O Level Elementary Mathematics Vectors Matrices Quiz

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O-Level Elementary Mathematics Quiz - Vectors Matrices

Name: ____________________
Class: ____________________
Date: ____________________
Score: ________ / 45

Duration: 60 Minutes
Total Marks: 45

Instructions:

1. Answer all questions in the spaces provided.
2. Show all essential working. Marks may be awarded for correct method even if the final answer is incorrect.
3. Use a calculator where necessary.
4. Give non-exact numerical answers to 3 significant figures.

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Section A: Matrices (Questions 1–10)

1. Given matrix $A = \begin{pmatrix} 3 & -1 \\ 2 & 5 \end{pmatrix}$ and matrix $B = \begin{pmatrix} 0 & 4 \\ -3 & 2 \end{pmatrix}$, find $A + B$.
   [2 marks]
   
   
   Answer: ____________________

2. If $M = \begin{pmatrix} 4 & 7 \\ -2 & 1 \end{pmatrix}$, find $3M$.
   [2 marks]
   
   
   Answer: ____________________

3. Given $P = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & -1 \\ 0 & 2 \end{pmatrix}$, calculate $P - Q$.
   [2 marks]
   
   
   Answer: ____________________

4. A matrix $R = \begin{pmatrix} x & 8 \\ 3 & y \end{pmatrix}$ is equal to $\begin{pmatrix} 5 & 8 \\ 3 & -2 \end{pmatrix}$. State the values of $x$ and $y$.
   [2 marks]
   
   
   Answer: $x =$ ______, $y =$ ______

5. Find the product of $\begin{pmatrix} 2 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.
   [2 marks]
   
   
   Answer: ____________________

6. Calculate $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix}$.
   [2 marks]
   
   
   Answer: ____________________

7. Given $C = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$, find $C^2$ (where $C^2 = C \times C$).
   [3 marks]
   
   
   Answer: ____________________

8. Matrix $D = \begin{pmatrix} a & 2 \\ 4 & b \end{pmatrix}$. If $2D = \begin{pmatrix} 6 & 4 \\ 8 & -10 \end{pmatrix}$, find the values of $a$ and $b$.
   [2 marks]
   
   
   Answer: $a =$ ______, $b =$ ______

9. The prices of two items, A and B, are given by matrix $P = \begin{pmatrix} 5.00 & 3.50 \end{pmatrix}$. The quantities bought by two customers are given by matrix $Q = \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix}$. Find the matrix $PQ$ representing the total cost for each customer.
   [3 marks]
   
   
   Answer: ____________________

10. If $\begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 13 \\ 4 \end{pmatrix}$, solve for $x$ and $y$.
    [3 marks]
    
    
    Answer: $x =$ ______, $y =$ ______

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Section B: Vectors (Questions 11–20)

11. Given $\vec{a} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$, find $2\vec{a} + \vec{b}$.
    [2 marks]
    
    
    Answer: ____________________

12. Find the magnitude of the vector $\vec{v} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}$.
    [2 marks]
    
    
    Answer: ____________________

13. Given $\vec{OA} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$, find the vector $\vec{AB}$.
    [2 marks]
    
    
    Answer: ____________________

14. Vector $\vec{p} = \begin{pmatrix} x \\ 12 \end{pmatrix}$ is parallel to vector $\vec{q} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$. Find the value of $x$.
    [2 marks]
    
    
    Answer: ____________________

15. Find the unit vector in the direction of $\vec{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.
    [3 marks]
    
    
    Answer: ____________________

16. Given $\vec{u} = 3\mathbf{i} - 2\mathbf{j}$ and $\vec{v} = \mathbf{i} + 4\mathbf{j}$, express $\vec{u} - 2\vec{v}$ in terms of $\mathbf{i}$ and $\mathbf{j}$.
    [2 marks]
    
    
    Answer: ____________________

17. A point $P(2, 3)$ is translated by the vector $\begin{pmatrix} -4 \\ 1 \end{pmatrix}$ to point $P'$. Find the coordinates of $P'$.
    [2 marks]
    
    
    Answer: ____________________

18. In $\triangle ABC$, $\vec{AB} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\vec{BC} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$. Find the vector $\vec{AC}$.
    [2 marks]
    
