The Manchester Grammar School 2019 Section B

IndependentBoys

11 questions

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Questions

The table below shows the highest and lowest temperatures in $\degree\text{C}$ recorded in six cities around the world on one particular day last year.

City	Highest temperature $(\degree \text{C})$	Lowest temperature $(\degree \text{C})$
Boston	$30$	$8$
Munich	$19$	$-6$
Los Angeles	$23$	$11$
Cairo	$32$	$19$
St Petersburg	$9$	$-8$
Budapest	$10$	$-9$

1.a
What is the difference between the highest and lowest temperatures in Boston?
1.b
What is the difference between the highest and lowest temperatures in St Petersburg?
1.c
Which city has the lowest recorded temperature of all?
1.d
Which city has the largest difference between its highest and lowest temperature?

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2
The gas meter reading on Andrew's smart meter in October was $5475$ units. Exactly three months later, in January, the reading was $6045$ units. M-power charge a fixed amount of $\rtqMathsSymbolPound 16.20$ each month plus $40 \rtqMathsUnitCurrencyPence$ for each unit used during the three months between the two readings.
1. 2.a
  How many units of gas has Andrew used in the three month period from October to January?
2. 2.b
  Calculate the total amount Andrew has to pay for that three month period.
3. 2.c
  In fact, M-power also have to add $5\%$ to Andrew's bill which is the tax known as VAT.
  
  What will be the final bill that Andrew has to pay for his gas?
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3
The “Blast” of a two digit number is obtained as follows:

The Blast of $63$ is $216$ because $6 \times 6 \times 6 = 216$

and the Blast of $27$ is $128$ because $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$
1. 3.a
  Write down the Blast of the two digit number $34$ .
2. 3.b
  Which two digit number has a Blast of $125$ ?
3. 3.c
  Work out another two digit number which has the same Blast as $24$ .
4. 3.d
  A particular two digit number is Blasted and then that answer is also Blasted.
  
  If the final answer is $9$ , what was the original number?
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4
A plant grows during the first month after it is planted. During the second month it grows by half the height it had already reached. During the third month it grows by one third of the height it had reached at the end of the second month. It then grows by one quarter and one fifth of the height reached in the same way in the next two months.
1. 4.a
  If a plant is $1 \rtqMathsUnit{metre}$ tall at the end of the first month, how tall will it be after five months?
2. 4.b
  If a plant grows $90 \rtqMathsUnit{cm}$ in the third month, how tall was it after one month?
3. 4.c
  If a plant is $2 \rtqMathsUnit{metres}$ tall at the end of the first month, how much will it have grown, in total, by the end of the fifth month?
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The table below shows four different makes of car each with four different engine sizes, in litres, and the Insurance Group for each. For example a Hissan $1.5$ car is in Insurance Group $\text{B}$ .

	Skoyota	Hissan	Foxhall	Jagley
Engine size
$1.1$	$\text{A}$	$\text{A}$	$\text{A}$	$\rtqMathsTableNoValue$
$1.2$	$\text{A}$	$\text{A}$	$\text{B}$	$\rtqMathsTableNoValue$
$1.5$	$\text{B}$	$\text{B}$	$\text{B}$	$\text{C}$
$1.9$	$\rtqMathsTableNoValue$	$\text{C}$	$\text{B}$	$\text{D}$

The second table shows the monthly insurance payment paid by drivers of different age bands for each of the insurance groups, so a person aged $37$ would be in the age band $21{-}45$ and so would pay $\rtqMathsSymbolPound 105$ each month for a car in Insurance group $\text{C}$ .

	Under $21$	$21{-}45$	Over $45$
Insurance Group
$\text{A}$	$\rtqMathsSymbolPound 90$	$\rtqMathsSymbolPound 80$	$\rtqMathsSymbolPound 70$
$\text{B}$	$\rtqMathsSymbolPound 110$	$\rtqMathsSymbolPound 95$	$\rtqMathsSymbolPound 85$
$\text{C}$	$\rtqMathsSymbolPound 122$	$\rtqMathsSymbolPound 105$	$\rtqMathsSymbolPound 98$
$\text{D}$	$\rtqMathsSymbolPound 190$	$\rtqMathsSymbolPound 160$	$\rtqMathsSymbolPound 170$

5.a
How much will a $19$ year old pay for a Foxhall $1.5$ car each month?
5.b
A $30$ year old pays $\rtqMathsSymbolPound 95$ for a Hissan car. What is the engine size of their car, in litres?
5.c
How much would a $50$ year old person save each month by using a $1.1$ Hissan car rather than a $1.9$ Hissan car?
5.d
If a $32$ year old pays $\rtqMathsSymbolPound 95$ for a Skoyota car, what would a $20$ year old pay for a Jagley car with the same size engine?

