O Level Elementary Mathematics Vectors Matrices Quiz

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O-Level Elementary Mathematics Quiz - Vectors Matrices

Name: ___________________________
Class: ___________________________
Date: ___________________________
Score: ___________ / 40

Duration: 60 minutes
Total Marks: 40
Topic: Vectors Matrices (Vectors & Matrices)

Instructions:

• Answer all 20 questions.
• Show your working clearly where required.
• Use vectors in column form or underlined notation as taught.
• Matrices should be written with rows clearly separated.
• Section A: Vectors basics (1–5). Section B: Matrices basics (6–10). Section C: Vectors & Matrices applications (11–20).

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Section A: Vectors (Questions 1–5)

1. Given $\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$, find $\mathbf{a} + \mathbf{b}$. [2]

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2. A vector $\mathbf{v}$ has magnitude 5 and is directed due east. Write $\mathbf{v}$ as a column vector using unit vectors $\mathbf{i}$ (east) and $\mathbf{j}$ (north). [1]

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3. Points $A(1, 2)$ and $B(4, 6)$ are given. Find the vector $\overrightarrow{AB}$. [2]

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4. Given $\mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$, find $3\mathbf{p}$. [1]

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5. State whether the vectors $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ are parallel. Give a reason. [2]

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Section B: Matrices (Questions 6–10)

6. Write the following information as a $2 \times 3$ matrix:
Row 1 (boys): 10, 12, 8
Row 2 (girls): 9, 11, 7 [2]

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7. Given $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ 1 & 5 \end{pmatrix}$, find $A + B$. [2]

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8. Given $M = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$, find $2M$. [2]

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9. A matrix $P = \begin{pmatrix} 4 & 5 \end{pmatrix}$ and $Q = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$. Find the product $PQ$. [2]

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10. The matrix $R = \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix}$ shows the number of books (fiction, non-fiction) borrowed by two students. State what the element $r_{21} = 1$ represents. [1]

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Section C: Vectors & Matrices Applications (Questions 11–20)

11. In the diagram below, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Point $C$ lies on $AB$ such that $AC : CB = 1 : 2$. Find $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Image pending generation: diagram for Q11.

[3]

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12. Given $\mathbf{u} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}$, find $2\mathbf{u} - \mathbf{v}$. [2]

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13. A transformation is represented by matrix $T = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Find the image of point $(1, 0)$ under $T$. [2]

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14. Two buses have seating: Bus X has 40 seats, Bus Y has 30 seats. A school hires 3 X and 2 Y for Monday, 2 X and 4 Y for Tuesday. Write a $2 \times 2$ matrix for the total seats hired per day. [3]

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15. Given $\overrightarrow{PQ} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$ and $\overrightarrow{QR} = \begin{pmatrix} -2 \\ 7 \end{pmatrix}$, find $\overrightarrow{PR}$. [2]

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16. The matrix $S = \begin{pmatrix} 1 & 3 \\ 2 & 5 \end{pmatrix}$. Find the product $S \begin{pmatrix} 2 \\ 1 \end{pmatrix}$. [2]

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17. Vectors $\mathbf{m} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}$ and $\mathbf{n} = \begin{pmatrix} -2 \\ -3 \end{pmatrix}$. Show that $\mathbf{m}$ and $\mathbf{n}$ are parallel. [2]

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18. A robot moves with velocity vector $\mathbf{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ cm/s. Find its speed. [2]

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19. Shop sales (in $100) for apples and oranges on two days:

Day	Apples	Oranges
1	5	3
2	4	6

Write this as a matrix and find total sales matrix for 2 such weeks (multiply by 2). [3]

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20. Given $\overrightarrow{XY} = 2\mathbf{a} - \mathbf{b}$ and $\overrightarrow{YZ} = \mathbf{a} + 3\mathbf{b}$, express $\overrightarrow{XZ}$ in simplest form. [2]

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

Total Marks: 40
Topic: Vectors Matrices

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Section A: Vectors

Q1. [2 marks]
$\mathbf{a} + \mathbf{b} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} + \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 3+1 \\ -2+4 \end{pmatrix} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$
Teaching note: Add corresponding components. Mark: 1 for setup, 1 for correct vector.

