Number and algebraIB Maths: Applications and Interpretation SL: Topic test

20 questions, 54 marks

IB Maths: Applications and Interpretation SL

Number and algebra topic test

Total 54 marks

Name

Class

Date

1
A single bacterium has a mass of $9.5\times10^{-13}$ g. A laboratory culture contains $3.8\times10^{9}$ bacteria.
(a)
Find the total mass of the bacteria in the culture, in grams. Give your answer in the form $a\times10^{k}$ , where $1\le a<10$ and $k$ is an integer, correct to 3 significant figures.
[1 mark]
A $3.61\times10^{-4}$ g
B $2.50\times10^{-22}$ g
C $3.61\times10^{-3}$ g
D $3.61\times10^{-21}$ g
(b)
Find the mass of the whole culture in kilograms. Give your answer in the form $a\times10^{k}$ , where $1\le a<10$ , correct to 3 significant figures.
[1 mark]
A $3.61\times10^{-6}$ kg
B $3.61\times10^{0}$ kg
C $3.61\times10^{-5}$ kg
D $3.61\times10^{-9}$ kg
(c)
Find the number of bacteria that have a combined mass of 1 g. Give your answer in the form $a\times10^{k}$ , where $1\le a<10$ and $k$ is an integer, correct to 3 significant figures.
[2 marks]
Total for question 1: 4 marks
2
A theatre has 22 rows of seats. The first row has 14 seats and each row after the first has 3 more seats than the row in front of it.
(a)
Find the number of seats in row 10.
[1 mark]
A $44$
B $41$
C $38$
D $30$
(b)
Find the total number of seats in the theatre.
[1 mark]
A $1034$
B $1694$
C $2002$
D $1001$
(c)
Find the number of the first row that has more than 60 seats.
[2 marks]
Total for question 2: 4 marks
3
A ball is dropped from a height of $3.2$ m onto a hard floor. After each bounce the ball rises to $70\%$ of the height it reached on the previous bounce.
(a)
Find the number of the first bounce after which the ball rises to a height of less than $0.5$ m. Use your GDC.
[3 marks]
(b)
Find the total distance travelled by the ball from the moment it is dropped until it hits the floor for the fifth time. Give your answer in metres, correct to 3 significant figures.
[4 marks]
Total for question 3: 7 marks
4
Layla invests $15\,000$ USD in an account that pays a nominal annual interest rate of $4.2\%$ , compounded quarterly. No money is added or withdrawn.
(a)
(i) Find the value of the investment after 8 years. Give your answer in USD, correct to 2 decimal places.
(ii) The inflation rate is $2.1\%$ per year, and this rate stays constant. Find the real value of the investment after 8 years, in USD to 2 decimal places, in terms of its value today.
(iii) Find the percentage increase in the real value of the investment over the 8 years, correct to 3 significant figures.
[6 marks]
(b)
Layla wants the value of the investment to double to $30\,000$ USD.
(i) Write down an equation for the number of years, $t$ , needed.
(ii) Use logarithms to solve your equation. Give $t$ correct to 3 significant figures.
(iii) Interest is added at the end of each quarter only. Find the least number of complete quarters needed for the value to be at least $30\,000$ USD, and justify your answer with values of the investment.
[6 marks]
Total for question 4: 12 marks
5
A yeast culture contains $N$ cells, where $N=500e^{0.3t}$ and $t$ is the time, in hours, since the culture was started.
(a)
Find the number of cells in the culture after 6 hours, correct to 3 significant figures.
[1 mark]
A $900$
B $6.05$
C $4050$
D $3020$
(b)
Find the time taken for the culture to reach 2000 cells.
[1 mark]
A $1.39$ hours
B $4.62$ hours
C $2.01$ hours
D $25.3$ hours
(c)
Find the time taken for the number of cells in the culture to double. Give your answer in hours, correct to 3 significant figures.
[2 marks]
Total for question 5: 4 marks
6
A market stall sells apples, oranges and bananas. The prices per kilogram, in AED, are $x$ for apples, $y$ for oranges and $z$ for bananas. Order 1 is 2 kg of apples, 1 kg of oranges and 3 kg of bananas, costing $28.65$ AED. Order 2 is 1 kg of apples, 3 kg of oranges and 2 kg of bananas, costing $26.85$ AED. Order 3 is 3 kg of apples, 2 kg of oranges and 1 kg of bananas, costing $31.80$ AED.
(a)
Which equation represents Order 3?
[1 mark]
A $3x+2y+z=31.80$
B $x+2y+3z=31.80$
C $3x+2y+z=6$
D $x+y+z=31.80$
(b)
Use your GDC to find the price of oranges per kilogram.
[1 mark]
A $6.50$ AED
B $3.80$ AED
C $4.25$ AED
D $5.30$ AED
(c)
Find the cost of an order of 4 kg of apples, 2 kg of oranges and 5 kg of bananas.
[2 marks]
Total for question 6: 4 marks
7
A solid metal cube has side length $4.5$ cm, correct to 1 decimal place, and a mass of $730$ g, correct to the nearest $10$ g.
(a)
Find the lower bound and the upper bound for the volume of the cube. Give each answer in cm³, correct to 3 significant figures.
[3 marks]
(b)
Find the greatest possible density of the metal, in g cm⁻³, correct to 3 significant figures. Use density $=\frac{\text{mass}}{\text{volume}}$ .
[4 marks]
Total for question 7: 7 marks
8
Priya takes out a loan of $18\,000$ EUR to buy equipment. Interest is charged at a nominal annual rate of $5.4\%$ , compounded monthly. The loan is repaid by equal payments made at the end of each month.
(a)
The loan is repaid over 4 years. Use your GDC finance solver.
(i) Find the monthly payment, in EUR to 2 decimal places.
(ii) Find the total interest paid over the 4 years, in EUR to 2 decimal places.
(iii) Find the balance still owed immediately after the 24th payment, in EUR to 2 decimal places.
[6 marks]
(b)
(i) Priya instead repays the loan over 3 years. Find her monthly payment, in EUR to 2 decimal places.
(ii) Find the interest she saves by repaying over 3 years instead of 4 years, in EUR to 2 decimal places.
(iii) Priya decides she can afford at most $400$ EUR a month. Find the largest loan she could repay in 4 years at the same interest rate, in EUR to 2 decimal places.
[6 marks]
Total for question 8: 12 marks

End of questions

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).