QCE Vault / Specialist Maths Vectors and matrices — Question 11 QCAA 2025, Paper 1 · 5 marks
Browse all questions Vectors revision Original practice exam Report an issue Q11 · 2025 · Technology-free Simple familiar 5 marks
QUESTION 11 (5 marks) Consider the system of linear equations represented using the augmented matrix shown.
[ 1 − 1 − 1 − 6 − 2 1 1 1 0 0 4 4 ] \left[\begin{array}{ccc|c}1&-1&-1&-6\\-2&1&1&1\\0&0&4&4\end{array}\right] 1 − 2 0 − 1 1 0 − 1 1 4 − 6 1 4
Key: R 1 R_1 R 1 represents the row 1 values. a) Modify the augmented matrix using the row operation shown.
R 2 ′ = R 2 + 2 R 1 . R_2'=R_2+2R_1. R 2 ′ = R 2 + 2 R 1 .
[ 1 − 1 − 1 − 6 □ □ □ □ 0 0 4 4 ] \left[\begin{array}{ccc|c}1&-1&-1&-6\\\square&\square&\square&\square\\0&0&4&4\end{array}\right] 1 □ 0 − 1 □ 0 − 1 □ 4 − 6 □ 4
Key: R 2 ′ = R 2 + 2 R 1 R_2'=R_2+2R_1 R 2 ′ = R 2 + 2 R 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 x-y-z=-6 x − y − z = − 6 , use your result from Question 11a) to determine the solution of the system of linear equations. [3 marks] c) The system of linear equations is geometrically represented by three planes. Use your result from Question 11b) to describe a geometrical interpretation of your solution of the system of linear equations.
[1 mark] WORKED SOLUTION
QCAA guide · typeset solution 5 marks ANSWER ( x , y , z ) = ( 5 , 10 , 1 ) (x,y,z)=(5,10,1) ( x , y , z ) = ( 5 , 10 , 1 ) ; the three planes intersect at a point. Worked solution
a) Applying R 2 ′ = R 2 + 2 R 1 R_2'=R_2+2R_1 R 2 ′ = R 2 + 2 R 1 gives
[ 1 − 1 − 1 − 6 0 − 1 − 1 − 11 0 0 4 4 ] . \left[\begin{array}{ccc|c}1&-1&-1&-6\\0&-1&-1&-11\\0&0&4&4\end{array}\right]. 1 0 0 − 1 − 1 0 − 1 − 1 4 − 6 − 11 4 . b) From R 3 R_3 R 3 , 4 z = 4 4z=4 4 z = 4 , so z = 1 z=1 z = 1 . From R 2 ′ R_2' R 2 ′ , − y − z = − 11 -y-z=-11 − y − z = − 11 , so y = 10 y=10 y = 10 . From R 1 R_1 R 1 , x − y − z = − 6 x-y-z=-6 x − y − z = − 6 , so x = 5 x=5 x = 5 . c) The planes intersect at a point.
correctly determines the new row 2 values in the augmented matrix
[1 mark] correctly determines the required value of z z z [1 mark] determines the required value of y y y [1 mark] determines the required value of x x x [1 mark] describes a geometrical interpretation based on prior mathematical reasoning
[1 mark] One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.
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