    
    Answer: ____________________

19. Given $\vec{OA} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}$, find the magnitude of $\vec{AB}$.
    [3 marks]
    
    
    Answer: ____________________

20. Point $M$ divides the line segment $AB$ in the ratio $1:2$. Given $\vec{OA} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 8 \\ 10 \end{pmatrix}$, find the position vector $\vec{OM}$.
    [4 marks]
    
    
    Answer: ____________________

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Answer Key - O-Level Elementary Mathematics Quiz (Vectors Matrices)

Section A: Matrices

1. $\begin{pmatrix} 3+0 & -1+4 \\ 2-3 & 5+2 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ -1 & 7 \end{pmatrix}$ [2 marks]
2. $\begin{pmatrix} 12 & 21 \\ -6 & 3 \end{pmatrix}$ [2 marks]
3. $\begin{pmatrix} 2-5 & 3-(-1) \\ 1-0 & 4-2 \end{pmatrix} = \begin{pmatrix} -3 & 4 \\ 1 & 2 \end{pmatrix}$ [2 marks]
4. $x = 5, y = -2$ [2 marks]
5. $(2 \times 3) + (1 \times 4) = 6 + 4 = 10$ [2 marks]
6. $\begin{pmatrix} (1 \times 5) + (2 \times 6) \\ (3 \times 5) + (4 \times 6) \end{pmatrix} = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$ [2 marks]
7. $\begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} (4+0) & (0+0) \\ (2+3) & (0+9) \end{pmatrix} = \begin{pmatrix} 4 & 0 \\ 5 & 9 \end{pmatrix}$ [3 marks]
8. $2a = 6 \rightarrow a = 3$; $2b = -10 \rightarrow b = -5$ [2 marks]
9. $\begin{pmatrix} 5.00 & 3.50 \end{pmatrix} \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} (10+3.5) & (20+10.5) \end{pmatrix} = \begin{pmatrix} 13.50 & 30.50 \end{pmatrix}$ [3 marks]
10. $y = 4$ (from 2nd row $0x + 1y = 4$). Substitute into 1st row: $2x + 3(4) = 13 \rightarrow 2x = 1 \rightarrow x = 0.5$ [3 marks]

Section B: Vectors

11. $2\begin{pmatrix} 4 \\ -3 \end{pmatrix} + \begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} 8 \\ -6 \end{pmatrix} + \begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} 10 \\ -1 \end{pmatrix}$ [2 marks]
12. $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ [2 marks]
13. $\vec{AB} = \vec{OB} - \vec{OA} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} - \begin{pmatrix} 3 \\ 7 \end{pmatrix} = \begin{pmatrix} -4 \\ -3 \end{pmatrix}$ [2 marks]
14. $\frac{x}{3} = \frac{12}{4} \rightarrow \frac{x}{3} = 3 \rightarrow x = 9$ [2 marks]
15. Magnitude $= \sqrt{3^2 + 4^2} = 5$. Unit vector $= \frac{1}{5} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 0.6 \\ 0.8 \end{pmatrix}$ [3 marks]
16. $(3\mathbf{i} - 2\mathbf{j}) - 2(\mathbf{i} + 4\mathbf{j}) = 3\mathbf{i} - 2\mathbf{j} - 2\mathbf{i} - 8\mathbf{j} = \mathbf{i} - 10\mathbf{j}$ [2 marks]
17. $\begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} -4 \\ 1 \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \end{pmatrix} \rightarrow P'(-2, 4)$ [2 marks]
18. $\vec{AC} = \vec{AB} + \vec{BC} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$ [2 marks]
19. $\vec{AB} = \begin{pmatrix} 1-5 \\ 6-2 \end{pmatrix} = \begin{pmatrix} -4 \\ 4 \end{pmatrix}$. $|\vec{AB}| = \sqrt{(-4)^2 + 4^2} = \sqrt{32} \approx 5.66$ [3 marks]
20. $\vec{OM} = \frac{2\vec{OA} + 1\vec{OB}}{1+2} = \frac{1}{3} \left[ 2\begin{pmatrix} 2 \\ 4 \end{pmatrix} + \begin{pmatrix} 8 \\ 10 \end{pmatrix} \right] = \frac{1}{3} \begin{pmatrix} 12 \\ 18 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$ [4 marks]