Systems of Equations - IB Math AA Question Bank

Question 1

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

Consider the system of equations

\begin{aligned} x+y+z&=2\\ 2x-y+3z&=13\\ x+4y-z&=-10. \end{aligned}

A

Write down the augmented matrix for the system.

[1]

B

Solve the system.

[4]

Question 2

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

The augmented matrix of a system of three linear equations in $x$ , $y$ and $z$ has been reduced to

\left[\begin{array}{ccc|c} 1&2&-1&3\\ 0&1&4&-2\\ 0&0&0&0 \end{array}\right]

A

Find the solution set of the system, writing your answer using a real parameter.

[3]

B

State the geometric meaning of this solution set for the three planes.

[1]

Question 3

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

A quadratic function $f$ is given by $f(x)=ax^2+bx+c$ . It is known that

f(0)=1,\quad f(1)=0,\quad f(2)=3

A

Form a system of three linear equations in $a$ , $b$ and $c$ .

[2]

B

Find $a$ , $b$ and $c$ .

[2]

Question 4

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

Consider the system of equations

\begin{aligned} 2x-y+3z&=14\\ x+4y-2z&=-3\\ 3x+y+z&=10. \end{aligned}

A

Write down the augmented matrix for this system.

[1]

B

Use your graphic display calculator to row-reduce the augmented matrix.

[2]

C

Hence find the solution and the value of $4x-2y+z$ .

[2]

Question 5

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

A system of three linear equations is given by

\begin{aligned} 2.4x-1.5y+0.8z&=6.65\\ -1.2x+3.1y+2.5z&=4.89\\ 4.0x+0.6y-1.7z&=-1.40. \end{aligned}

A

Write down the augmented matrix for the system.

[1]

B

Use your graphic display calculator to solve the system.

[2]

C

Calculate $5x-2y+z$ .

[2]

Question 6

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

The augmented matrix of a system in $x$ , $y$ and $z$ is row-reduced using a graphic display calculator. The result is

\left[\begin{array}{ccc|c} 1&0&-2&4\\ 0&1&3&-1\\ 0&0&0&0 \end{array}\right]

A

State the number of solutions of the system. Give a reason.

[1]

B

Write the solution set in terms of a parameter.

[3]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the system, where $k\in\mathbb{R}$ ,

\begin{aligned} x+2y-z&=3\\ 2x+5y+z&=7\\ 3x+7y+kz&=10. \end{aligned}

A

Use row reduction to obtain an equation involving only $z$ and $k$ .

[2]

B

Find the value of $k$ for which the system does not have a unique solution, and state the number of solutions in this case.

[2]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the system, where $k\in\mathbb{R}$ ,

\begin{aligned} x-y+z&=2\\ 2x+y-z&=1\\ 3x+ky+z&=5. \end{aligned}

A

Find the value of $k$ for which the system has no solution.

[3]

B

Solve the system when $k=1$ .

[3]

Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the system of two equations in $x$ and $y$ , where $k\in\mathbb{R}$ ,

\begin{aligned} kx+2y&=6\\ 2x+ky&=6. \end{aligned}

A

Find the values of $k$ for which the system does not have a unique solution.

[2]

B

Classify the number of solutions for all real values of $k$ .

[3]

Question 10

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A theatre sells adult, child and senior tickets. Let $a$ , $c$ and $s$ be the prices, in dollars, of an adult, child and senior ticket respectively. The ticket sales for three shows are shown below.

Show 1: $80$ adult, $60$ child and $40$ senior tickets, total revenue $1780$ dollars.

Show 2: $100$ adult, $45$ child and $30$ senior tickets, total revenue $1835$ dollars.

Show 3: $60$ adult, $90$ child and $50$ senior tickets, total revenue $1830$ dollars.

A

Form a system of three linear equations in $a$ , $c$ and $s$ .

[2]

B

Find the price of each type of ticket.

[2]

C

A fourth show sells $25$ adult, $40$ child and $20$ senior tickets. Calculate its total revenue.

[1]

Question 11

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A quadratic function has the form $f(x)=ax^2+bx+c$ . It passes through the points $(1,3.45)$ , $(3,4.65)$ and $(5,11.85)$ .

A

Form a system of three linear equations in $a$ , $b$ and $c$ .

[2]

B

Use your graphic display calculator to find $a$ , $b$ and $c$ .

[2]

C

Find $f(4)$ .

[1]

Question 12

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

After valid row operations, the augmented matrix of a system in $x$ , $y$ and $z$ is

\left[\begin{array}{ccc|c} 1&2&-1&3\\ 0&1&4&-2\\ 0&0&h-5&2h-10 \end{array}\right]

where $h\in\mathbb{R}$ .

A

Find the value of $h$ for which the system has infinitely many solutions.

[2]

B

For this value of $h$ , write the solution set in terms of a parameter.

