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Number and Algebra - IB Questionbank | RevisionDojo

Practice Number and Algebra with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

Let $M=\left(\begin{array}{cc}a & 2 \\ 2 & -1\end{array}\right)$ , where $a \in \mathbf{Z}$

Find $M^{2}$ in terms of $a$

[2]

If $M^{2}$ is equal to $\left(\begin{array}{cc}5 & -4 \\ -4 & 5\end{array}\right)$ , find the value of $a$ .

[2]

Using this value of $a$ , find $M^{-1}$ and hence solve the system of equations : $\left\{\begin{aligned}-x+2 y & =-3 \\ 2 x-y & =3\end{aligned}\right.$

[5]

Question 2

HLPaper 1

Let $z=x+y i$ be a complex number.

Given that $w=z+16$ , write down the conjugate of $w$ .

[2]

Express $|w|^{2}$ in the form $x^{2}+y^{2}+a x+b$

[2]

Express $|z+1|^{2}$ in terms of $x$ and $y$ .

[2]

Given that $|z+16|=4|z+1|$ , find the value of $|z|$ .

[4]

Question 3

SLPaper 2

Use technology to solve for $x$ :

$2^{x}=x+3$

[3]

$x^{2}-\frac{5}{x}=0$

[2]

$x=\sqrt{5}-\sqrt{x}$

[2]

Question 4

HLPaper 1

Let $\ln a=p, \ln b=q$ . Write the following expressions in terms of $p$ and $q$

$\ln a^{3} b$

[2]

$\ln \frac{\sqrt{a}}{b}$

[2]

Question 5

HLPaper 1

Solve the equation $2-x \sqrt{3}=\frac{7 x}{\sqrt{3}}$ , giving your answer in the form $\frac{\sqrt{a}}{b}$ , where $a$ and $b$ are integers.

Find the value of $a$ and $b$ .

[3]

Given that $8 \sqrt{2}=2^{a}$ , find the value of $a$ .

[2]

Question 6

SLPaper 2

Terrence has $800000$ in his savings fund. He rolls his money into an annuity fund which earns $4.7 \%$ compounded monthly. He wants to withdraw $7000$ each month to live on.

How long will his money last.

[2]

How much longer would his money last if he only withdrew $6000$ each month.

[3]

Question 7

HLPaper 1

Solve the following equation for $z$ , where $z$ is a complex number.

Give your answer in the form $a+b i$ , where $a$ and $b$ are both real numbers.

$\frac{z}{3+4 i}+\frac{z-1}{5 i}=\frac{5}{3-4 i}$

[4]

Question 8

SLPaper 2

Cary decides to buy a new boat at a cost of $\$ 20000$, but cannot afford the full amount. The boat dealership offers him to finance a loan.

A 5 year loan at a nominal annual interest rate of $15 \%$ , compounded monthly. No deposit required and repayments are made each month.

Find the repayment made each month.

[2]

Question 9

HLPaper 1

Find the value of the integer $k$ in each of the following cases:

$\log 3 k=\log 3+\log 5$

[1]

$\log 21=\log 3+\log k$

[1]

$\ln 21=\ln 3+\ln k$

[1]

Question 10

HLPaper 1

Let $A(t)=5.1 \sin (\omega t)$ and $B(t)=3.2 \sin (\omega t+5)$

Find $A(t)+B(t)$ in the form $a \sin (\omega t+B)$ .

[4]

Number and Algebra Questionbank