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GIRARD’S SUMS (30 marks)
You are advised to spend no more than 50 minutes on this part.
Albert Girard, a 17th century French mathematician, investigated numbers, $N$, that can be written as the sum of two squares, $a^2 + b^2$.
This task is about these numbers.
For this task, $a$ and $b$ are integers where $a \geq 0$ and $b \geq 0$.
(a) Complete the table.
| $a$ | $a^2$ | $b$ | $b^2$ | $N = a^2 + b^2$ | $N \div 4$ |
|----|-----|----|-----|---------------|-------|
| 2 | 4 | 6 | 36 | 40 | 10 remainder 0 |
| | 18 | 10 | | | 106 remainder 0 |
| | 28 | | | | remainder 0 |
| 4 | | | 64 | 20 | remainder 0 |
| | | 144 | 196 | 85 | remainder 0 |
| 20 | 400 | | | 884 | 221 remainder 0 |
| 0 | 0 | | | 900 | 225 remainder 0 |
[5]
(b) (i) When $a = 2$ and $b = 4$ then $N = 4k$, so $N$ is a multiple of 4.
Find the value of $k$.
[2]
(ii) The values of $a$ and $b$ in the table are all even numbers.
When $a = 2m$ and $b = 2n$ then $N = 4k$.
Find an expression for $k$ in terms of $m$ and $n$.
[3]
(c) Not all multiples of 4 can be written as the sum of two square numbers.
Show that there are no values of $a$ and $b$ that give $k = 11$.
[2]
507
563
588
(a) Complete the table.
| a | a^2 | b | b^2 | N = a^2 + b^2 | N \div 4 |
|---|---|---|---|---|---|
| 7 | 49 | 5 | 25 | 74 | 18 remainder 2 |
| 21 | 19 | 802 | 200 remainder 2 | ||
| 17 | 289 | 914 | remainder 2 | ||
| 49 | 170 | remainder 2 | |||
| 1 | 1 | remainder |
(b) When $a$ is an odd number, $a = 2n - 1$.
(i) Use algebra to explain why, when $a$ is an odd number, $a^2 \div 4$ has a remainder of 1. .................................................................................................................... [3]
(ii) Explain why, for the values in the table in part (a), $N$ is always $4k + 2$. .................................................................................................................... [2]
(c) When $a$ and $b$ are both odd, $N = 4k + 2$, so $N$ is a multiple of 4 plus 2. Not all multiples of 4 plus 2 can be written as the sum of two square numbers.
Find all the values of $k$ from 1 to 9 where $N = a^2 + b^2$. .................................................................................................................... [3]
568
503
514
The values of $N$ that can be written as the sum of two square numbers are of the form $4k + r$, where the remainder $r$ is a constant.
(a) Explain why $r$ can be 0, 1 or 2 but cannot be 3. [3]
(b) $N = a^2 + b^2$
Find all the values of $N$, where $10 < N < 30$, that are of the form $4k + 1$. [3]
573
542
578
PRODUCTION BOUNDARIES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task is about the number of computer tablets and mobile phones a company makes and sells.
The company owns two factories, A and B.
Factory A makes A-tablets and A-phones.
Factory B makes B-tablets and B-phones.
A production boundary is a curve or line.
Points on the curve or line are the maximum numbers of the two items a factory can make when all resources are used.
It is the boundary of the region which shows all the combinations of the two items a factory can make.
Factory A makes $t$ A-tablets and $p$ A-phones each day.
The manager of factory A uses the model $p = 9000 - \frac{t^2}{1000}$ where $t \geq 0$, as the production boundary for the output of A-tablets and A-phones.
(a) On the axes below, sketch this model.
[2]
(b) When factory A makes 9000 A-phones it cannot make any A-tablets.
Write down the maximum number of A-tablets it can make when it does not make any A-phones.
.............................................. [1]
(c) On Monday, factory A makes 1000 A-tablets.
On Tuesday, factory A makes 1500 A-tablets.
Find the decrease in the maximum number of A-phones it can make from Monday to Tuesday.
.............................................. [3]
(d) (i) On Wednesday, factory A makes 5000 A-phones.
Use your graph from part (a) to explain why it is not possible for it to make 2500 A-tablets on Wednesday.
........................................................................................................
........................................................................................................ [1]
(ii) On the graph in part (a) shade the region that represents the numbers of A-phones and A-tablets that factory A can make.
[1]
(e) The company sells all the A-phones and A-tablets that factory A makes each day.
The company makes $160 profit for each A-tablet and $100 profit for each A-phone it sells.
The greatest possible daily profit at factory A is $\$964\ 000$.
(i) Write down a linear equation for this profit in terms of $p$ and $t$.
Give your answer in the form $p = mt + c$.
.............................................. [2]
(ii) Find the number of A-tablets and A-phones that factory A should sell in order to make a profit of $\$964\ 000$.
$t =$ ..............................................
$p =$ .............................................. [3]
526
576
523
Factory B makes $t$ B-tablets and $p$ B-phones.
The table shows the maximum numbers of B-phones that factory B can make each day for some numbers of B-tablets.
\[ \begin{array}{|c|c|} \hline \text{Number of B-tablets } t & \text{Number of B-phones } p \\ \hline 1000 & 8000 \\ 2000 & 6000 \\ 3000 & 4000 \\ 4000 & 2000 \\ \hline \end{array} \]
As the number of B-tablets increases, the number of B-phones decreases at a constant rate.
(a) (i) Draw the production boundary for factory B on the axes below.
[Image]
[2]
(ii) Find the equation which models this production boundary, giving $p$ as a function of $t$.
.................................................................[2]
(iii) Factory B makes at least 1000 B-tablets but no more than 4000 B-tablets each day.
Write down the domain of the model in part (a)(ii).
.................................................................[1]
(b) The company sells all the B-tablets and B-phones factory B makes each day.
The company makes $200 profit for each B-tablet and $190 profit for each B-phone it sells.
Each day, the manager of factory B expects to make the greatest possible profit.
(i) Find the greatest possible profit each day.
.................................................................[3]
(ii) One day factory B has to make 2500 B-tablets.
On this day the profit is 73.3% of the greatest possible profit.
Work out the number of B-phones factory B makes on this day.
.................................................................[4]
567
535
577
The company puts new machinery to make phones in factory A and factory B.
Factory A can now make double the number of A-phones.
Factory B can now make 10% more B-phones.
All other conditions remain the same.
(a) Complete the following models for the production boundaries at each factory after the changes.
Use the models in Question 4 and Question 5(a).
Factory A: $p = \text{...............................................................}$ for $t \ge 0$
Factory B: $p = -2.2t + 11000$ for $............ \le t \le ............$
[2]
(b) After the changes, the greatest possible profit made each day by factory A is $1\,830\,000$.
Find the total greatest possible profit made each day by the company.
................................................
[3]
528
599
540
