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AREA OF A PARALLELOGRAM (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the area of a parallelogram drawn on a unit square grid.
Find the area of the rectangle.
Write your answer inside the rectangle.
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Find the area of each right-angled triangle. Write your answer inside each triangle.
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(a) Find the area of the rectangle that goes around the parallelogram.
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(b) (i) Use the results in Question 1 and Question 2 to write the areas of the small rectangles and right-angled triangles inside each shape. [1]
(ii) Use your answer to part (a) to show that the area of the parallelogram is 28. [1] [Images: Upper and lower parallelogram]
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Use the method of Question 3 to find the area of parallelogram $OPQR$.
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(a)
On the diagram, complete parallelogram $OPQR$. [1]
(b) Use the method of Question 3 to find the area of parallelogram $OPQR$. [3]
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In parallelogram $OPQR$, $O$ is $(0, 0)$, $P$ has coordinates $(a, b)$ and $R$ has coordinates $(c, d)$.
(a) Complete the table using your answers to Question 4 and Question 5 and any patterns you notice.
[Table_1]
\begin{array}{c|c|c|c|c|c}
P(a, b) & Q & R(c, d) & ad & bc & \text{Area of parallelogram } OPQR \\\hline
\text{Question 3} & (6, 1) & (8, 6) & (2, 5) & 30 & 2 & 28 \\
\text{Question 4} & (3, 4) & (-1, 3) & (-4, -1) & & & \\
\text{Question 5} & (5, 2) & (-2, 2) & & & & \\
& (9, 1) & (1, 8) & & & 71 \\
& (3, -1) & & & 18 & -4\end{array}
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(b) Using the table, write down, in terms of $a$, $b$, $c$ and $d$,
(i) the coordinates of $Q$ $\;(\text{.................. , ..................} ) [1]$
(ii) the area of parallelogram $OPQR$. $\;\text{...........................................................} [1]$
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In parallelogram $OPQR$, $O$ is $(0, 0)$ and $P$ is $(5, 2)$. The coordinates of $Q$ and $R$ are positive integers. The area of the parallelogram is 12.
Use your answers to Question 6(b) to find three possible parallelograms. Write the coordinates of the points $Q$ and $R$ in each case.
$Q$(............, ............) $R$(............, ............)
$Q$(............, ............) $R$(............, ............)
$Q$(............, ............) $R$(............, ............) [4]
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(a) Draw a sketch showing that two congruent triangles make a parallelogram. [1]
(b) The integers $g$ and $h$ are between $-50$ and $50$.
The vertices of triangle $OGH$ are $O(0, 0)$, $G(7, g)$ and $H(h, 8)$.
The area of the triangle is $25$.
Find how many triangles are possible. [4]
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FORGETTING CURVES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at how memory decreases with time after we have memorised something.
Research has shown that after a day we remember $\frac{1}{2}$ of what we memorised the day before.
This decrease in memory continues for the following days.
Retention is the fraction that we still remember.
Complete the retention at the end of day \(d\) as a power of 2.
[Table_1]
Table:
| Time | At the start | At the end of |
|--------------|--------------|----------------------|
| Retention | 1 | day 1 | day 2 | day 3 | day 4 | day d |
| | | $\frac{1}{2}$ | $\frac{1}{4}$ | $\frac{1}{8}$ | $\frac{1}{16}$ | $2^{\ldots}$ |
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Some people have more powerful memories than other people.
A possible model for the retention, $R$, is $$R = 2^{-\frac{d}{p}}$$ where $d$ is the time in days and $p$ is the power of a person's memory.
(a) On the grid sketch the graphs of $R$ for
• $p = 1$
• $p = 5$.
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(b) Find the difference in the retention, $R$, for $p = 1$ and $p = 5$ at the end of 5 days.
Give your answer correct to two decimal places.
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(c) When $p = 5$, the graph is nearly a straight line.
A model for this straight line is $R = ad + b$ where $a$ and $b$ are constants.
(i) Use the values of $R$ when $d = 0$ and $d = 5$ to find the model.
[3]
(ii) Make a statement about the validity of your model after more than 10 days.
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One method to improve your memory is to memorise the material again each day. The graphs show the retention over 5 days when you use this method.
At the end of each day the graph is translated 1 unit to the right and the power, $p$, of your memory increases by 1.
The general model is $R = 2^{-\frac{d}{p}}$.
(a) (i) At the start, $p = 1$.
Write down why $p = 5$ after 4 days.
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(ii) When a graph is translated by $k$ units to the right, its equation changes from $y = f(x)$ to $y = f(x - k)$.
Show that $R = 2^{\frac{4 - d}{5}}$ gives the retention after 4 days.
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(b) At the end of 1 day the retention is $\frac{1}{2}$, as seen on the graph.
After 5 days there is no more memorising again and $R = 2^{\frac{4 - d}{5}}$.
Find the next time, $d$, that the retention, $R$, is $\frac{1}{2}$.
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The German psychologist Hermann Ebbinghaus did important research into retention. He learned a list of nonsense words and measured how much he could remember at the end of certain times.
The table shows his results.
\[\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline & S & T & U & V & W & X & Y & Z \\ \hline \text{Time since learning} & 0 \text{ minutes} & 20 \text{ minutes} & 1 \text{ hour} & 9 \text{ hours} & 1 \text{ day} & 2 \text{ days} & 6 \text{ days} & 1 \text{ month} (31 \text{ days}) \\ \hline \text{Retention} & 1 & 0.58 & 0.44 & 0.36 & 0.33 & 0.28 & 0.25 & 0.21 \\ \hline \end{array}\]
(a) Complete the table by changing the time to hours.
\[\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline & S & T & U & V & W & X & Y & Z \\ \hline \text{Time in hours since learning} (h) & 0 & 1 & 9 & 24 & 48 & 744 \\ \hline \text{Retention} (R) & 1 & 0.58 & 0.44 & 0.36 & 0.33 & 0.28 & 0.25 & 0.21 \\ \hline \end{array}\] [1]
(b) The graph shows the points \(S, T, W\) and \(X\) from the table in part (a).
Plot the points \(U\) and \(V\).
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(c) A possible model for Ebbinghaus’s data is \( R = a \times 2^{-h} + b \).
(i) Use points \(S\) and \(U\) to write two equations in terms of \(a\) and \(b\).
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(ii) Use the method of simultaneous equations to find the value of \(a\) and the value of \(b\).
\(a = ..................................................\)
\(b = ..................................................\) [3]
(iii) Write down the model and sketch it on the graph on page 12.
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(iv) Make a statement about the validity of the model in part (iii).
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(d) In 1885 Ebbinghaus wrote that a good model for his results was \( R = \frac{1}{k \log (h + 1)} \) where \( k \) is a positive constant.
(i) Give algebraic reasons why this model is not valid for \( 0 < h < 1 \).
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(ii) Use the point \(W(24, 0.33)\) to find the value of \(k\) correct to two decimal places.
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(iii) Show that at the end of one month the difference between the model and Ebbinghaus’s results is less than 0.05.
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