These Ten-Year-Series (TYS) worked solutions with video explanations for 2022 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-516823@troppus.
A curve $~C$ has equation $y=ax+b+\frac{a+2b}{x-1}$, where $a$ and $b$ are real constants such that $a>0$, $b\ne -\frac{1}{2}a$ and $x\ne 1$.
(a)
Given that $~C$ has no stationary points, use differentiation to find the relationship between $a$ and $b$.
[3]
(a) Given that $~C$ has no stationary points, use differentiation to find the relationship between $a$ and $b$.
[3]
It is now given that $b=-2a$.
(b)
Sketch $~C$ on the axes below stating the equations of any asymptotes and the coordinates of the points where$~C$ crosses the axes.
[4]
(b) Sketch $~C$ on the axes below stating the equations of any asymptotes and the coordinates of the points where$~C$ crosses the axes.
[4]
(c)
On the same axes, sketch the graph of $y=ax-a$.
[1]
(c) On the same axes, sketch the graph of $y=ax-a$.
[1]
(d)
Hence solve the inequality $x-2-\frac{3}{x-1}\le x-1$.
[2]
(d) Hence solve the inequality $x-2-\frac{3}{x-1}\le x-1$.
[2]
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A gas company has plans to install a pipeline from a gas field to a storage facility. One part of the route for the pipeline has to pass under a river. This part of the pipeline is in a straight line between two points, $P$ and $Q$.
Points are defined relative to an origin $\left( 0,0,0 \right)$ at the gas field. The $x$-, $y$- and $z$- axes are in the directions east, north and vertically upwards respectively, with units in metres. $P$ has coordinates $\left( 1136,\,\,92,\,\,p \right)$ and $Q$ has coordinates $\left( 200,\,\,20,\,\,-15 \right)$.
(a)
The length of the pipeline $PQ$ is $939$ m. Given that the level of $P$ is below that of $Q$, find the value of $p$.
[3]
(a) The length of the pipeline $PQ$ is $939$ m. Given that the level of $P$ is below that of $Q$, find the value of $p$.
[3]
A thin layer of rock lies below the ground. This layer is modelled as a plane. Three points in this plane are $\left( 400,\,600,\,-20 \right)$, $\left( 500,\,200,\,-70 \right)$ and $\left( 600,\,-340,\,-50 \right)$.
(b)
Find the cartesian equation of this plane.
[4]
(b) Find the cartesian equation of this plane.
[4]
(c)
Hence find the coordinates of the point where the pipeline meets the rock.
[4]
(c) Hence find the coordinates of the point where the pipeline meets the rock.
[4]
(d)
Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal.
[2]
(d) Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal.
[2]
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Scientists are interested in the population of a particular species. They attempt to model the population $P$ at time $t$ days using a differential equation. Initially the population is observed to be $50$ and after $10$ days the population is $100$.
The first model the scientists use assumes that the rate of change of the population is proportional to the population.
(a)
Write down a differential equation for this model and solve it for $P$ in terms of $t$.
[5]
(a) Write down a differential equation for this model and solve it for $P$ in terms of $t$.
[5]
To allow for constraints on population growth, the model is refined to
$\frac{\text{d}P}{\text{d}t}=\lambda P\left( 500-P \right)$
where $\lambda $ is a constant.
(b)
Solve this differential equation to find $P$ in terms of $t$.
[6]
(b) Solve this differential equation to find $P$ in terms of $t$.
[6]
(c)
Using the refined model, state the population of this species in the long term. Comment on how this value suggests the refined model is an improvement on the first model.
[2]
(c) Using the refined model, state the population of this species in the long term. Comment on how this value suggests the refined model is an improvement on the first model.
[2]
In the long run, the population of this species will increase from $50$ and stabalise at $500$ in the refined model whereas the first model would suggest that the population will increase indefinitely. Thus, the refined model is better as in real life a population growth of a species will be limited by external factors such as death rate and competition to survive.
In the long run, the population of this species will increase from $50$ and stabalise at $500$ in the refined model whereas the first model would suggest that the population will increase indefinitely. Thus, the refined model is better as in real life a population growth of a species will be limited by external factors such as death rate and competition to survive.