[3]

Question 13

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A manufacturer blends three liquid concentrates, $A$ , $B$ and $C$ , to make $120$ kg of a mixture. Concentrate $A$ contains $10\%$ active ingredient, $B$ contains $25\%$ , and $C$ contains $40\%$ . The final mixture must contain $30$ kg of active ingredient. The costs of $A$ , $B$ and $C$ are $8$ , $5$ and $3$ dollars per kg respectively, and the total cost is $645$ dollars.

A

Form a system of three linear equations for the masses $x$ , $y$ and $z$ of concentrates $A$ , $B$ and $C$ respectively.

[2]

B

Use your graphic display calculator to calculate $x$ , $y$ and $z$ .

[2]

C

Find the percentage of the mixture that is concentrate $C$ .

[1]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The system of equations

\begin{aligned} x+y+z&=4\\ 2x-y+3z&=7\\ 3x+ay+4z&=b \end{aligned}

represents three planes, where $a,b\in\mathbb{R}$ .

A

Find the set of values of $a$ and $b$ for which the three planes have no common point of intersection.

[5]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A system of three linear equations has augmented matrix

\left[\begin{array}{ccc|c} 1&1&1&1\\ 2&3&4&5\\ 3&4&p&q \end{array}\right]

where $p,q\in\mathbb{R}$ .

A

Determine the conditions on $p$ and $q$ for the system to have a unique solution, no solution, and infinitely many solutions.

[6]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The planes $\Pi_1$ , $\Pi_2$ and $\Pi_3$ have equations

\begin{aligned} \Pi_1:\quad x+y+z&=1\\ \Pi_2:\quad 2x+3y+z&=4\\ \Pi_3:\quad x+2y+\lambda z&=\mu, \end{aligned}

where $\lambda,\mu\in\mathbb{R}$ .

A

Find the values of $\lambda$ and $\mu$ for which the three planes intersect in a line.

[4]

B

For these values of $\lambda$ and $\mu$ , find a parametric form of the line of intersection.

[2]

Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the system, where $k,m\in\mathbb{R}$ ,

\begin{aligned} x+2y-z&=5\\ 2x+5y+z&=12\\ 3x+7y+kz&=m. \end{aligned}

A

Find the values of $k$ and $m$ for which the system has infinitely many solutions.

[4]

B

For these values of $k$ and $m$ , write the solution set using a real parameter.

[2]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

At three junctions in a one-way traffic network, the unknown hourly traffic flows $x$ , $y$ and $z$ satisfy

\begin{aligned} x+y&=620\\ y+z&=480\\ x-z&=140. \end{aligned}

A

Solve the system, giving the solution set in terms of a parameter.

[3]

B

Traffic flows cannot be negative. Determine the possible range of values of $z$ .

[1]

C

If $x=350$ , find $y$ and $z$ .

[2]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the system

\begin{aligned} 2x-y+z&=1\\ x+3y-2z&=4\\ 5x+y+kz&=11, \end{aligned}

where $k\in\mathbb{R}$ .

A

Determine the values of $k$ for which the system has a unique solution.

[2]

B

Classify the system when $k=0$ .

[2]

C

For $k=7$ , solve the system.

[2]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The point $P(1,-2,3)$ lies on each of the three planes

\begin{aligned} mx+2y-z&=n\\ 3x-y+mz&=14\\ x+ny+2z&=15, \end{aligned}

where $m,n\in\mathbb{R}$ .

A

Determine the values of $m$ and $n$ .

[4]

B

For these values of $m$ and $n$ , determine whether the system has a unique solution.

[2]

Question 21

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

In a small directed network, the unknown flows $x$ , $y$ and $z$ satisfy

\begin{aligned} x+y-z&=8\\ 2x-y+3z&=10\\ 3x+2z&=18. \end{aligned}

All flows are measured in hundreds of units per hour.

A

I.

Show that the third equation is dependent on the first two equations.

[2]

II.

Find the general solution of the system in terms of $z=t$ .

[3]

B

Given that no flow can be negative, determine the possible values of $t$ .

[2]

C

I.

An additional observation gives $y=2x$ . Find $t$ .

[2]

II.

Hence find the three flows.

[1]

Question 22

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The same coefficient matrix is used in two different systems:

\begin{aligned} x+y+z&=2\\ 2x-y+3z&=-3\\ x+2y-z&=6 \end{aligned} \qquad\text{and}\qquad \begin{aligned} x+y+z&=3\\ 2x-y+3z&=5\\ x+2y-z&=0. \end{aligned}

A

I.

Solve the first system.

[3]

II.

Solve the second system.

[2]

B

Let $\mathbf{b}_1$ and $\mathbf{b}_2$ denote the right-hand side vectors of the first and second systems respectively. Without repeating elimination, solve the system with the same coefficient matrix and right-hand side vector $2\mathbf{b}_1-\mathbf{b}_2$ .

[2]

C

Explain why the conclusion in part (b) would not necessarily be valid if the coefficient matrix were different in the two original systems.

