Use this model to calculate the circumference of the Moon in kilometres. Give your full calculator display.

Give your answer to part (a) correct to three significant figures.

Write your answer to part (b) in the form $a\times {10}^k$, where 1 ≤ $a$ < 10 , $k\in \mathbb{Z}$.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\frac{3474000\times \pi }{1000}$      (M1)(M1)

Note: Award (M1) for correct numerator and (M1) for dividing by 1000 OR equivalent, such as $\frac{3474000\times 2\times \pi }{2000}$ ie diameter.
Do not accept use of area formula ie $\pi r^2$.

10 913.89287… (km)      (A1)  (C3)

[3 marks]

10 900 (km)      (A1)(ft)  (C1)

Note: Follow through from part (a).

[1 mark]

1.09 × 10⁴      (A1)(ft)(A1)(ft)  (C2)

Note: Follow through from part (b) only. Award (A1)(ft) for 1.09, and (A1)(ft) × 10⁴. Award (A0)(A0) for answers of the type: 10.9 × 10³.

[2 marks]

Calculate the value of $p$. Write down your full calculator display.

Write your answer to part (a)

(i)     correct to two decimal places;

(ii)     correct to three significant figures.

Write your answer to part (b)(ii) in the form $a\times {10}^k$, where .

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\frac{\cos {36}^{\circ }+\sin {18}^{\circ }}{\sqrt{{29}^2-21.8}}$    (M1)

Note:     Award (M1) for correct substitution into formula.

$=0.0390625$    (A1)     (C2)

Note:     Accept $\frac{5}{128}$.

[2 marks]

(i)     0.04     (A1)(ft)

(ii)     0.0391     (A1)(ft)     (C2)

Note:     Follow through from part (a).

[2 marks]

$3.91\times {10}^{-2}$    (A1)(ft)(A1)(ft)     (C2)

Note:     Answer should be consistent with their answer to part (b)(ii). Award (A1)(ft) for 3.91, and (A1)(ft) for ${10}^{-2}$. Follow through from part (b)(ii).

[2 marks]

Write down the median length of these leaves.

Write down the number of leaves with a length less than or equal to 8 cm.

Use the graph to find the value of $k$.

Before measuring, the researcher estimated $k$ to be approximately 9.5 cm. Find the percentage error in her estimate.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

9 (cm)     (A1)     (C1)

[1 mark]

40 (leaves)     (A1)     (C1)

[1 mark]

$(200\times 0.90=)180$ or equivalent     (M1)

Note:     Award (M1) for a horizontal line drawn through the cumulative frequency value of 180 and meeting the curve (or the corresponding vertical line from 10.5 cm).

$(k=)10.5\text{ (cm)}$     (A1)     (C2)

Note:     Accept an error of ±0.1.

[2 marks]

$\left|\frac{9.5-10.5}{10.5}\right|\times 100\mathrm{\%}$     (M1)

Notes:     Award (M1) for their correct substitution into the percentage error formula.

$\text{9.52 (\% ) }\left(\text{9.52380}\dots \text{ (\% )}\right)$     (A1)(ft)     (C2)

Notes:     Follow through from their answer to part (c)(i).

Award (A1)(A0) for an answer of $-9.52$ with or without working.

[2 marks]

Calculate the volume of this hemisphere in cm³.

Give your answer correct to one decimal place.

Write down your answer to part (a) correct to the nearest integer.

Write down your answer to part (b) in the form a × 10^k , where 1 ≤ a < 10 and k ∈ $\mathbb{Z}$.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\sqrt{\frac{4{\left(350\right)}^3}{243\pi }}$  OR $\sqrt{\frac{171500000}{763.407\dots }}$     (M1)

Note: Award (M1) for substitution of 350 into volume formula.

= 473.973…    (A1) 

= 474 (cm³)    (A1)(ft)   (C3)

Note: The final (A1)(ft) is awarded for rounding their answer to 1 decimal place provided the unrounded answer is seen.

[3 marks]

474 (cm³)      (A1)(ft) (C1)

Note: Follow through from part (a).

[1 mark]

4.74 × 10² (cm³)     (A1)(ft)(A1)(ft)   (C2)

Note: Follow through from part (b) only.