Q2. [1 mark]
$\mathbf{v} = 5\mathbf{i} = \begin{pmatrix} 5 \\ 0 \end{pmatrix}$
Teaching note: Due east means no north component. Magnitude 5 → (5,0).

Q3. [2 marks]
$\overrightarrow{AB} = \begin{pmatrix} 4-1 \\ 6-2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$
Teaching note: Subtract coordinates of A from B. Mark: 1 for method, 1 for answer.

Q4. [1 mark]
$3\mathbf{p} = 3 \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}$

Q5. [2 marks]
Yes, they are parallel because $\begin{pmatrix} 4 \\ 6 \end{pmatrix} = 2 \begin{pmatrix} 2 \\ 3 \end{pmatrix}$.
Teaching note: One is scalar multiple of the other. Mark: 1 for yes, 1 for reason.

Section B: Matrices

Q6. [2 marks]
$\begin{pmatrix} 10 & 12 & 8 \\ 9 & 11 & 7 \end{pmatrix}$
Mark: 1 for rows, 1 for columns correct.

Q7. [2 marks]
$A+B = \begin{pmatrix} 1+2 & 2+0 \\ 3+1 & 4+5 \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ 4 & 9 \end{pmatrix}$

Q8. [2 marks]
$2M = \begin{pmatrix} 6 & 2 \\ 0 & 4 \end{pmatrix}$
Multiply each element by 2.

Q9. [2 marks]
$PQ = \begin{pmatrix} 4 & 5 \end{pmatrix} \begin{pmatrix} 1 \\ 3 \end{pmatrix} = (4\times1 + 5\times3) = (19)$
Result is 1×1 matrix: $\begin{pmatrix} 19 \end{pmatrix}$.

Q10. [1 mark]
$r_{21}=1$ means student 2 borrowed 1 fiction book.
Row 2 = student 2, column 1 = fiction.

Section C: Applications

Q11. [3 marks]
$\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{AC}$
$\overrightarrow{AC} = \frac{1}{3}\overrightarrow{AB} = \frac{1}{3}(\mathbf{b}-\mathbf{a})$
$\overrightarrow{OC} = \mathbf{a} + \frac{1}{3}(\mathbf{b}-\mathbf{a}) = \frac{2}{3}\mathbf{a} + \frac{1}{3}\mathbf{b}$
Mark: 1 position ratio, 1 substitution, 1 final.

Q12. [2 marks]
$2\mathbf{u} - \mathbf{v} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} - \begin{pmatrix} -3 \\ 4 \end{pmatrix} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}$

Q13. [2 marks]
$T \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0\times1 + (-1)\times0 \\ 1\times1 + 0\times0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$
Image point (0,1) – 90° rotation.

Q14. [3 marks]
Mon: 3×40+2×30 = 180; Tue: 2×40+4×30 = 200
Matrix = $\begin{pmatrix} 180 & 120 \\ 80 & 120 \end{pmatrix}$? Wait: rows = days, cols = bus type:
$\begin{pmatrix} 3\times40 & 2\times30 \\ 2\times40 & 4\times30 \end{pmatrix} = \begin{pmatrix} 120 & 60 \\ 80 & 120 \end{pmatrix}$ total seats.
Mark: 1 layout, 2 calculations.

Q15. [2 marks]
$\overrightarrow{PR} = \begin{pmatrix} 5-2 \\ -3+7 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$

Q16. [2 marks]
$S \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 1\times2+3\times1 \\ 2\times2+5\times1 \end{pmatrix} = \begin{pmatrix} 5 \\ 9 \end{pmatrix}$

Q17. [2 marks]
$\mathbf{m} = -3 \begin{pmatrix} -2 \\ -3 \end{pmatrix} = -3\mathbf{n}$ → parallel (scalar multiple).

Q18. [2 marks]
Speed = $\sqrt{3^2+4^2} = 5$ cm/s.

Q19. [3 marks]
Matrix = $\begin{pmatrix} 5 & 3 \\ 4 & 6 \end{pmatrix}$; ×2 = $\begin{pmatrix} 10 & 6 \\ 8 & 12 \end{pmatrix}$
Mark: 1 matrix, 2 scaled.

Q20. [2 marks]
$\overrightarrow{XZ} = (2\mathbf{a}-\mathbf{b}) + (\mathbf{a}+3\mathbf{b}) = 3\mathbf{a} + 2\mathbf{b}$