Senior Secondary (HKDSE) · Mathematics M2 (Algebra and Calculus)

Systems of linear equations : Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Systems of linear equations .

10 questions27 marksFree, no account

Question 1

1 mark

Which of the following systems of linear equations has a unique solution?

Question 2

1 mark

If the system of linear equations $kx + 3y = 1$ and $3x + ky = 1$ has no solution, what is the value of $k$?

Question 3

1 mark

Consider the following system of linear equations:
$x + y + z = 1$
$2x + y - z = 2$
$x + 2y + 4z = p$
Find the value of $p$ such that the system is consistent.

Question 4

1 mark

Consider a system of homogeneous linear equations:
$ax + by = 0$
$cx + dy = 0$
Which of the following conditions ensures that the system has only the trivial solution $(x, y) = (0, 0)$?

Question 5

1 mark

For what value of $m$ does the following system of homogeneous linear equations have non-trivial solutions?
$x + y + z = 0$
$2x - y + 3z = 0$
$5x + 2y + mz = 0$

Question 6

2 marks

If the augmented matrix of a system of linear equations in variables $x_1, x_2, x_3$ is row equivalent to the following matrix in row echelon form:

$$\begin{pmatrix} 1 & 2 & 1 & | & 5 \ 0 & 1 & -1 & | & 1 \ 0 & 0 & 1 & | & 2 \ \end{pmatrix}$$

What is the value of the variable $x_2$?

Write your answer out first, then check it against the worked solution.

Question 7

3 marks

A system of linear equations is represented by the matrix equation $A\mathbf{x} = \mathbf{b}$, where $A = \begin{pmatrix} 3 & -1 \\ 1 & 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 7 \\ 0 \end{pmatrix}$. Find the value of $x$ (the first component of $\mathbf{x}$).

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Question 8

4 marks

Find the general solution to the following system of homogeneous linear equations, expressing the solution in terms of a parameter $t$:

$$\begin{aligned} x + y + z &= 0 \\ 2x + y + z &= 0 \\ 3x + 2y + 2z &= 0 \end{aligned}$$

Write your answer out first, then check it against the worked solution.

Question 9

5 marks

Consider the following system of linear equations in $x, y, z$:
$ \begin{cases} x + 2y + z = 4 \\ 2x + y + 3z = 7 \\ 3x + 3y + az = 11 \end{cases} $
where $a$ is a real constant.

(a) Find the value of $a$ such that the system does not have a unique solution.

(b) Suppose $a = 5$. Solve the system of equations for $x, y, z$ using Gaussian elimination.

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Question 10

8 marks

Consider the system of linear equations:

$$ (k+1)x + y + z = 1 \\ x + (k+1)y + z = 1 \\ x + y + (k+1)z = 1 $$
where $k$ is a real constant.

(a) Find the value(s) of $k$ for which the system does not have a unique solution.

(b) For the value(s) of $k$ found in (a), determine the nature of the solution (no solution or infinitely many solutions) for each case. Hence, find the general solution of the system for the value of $k$ that yields infinitely many solutions.

Write your answer out first, then check it against the worked solution.

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