Hence find
(a) the exact time taken for the mass of the substance present in the chemical reaction to become half of its initial value,
[1]
(a) the exact time taken for the mass of the substance present in the chemical reaction to become half of its initial value,
[1]
(b) the time taken for there to be none of the substance present in the chemical reaction, giving your answer correct to $3$ decimal places.
[1]
(b) the time taken for there to be none of the substance present in the chemical reaction, giving your answer correct to $3$ decimal places.
[1]
(iv)
Express the solution of the differential equation in the form $x=\text{f}\left( t \right)$ and sketch the part of the curve with this equation which is relevant in this context.
[5]
(iv) Express the solution of the differential equation in the form $x=\text{f}\left( t \right)$ and sketch the part of the curve with this equation which is relevant in this context.
[5]
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[It is given that the volume of a sphere of radius $r$ is $\frac{4}{3}\pi {{r}^{3}}$ and that the volume of a circular cone with base radius $r$ and height $h$ is $\frac{1}{3}\pi {{r}^{2}}h$.]
A toy manufacturer makes a toy which consists of a hemisphere of radius $r$ cm joined to a circular cone of base radius $r$ cm and height $h$ cm (see diagram). The manufacturer determines that the length of the slant edge of the cone must be $4$ cm and that the total volume of the toy, $V$ cm$^{3}$, should be as large as possible.
(i)
Find a formula for $V$ in terms of $r$. Given that $r={{r}_{1}}$ is the value of $r$ which gives the maximum value of $V$, show that ${{r}_{1}}$ satisfies the equation $45{{r}^{4}}-768{{r}^{2}}+1024=0$.
[6]
(i) Find a formula for $V$ in terms of $r$. Given that $r={{r}_{1}}$ is the value of $r$ which gives the maximum value of $V$, show that ${{r}_{1}}$ satisfies the equation $45{{r}^{4}}-768{{r}^{2}}+1024=0$.
[6]
(ii)
Find the two solutions to the equation in part (i) for which $r>0$, giving your answers correct to $3$ decimal places.
[2]
(ii) Find the two solutions to the equation in part (i) for which $r>0$, giving your answers correct to $3$ decimal places.
[2]
(iii)
Show that one of the solutions found in part (ii) does not give a stationary value of $V$. Hence write down the value of ${{r}_{1}}$ and find the corresponding value of $h$.
[3]
(iii) Show that one of the solutions found in part (ii) does not give a stationary value of $V$. Hence write down the value of ${{r}_{1}}$ and find the corresponding value of $h$.
[3]
(iv)
Sketch the graph showing the volume of the toy as the radius of the hemisphere varies.
[3]
(iv) Sketch the graph showing the volume of the toy as the radius of the hemisphere varies.
[3]
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In a training exercise, athletes run from a starting point $O$ to and from a series of points, ${{A}_{1}}$, ${{A}_{2}}$, ${{A}_{3}}$,….., increasingly far away in a straight line. In the exercise, atheletes start at $O$ and run stage 1 from $O$ to ${{A}_{1}}$ and back to $O$, then stage 2 from $O$ to ${{A}_{2}}$ and back to $O$, and so on.
(i)
In Version 1 of the exercise, the distances between adjacent points are all $4$m (see Fig. 1).
(a) Find the distance run by an athlete who completes the first $10$ stages of Version 1 of the exercise.
[2]
In Version 1 of the exercise, the distances between adjacent points are all $4$m (see Fig. 1).
(a) Find the distance run by an athlete who completes the first $10$ stages of Version 1 of the exercise.
[2]
(b) Write down an expression for the distance run by an athlete who completes $n$ stages of Version I. Hence find the least number of stages that the athlete needs to complete to run at least $5$km.
[4]
(b) Write down an expression for the distance run by an athlete who completes $n$ stages of Version I. Hence find the least number of stages that the athlete needs to complete to run at least $5$km.
[4]
(ii)
In Version 2 of the exercise, the distances between the points are such that $O{{A}_{1}}=4$m, ${{A}_{1}}{{A}_{2}}=4$m, ${{A}_{2}}{{A}_{3}}=8$m, ${{A}_{n}}{{A}_{n+1}}=2{{A}_{n-1}}{{A}_{n}}$ (see Fig. 2). Write down an expression for the distance run by an athlete who completes $n$ stages of Version 2.
In Version 2 of the exercise, the distances between the points are such that $O{{A}_{1}}=4$m, ${{A}_{1}}{{A}_{2}}=4$m, ${{A}_{2}}{{A}_{3}}=8$m, ${{A}_{n}}{{A}_{n+1}}=2{{A}_{n-1}}{{A}_{n}}$ (see Fig. 2). Write down an expression for the distance run by an athlete who completes $n$ stages of Version 2.
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It is given that $w=\sqrt{3}-\text{i}$.
(i)
Without using a calculator, find an exact expression for ${{w}^{6}}$. Give your answer in the form $r{{\text{e}}^{\text{i}\theta }}$, where $r>0$ and $0\le \theta <2\pi .$
(i) Without using a calculator, find an exact expression for ${{w}^{6}}$. Give your answer in the form $r{{\text{e}}^{\text{i}\theta }}$, where $r>0$ and $0\le \theta <2\pi .$
(ii)
Without using a calculator, find the three smallest positive whole number values for $n$ for which $\frac{{{w}^{n}}}{{{w}^{*}}}$ is a real number.
(ii) Without using a calculator, find the three smallest positive whole number values for $n$ for which $\frac{{{w}^{n}}}{{{w}^{*}}}$ is a real number.
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A team in a particular sport consists of 1 goalkeeper, 4 defenders, 2 midfielders and 4 attackers.
A certain club has 3 goalkeepers, 8 defenders, 5 midfielders and 6 attackers.
(i)
How many different teams can be formed by the club?
[2]
(i) How many different teams can be formed by the club?
[2]
One of the midfielders in the club is the brother of one of the attackers in the club.
(ii)
How many different teams can be formed which include exactly one of the two brothers?
[3]
(ii) How many different teams can be formed which include exactly one of the two brothers?
[3]
The two brothers leave the club. The club manager decides that one of the remaining midfielders can play as either a midfielder or as a defender.
(iii)
How many different teams can now be formed by the club?
[3]
(iii) How many different teams can now be formed by the club?
[3]
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