[2]

Question 23

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A shop sells three items $A$ , $B$ and $C$ at prices $p$ , $q$ and $r$ dollars respectively. Three bundle prices give the system

\begin{aligned} 2p+q+r&=17\\ p+3q+2r&=26\\ 3p+4q+3r&=43. \end{aligned}

A

I.

Show that the system does not determine a unique set of prices.

[2]

II.

Let $r=t$ . Find $p$ and $q$ in terms of $t$ .

[3]

B

Assuming that all prices are non-negative, determine the possible values of $t$ .

[2]

C

I.

If items $A$ and $B$ have the same price, find $t$ .

[2]

II.

Hence find the three prices.

[1]

Question 24

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the system

\begin{aligned} x+y-z&=2\\ 2x+3y+z&=7\\ -x+py+qz&=r, \end{aligned}

where $p,q,r\in\mathbb{R}$ .

A

Determine the conditions on $p$ , $q$ and $r$ for the system to be inconsistent.

[5]

B

For $p=0$ , $q=4$ and $r=1$ , find the solution set.

[2]

Question 25

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the symmetric system, where $a\in\mathbb{R}$ ,

\begin{aligned} x+y+az&=2\\ x+ay+z&=2\\ ax+y+z&=2. \end{aligned}

A

Find the values of $a$ for which the system does not have a unique solution.

[4]

B

Classify the number of solutions for each of the values found in part (a).

[3]

Question 26

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

An engineer calibrates three sensors, $A$ , $B$ and $C$ . The unknown output contributions, in millivolts, from one unit of each sensor are $a$ , $b$ and $c$ respectively. Three calibration trials are shown in the table.

Trial	Sensor A [units]	Sensor B [units]	Sensor C [units]	Output [mV]
1	1	2	1	18.2
2	3	1	2	34.4
3	2	4	3	47.6
4	4	0	1	25.1

A

AI.

Form an augmented matrix which could be used to find $a$ , $b$ and $c$ .

[2]

AII.

Use your graphic display calculator to find $a$ , $b$ and $c$ .

[3]

B

The fourth-trial sensor quantities are $4$ units of sensor $A$ , $0$ units of sensor $B$ , and $1$ unit of sensor $C$ .

BI.

Calculate the predicted output for the fourth trial.

[2]

BII.

The actual output in the fourth trial is $25.1\ \text{mV}$ . Determine the percentage error between the predicted and actual outputs, relative to the predicted output.

[3]

Question 27

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The height $h$ metres of a moving platform at time $t$ seconds is modelled by

h(t)=at^2+bt+c

Measurements give $h(0)=1.80$ , $h(2)=5.00$ and $h(5)=0.500$ .

Scatter plot of the measured platform height against time at three observed times.

A

I.

Write down a system of three linear equations in $a$ , $b$ and $c$ .

[2]

II.

Use your graphic display calculator to find $a$ , $b$ and $c$ .

[3]

B

Use the model found in part (a).

I.

Find the maximum height of the platform and the time at which it occurs.

[3]

II.

Determine the two times at which the platform is at a height of $3.00\ \text{m}$ .

[2]

Question 28

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

After valid row operations, the augmented matrix of a system of three linear equations in $x$ , $y$ and $z$ is

\left[\begin{array}{ccc|c} 1&0&2&5\\ 0&1&-1&-3\\ 0&0&k-4&2k-8 \end{array}\right]

where $k\in\mathbb{R}$ .

A

I.

Determine the values of $k$ for which the system has a unique solution.

[2]

II.

For $k\neq 4$ , find the solution in terms of $x$ , $y$ and $z$ .

[3]

B

I.

Classify the system when $k=4$ .

[2]

II.

For $k=4$ , write the solution set in terms of a real parameter.

[2]

Question 29

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A nutritionist prepares a $200$ g mixture using ingredients $A$ , $B$ and $C$ . Let $x$ , $y$ and $z$ be the masses, in grams, of ingredients $A$ , $B$ and $C$ respectively. The mixture must contain $39$ g of protein and have total cost $790$ cents. The following information is available.

Ingredient	Protein per gram / $g\,g^{-1}$	Cost / cents $g^{-1}$
A	0.20	4
B	0.10	3
C	0.30	5

A

(a)

I.

Form a system of three linear equations in $x$ , $y$ and $z$ .

[3]

II.

Use your graphic display calculator to determine the type of solution and express the solutions parametrically.

[3]

B

A second mixture must still have total mass $200$ g and protein mass $39$ g, but the total cost is not specified.

I.

Write the possible masses in terms of $z=t$ .

[2]

II.

Determine the range of $t$ for which all masses are non-negative.

[2]

Question 30

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The system

\begin{aligned} x+2y-z&=4\\ 2x+ay+z&=5\\ 3x+4y+(a-2)z&=9 \end{aligned}

has parameter $a\in\mathbb{R}$ .

A

Determine the values of $a$ for which the system has a unique solution.

[3]

B

For the remaining values of $a$ , classify the system and, where appropriate, give the solution set.