Award (A0)(A0) for answers of the type 0.474 × 10³.

[2 marks]

Write down the elements that belong to $A\cap B$.

Write down the elements that belong to $A\cap B^{\prime }$.

Write down $n\left(A\cap B^{\prime }\right)$.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$A$ = {3, 6, 9, 12}  AND  $B$ = {2, 4, 6, 8, 10, 12, 14}      (M1)

Note: Award (M1) for listing all elements of sets $A$ and $B$. May be seen in part (b). Condone the inclusion of 15 in set $A$ when awarding the (M1).

6, 12     (A1)(A1)   (C3)  

Note: Award (A1) for each correct element. Award (A1)(A0) if one additional value seen. Award (A0)(A0) if two or more additional values are seen.

[3 marks]

3, 9     (A1)(ft)(A1)(ft)   (C2)  

Note: Follow through from part (a) but only if their $A$ and $B$ are explicitly listed.
Award (A1)(ft) for each correct element. Award (A1)(A0) if one additional value seen. Award (A0)(A0) if two or more additional values are seen.

[2 marks]

2     (A1)(ft)   (C1)  

Note: Follow through from part (b)(i).

[1 mark]

Calculate the radius of the base of the cone which has been removed.

Calculate the curved surface area of the cone which has been removed.

Calculate the curved surface area of the remaining solid.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\sqrt{{15}^2-{12}^2}$     (M1)

Note: Award (M1) for correct substitution into Pythagoras theorem.

OR

$\frac{\text{radius}}{21}=\frac{15}{35}$     (M1)

Note: Award (M1) for a correct equation.

= 9 (cm)     (A1) (C2)

[2 marks]

$\pi \times 9\times 15$      (M1)

Note: Award (M1) for their correct substitution into curved surface area of a cone formula.

$=424\text{c}{\text{m}}^2\left(135\pi ,\mathrm{424.115...}\text{c}{\text{m}}^2\right)$     (A1)(ft) (C2)

Note: Follow through from part (a).

[2 marks]

$\pi \times 21\times 35-\mathrm{424.115...}$     (M1)

Note: Award (M1) for their correct substitution into curved surface area of a cone formula and for subtracting their part (b).

$=1880\text{c}{\text{m}}^2\left(600\pi ,\mathrm{1884.95...}\text{c}{\text{m}}^2\right)$     (A1)(ft) (C2)

Note: Follow through from part (b).

[2 marks]

Calculate the time, in minutes, it takes for light from the Sun to reach the Earth.

Find the distance from the Earth to Polaris in millions of km. Give your answer in the form $a\times {10}^k$ with and $k\in \mathbb{Z}$.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\frac{149600000}{300000\times 60}$     (M1)(M1)

Note:     Award (M1) for dividing the correct numerator (which can be presented in a different form such as $149.6\times {10}^6$ or $1.496\times {10}^8$) by $\text{300}\text{000}$ and (M1) for dividing by 60.

$=8.31(\text{minutes})(8.31111\dots \text{, 8 minutes 19 seconds})$     (A1)     (C3)

[3 marks]

$323\times 9467280$     (M1)

Note:     Award (M1) for multiplying 323 by $9467280$, seen with any power of 10; therefore only penalizing incorrect power of 10 once.

$=3.06\times {10}^9\text{ ( = }3.05793\dots \times {10}^9)$     (A1)(A1)     (C3)

Note:     Award (A1) for 3.06.

Award (A1) for $\times {10}^9$

Award (A0)(A0) for answers of the type: $30.6\times {10}^8$

[3 marks]

Find the radius, $r$, of the circular base of the cone.

Find the slant height, $l$, of the cone.

Find the curved surface area of the cone.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$100=\frac{1}{3}\pi r^2(8)$     (M1)

Note:     Award (M1) for correct substitution into volume of cone formula.

$r=3.45\text{ (cm) }\left(3.45494\dots \text{ (cm)}\right)$     (A1)     (C2)

[2 marks]

$l^2=8^2+(3.45494\dots {)}^2$     (M1)

Note:     Award (M1) for correct substitution into Pythagoras’ theorem.