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The complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are such that ${{z}_{1}}=3-\mathbf{i}\sqrt{3}$, ${{z}_{2}}=\frac{1}{2}{{\text{e}}^{\mathbf{i}\frac{2\pi }{5}}}$ and ${{z}_{3}}={{z}_{1}}\times {{z}_{2}}$.
(a)
Find exactly the modulus and argument of ${{z}_{3}}$.
[3]
(a) Find exactly the modulus and argument of ${{z}_{3}}$.
[3]
(b)
Sketch an Argand diagram showing ${{z}_{1}}$, ${{z}_{2}}$ and $x$.
[2]
(b) Sketch an Argand diagram showing ${{z}_{1}}$, ${{z}_{2}}$ and $x$.
[2]
(c)
Find the smallest positive integer value of $n$ for which ${{z}_{3}}^{n}$ is purely imaginary. State the modulus and argument of ${{z}_{3}}^{n}$ in this case, giving the modulus in the form $k\sqrt{3}$, where $k$ is an integer.
[4]
(c) Find the smallest positive integer value of $n$ for which ${{z}_{3}}^{n}$ is purely imaginary. State the modulus and argument of ${{z}_{3}}^{n}$ in this case, giving the modulus in the form $k\sqrt{3}$, where $k$ is an integer.
[4]
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The diagram shows part of the circle ${{x}^{2}}+{{y}^{2}}={{r}^{2}}$ and the line $y=r-h$, where $0<h<r$. The shaded region between the circle and the line is rotated about the $y$-axis to form a solid, which is called a spherical cap. The height of the spherical cap is $h$.
(a)
Show by integration that the volume of the spherical cap is $\frac{1}{3}\pi {{h}^{2}}\left( 3r-h \right)$.
[5]
(a) Show by integration that the volume of the spherical cap is $\frac{1}{3}\pi {{h}^{2}}\left( 3r-h \right)$.
[5]
An ornament is made from a solid sphere of radius $15$ cm by removing two spherical caps, one of height $p$ cm from the top of the sphere and the other height $3p$ cm from the bottom of the sphere. The plane faces of the ornament are parallel (see diagram). The volume of the ornament, shown shaded, is $3402\pi $ cm$^{2}$.
[It is given that the volume of a sphere of radius is $\frac{4}{3}\pi {{r}^{3}}$.]
(b)
Find the cubic equation satisfied by $p$ , and hence find the value of $p$.
[4]
(b) Find the cubic equation satisfied by $p$ , and hence find the value of $p$.
[4]
A different ornament is made by making two parallel cuts to another sphere of radius $15$ cm.
(c)
Find the volume of this second ornament. Give your answer as an exact multiple of $\pi $.
[2]
(c) Find the volume of this second ornament. Give your answer as an exact multiple of $\pi $.
[2]
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A large company is organised into $4$ departments: Administration, Production, Marketing and Staffing.
(a)
The company owner wishes to find out about the qualifications of the employees in Staffing, so she gives a questionnaire to all $75$ employees in this department. She receives replies from $53$ of these employees. Explain whether these $53$ employees form a population or a sample.
[1]
(a) The company owner wishes to find out about the qualifications of the employees in Staffing, so she gives a questionnaire to all $75$ employees in this department. She receives replies from $53$ of these employees. Explain whether these $53$ employees form a population or a sample.
[1]
(b)
The company owner also wishes to investigate the views of employees about the refreshment facilities but does not have time to ask every employee. Explain how she should carry out her investigation, and why she should do the investigation this way.
[2]
(b) The company owner also wishes to investigate the views of employees about the refreshment facilities but does not have time to ask every employee. Explain how she should carry out her investigation, and why she should do the investigation this way.
[2]
The company operates with a system of Team Leaders. There are $7$ Team Leaders in Administration, $6$ in Production, $4$ in Marketing and $3$ in Staffing.
