IB Maths AI SL 1 Number and Algebra

Practise IB Maths AI SL number and algebra through sequences, financial models, logarithms and technology-supported calculations with clear interpretation.

Covered syllabus points

1.1 Number and algebra - SL content

[Maximum number: 3]

The Scheveningen Ferris wheel's lowest point is 8 m above sea level, and its highest point is 45 m above sea level.

Figure for Question 1 Number and algebra question 1 — IB Maths AI SL

Calculate the total number of revolutions that the Ferris wheel has made. Give your answer in the form $a \times 10^{k}$ where $1 \leq a<10$ and k is an integer.

4 (revolutions per hour) seen OR 32 (revolutions per day) seen $(4 \times 8 \times 3000=) 96000$

[Maximum number: 18]

A marathon is a race over a distance of 42.19 km .

(a)

Write down 42.19 km correct to the nearest kilometre.

[ 3 ]

$42(\mathrm{~km})$

(b)

Find the percentage error if 42.19 km is rounded to the nearest kilometre.

Rolando and Davi are training to run a marathon. They both start their training in the same week.
Rolando runs 6 km in the first week, 8 km in the second week and 10 km in the third week. In each subsequent week he runs 2 km further than the previous week.

[ 3 ]

$\quad(\varepsilon=)\left|\frac{42-42.19}{42.19}\right| \times 100 \%$
0.450 (\%) (0.450343...(\%))

Note: Award (M1) for correct substitution of their part (a)(i) into percentage error formula.

(c)

Find the week during which Rolando runs 42 km .

[ 3 ]

METHOD 1 attempt to substitute into arithmetic sequence formula
$42=6+(n-1) \times 2$ OR 42=2 n+4
( n= ) 19

METHOD 2
list of any consecutive terms ending on/above 42
at least two correct terms (not including 6, 8 and 10)
( n= ) 19

(d)

Find the total distance that Rolando runs in the first 10 weeks.

Davi runs 3 km in the first week. In each subsequent week he runs 20 % further than the previous week. These distances form the terms of a geometric sequence.

[ 2 ]

METHOD 1
$\left(S_{10}=\right) \frac{10}{2}(2 \times 6+(10-1) 2)$ $150(\mathrm{~km})$

METHOD 2
summing (some) correct terms up to 24 OR list of correct cumulative values $6+8+10+\ldots+24$ OR $6,14,24,36 \ldots$ $150(\mathrm{~km})$

(e)

Find the first week during which Davi runs more than 42 km .

[ 4 ]

(d) METHOD 1
$3(1.2)^{n-1}$

Note: Award (A1) for correct substitution into geometric sequence formula.
comparing their geometric expression to 42
e.g. $3(1.2)^{n-1}>42$ OR $3(1.2)^{n-1}=42$ $(n=) 15.4747 \ldots$ seen OR $38.5175 \ldots$ and $46.2210 \ldots$ seen (from table)

16 (weeks)

METHOD 2

List of values see. At least 3, 3.6, 4.32 etc
$38.5175 \ldots$ and $46.2210 \ldots$ seen

16 (weeks)

(f)

Find the first week that the total distance that Davi has run, since the start of training, exceeds 150 km .

[ 3 ]

METHOD 1 attempt to substitute into geometric series formula

Note: FT their r from (d).

150<\frac{3\left(1-1.2^{n}\right)}{1-1.2} \text { OR } 150=\frac{3\left(1-1.2^{n}\right)}{1-1.2} \text { OR } 13.1520 \ldots

14 (weeks)

METHOD 2
evidence using tables in GDC or adding the first 14 terms both crossover values seen, e.g $145.589 \ldots$ and $177.587 \ldots$

[Maximum number: 21]

Give all answers in this question to the nearest dollar.
Anika invested 10000 Canadian dollars (CAD) in an account which earns a nominal annual interest rate of 3.4 %, compounded monthly.

(a)

Calculate the value of Anika's investment after 6 years.

Anika wants to purchase a house that costs 225000 CAD. Anika uses the value obtained in part (a) to make an initial payment on the house and takes out a loan to pay the rest.

[ 3 ]

EITHER
N=72
(M1)(A1)
I %=3.4
$P V= \pm 10000$
P / Y=12
C / Y=12
Note: Award M1 for an attempt to use a financial app in their technology (i.e., at least three entries seen, but not necessarily correct), A1 for all entries correct.

OR
N=72
I %=3.4 / 12
$P V= \pm 10000$
P / Y=1
C / Y=1
Note: Award M1 for an attempt to use a financial app in their technology (i.e., at least three entries seen, but not necessarily correct), A1 for all entries correct.

OR
attempt to substitute into the compound interest formula
$F V=10000 \times\left(1+\frac{3.4}{100 \times 12}\right)^{6 \times 12}$

THEN
(FV =) 12259 (CAD)

(b)

Determine the amount of Anika's loan.

Anika amortizes her loan over 15 years. She arranges to make a loan payment at the end of each month. The nominal annual interest rate for the loan is 6.4 %, compounded half-yearly.

[ 1 ]

(c)

Determine Anika's monthly loan payment.

After 2 complete years, Anika decides to increase her monthly loan payment to 2200 CAD. At the time, the remaining amount of the loan is 194572 CAD.

[ 5 ]

$\quad N=180$
F V=0
I %=6.4
$P V= \pm 212741$
P / Y=12
C / Y=2

Note: Award M1 for attempt to use a financial app with at least three entries seen, but not necessarily correct, A1 for N and I, A1 for P / Y and C / Y, A1 for their P V from (b) and F V=0. Approaches that use the compound interest formula receive no marks.
( P M T= ) 1832 (CAD)

Note: Accept -1832 (CAD).

(d)

Calculate the amount by which the loan will decrease during the next 3 complete years.

[ 6 ]

$\quad N=36$ $P V= \pm 194572$
PMT = 2200
Note: Award M1 for adjusting at least one of N, P V, or P M T; A1 for correct entries ( P V and P M T must have opposite signs). Allow FT from incorrect entries for P / Y and/or C / Y from part (c).
( F V= )148099 (CAD)
(194572-148099)
46473 (CAD)

(e)

Determine the total interest that Anika will pay over this 3 -year period.

[ 6 ]

$2200 \times 12 \times 3(=79200)$
(79200-46473=)
32727 (CAD)

All question bank results loaded

Covered syllabus points

1 Number and algebra question 1

1 Number and algebra question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (f)

1 Number and algebra question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)