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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 $M=\begin{pmatrix}a & 2 \\ 2 & -1\end{pmatrix}$ , where $a \in \mathbb{Z}$

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

[2]

If $M^{2}$ is equal to $\begin{pmatrix}5 & -4 \\ -4 & 5\end{pmatrix}$ , 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 2

In a geometric sequence, the sum to infinity is 25 and the second term is 6.

Find the first term and the common ratio of this sequence.

[5]

Question 5

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 6

SLPaper 2

Yejin plans to retire at age 60. She wants to create an annuity fund, which will pay her a monthly allowance of $4000 at the end of each month during her retirement. She wants to save enough money so that the payments last for 30 years. A financial advisor has told her that she can expect to earn 5% interest on her funds, compounded annually.

Calculate the amount Yejin needs to have saved into her annuity fund, in order to meet her retirement goal. Give your answer to the nearest dollar.

[3]

Yejin has just turned 28 years old. She currently has no retirement savings. She wants to save part of her salary at the end of each month into her annuity fund.

Calculate the amount Yejin needs to save each month, to meet her retirement goal given the same annual interest and annual compounding. Give your answer to the nearest dollar.

[3]

Question 7

HLPaper 1

Questions on algebra and surds.

Solve the equation $2-x \sqrt{3}=\frac{7 x}{\sqrt{3}}$ , giving your answer in the form $\frac{\sqrt{a}}{b}$ , where $a, b \in \mathbb{Z}$ . Find the value of $a$ and the value of $b$ .

[3]

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

[2]

Question 8

SLPaper 2

Terrence has $800,000 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 9

HLPaper 2

Consider the following matrices: $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 2 \\ -1 & 3 \end{pmatrix}$

Calculate $A + B$ and $B + A$ to verify that matrix addition is commutative.

[3]

Find $2A$ and verify that $(2A) + B = A + A + B$ .

[4]

Paper

Level

Question 1

HLPaper 2

Let $M=\begin{pmatrix}a & 2 \\ 2 & -1\end{pmatrix}$ , where $a \in \mathbb{Z}$

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

[2]

If $M^{2}$ is equal to $\begin{pmatrix}5 & -4 \\ -4 & 5\end{pmatrix}$ , 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 2

In a geometric sequence, the sum to infinity is 25 and the second term is 6.

Find the first term and the common ratio of this sequence.

[5]

Question 5

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 6

SLPaper 2

Calculate the amount Yejin needs to have saved into her annuity fund, in order to meet her retirement goal. Give your answer to the nearest dollar.

[3]

Yejin has just turned 28 years old. She currently has no retirement savings. She wants to save part of her salary at the end of each month into her annuity fund.

Calculate the amount Yejin needs to save each month, to meet her retirement goal given the same annual interest and annual compounding. Give your answer to the nearest dollar.

[3]

Question 7

HLPaper 1

Questions on algebra and surds.

Solve the equation $2-x \sqrt{3}=\frac{7 x}{\sqrt{3}}$ , giving your answer in the form $\frac{\sqrt{a}}{b}$ , where $a, b \in \mathbb{Z}$ . Find the value of $a$ and the value of $b$ .

[3]

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

[2]

Question 8

SLPaper 2

Terrence has $800,000 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 9

HLPaper 2

Consider the following matrices: $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 2 \\ -1 & 3 \end{pmatrix}$

Calculate $A + B$ and $B + A$ to verify that matrix addition is commutative.

[3]

Find $2A$ and verify that $(2A) + B = A + A + B$ .

[4]

Paper

Level

Question 1

HLPaper 2

Let $M=\begin{pmatrix}a & 2 \\ 2 & -1\end{pmatrix}$ , where $a \in \mathbb{Z}$

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

[2]

If $M^{2}$ is equal to $\begin{pmatrix}5 & -4 \\ -4 & 5\end{pmatrix}$ , 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 2

In a geometric sequence, the sum to infinity is 25 and the second term is 6.

Find the first term and the common ratio of this sequence.

[5]

Question 5

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 6

SLPaper 2

Calculate the amount Yejin needs to have saved into her annuity fund, in order to meet her retirement goal. Give your answer to the nearest dollar.

[3]

Yejin has just turned 28 years old. She currently has no retirement savings. She wants to save part of her salary at the end of each month into her annuity fund.

Calculate the amount Yejin needs to save each month, to meet her retirement goal given the same annual interest and annual compounding. Give your answer to the nearest dollar.

[3]

Question 7

HLPaper 1

Questions on algebra and surds.

Solve the equation $2-x \sqrt{3}=\frac{7 x}{\sqrt{3}}$ , giving your answer in the form $\frac{\sqrt{a}}{b}$ , where $a, b \in \mathbb{Z}$ . Find the value of $a$ and the value of $b$ .

[3]

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

[2]

Question 8

SLPaper 2

Terrence has $800,000 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 9

HLPaper 2

Consider the following matrices: $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 2 \\ -1 & 3 \end{pmatrix}$

Calculate $A + B$ and $B + A$ to verify that matrix addition is commutative.

[3]

Find $2A$ and verify that $(2A) + B = A + A + B$ .

[4]

Number and Algebra Questionbank