[3]

Question 31

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Three planes are represented by

\begin{aligned} x+y+z&=2\\ 2x-y+3z&=5\\ 3x+ay+5z&=b, \end{aligned}

where $a,b\in\mathbb{R}$ .

A

Show that the determinant of the coefficient matrix is $-a-3$ .

[2]

B

Determine the values of $a$ and $b$ for which the system has infinitely many solutions.

[3]

C

For these values of $a$ and $b$ , write the solution set in terms of a parameter.

[2]

Question 32

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A row-reduction process is applied to the parameter system

\begin{aligned} x+2y-z&=3\\ 2x+5y+hz&=8\\ 3x+7y+4z&=m, \end{aligned}

where $h,m\in\mathbb{R}$ .

A simple schematic flowchart showing only the initial augmented matrix and arrows leading to an unlabeled echelon-form placeholder. Do not display any intermediate or final reduced equations, matrix entries, or answers from parts (i) or (ii). Include a label stating that elementary row operations preserve the solution set.

A

I.

Use the operations $R_2\to R_2-2R_1$ and $R_3\to R_3-3R_1$ to obtain two equations in $y$ and $z$ .

[2]

II.

Hence write down an equation involving only $z$ , $h$ and $m$ .

[2]

B

For $h=3$ and $m=15$ , solve the system.

[2]

C

Classify the number of solutions when $h=5$ , in terms of $m$ .

[3]

Question 33

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the system of equations

\begin{aligned} x+y+z&=3\\ 2x+3y+z&=7\\ x+ay+(a-1)z&=b, \end{aligned}

where $a,b\in\mathbb{R}$ .

A

I.

Use the first two equations to express $x$ and $z$ in terms of $y$ .

[3]

II.

Determine the conditions on $a$ and $b$ for which the system has a unique solution, no solution, and infinitely many solutions.

[3]

B

Find the solution when $a=2$ and $b=5$ .

[2]

C

In the case where the system has infinitely many solutions, write the solution set in parametric form.

[2]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the system

\begin{aligned} x+y+kz&=1\\ 2x+(k+1)y+3z&=4\\ x+2y+(k+2)z&=3, \end{aligned}

where $k\in\mathbb{R}$ .

A

I.

Show that the determinant of the coefficient matrix is $4k-5$ .

[3]

II.

Hence state the value of $k$ for which the system does not have a unique solution.

[2]

B

Determine the number of solutions when $k=\dfrac{5}{4}$ .

[2]

C

I.

Solve the system when $k=1$ .

[2]

II.

For $k\neq\dfrac54$ , find the value of $k$ for which the solution has $z=0$ .

[1]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the system

\begin{aligned} x+2y-z&=3\\ 2x+5y+hz&=7\\ -x+y+(h-2)z&=q, \end{aligned}

where $h,q\in\mathbb{R}$ .

A

I.

Use row operations to show that the final row may be written in the form $0x+0y+(-2h-9)z=q$ .

[3]

II.

Hence classify the system according to the values of $h$ and $q$ .

[3]

B

For the values of $h$ and $q$ giving infinitely many solutions, find the solution set in parametric form.

[3]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A function has the form

f(x)=\frac{ax+b}{x+c}

where $a,b,c\in\mathbb{R}$ and $x\neq -c$ . It is known that

f(0)=2,\qquad f(1)=1,\qquad f(3)=\frac53

A

I.

Form a system of three linear equations in $a$ , $b$ and $c$ .

[3]

II.

Explain why these equations are linear in $a$ , $b$ and $c$ .

[1]

B

Solve the system to find $a$ , $b$ and $c$ .

[3]

C

Find the value of $x$ for which $f(x)=0$ .

[2]

Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the symmetric system

\begin{aligned} x+y+az&=1\\ x+ay+z&=1\\ ax+y+z&=1, \end{aligned}

where $a\in\mathbb{R}$ .

A

I.

Show that the determinant of the coefficient matrix is $-(a-1)^2(a+2)$ .

[3]

II.

Hence state the values of $a$ for which the system does not have a unique solution.

[2]

B

Classify the system for the values of $a$ found in part (a).

[3]

C

Solve the system when $a=0$ .

[2]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the system

\begin{aligned} x+2y-z&=4\\ 2x+5y+az&=9\\ 3x+8y+(a+1)z&=b, \end{aligned}

where $a,b\in\mathbb{R}$ .

A

I.

Use row reduction to obtain the final row $0x+0y-az=b-14$ .

[3]

II.

Hence classify the system according to $a$ and $b$ .

[2]

B

For $a=0$ and $b=14$ , write the solution set in parametric form.

[3]

C

Solve the system when $a=1$ and $b=9$ .

[2]

Question 39

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A plane is written in the form

ax+by+cz=1

It is required to pass through the points $P(1,0,1)$ , $Q(0,1,1)$ and $R(1,1,k)$ , where $k\in\mathbb{R}$ .

A

I.