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6
If the weight of straight bones in the human body depends on the length, radius and density of the bone using the following formula:

$\begin{aligned} \text{Weight} &= \text{Length} \times \text{Radius} \times \text{Radius} \times \text{Density} \\\\ \text{or } W &= L \times R \times R \times D \end{aligned}$

then a shin bone of length $40 \rtqMathsUnit{cm}$ , radius $3 \rtqMathsUnit{cm}$ and density $5$ would have a weight of $1800 \rtqMathsUnit{g}$ because:

$W = 40 \times 3 \times 3 \times 5 = 1800$

Using this formula,
1. 6.a
  Work out the weight of a finger bone of length $3 \rtqMathsUnit{cm}$ , with a radius of $0.5 \rtqMathsUnit{cm}$ and density $2$ .
2. 6.b
  Work out the density of a forearm of length $30 \rtqMathsUnit{cm}$ , with a radius of $2 \rtqMathsUnit{cm}$ which weighs $480 \rtqMathsUnit{g}$ .
3. 6.c
  Work out the radius of a thigh bone of length $50 \rtqMathsUnit{cm}$ which has a density of $8$ and weighs $3600 \rtqMathsUnit{g}$ .
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7
A top international bowler playing in a cricket match bowls a cricket ball at $90 \rtqMathsUnit{miles per hour}$ .
1. 7.a
  How many miles would the ball travel in one minute?
2. 7.b
  If a mile is $1760 \rtqMathsUnit{yards}$ , how many yards would the ball travel in one minute?
3. 7.c
  How many yards does the ball travel in one second?
4. 7.d
  If the distance the ball travels through the air from the point where the bowler bowls the ball to the point where it is hit by the batsman is $22 \rtqMathsUnit{yards}$ , how long is the ball travelling for?
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8
Complete all the entries in the table below showing the sum, difference, product and quotient for pairs of numbers from the first two columns. The first row is completed as an example to help you.

1st number 2nd number SUM DIFFERENCE PRODUCT QUOTIENT
$6$ $2$ $8$ $4$ $12$ $3$
$12$ $3$
$15$ $5$
$2$ $8$
$18$ $2$
$21$ $6$
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9
Megaprint make small business cards with names and contact details on, that people can give to clients as part of their work. The total cost of buying the cards they sell is made up of two amounts.

There is a fixed cost that people must pay no matter how many cards they order and this amount is always the same. Then there is a second amount which depends on the number of cards ordered. These two amounts are added together to give the total cost of the order and the graph below shows what the total cost would be for the number of cards ordered.

Using the graph,
1. 9.a
  Write down the total cost of having $100$ cards printed.
2. 9.b
  Andy's order costs him $\rtqMathsSymbolPound 16.00$ .
  
  Write down the number of cards he ordered.
3. 9.c
  If $200$ cards are ordered, using the total cost what is the cost of one card?
4. 9.d
  Write down the fixed cost that Megaprint charge for each order, regardless of the number of cards ordered.
5. 9.e
  If you exclude the fixed cost, how much is charged for one card to be printed?
6. 9.f
  Maxicard also make business cards but they charge a fixed cost of $\rtqMathsSymbolPound 6.00$ for an order and the total cost of $500$ cards is $\rtqMathsSymbolPound 16.00$ .
  
  Mark two points on the graph and join them with a straight line to show this information.
7. 9.g
  Using the graph, or otherwise, work out the number of cards for which the total cost would be the same from both companies.
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10
A two digit number is called a multisum if it is a multiple of the sum of its digits.

So $84$ is a multisum since $8 + 4 = 12$ and $84$ is a multiple of $12$ .
1. 10.a
  Work out and write down the smallest number, greater than $50$ , which is a multisum.
2. 10.b
  Work out and write down the smallest multiple of $7$ which is a multisum.
3. 10.c
  Work out and write down the only multiple of $9$ which is NOT a multisum.
4. 10.d
  Work out and write down all the numbers between $20$ and $30$ which are NOT multisums.
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11
A pair of numbers written as a column like this $\begin{pmatrix} 5 \\ 2 \end{pmatrix}$ is known as a VEC.

