IB Maths AI HL 1 Number and Algebra

Practise IB Maths AI HL number and algebra through shared-core and HL sequences, finance, logarithms and technology-supported models with clear interpretation.

[Maximum number: 6]

Zaha is designing a bridge to cross a river. She believes that the weight of the steel needed for this bridge is approximately 53632000 kg .
The exact weight of the steel needed for the bridge is 55625000 kg .

(a)

Find the percentage error in Zaha's approximation.

Zaha's design is used to build five identical bridges.

[ 2 ]

attempt to substitute into percentage error formula

\left|\frac{53632000-55625000}{55625000}\right| \times 100

3.58 (\%) (3.58292…(\%))

Note: Award (M1)A0 for a final answer of -3.58 (\%) or 0.0358 .

(b)

Find the weight of the steel needed for these five bridges, to three significant figures.

[ 1 ]

(c)

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

[ 3 ]

$2.78 \times 10^{8}$

Note: Award A1 for correct mantissa, consistent with their answer in part (b)(i).

Marking guidance:

Award A1 for a correct exponent, consistent with their answer in part (b)(i).
Award AOAO for answers such as $27.8 \times 10^{7}$ .

[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
(A1)
A1

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

[Maximum number: 8]

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 ]

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

(b)

Show that Nanthana's claim is correct, stating the exact value of the common ratio for this sequence.

[ 4 ]

attempt to find |r|

\begin{aligned} & |r|=\sqrt{\left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{2}} \\ & |r|=\frac{2 \sqrt{2}}{3} \end{aligned}

statement that multiplying a number by a complex number changes the modulus of the number by a factor of |r| (may be seen at any stage)

Note: Award at most (M1)A1A0R1 if common ratio is given as $0.942809 \ldots \ldots$ .
therefore it is a geometric sequence

(c)

Hence, find the sum of the infinite sequence $\left|u_{1}\right|,\left|u_{2}\right|,\left|u_{3}\right|, \ldots$ .

[ 2 ]

substitution of their r and 9 into the sum formula for infinite sequence
$S_{\infty}=\frac{9}{1-\frac{2 \sqrt{2}}{3}}$ $\left(S_{\infty}=\right) 157(157.367 \ldots, 81+54 \sqrt{2})$

Note: No FT if their $|r| \geq 1$ .

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Covered syllabus points

1 Number and algebra question 1

Question (a)

Question (b)

Question (c)

1 Number and algebra question 2

Question (a)

Question (b)

Question (c)

1 Number and algebra question 3

Question (a)

Question (i)

Question (ii)

Question (b)

Question (c)