IB Maths AI HL 1.1 Number and Algebra Sl Content Questions

Practise IB Mathematics AI HL 1.1 by extending number-and-algebra models, sequences, finance and symbolic reasoning with technology.

Exam points

Represent, convert and check numerical quantities using scientific notation, powers, logarithms, accuracy, bounds and percentage error.
Model arithmetic and geometric sequences or series, calculating terms, finite sums, thresholds, convergence and long-term behaviour.
Apply compound-interest, depreciation, annuity and loan models, comparing rates, payment timing, balances and target periods.
Use integer exponent and logarithm laws to solve equations and interpret logarithmic scales in contextual models.
Use graphing or numerical technology to solve permitted systems and polynomial equations, then select and justify valid roots.
Interpret numerical output with correct units, domains, rounding, parameter meaning and practical reasonableness.

Covered syllabus points

SL 1.1—Scientific notation SL 1.2—Arithmetic sequences and series SL 1.3—Geometric sequences and series SL 1.4—Financial applications SL 1.5—Integer exponents and logarithms SL 1.6—Approximation and error SL 1.7—Amortization and annuities SL 1.8—Technology for equations

[Maximum number: 5]

A bakery sells egg tarts in boxes. Each box contains 6 tarts. On a particular morning, the bakery has 400 freshly baked tarts. The number of tarts remaining, after n boxes have been sold, forms an arithmetic sequence, $u_{n}=u_{1}+(n-1) d$ .

(a)

Write down the value of d.

[ 1 ]

(b)

Find the value of $u_{1}$ .

[ 1 ]

(c)

Find the greatest number of boxes the bakery can sell that morning.

[ 3 ]

attempt to solve $394-6(n-1) \geq 0$ OR $400 \div 6$

Note: Accept equality or any inequality with zero seen, graphical approach to find zero, use of table/list to find zero.
( n< ) $66.6666 \ldots$
( n= ) 66

Note: If 400 is given as the answer in (b) then n=67 in part (c).

[Maximum number: 2]

Nanthana is completing an exploration on multiplication using complex numbers. She considers a geometric sequence where $u_{1}=9$ and $r=\frac{2}{3}+\frac{2}{3} \mathrm{i}$ .

(a)

Write down the value of

[ 2 ]

(i)

$\quad u_{2}$ .

[ 1 ]

$\left(u_{2}=\right) 6+6 \mathrm{i} \quad \boldsymbol{A} \mathbf{1}$

(ii)

$u_{3}$ .

Nanthana claims that the sequence $\left|u_{1}\right|,\left|u_{2}\right|,\left|u_{3}\right|, \ldots$ also forms a geometric sequence.

[ 1 ]

$\quad\left(u_{3}=\right) 8 \mathrm{i} \quad \boldsymbol{A} \mathbf{1}$

[Maximum number: 19]

Give your answers in parts (a), (d)(i), (e) and (f) to the nearest dollar.
Daisy invested 37000 Australian dollars (AUD) in a fixed deposit account with an annual interest rate of 6.4 % compounded quarterly.

(a)

Calculate the value of Daisy's investment after 2 years.

[ 3 ]

EITHER
N=2
P V=-37000
I %=6.4
P / Y=1
C / Y=4
Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct.

OR
N=8
P V=-37000
I %=6.4
P / Y=4
C / Y=4

Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct.

OR
$F V=37000 \times\left(1+\frac{6.4}{100 \times 4}\right)^{4 \times 2}$
Note: Award M1 for substitution into compound interest formula, (A1) for correct substitution.
= 42010 AUD

Note: Award (M1)(A1)A0 for unsupported 42009.87.

(b)

After m months, the amount of money in the fixed deposit account has appreciated to more than 50000 AUD.

Find the minimum value of m, where $m \in \mathbb{N}$ .

[ 4 ]

EITHER
P V=-37000
F V=50000
I %=6.4
P / Y=1
C / Y=4
Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct. The final mark can still be awarded for the correct number of months (multiple of 3 ).
OR
P V=-37000
F V=50000
I %=6.4
P / Y=4
C / Y=4
Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct.
OR

50000<37000 \times\left(1+\frac{6.4}{100 \times 4}\right)^{4 \times n} \text { OR } 50000<37000 \times\left(1+\frac{6.4}{100 \times 4}\right)^{n} \quad \text { (M1)(A1) }

Note: Award M1 for the correct inequality, 50000 and substituted compound interest formula. Allow an equation. Award A1 for correct substitution.

THEN

\begin{aligned} & N=4.74(\text { years })(4.74230 \ldots) \text { OR } N=18.9692 \ldots \text { (quarters) } \\ & m=57 \text { months } \end{aligned}

Note: Award A1 for rounding their m to the correct number of months. The final answer must be a multiple of 3 . Follow through within this part.

(c)

Daisy is saving to purchase a new apartment. The price of the apartment is 200000 AUD.
Daisy makes an initial payment of 25 % and takes out a loan to pay the rest.