Form a system of three linear equations in $a$ , $b$ and $c$ .

[3]

II.

Show that this system reduces to $(k-2)c=-1$ .

[1]

B

Determine the values of $k$ for which such a plane exists in the form $ax+by+cz=1$ .

[2]

C

Find the equation of the plane when $k=3$ .

[2]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the system of equations

\begin{aligned} x+y+z&=4\\ 2x+3y+z&=7\\ x+ky+2z&=m, \end{aligned}

where $k,m\in\mathbb{R}$ .

A

I.

Show that the determinant of the coefficient matrix is $k$ .

[3]

II.

Determine the values of $k$ and $m$ for which the system has no solution, exactly one solution, or infinitely many solutions.

[4]

B

For the fixed values $k=2$ and $m=8$ , answer the following.

I.

Solve the system.

[2]

II.

Interpret this solution geometrically.

[2]

Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Water flows through three pipes with unknown hourly flow rates $x$ , $y$ and $z$ , measured in kilolitres per hour. Conservation of flow at three junctions gives

\begin{aligned} x+y&=120\\ y+z&=95\\ x-z&=25. \end{aligned}

A simple directed network diagram with three labelled internal pipe flows $x$, $y$ and $z$ and three junction labels. The arrows must be oriented so that at J1, $x$ and $y$ enter and 120 exits; at J2, $y$ and $z$ enter and 95 exits; and at J3, $x$ enters while $z$ and 25 exit. The diagram should communicate that the three listed conservation equations arise from balancing inflows and outflows at junctions, without showing any solution values.

A

In practice, each flow rate must be non-negative. The hourly pumping cost, measured in dollars per hour, is defined by

C=0.12x+0.08y+0.15z

Here the coefficients are costs in dollars per kilolitre, and $x$ , $y$ and $z$ are measured in $\text{kL h}^{-1}$ .

I.

Use row reduction to show that the system does not have a unique solution.

[2]

II.

Find the maximum value of $z$ if the hourly pumping cost must not exceed $20.00 per hour.

[3]

B

In practice, each flow rate must be non-negative. The hourly pumping cost is

C=0.12x+0.08y+0.15z

I.

Determine the possible range of values of $t$ .

[2]

II.

Find the maximum value of $z$ if the hourly pumping cost must not exceed $20.00$ .

[3]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

In an electrical network, the currents $I_1$ , $I_2$ and $I_3$ , measured in amperes, satisfy

\begin{aligned} I_1+I_2-I_3&=0\\ 12I_1+4I_3&=18\\ 8I_2-4I_3&=6. \end{aligned}

A decorative circuit outline with a junction and two loops, labelled $I_1$, $I_2$ and $I_3$. Do not show current arrows, battery polarities, voltage signs, or solved current values; the displayed equations, rather than the illustration, define the current sign conventions.

A

I.

Write down the coefficient matrix and constant vector for this system.

[2]

II.

Solve for $I_1$ , $I_2$ and $I_3$ .

[2]

B

For the following parts, replace $12$ in the second equation by the parameter $R$ . The system is

\begin{aligned} I_1+I_2-I_3&=0\\ RI_1+4I_3&=18\\ 8I_2-4I_3&=6. \end{aligned}

The constant vector is unchanged and is $\begin{pmatrix}0\\18\\6\end{pmatrix}$ .

I.

Show that the determinant of the new coefficient matrix is $-4(R+8)$ .

[3]

II.

Determine the values of $R$ for which the system has a unique solution.

[1]

III.

Classify the system when $R=-8$ .

[2]

Question 43

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A population moves between three regions $P$ , $Q$ and $R$ each year. In the long term, the proportions in the three regions are $p$ , $q$ and $r$ . They satisfy

\begin{aligned} p&=0.7p+0.2q+0.1r\\ q&=0.2p+0.6q+0.3r\\ p+q+r&=1. \end{aligned}

The total population is expected to remain $6000$ people.

to \ from	P	Q	R
P	0.7	0.2	0.1
Q	0.2	0.6	0.3
R	0.1	0.2	0.6

A

I.

Rewrite the system in standard linear form.

[3]

II.

Use your graphic display calculator to solve for $p$ , $q$ and $r$ .

[3]

B

The total population is expected to remain $6000$ .

I.

Calculate the long-term number of people in each region.

[2]

II.

A planning target requires region $Q$ to contain at least $2250$ people in the long term. Determine whether this target is met.

[2]

Question 44

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Three forces $A$ , $B$ and $C$ , measured in newtons, act on a mechanical joint. Equilibrium conditions lead to

\begin{aligned} A+B+C&=14\\ 2A-B+3C&=28\\ -A+4B+2C&=22. \end{aligned}

A mechanical joint diagram with three labelled forces A, B and C acting in different directions. The diagram should communicate that equilibrium produces three simultaneous linear equations, without including solution values.

A

I.

Use your graphic display calculator to solve the system.

[3]

II.

State which force is the greatest.

