Specialist Mathematics · Past QCAA questions All simple familiar questions
2025 · Paper 1 · Simple familiar

Question 11

[5 marks] Technology-free
Topic guidePremiumUnit 3 Topic 5.1 · Matrix algebra and systems of equations
Game plansPremium5.2 Choosing the route5.3 Identifying the case
The question

Consider the system of linear equations represented using the augmented matrix shown.

$$\left[\begin{array}{rrr|r}1&-1&-1&-6\\-2&1&1&1\\0&0&4&4\end{array}\right]\begin{array}{l}R_1\\R_2\\R_3\end{array}$$

Key: $R_1$ represents the row 1 values.

a)

Modify the augmented matrix using the row operation shown.

$$R_2'=R_2+2R_1$$

Key: $R_2'=R_2+2R_1$ indicates that the new row 2 values are equal to the sum of the existing row 2 and twice row 1 values.

[1 mark] b)

Given the row 1 values represent the equation $x-y-z=-6$, use your result from Question 11a) to determine the solution of the system of linear equations.

[3 marks]

The system of linear equations is geometrically represented by three planes.

c)

Use your result from Question 11b) to describe a geometrical interpretation of your solution of the system of linear equations.

[1 mark]
Watch the situation first

One row operation, then read upwards

Watch the first entry of row 2. Once it is zero, the bottom two rows give you the answer one variable at a time.

Row 2 starts with −2. Clear it using row 1. Add twice row 1 to every entry of row 2 Row 2 now has a zero in the first column Read the rows from the bottom up Each row is a plane. All three meet at one point. 1−1−1−6 0044 R1 R3 −2111 R2 0−1−1−11 R′2 2−2−2−12 + 2R1 From R3:4z = 4, so z = 1 From R′2:−y − z = −11, so y = 10 From R1:x − y − z = −6, so x = 5 (5, 10, 1) R1x − y − z = −6 R2−2x + y + z = 1 R34z = 4

The planes are the three original rows. The row operation changes the equations you work with, but it never moves the point where they meet.

Now for the mathematics

Choose the multiple of row 1 that clears row 2

Move the slider to try $R_2'=R_2+kR_1$ for different values of $k$. The blue plane is your new row 2. It turns about the dashed line it shares with plane 1, so the orange point stays put whatever you choose.

R1 R2′ R3 (5, 10, 1)

Drag the picture or use the sliders to turn it. The darker plane is row 2, which changes with . Row 1 has a dashed outline and row 3 a dotted one.

Your matrix after the row operation
Row 2 as a plane

The first entry is zero, so row 2 has no $x$ term. This is the operation in the question, and the blue plane now runs parallel to the $x$-axis.

This is the original row 2. Its first entry is $-2$, so it still has an $x$ term.

The first entry is not zero yet, so row 2 still has an $x$ term. Look at $-2+k$ and choose $k$ to make it zero.

QCAA marking guide · 5 marks

Work through the solution one mark at a time

Try each step yourself before you reveal it. This follows the QCAA’s first method for part b), which back-substitutes from the matrix in part a).

Step 1 · part a)1 mark

Double every entry of row 1, including the constant past the line, and add it to row 2. Rows 1 and 3 stay as they are.

$$\begin{aligned}R_2+2R_1&=\left[\begin{array}{rrr|r}-2+2&1-2&1-2&1-12\end{array}\right]\\&=\left[\begin{array}{rrr|r}0&-1&-1&-11\end{array}\right]\end{aligned}$$

$$\left[\begin{array}{rrr|r}1&-1&-1&-6\\0&-1&-1&-11\\0&0&4&4\end{array}\right]\begin{array}{l}R_1\\R_2'=R_2+2R_1\\R_3\end{array}$$

Marker: correctly determines the new row 2 values in the augmented matrix

Step 2 · part b)1 mark

Row 1 tells you the columns are $x$, $y$ and $z$ in that order. Start with the bottom row, because it has only one unknown.

$$\begin{aligned}R_3:\quad 4z&=4\\z&=1\end{aligned}$$

Marker: correctly determines required value of $z$

Step 3 · part b)1 mark

Your new row 2 has no $x$ in it, so it gives you $y$ as soon as you substitute $z=1$.

$$\begin{aligned}R_2':\quad -y-z&=-11\\-y-1&=-11\\y&=10\end{aligned}$$

If you made a slip in part a), you can still earn this mark and the next one by follow-through.

Marker: determines required value of $y$

Step 4 · part b)1 mark

Now use row 1, the equation you were given, with both values.

$$\begin{aligned}R_1:\quad x-y-z&=-6\\x-10-1&=-6\\x&=5\end{aligned}$$

$x=5,\ y=10,\ z=1$

The QCAA also accept a second method. Keep reducing until you reach reduced row echelon form, using $R_3'=R_3\div4$, then $R_1'=R_1-R_2'$, then $R_2''=-R_2'-R_3'$. The three marks are then for $z$, $x$ and $y$ in that order.

Marker: determines required value of $x$

Step 5 · part c)1 mark

Part b) gave you exactly one solution, so each equation is a plane and all three pass through one point. Say so in a sentence and give the point.

The three planes intersect at a single point, $(5,10,1)$.

The QCAA also accept a sketch of three planes meeting at a point. A line of intersection or no intersection would not match your answer to part b), so it would not earn the mark.

Marker: describes a geometrical interpretation based on prior mathematical reasoning

Putting it all together

Each phrase of the question gave you something

“using the row operation shown”
You use $R_2'=R_2+2R_1$ exactly as written, on all four entries of the row.
“the row 1 values represent the equation $x-y-z=-6$”
The columns are $x$, $y$ and $z$ in that order, so your new row 2 reads $-y-z=-11$.
“use your result from Question 11a)”
You back-substitute from your new matrix. There is no need to start again with a different method.
“geometrically represented by three planes”
Each row is the equation of a plane in three dimensions.
“Use your result from Question 11b)”
One solution means one point, so the planes meet at $(5,10,1)$.
What makes this simple familiar

You have seen this routine before. The question hands you the row operation, and each part tells you which earlier result to use. Row 3 was already reduced, so one operation was enough.

If I had to guess where you might lose marks, it would be the constant column in part a). Twice $-6$ added to $1$ is $-11$, and a slip there loses you the part a) mark. You would still keep follow-through marks for $y$ and $x$. In part c), write the sentence. A bare point with no mention of planes is a risk.

Only mark this done when you could do it without help. Reading the solution does not count.

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Question wording and marking-guide steps are from the 2025 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

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