$l=8.71\text{ (cm) }\left(8.71416\dots \text{ (cm)}\right)$     (A1)(ft)     (C2)

Note:     Follow through from part (a).

[2 marks]

$\pi \times 3.45494\dots \times 8.71416\dots$     (M1)

Note:     Award (M1) for their correct substitutions into curved surface area of a cone formula.

$=94.6\text{ c}{\text{m}}^2(94.5836\dots \text{ c}{\text{m}}^2)$     (A1)(ft)     (C2)

Note:     Follow through from parts (a) and (b). Accept $94.4\text{ c}{\text{m}}^2$ from use of 3 sf values.

[2 marks]

Place the numbers $2\pi \text{,}-5\text{,}{3}^{-1}$ and $2^{\frac{3}{2}}$ in the correct position on the Venn diagram.

In the table indicate which two of the given statements are true by placing a tick (✔) in the right hand column.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

    (A1)(A1)(A1)(A1)   (C4)

Note: Award (A1) for each number in the correct position.

[4 marks]

     (A1)(A1)   (C2)

Note: Award (A1) for each correctly placed tick.

[2 marks]

Find an expression for the slant height of the cone in terms of r.

The total external surface area of the pencil case rounded to 3 significant figures is 570 cm².

Using your graphic display calculator, calculate the value of r.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

(slant height² =) 10² + r ²     (M1)

Note: For correct substitution of 10 and r into Pythagoras’ Theorem.

$\sqrt{{10}^2+r^2}$     (A1) (C2)

[2 marks]

$\pi r^2+2\pi r\times 12+\pi r\sqrt{100+r^2}=570$     (M1)(M1)(M1)

Note: Award (M1) for correct substitution in curved surface area of cylinder and area of the base, (M1) for their correct substitution in curved surface area of cone, (M1) for adding their 3 surface areas and equating to 570. Follow through their part (a).

= 4.58   (4.58358...)      (A1)(ft) (C4)

Note: Last line must be seen to award final (A1). Follow through from part (a).

[4 marks]

Find the volume, in cm³, of this calculator case.

evidence of 10 mm = 1 cm     (A1)

Note: Award (A1) for dividing their volume from part (a) or part (b) by 1000.

529 (cm³)  (528.612 (cm³))    (A1)(ft) (C2)

Note: Follow through from parts (a) or (b). Accept answers written in scientific notation.

[2 marks]

Write down the value of x.

Calculate the volume of the paperweight.

1 cm³ of glass has a mass of 2.56 grams.

Calculate the mass, in grams, of the paperweight.

3 (cm)    (A1) (C1)

[1 mark]

units are required in part (a)(ii)

$\frac{1}{2}\times \frac{4\pi \times {\left(3\right)}^3}{3}+3\times {\left(6\right)}^2$      (M1)(M1)

Note: Award (M1) for their correct substitution in volume of sphere formula divided by 2, (M1) for adding their correctly substituted volume of the cuboid.

= 165 cm³   (164.548…)      (A1)(ft) (C3)

Note: The answer is 165 cm³; the units are required. Follow through from part (a)(i).

[3 marks]

their 164.548… × 2.56      (M1)

Note: Award (M1) for multiplying their part (a)(ii) by 2.56.

= 421 (g)   (421.244…(g))      (A1)(ft) (C2)

Note: Follow through from part (a)(ii).

[2 marks]

Calculate the length of side AC in km.

Calculate the area of the park.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

(AC² =) 62² + 79² − 2 × 62 × 79 × cos(52°)     (M1)(A1)

Note: Award (M1) for substituting in the cosine rule formula, (A1) for correct substitution.

63.7  (63.6708…) (km)     (A1) (C3)

[3 marks]

$\frac{1}{2}$ × 62 × 79 × sin(52°)     (M1)(A1)

Note: Award (M1) for substituting in the area of triangle formula, (A1) for correct substitution.

1930 km²  (1929.83…km²)     (A1) (C3)

[3 marks]

Use the above information to write down an equation in $x$ and $y$.

Use the information about the cost of tickets to write down a second equation in $x$ and $y$.

Find the value of $x$ and the value of $y$.