[1]

B

The final constant $22$ is replaced by a variable load $L$ , so the new system is

\begin{aligned} A+B+C&=14\\ 2A-B+3C&=28\\ -A+4B+2C&=L. \end{aligned}

I.

Solve the new system in terms of $L$ .

[4]

II.

If the load $L$ is non-negative, determine the range of $L$ for which all three forces are non-negative.

[2]

Question 45

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A digital filter transforms an original colour vector $(R,G,B)$ into an observed colour vector. For one pixel, the filter gives

\begin{aligned} 1.2R+0.3G+0.1B&=70\\ 0.2R+1.1G+0.4B&=91\\ 0.1R+0.2G+1.3B&=105. \end{aligned}

Filter	Coefficient matrix	Observed vector
Original $M$	$\begin{pmatrix}1.2&0.3&0.1\\0.2&1.1&0.4\\0.1&0.2&1.3\end{pmatrix}$	$\mathbf{v}=\begin{pmatrix}70\\91\\105\end{pmatrix}$
Modified $M_k$ ( $1.3\to k$ )	$\begin{pmatrix}1.2&0.3&0.1\\0.2&1.1&0.4\\0.1&0.2&k\end{pmatrix}$	$\mathbf{v}=\begin{pmatrix}70\\91\\105\end{pmatrix}$

A

I.

Write the system in matrix form $M\mathbf{x}=\mathbf{v}$ .

[2]

II.

Use your graphic display calculator to find the original colour vector.

[2]

B

The coefficient $1.3$ in the third equation is replaced by a parameter $k$ , so that the new coefficient matrix is

M_k=\begin{pmatrix}1.2&0.3&0.1\\0.2&1.1&0.4\\0.1&0.2&k\end{pmatrix}

I.

Show that the determinant of the new coefficient matrix is $1.26k-0.091$ .

[3]

II.

Find the value of $k$ for which the original colour vector cannot be determined uniquely from the observed colour vector.

[2]

III.

State why this value of $k$ causes a problem for reversing the filter.

[1]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The three planes $\Pi_1$ , $\Pi_2$ and $\Pi_3$ are represented by the system

\begin{aligned} x+y+z&=1\\ 2x+3y+kz&=4\\ x+ky+3z&=m, \end{aligned}

where $k,m\in\mathbb{R}$ . This question investigates how the values of $k$ and $m$ affect the common intersection of the planes.

A schematic three-dimensional diagram showing three labelled planes Pi1, Pi2 and Pi3. The diagram should suggest that, depending on parameters, three planes may meet at a point, share a line, or have no common point. No specific coordinates or equations are shown in the diagram.

A

I.

Show that the determinant of the coefficient matrix is $k(3-k)$ .

[3]

II.

Hence state the values of $k$ for which the system has a unique solution.

[2]

B

For the case $k=0$ , determine the values of $m$ for which the system is consistent. If it is consistent, write the solution set using a real parameter.

[4]

C

For the case $k=3$ , determine the values of $m$ for which the system is consistent. If it is consistent, write the solution set using a real parameter.

[4]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A laboratory calibrates three constants $a$ , $b$ and $c$ in a linear model. Three test readings lead to the system

\begin{aligned} 2a-b+c&=5\\ a+3b+2c&=14\\ 4a+b+kc&=d, \end{aligned}

where $k,d\in\mathbb{R}$ .

Test reading	coefficient of a	coefficient of b	coefficient of c	reading
1	2	-1	1	5
2	1	3	2	14
3	4	1	k	d

A

I.

Calculate the determinant of the coefficient matrix.

[3]

II.

State the value of $k$ for which the system does not have a unique solution.

[2]

B

For $k=2$ and $d=17$ , solve the system.

[2]

C

When $k=\frac{23}{7}$ , determine the value of $d$ for which the system has infinitely many solutions, and write the solution set for this value of $d$ .

[3]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A company models three unknown daily subscription numbers $u$ , $v$ and $w$ . A parameter $p$ represents a change in the weighting of two survey questions. The model is

\begin{aligned} u+v+w&=6\\ u+pv+2w&=8\\ u+2v+pw&=q, \end{aligned}

where $p,q\in\mathbb{R}$ .

Equation	$u$	$v$	$w$	RHS
1	1	1	1	6
2	1	p	2	8
3	1	2	p	q

A

I.

Show that the determinant of the coefficient matrix is $p(p-2)$ .

[3]

II.

State the values of $p$ for which the system may fail to have a unique solution.

[1]

B

For $p=0$ , determine the value of $q$ for which the system has infinitely many solutions. For this value of $q$ , write the solution set.

[3]

C

For $p=2$ , classify the system for all real values of $q$ .

[3]

Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Three pumps $X$ , $Y$ and $Z$ supply water to different parts of an irrigation system. Let $x$ , $y$ and $z$ be the numbers of minutes that the pumps run during a control period. Operational requirements give

\begin{aligned} x+y+z&=120\\ 2x+y&=150\\ x+2y+3z&=d, \end{aligned}

where $d$ is the total weighted delivery index.