Write down the amount of the loan.

[ 1 ]

(d)

The loan is for 10 years, compounded monthly, with equal monthly payments of 1700 AUD made by Daisy at the end of each month.

For this loan, find

[ 5 ]

(i)

the amount of interest paid by Daisy.

[ 2 ]

$120 \times 1700-150000$
=54000 AUD

(ii)

the annual interest rate of the loan.

[ 3 ]

$\quad N=120$
P V=-150000
P M T=1700
F V=0
P / Y=12
C / Y=12
(M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology or an attempt to use an annuity formula or F V=0 seen. If a compound interest formula is equated to zero, award M1, otherwise award M0 for a substituted compound interest formula.

Award A1 for all entries correct in financial app or correct substitution in annuity formula, but award A0 for a substituted compound interest formula. Follow through marks in part (d)(ii) are contingent on working seen.

$r=6.46\%\quad(6.45779\ldots)$

(e)

After 5 years of paying off this loan, Daisy decides to pay the remainder in one final payment.

Find the amount of Daisy's final payment.

[ 3 ]

$\quad N=60$
I=6.46 (6.45779...)
P V=-150000
P M T=1700
P / Y=12
C / Y=12
(M1)(A1)
Note: Award M1 for an attempt to use a financial app in their technology or an attempt to use an annuity formula. Award (M0) for a substituted compound interest formula. Award A1 for all entries correct. Follow through marks in part (e) are contingent on working seen.

$FV=86973\ \mathrm{AUD}$

(f)

Find how much money Daisy saved by making one final payment after 5 years.

[ 3 ]

$204000-(60 \times 1700+86973)$ OR 204000-188973
(M1)(M1)

Note: Award M1 for $60 \times 1700$ . Award M1 for subtracting their $(60 \times 1700+86973)$ from their (204000). Award at most M1M0 for their 204000- ( $60 \times 1700$ ) or M0M0 for their 204000-(86973). Follow through from parts (d)(i) and (e). Follow through marks in part (f) are contingent on working seen.

15027 AUD

[Maximum number: 1]

This question is about applying ideas from logarithms, calculus and probability to an unfamiliar mathematical theory called information theory.
Claude Shannon developed a mathematical theory called information theory to measure the information gained when random events occur. He defined the information, I, that is gained when an event with probability p occurs as

I=-\ln p

where $0<p \leq 1$ . For example, no information is gained ( I=0 ) when an event is certain to $\operatorname{occur}(p=1)$ .

When a coin is flipped, the outcome is either heads or tails. The coin may be biased. Let p be the probability of the outcome being heads.

Find, in terms of p, the information gained when the outcome is tails.

$(I=)-\ln (1-p)$

[Maximum number: 3]

Linda owns a field, represented by the shaded region R. The plan view of the field is shown in the following diagram, where both axes represent distance and are measured in metres.

The segments [AB], [CD] and [AD] respectively represent the western, eastern and southern boundaries of the field. The function, f(x), models the northern boundary of the field between points B and C and is given by

f(x)=\frac{-x^{2}}{50}+2 x+30, \text { for } 0 \leq x \leq 70

Calculate the percentage error in Linda's estimate.

$\frac{11.4}{4713.33 \ldots} \times 100 \%$ OR $\left|\frac{4701.93 \ldots-4713.33 \ldots}{4713.33 \ldots}\right| \times 100 \% \quad$ (M1)

Note: Award (M1) for their correct substitution into the percentage error formula.
$0.242 \%(0.241867 \ldots \%)$ A1
Note: Percentage sign is required. Accept $0.242038 \ldots \%$ if 4710 is used.

[Maximum number: 3]

A sports player on a horizontal athletic field hits a ball. The height of the ball above the field, in metres, after it is hit can be modelled using a quadratic function of the form $f(x)=a x^{2}+b x+c$ , where x represents the horizontal distance, in metres, that the ball has travelled from the player.
A specialized camera tracks the initial path of the ball after it is hit by the player. The camera records that the ball travels through the three points (3,2.36),(10,5) and (17,7.16), as shown in Diagram 1.

Diagram 1

Use your answer to part (a) and two similar equations to find the equation of the quadratic model for the height of the ball.

finding other equations to solve simultaneously
$5=a(10)^{2}+b(10)+c$ AND 7.16 $=a(17)^{2}+b(17)+c$
OR 5=100 a+10 b+c AND 7.16 =289 a+17 b+c
any one coefficient in equation correct
$f(x)=-0.00490 x^{2}+0.441 x+1.08$ $\left(f(x)=-0.00489795 \ldots x^{2}+0.440816 \ldots x+1.08163 \ldots\right)$ $\left(f(x)=-\frac{6}{1225} x^{2}+\frac{108}{245} x+\frac{53}{49}\right)$
Note: Award at most (M1)(A1)A0 if answer is not expressed as an equation.

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (i)

Question (ii)

Question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question (i)

Question (ii)

Question (e)

Question (f)

Question 4

Question 5

Question 6