Find the percentage error.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$x+y=154$    (A1)     (C1)

[1 mark]

$320x+85y=14970$    (A1)     (C1)

[1 mark]

$x=8,y=146$    (A1)(ft)(A1)(ft)     (C2)

Note:     Follow through from parts (a) and (b) irrespective of working seen, but only if both values are positive integers.

Award (M1)(A0) for a reasonable attempt to solve simultaneous equations algebraically, leading to at least one incorrect or missing value.

[2 marks]

$\left|\frac{14270-14970}{14970}\right|\times 100$    (M1)

Note:     Award (M1) for correct substitution into percentage error formula.

$=4.68(\mathrm{\%})(4.67601\dots )$    (A1) (C2)

[2 marks]

Write down an expression for the width of the garden in terms of $x$.

Write down an equation for the perimeter of the garden in terms of $x$.

Find the value of $x$.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$x-4.5$      (A1) (C1)

Note: Accept $w=x-4.5$  OR  width$=x-4.5$.

[1 mark]

$2x+2\left(x-4.5\right)=111$     (A1)(ft) (C1)

Note: Follow through from their expression for the width from part (a).

[1 mark]

$4x-9=111$ (or equivalent)     (M1)

Note: Award (M1) for correctly removing the parentheses and collecting $x$ terms. This may be seen in part (b).

($x=$) 30      (A1)(ft) (C2)

Note: Follow through from their equation from part (b) provided $x>4.5$.

[2 marks]

Find the value of $a$.

Find the time, in years, for the population to decrease to 20 turtles.

There is a number $m$ beyond which the turtle population will not decrease.

Find the value of $m$. Justify your answer.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$55=a\times 2^0+10$   (M1)

Note: Award (M1) for correct substitution of zero and 55 into the function.

45      (A1)   (C2)  

[2 marks]

$45\times 2^{-t}+10⩽20$       (M1)

Note: Award (M1) for comparing correct expression involving 20 and their 45. Accept an equation.

$t=2.17$  (2.16992…)      (A1)(ft)   (C2)

Note: Follow through from their part (a), but only if positive.
Answer must be in years; do not accept months for the final (A1).  

[2 marks]

$m=$10       (A1)

because as the number of years increases the number of turtles approaches 10      (R1)   (C2)

Note: Award (R1) for a sketch with an asymptote at approximately $y=10$,
OR for table with values such as 10.003 and 10.001 for $t=14$ and $t=15$, for example,
OR when $t$ approaches large numbers $y$ approaches 10. Do not award (A1)(R0). 

[2 marks]

Find the amount of Euros that Claudia receives. Give your answer correct to two decimal places.

Find the value of $r$.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$8000\times 0.09819\times 0.98$     (M1)(M1)

Note:     Award (M1) for multiplying 8000 by 0.09819, (M1) for multiplying by 0.98 (or equivalent).

769.81 (EUR)     (A1)     (C3)

[3 marks]

$r\mathrm{\%}\times \frac{85}{0.08753}=14.57$     (M1)(M1)

Note:     Award (M1) for dividing 85 by 0.08753, and (M1) for multiplying their $\frac{85}{0.08753}$ by $r\mathrm{\%}$ and equating to 14.57.

OR

$\frac{85}{0.08753}=\text{971.095}\dots$     (M1)

Note:     Award (M1) for dividing 85 by 0.08753.

$\frac{14.57}{9.71095\dots }$$$OR$$$\frac{14.57}{971.095\dots }\times 100$     (M1)

Note:     Award (M1) for dividing 14.57 by 9.71095… or equivalent.

$r=1.50(1.50036\dots )$     (A1)     (C3)

[3 marks]

Use this exchange rate to calculate the amount of ARS that is equal to 350 AUD.

Calculate the total amount of ARS Jose paid to get 350 AUD.

Find the value of $x$.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

Note:     In this question, the first time an answer is not to 2 dp the final (A1) is not awarded.

$\frac{350}{0.1559}$     (M1)

Note:     Award (M1) for dividing 350 by 0.1559.