A schematic irrigation network with three labelled pumps X, Y and Z feeding a common system. The diagram shows that different constraints combine pump running times, but it does not include numerical results.

A

I.

Use the first two equations to express $x$ and $y$ in terms of $z$ .

[2]

II.

Hence show that the system is consistent only when $d=210$ .

[2]

B

For $d=210$ , write the solution set and determine the possible range of $z$ if no pump can run for a negative time.

[3]

C

The cost, in dollars, of running the pumps is $C=5x+4y+6z$ . Determine the minimum possible cost.

[4]

Question 50

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A plane surface is modelled by

z=Ax+By+C

It is required to pass through the points $(0,0,2)$ , $(1,2,7)$ and $(t,1,s)$ , where $t,s\in\mathbb{R}$ .

A neutral schematic listing the three points $(0,0,2)$, $(1,2,7)$ and $(t,1,s)$ alongside the plane equation, with no plotted coordinate positions, triangle, or indication of collinearity.

A

I.

Form the system of equations for $A$ , $B$ and $C$ .

[2]

II.

Show that this system has a unique solution unless $t=\frac{1}{2}$ .

[2]

B

Find $A$ , $B$ and $C$ when $t=2$ and $s=9$ .

[2]

C

For $t=\frac{1}{2}$ , determine the value of $s$ for which there are infinitely many possible planes of the form $z=Ax+By+C$ .

[3]

Question 51

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A quadratic expression is written in the form

f(r)=x+ry+r^2z

Values of $f(r)$ give a system of linear equations for $x$ , $y$ and $z$ . This question investigates when the coefficients are uniquely determined.

Case	$r$	$f(r)$
general	$0$	$a$
general	$1$	$b$
general	$k$	$c$
numerical	$0$	$1$
numerical	$1$	$4$
numerical	$3$	$22$

A

I.

Given $f(0)=1$ , $f(1)=4$ and $f(3)=22$ , form the system of equations for $x$ , $y$ and $z$ .

[2]

II.

Solve the system.

[2]

B

For the general values $f(0)=a$ , $f(1)=b$ and $f(k)=c$ , show that the determinant of the coefficient matrix is $k(k-1)$ .

[3]

C

Deduce the values of $k$ for which $x$ , $y$ and $z$ are not uniquely determined, and interpret this in terms of the input values of $r$ .

[3]

Question 52

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A simple three-state model has long-term proportions $p$ , $q$ and $r$ . At equilibrium,

\begin{aligned} p&=0.6p+0.2q+0.1r\\ q&=0.3p+0.5q+0.2r\\ p+q+r&=1. \end{aligned}

For a formal coefficient perturbation of this equilibrium system, replace $0.6$ in the first equation by $1-t$ , giving the coefficient matrix

\begin{pmatrix} t&-0.2&-0.1\\ -0.3&0.5&-0.2\\ 1&1&1 \end{pmatrix}

No visual is required for this formal coefficient-perturbation question; remove the attached transition diagram.

A

I.

Rewrite the first two equilibrium equations as linear equations equal to zero.

[2]

II.

Solve the original equilibrium system.

[2]

B

For the formal coefficient perturbation, show that the determinant of the coefficient matrix is $0.7t+0.06$ .

[3]

C

Justify that the formal perturbed system has a unique solution for all $0\le t\le 1$ .

[2]

Question 53

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Three correction values $x$ , $y$ and $z$ are applied to a navigation system. They satisfy

\begin{aligned} x+y+z&=30\\ 2x-y+z&=18\\ x+ay+3z&=b, \end{aligned}

where $a,b\in\mathbb{R}$ .

Eqn	x	y	z	RHS
1	1	1	1	30
2	2	-1	1	18
3	1	a	3	b

A

I.

Show that the determinant of the coefficient matrix is $a-7$ .

[3]

II.

State the condition on $a$ for a unique solution.

[1]

B

For $a=4$ and $b=100$ , solve the system.

[2]

C

For $a=7$ , classify the system for all real values of $b$ .

[3]

Question 54

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The planes $\Pi_1$ and $\Pi_2$ are given by

\Pi_1:x-y+z=2,\qquad \Pi_2:2x+y-z=1

A third plane is given by

\Pi_3:\lambda x+3y+\mu z=\nu

where $\lambda,\mu,\nu\in\mathbb{R}$ .

A

I.

Find the line of intersection of $\Pi_1$ and $\Pi_2$ in parametric form.

[3]

II.

Hence find the conditions on $\lambda$ , $\mu$ and $\nu$ for which all three planes intersect in this line.

[2]

B

Suppose $\lambda=2$ and $\nu=0$ . Classify the intersection of the three planes for all real values of $\mu$ .

[4]

C

For $\lambda=2$ , $\mu=-3$ and $\nu=-1$ , write down the common line of intersection.