(c)
Find the number of ways in which a working party of $8$ Team Leaders can be formed so that there is at least one Team Leader from each of the $4$ departments AND there are more Team Leaders from Administration in the working party from any other single department.
[5]
(c)
Find the number of ways in which a working party of $8$ Team Leaders can be formed so that there is at least one Team Leader from each of the $4$ departments AND there are more Team Leaders from Administration in the working party from any other single department.
[5]
The $53$ employees who responded do not represent the entire Staffing department (which has 75 employees) but rather a subset of it. Therefore, the $53$ employees form a sample of the population (the $75$ employees in the Staffing department).
She needs to collect a random sample of employees, ensuring each employee has an equal chance of being selected independently of each other. She can achieve this by numbering the employees and using a random number generator to select the corresponding employees. This method helps avoid bias and provides a representative sample.
The $53$ employees who responded do not represent the entire Staffing department (which has 75 employees) but rather a subset of it. Therefore, the $53$ employees form a sample of the population (the $75$ employees in the Staffing department).
She needs to collect a random sample of employees, ensuring each employee has an equal chance of being selected independently of each other. She can achieve this by numbering the employees and using a random number generator to select the corresponding employees. This method helps avoid bias and provides a representative sample.
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(a)
The independent random variables $X$ and $Y$ are such that $X\sim$$N\left( p,{{q}^{2}} \right)$ and $Y\sim$$N\left( s,{{t}^{2}} \right)$. Write down an expression for the distribution of the random variable $aX-bY$, where $a$ and $b$ are constants.
[2]
(a) The independent random variables $X$ and $Y$ are such that $X\sim$$N\left( p,{{q}^{2}} \right)$ and $Y\sim$$N\left( s,{{t}^{2}} \right)$. Write down an expression for the distribution of the random variable $aX-bY$, where $a$ and $b$ are constants.
[2]
(b)
The random variable $V$ is normally distributed with standard deviation $2$. The probability that $V>8$ is equal to the probability that $V<4$.
(b) The random variable $V$ is normally distributed with standard deviation $2$. The probability that $V>8$ is equal to the probability that $V<4$.
(i) Draw a sketch to show the distribution of $V$, including the main features of the curve.
[2]
(i) Draw a sketch to show the distribution of $V$, including the main features of the curve.
[2]
(ii) On your sketch, shade the area represented by $\text{P}\left( V>10 \right)$ and state its value.
[2]
(ii) On your sketch, shade the area represented by $\text{P}\left( V>10 \right)$ and state its value.
[2]
(c)
The random variable $W$ is such that $W\sim\text{B}\left( 8,p \right)$. The mean of $W$ is $1.2$ times the variance of $W$.
Find $\text{P}\left( W<2 \right)$.
[3]
(c) The random variable $W$ is such that $W\sim\text{B}\left( 8,p \right)$. The mean of $W$ is $1.2$ times the variance of $W$.
Find $\text{P}\left( W<2 \right)$.
[3]
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A game is played using a counter on a board.
• The counter starts from a point S at the bottom of the board and moves upwards (see diagram).
• When leaving S the counter is equally likely to move up to the left to junction L or up to the right to junction R.
• At every junction after S, the counter moves up to the left with probability $p$ or up the right with probability $q$, where $p+q=1$.
• The counter eventually arrives at one of the endpoints A, B, C, D, E or F.
(a)
Show that the probability the counter arrives at B is $2{{p}^{3}}q+\frac{1}{2}{{p}^{4}}$.
[2]
(a) Show that the probability the counter arrives at B is $2{{p}^{3}}q+\frac{1}{2}{{p}^{4}}$.
[2]
John and Kath each play this game, and their counters both arrive at B.
(b)
Find, in terms of $p$, a simplified fraction for the probability that Jon’s and Kath’s counters followed exactly the same route.
[4]
(b) Find, in terms of $p$, a simplified fraction for the probability that Jon’s and Kath’s counters followed exactly the same route.
[4]
The probability that a counter arrives at B is the same as the probability that a counter arrives at C.
(c)
Find the value of $p$.
[4]
(c) Find the value of $p$.
[4]
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