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 support@timganmath.edu.sg.
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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(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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