[2]

Question 55

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the system

\begin{aligned} kx+y+z&=1\\ x+ky+z&=1\\ x+y+kz&=m, \end{aligned}

where $k,m\in\mathbb{R}$ .

A

I.

Show that the determinant of the coefficient matrix is $(k-1)^2(k+2)$ .

[3]

II.

State the values of $k$ for which further investigation is needed.

[1]

B

Classify the system for all values of $k$ and $m$ .

[5]

C

I.

For $k=1$ and $m=1$ , write the solution set.

[1]

II.

For $k=-2$ and $m=-2$ , write the solution set in terms of one parameter.

[2]

Question 56

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two planes are given by

\Pi_1: x+y+z=6,\qquad \Pi_2: 2x-y+z=3

A third plane is of the form

\Pi_3: x+py+qz=9

where $p,q\in\mathbb{R}$ .

A three-dimensional sketch showing two planes intersecting in a line, with a third plane that may either contain that line or cut it at a single point. The line of intersection should be labelled, but no coordinates or parameter values should be shown.

A

I.

Find a parametric form of the line of intersection of $\Pi_1$ and $\Pi_2$ .

[3]

II.

State the direction vector of this line.

[1]

B

I.

Determine $p$ and $q$ so that $\Pi_3$ contains the whole line of intersection of $\Pi_1$ and $\Pi_2$ .

[3]

II.

Describe the number of common points of the three planes for these values of $p$ and $q$ .

[1]

C

Instead let $p=2$ and $q=2$ .

I.

Find the common point of the three planes.

[2]

II.

Explain why the three planes now have exactly one common point.

[1]

Question 57

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the system

\begin{aligned} x+y+z&=1\\ 2x+3y+az&=4\\ x+ay+3z&=b, \end{aligned}

where $a,b\in\mathbb{R}$ .

A

I.

Show that the determinant of the coefficient matrix is $a(3-a)$ .

[3]

II.

State the values of $a$ for which the system may fail to have a unique solution.

[2]

B

I.

Classify the system when $a=0$ , in terms of $b$ .

[2]

II.

Classify the system when $a=3$ , in terms of $b$ .

[2]

C

I.

For $a=3$ and $b=5$ , write the solution set in terms of a real parameter.

[2]

II.

For $a\neq 0,3$ , state the number of solutions and justify your answer.

[1]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An input-output model for three sectors has total outputs $x$ , $y$ and $z$ . The equations are

\begin{aligned} 0.8x-0.1y&=20\\ -0.3x+y-0.2z&=30\\ -ky+z&=40, \end{aligned}

where $k$ is a parameter representing the dependence of sector $Z$ on sector $Y$ .

A directed network with three labelled sectors X, Y and Z. Arrows indicate dependencies among the sectors, including a parameter k from Y to Z. External demand arrows are shown but without extra numerical calculations.

A

I.

Find the determinant of the coefficient matrix in terms of $k$ .

[2]

II.

Explain why the system has a unique solution for all $0\le k\le 1$ .

[2]

B

For $k=0.5$ , solve the system, giving each output to three significant figures.

[3]

C

The model is considered realistic only if all outputs are positive. State whether the solution in part (b) is realistic, giving a reason.

[2]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A point $P$ in the plane is represented using weights $\alpha$ , $\beta$ and $\gamma$ applied to three reference points

A(1,0),\quad B(0,2),\quad C(t,1)

The weights satisfy

\alpha A+\beta B+\gamma C=P,\qquad \alpha+\beta+\gamma=1

A genuinely schematic coordinate diagram showing labels $A$, $B$, $C(t,1)$, and $P$ without assigning a numerical position to $C$ or $P$; no triangle, collinearity, or interior relationship is shown. The diagram is illustrative only and does not encode any particular part.

A

I.

For $t=3$ and $P(2,1)$ , form the system for $\alpha$ , $\beta$ and $\gamma$ .

[2]

II.

Solve the system and interpret the signs of the weights.

[3]

B

Show that the system for the weights has a unique solution unless $t=\frac{1}{2}$ .

[3]

C

For $t=\frac{1}{2}$ , explain geometrically why some points $P$ cannot be represented by any weights satisfying $\alpha+\beta+\gamma=1$ .

[3]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Consider the family of systems

\begin{aligned} x+py+z&=1\\ 2x-y+pz&=3\\ 3x+(p-1)y+(p+1)z&=\lambda, \end{aligned}

where $p,\lambda\in\mathbb{R}$ . This question investigates a situation in which a third equation may add no new information.

A conceptual diagram of three planes where the third plane is shown as possibly related to the first two. Labels indicate first equation, second equation and combined third equation, without numerical intersections.

A

I.

Show that the left-hand side of the third equation is the sum of the left-hand sides of the first two equations.

[2]

II.

Deduce the value of $\lambda$ for which the system is consistent.

[1]

B

For $p=2$ and $\lambda=4$ , write the solution set using a real parameter.

[3]

C

Prove that this family of systems never has a unique solution.