$=2245.03\text{ (ARS)}$     (A1)     (C2)

[2 marks]

$2245.03\times 1.02$     (M1)

Note:     Award (M1) for multiplying their answer to part (a) by 1.02.

$=2289.93\text{ (ARS)}$     (A1)(ft)     (C2)

OR

$2245.03\times 0.02$     (M1)

Note:     Award (M1) for multiplying their answer to part (a) by 0.02.

$=44.9006$

$2245.03+44.90$

$=2289.93\text{ (ARS)}$     (A1)(ft)     (C2)

Note:     Follow through from part (a).

[2 marks]

$\frac{4228.38}{585}$     (M1)

Note:     Award (M1) for dividing 4228.38 by 585.

$=7.23$     (A1)     (C2)

[2 marks]

Calculate the amount of PEN she receives.

Calculate the amount of USD she receives.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$(85-5)\times 3.25$     (M1)(M1)

Note:     Award (M1) for subtracting 5 from 85, (M1) for multiplying by 3.25.

Award (M1) for $85\times 3.25$, (M1) for subtracting $5\times 3.25$.

$=260\text{ (PEN)}$     (A1)     (C3)

[3 marks]

$\frac{(370\times 0.96)}{9.6}$     (M1)(M1)

Note:     Award (M1) for multiplying by 0.96 (or equivalent), (M1) for dividing by 9.6. If division by 3.25 seen in part (a), condone multiplication by 9.6 in part (b).

$=37\text{ (USD)}$     (A1)     (C3)

[3 marks]

Calculate the amount of MXN Harry received.

Calculate the value of the commission, in MXN, that Harry paid.

The exchange rate for this exchange was 1 USD = 17.24 MXN.

Calculate the amount of USD Harry received. Give your answer correct to the nearest cent.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

700 × 18.86      (M1)

Note: Award (M1) for multiplication by 18.86.

= 13 200 (13 202) (MXN)      (A1) (C2)

[2 marks]

2400 × 0.035       (M1)

Note: Award (M1) for multiplication by 0.035.

= 84 (MXN)     (A1) (C2)

[2 marks]

$\frac{2400-\text{their part (b)}}{17.24}$      (M1)

Note: Award (M1) for dividing 2400 minus their part (b), by 17.24. Follow through from part (b).

= 134.34 (USD)       (A1)(ft) (C2)

Note: Award at most (M1)(A0) if final answer is not given to nearest cent.

[2 marks]

Calculate the amount of VEF that Javier receives.

Calculate the total amount, in USD, that Javier has remaining from his 5000 USD after his trip to Venezuela.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

The first answer not given correct to two decimal places is not awarded the final (A1).

Incorrect rounding is not penalized thereafter.

$3000\times 6.3021$    (M1)

Note:     Award (M1) for multiplying 3000 by 6.3021.

$=18906.30$     (A1)     (C2)

[2 marks]

$\frac{18906.30-12000}{8.7268}\text{ + }(2000-1250)$    (M1)(M1)(M1)

Note:     Award (M1) for subtracting 12 000 from their answer to part (a) OR for 6906.30 seen, (M1) for dividing their amount by 8.7268 (can be implied if 791.389… seen) and (M1) for $2000-1250$ OR 750 seen.

$=1541.39$    (A1)(ft)     (C4)

Note:     Follow through from part (a).

[4 marks]

Calculate the amount that Velina receives in DKK.

Calculate the amount, in DKK, that will be left to exchange after commission.

Hence, calculate the amount of USD she receives.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

1200 × 7.0208       (M1)

Note: Award (M1) for multiplying by 7.0208.

8424.96 (DKK)       (A1) (C2)

[2 marks] 

0.95 × 3450       (M1)

Note: Award (M1) for multiplying 3450 by 0.95 (or equivalent).

3277.50 (DKK)       (A1) (C2)

Note: The answer must be given to two decimal places unless already penalized in part (a).

[2 marks] 

$\frac{3277.50}{7.0208}$      (M1)

Note: Follow through from part (b)(i). Award (M1) for dividing their part (b)(i) by 7.0208.

466.83 (USD)       (A1)(ft) (C2)

Note: The answer must be given to two decimal places unless already penalized in parts (a) or (b)(i).

[2 marks]