Systems of Equations - IB Math AA Question Bank

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Systems of Equations

Practice exam-style IB Math AA questions for Systems of Equations, aligned with the syllabus and grouped by topic.

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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}

Write down the augmented matrix for the system.

[1]

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]

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

[3]

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

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

[2]

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}

Write down the augmented matrix for this system.

[1]

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

[2]

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

[2]

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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}

Write down the augmented matrix for the system.

[1]

Use your graphic display calculator to solve the system.

[2]

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]

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

[1]

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}

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

[2]

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}

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

[3]

Solve the system when $k=1$ .

[3]

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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}

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

[2]

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.

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

[2]

Find the price of each type of ticket.

[2]

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 $(5,11.85)$ .

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

[2]

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

[2]

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 $h\in\mathbb{R}$ .

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

[2]

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

[3]

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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.

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

[2]

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

[2]

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

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

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

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

[4]

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

[2]

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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}

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

[4]

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}

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

[3]

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

[1]

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 $k\in\mathbb{R}$ .

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

[2]

Classify the system when $k=0$ .

[2]

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

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

[4]

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

[2]

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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.

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]

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

[2]

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}

I.

Solve the first system.

[3]

II.

Solve the second system.

[2]

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]

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}

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]

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

[2]

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

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

[5]

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

[2]

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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}

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

[4]

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

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]

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 $h(5)=0.500$ .

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]

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 $k\in\mathbb{R}$ .

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]

I.

Classify the system when $k=4$ .

[2]

II.

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

[2]

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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)

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]

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 $a\in\mathbb{R}$ .

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

[3]

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

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

[2]

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

[3]

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

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]

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

[2]

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

[3]

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

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]

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

[2]

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 $k\in\mathbb{R}$ .

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]

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

[2]

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

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]

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

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]

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

[3]

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

[2]

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Question 37

HL • Paper 1

Hard

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HL • Paper 1

Hard

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Consider the symmetric system

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

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

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]

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

[3]

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

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]

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

[3]

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 $k\in\mathbb{R}$ .

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]

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

[2]

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

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]

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

I.

Solve the system.

[2]

II.

Interpret this solution geometrically.

[2]

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Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

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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}

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]

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}

I.

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

[2]

II.

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

[2]

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

I.

Rewrite the system in standard linear form.

[3]

II.

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

[3]

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}

I.

Use your graphic display calculator to solve the system.

[3]

II.

State which force is the greatest.

[1]

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]

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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 $M_k$ ( $1.3\to k$ )	$\begin{pmatrix}1.2&0.3&0.1\\0.2&1.1&0.4\\0.1&0.2&k\end{pmatrix}$	$\mathbf{v}=\begin{pmatrix}70\\91\\105\end{pmatrix}$

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]

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.

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]

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]

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

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]

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

[2]

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

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]

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]

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

[3]

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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.

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]

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

[3]

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 $t,s\in\mathbb{R}$ .

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]

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

[2]

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	$3$	$22$

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]

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]

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}

I.

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

[2]

II.

Solve the original equilibrium system.

[2]

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

[3]

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

[2]

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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

I.

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

[3]

II.

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

[1]

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

[2]

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

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]

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

[4]

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

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]

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

[5]

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

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]

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]

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]

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

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]

I.

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

[2]

II.

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

[2]

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 $Y$ .

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]

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

[3]

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

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]

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

[3]

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.

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]

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

[3]

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

[2]

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