NumberIB MYP Maths Extended: Topic test

1. 1
   Priya is sorting numbers for a class poster on the real number system. She uses $\mathbb{Z}$ for the set of integers.

   (a)

   Which of these numbers is irrational?

   [1 mark]

   • A$\sqrt{81}$
   • B$\frac{22}{7}$
   • C$\sqrt{20}$
   • D$0.\dot{7}$

   (b)

   Which of these numbers is an integer but not a natural number?

   [1 mark]

   • A$\frac{1}{2}$
   • B$-5$
   • C$7$
   • D$\sqrt{3}$

   (c)

   The set $A=\{x\in\mathbb{Z}:\ -3<x\le2\}$. List the elements of $A$ and state how many elements it has.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Tariq checks his index-law homework. Assume that $a\neq0$ throughout.

   (a)

   Simplify $a^{5}\times a^{-2}$.

   [1 mark]

   • A$a^{3}$
   • B$a^{7}$
   • C$a^{-10}$
   • D$a^{-3}$

   (b)

   Evaluate $\left(\frac{2}{3}\right)^{-2}$.

   [1 mark]

   • A$\frac{4}{9}$
   • B$-\frac{4}{9}$
   • C$-\frac{9}{4}$
   • D$\frac{9}{4}$

   (c)

   Simplify $\dfrac{(3a^{2})^{3}}{9a^{4}}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A laboratory in Seoul studies a virus particle of diameter $1.2\times10^{-7}$ m and a grain of table salt of diameter $3\times10^{-4}$ m. A sample is described as containing $6.4\times10^{9}$ virus particles in each millilitre.

   (a)

   How many virus particles, placed side by side in a line, would span the diameter of one grain of salt? Give your answer as an ordinary number.

   [3 marks]

   (b)

   A sample of $5\times10^{2}$ mL is tested. (i) Find the total number of virus particles in the sample, in standard form. [2] (ii) A second sample contains $4\times10^{12}$ virus particles. Find, in standard form, how many more particles the second sample contains than the first. [2]

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A café in Accra buys coffee beans. Supplier A sells $8$ kg for GHS $144$. Supplier B sells $5$ kg for GHS $95$ and takes $12\%$ off the price of every order of more than $20$ kg. The café needs $30$ kg each month and has a monthly budget of GHS $520$.

   (a)

   Calculate the cost of $30$ kg of beans from each supplier, and state which supplier is cheaper for this order.

   [6 marks]

   (b)

   Next year, Supplier B raises its price per kilogram (before the discount) by $5\%$ and then raises it by a further $3\%$. The $12\%$ discount still applies. Find the largest whole number of kilograms the café can buy for GHS $520$, and state whether this is enough for the $30$ kg the café needs.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A ferry sails $54$ km from a port to an island, leaving at 08:15 and arriving at 10:45.

   (a)

   Find the average speed of the ferry in km/h.

   [1 mark]

   • A$23.5$
   • B$21.6$
   • C$27$
   • D$135$

   (b)

   The ferry's average speed is $21.6$ km/h. Express this speed in m/s.

   [1 mark]

   • A$0.36$
   • B$77.76$
   • C$6$
   • D$360$

   (c)

   The return journey from the island to the port is at an average speed of $18$ km/h. Find the average speed for the whole round trip, to $3$ significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A conservation team studies a leatherback turtle. Its mass is $412$ kg, correct to the nearest kilogram, and its shell length is $1.4562$ m.

   (a)

   What is the upper bound of the turtle's mass?

   [1 mark]

   • A$412.9$ kg
   • B$413$ kg
   • C$411.5$ kg
   • D$412.5$ kg

   (b)

   Round the shell length $1.4562$ m to $3$ significant figures.

   [1 mark]

   • A$1.46$ m
   • B$1.45$ m
   • C$1.456$ m
   • D$1.5$ m

   (c)

   A second turtle has mass $398$ kg, correct to the nearest kilogram. Find the greatest possible difference between the masses of the two turtles.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The first row of seats in a stadium in Doha has $18$ seats, and each row after it has $4$ more seats than the row in front.

   (a)

   Find the number of seats in the $15$th row.

   [3 marks]

   (b)

   Find a formula for the number of seats in row $n$. Hence find the first row that has more than $100$ seats.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A fish farm in Norway starts with $2000$ salmon. Each year the population grows by $12\%$ and no fish are removed, so after $n$ years the number of salmon is modelled by $P=2000\times1.12^{n}$.

   (a)

   (i) Find the number of salmon after $3$ years, to the nearest whole number. [2] (ii) Use logarithms to find how many whole years it takes for the population to first exceed $5000$. [4]

   [6 marks]

   (b)

   A neighbouring farm also starts with $2000$ salmon but adds a fixed $260$ salmon each year. Find the difference between the two populations after $10$ years, and explain why the first farm's population will keep pulling ahead of the second farm's.

   [6 marks]

   Total for question 8: 12 marks

End of questions