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SL Paper 1

Let $p = \frac{{\cos x + \sin y}}{{\sqrt {{w^2} - z} }}$ ,

where $x = 36^\circ ,{\text{ }}y = 18^\circ ,{\text{ }}w = 29$ and $z = 21.8$ .

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

The following table shows four different sets of numbers: $\mathbb{N}$ , $\mathbb{Z}$ , $\mathbb{Q}$ and $\mathbb{R}$ .

Complete the second column of the table by giving one example of a number from each set.

[4]

Josh states: “Every integer is a natural number”.

Write down whether Josh’s statement is correct. Justify your answer.

[2]

In this question, give all answers to two decimal places.

Karl invests 1000 US dollars (USD) in an account that pays a nominal annual interest of 3.5%, compounded quarterly. He leaves the money in the account for 5 years.

Calculate the amount of money he has in the account after 5 years.

[3]

a.i.

Write down the amount of interest he earned after 5 years.

[1]

a.ii.

Karl decides to donate this interest to a charity in France. The charity receives 170 euros (EUR). The exchange rate is 1 USD = t EUR.

Calculate the value of t.

[2]

For a study, a researcher collected 200 leaves from oak trees. After measuring the lengths of the leaves, in cm, she produced the following cumulative frequency graph.

M17/5/MATSD/SP1/ENG/TZ2/06

Write down the median length of these leaves.

[1]

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

[1]

The speed of light is ${\text{300}}\,{\text{000}}$ kilometres per second. The average distance from the Sun to the Earth is 149.6 million km.

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

Little Green island originally had no turtles. After 55 turtles were introduced to the island, their population is modelled by

$N\left( t \right) = a \times {2^{ - t}} + 10{\text{,}}\,\,\,t \geqslant 0{\text{,}}$

where $a$ is a constant and $t$ is the time in years since the turtles were introduced.

Find the value of $a$ .

[2]

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

[2]

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

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

[2]

Consider the following sets:

The universal set $U$ consists of all positive integers less than 15;
$A$ is the set of all numbers which are multiples of 3;
$B$ is the set of all even numbers.

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

[3]

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

[2]

b.i.

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

[1]

b.ii.

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

[4]

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

[2]

A solid right circular cone has a base radius of 21 cm and a slant height of 35 cm.
A smaller right circular cone has a height of 12 cm and a slant height of 15 cm, and is removed from the top of the larger cone, as shown in the diagram.

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

A solid glass paperweight consists of a hemisphere of diameter 6 cm on top of a cuboid with a square base of length 6 cm, as shown in the diagram.

The height of the cuboid, x cm, is equal to the height of the hemisphere.

Write down the value of x.

[1]

a.i.

Calculate the volume of the paperweight.

[3]

a.ii.

1 cm³ of glass has a mass of 2.56 grams.

Calculate the mass, in grams, of the paperweight.

[2]

Harry travelled from the USA to Mexico and changed 700 dollars (USD) into pesos (MXN).

The exchange rate was 1 USD = 18.86 MXN.

On his return, Harry had 2400 MXN to change back into USD.

There was a 3.5 % commission to be paid on the exchange.

Calculate the amount of MXN Harry received.

[2]

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

[2]

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.

[2]

In this question give all answers correct to two decimal places.

Javier takes 5000 US dollars (USD) on a business trip to Venezuela. He exchanges 3000 USD into Venezuelan bolívars (VEF).

The exchange rate is 1 USD $=$ 6.3021 VEF.

During his time in Venezuela, Javier spends 1250 USD and 12 000 VEF. On his return home, Javier exchanges his remaining VEF into USD.

The exchange rate is 1 USD $=$ 8.7268 VEF.

Calculate the amount of VEF that Javier receives.

[2]

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

[4]

In this question, give all answers correct to 2 decimal places.

Jose travelled from Buenos Aires to Sydney. He used Argentine pesos, ARS, to buy 350 Australian dollars, AUD, at a bank. The exchange rate was 1 ARS = 0.1559 AUD.

The bank charged Jose a commission of 2%.

Jose used his credit card to pay his hotel bill in Sydney. The bill was 585 AUD. The value the credit card company charged for this payment was 4228.38 ARS. The exchange rate used by the credit card company was 1 AUD = $x$ ARS. No commission was charged.

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

[2]

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

[2]

Find the value of $x$ .

[2]

Complete the following table by placing ticks (✓) to show which of the number sets $\mathbb{N}$ , $\mathbb{Z}$ , $\mathbb{Q}$ , and $\mathbb{R}$ these numbers belong to. The first row has been completed as an example.

Claudia travels from Buenos Aires to Barcelona. She exchanges 8000 Argentine Pesos (ARS) into Euros (EUR).

The exchange rate is 1 ARS = 0.09819 EUR. The bank charges a 2% commission on the exchange.

When Claudia returns to Buenos Aires she has 85 EUR left and exchanges this money back into ARS. The exchange rate is 1 ARS = 0.08753 EUR. The bank charges $r$ % commission. The commission charged on this exchange is 14.57 ARS.

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

[3]

Find the value of $r$ .

[3]

In this question, give all answers to two decimal places.

Velina travels from New York to Copenhagen with 1200 US dollars (USD). She exchanges her money to Danish kroner (DKK). The exchange rate is 1 USD = 7.0208 DKK.

At the end of her trip Velina has 3450 DKK left that she exchanges to USD. The bank charges a 5 % commission. The exchange rate is still 1 USD = 7.0208 DKK .

Calculate the amount that Velina receives in DKK.

[2]

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

[2]

b.i.

Hence, calculate the amount of USD she receives.

[2]

b.ii.

Daniela is going for a holiday to South America. She flies from the US to Argentina stopping in Peru on the way.

In Peru she exchanges 85 United States dollars (USD) for Peruvian nuevo sol (PEN). The exchange rate is 1 USD = 3.25 PEN and a flat fee of 5 USD commission is charged.

At the end of Daniela’s holiday she has 370 Argentinean peso (ARS). She converts this back to USD at a bank that charges a 4% commission on the exchange. The exchange rate is 1 USD = 9.60 ARS.

Calculate the amount of PEN she receives.

[3]

Calculate the amount of USD she receives.

[3]