VECs can be combined together in two different ways, as follows:

$\begin{aligned} \begin{pmatrix} 5 \\ 2 \end{pmatrix} & \rtqMathsBinaryOperatorLowDot \begin{pmatrix} 6 \\ 7 \end{pmatrix} &= 5 \times 6 + 2 \times 7 = 30 + 14 = 44 \\\\ \begin{pmatrix} 5 \\ 2 \end{pmatrix} & \rtqMathsBinaryOperatorWedge \begin{pmatrix} 6 \\ 7 \end{pmatrix} &= 5 \times 7 - 2 \times 6 = 35 - 12 = 23 \end{aligned}$
1. 11.a
  Work out the value of $\begin{pmatrix} 7 \\ 5 \end{pmatrix} \rtqMathsBinaryOperatorLowDot \begin{pmatrix} 8 \\ 9 \end{pmatrix}$
2. 11.b
  Work out the value of $\begin{pmatrix} 7 \\ 5 \end{pmatrix} \rtqMathsBinaryOperatorWedge \begin{pmatrix} 8 \\ 9 \end{pmatrix}$
3. 11.c
  Find the value of $p$ if $\begin{pmatrix} 3 \\ 11 \end{pmatrix} \rtqMathsBinaryOperatorLowDot \begin{pmatrix} 5 \\ p \end{pmatrix} = 37$
4. 11.d
  Find the value of $q$ if $\begin{pmatrix} 3 \\ 9 \end{pmatrix} \rtqMathsBinaryOperatorWedge \begin{pmatrix} 2 \\ q \end{pmatrix} = 6$ .
5. 11.e
  Find the value of $r$ if $\begin{pmatrix} 6 \\ r \end{pmatrix} \rtqMathsBinaryOperatorLowDot \begin{pmatrix} r \\ 4 \end{pmatrix} = 40$
6. 11.f
  Find the value of $s$ if $\begin{pmatrix} 8 \\ s \end{pmatrix} \rtqMathsBinaryOperatorWedge \begin{pmatrix} 2 \\ s \end{pmatrix} = 30$ .
7. 11.g
  Find the value of $t$ if $\begin{pmatrix} 12 \\ t \end{pmatrix} \rtqMathsBinaryOperatorWedge \begin{pmatrix} t \\ 8 \end{pmatrix} = 47$
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City

Highest temperature

(\degree \text{C})

Lowest temperature

(\degree \text{C})

Boston

30

8

Munich

19

-6

Los Angeles

23

11

Cairo

32

19

St Petersburg

9

-8

Budapest

10

-9

Car type

Skoyota

Hissan

Foxhall

Jagley

Engine size

1.1

\text{A}

\text{A}

\text{A}

\rtqMathsTableNoValue

1.2

\text{A}

\text{A}

\text{B}

\rtqMathsTableNoValue

1.5

\text{B}

\text{B}

\text{B}

\text{C}

1.9

\rtqMathsTableNoValue

\text{C}

\text{B}

\text{D}

Age band

Under

21

21{-}45

Over

45

Insurance Group

\text{A}

\rtqMathsSymbolPound 90

\rtqMathsSymbolPound 80

\rtqMathsSymbolPound 70

\text{B}

\rtqMathsSymbolPound 110

\rtqMathsSymbolPound 95

\rtqMathsSymbolPound 85

\text{C}

\rtqMathsSymbolPound 122

\rtqMathsSymbolPound 105

\rtqMathsSymbolPound 98

\text{D}

\rtqMathsSymbolPound 190

\rtqMathsSymbolPound 160

\rtqMathsSymbolPound 170

1st number

2nd number

SUM

DIFFERENCE

PRODUCT

QUOTIENT

6

2

8

4

12

3

12

3

15

5

2

8

18

2

21

6

1st number	2nd number	SUM	DIFFERENCE	PRODUCT	QUOTIENT
$6$	$2$	$8$	$4$	$12$	$3$
$12$	$3$
		$15$	$5$
			$2$	$8$
				$18$	$2$
		$21$			$6$

1st number	2nd number	SUM	DIFFERENCE	PRODUCT	QUOTIENT
$6$	$2$	$8$	$4$	$12$	$3$
$12$	$3$
		$15$	$5$
			$2$	$8$
				$18$	$2$
		$21$			$6$

The Manchester Grammar School 2019 · Section B

Questions

The Manchester Grammar School 2019 · Section B

Questions

The Manchester Grammar School 2019 Section B

The Manchester Grammar School 2019 Section B