Secondary 3 Additional Mathematics Vectors Matrices Quiz

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Secondary 3 Additional 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.
• Use vectors in column form or $\mathbf{i}, \mathbf{j}, \mathbf{k}$ notation where appropriate.
• Matrices should be written clearly with brackets.
• Section A: 1–5 (2 marks each), Section B: 6–13 (2 marks each), Section C: 14–20 (2 marks each).

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Section A: Vector Basics (Questions 1–5)

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

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2. Vector $\mathbf{p} = 3\mathbf{i} - 2\mathbf{j}$ and $\mathbf{q} = -\mathbf{i} + 5\mathbf{j}$. Find $2\mathbf{p} - \mathbf{q}$.
[2]

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3. Find the magnitude of vector $\mathbf{v} = \begin{pmatrix} -5 \\ 12 \end{pmatrix}$.
[2]

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4. Given points $A(1, 2)$ and $B(4, 6)$, express the vector $\overrightarrow{AB}$ as a column vector.
[2]

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5. A vector has magnitude $10$ and direction angle $30^\circ$ above the positive $x$-axis. Write it in the form $x\mathbf{i} + y\mathbf{j}$, correct to 1 decimal place.
[2]

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Section B: Vector Geometry and Matrices (Questions 6–13)

6. Points $P(0,0)$, $Q(3,1)$, $R(1,4)$. Show that $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ are not parallel by comparing their directions.
[2]

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7. Given $\mathbf{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$, state whether they are parallel and give a reason.
[2]

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8. Let $M = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $N = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. Find $M + N$.
[2]

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9. Find the product $MN$ for $M = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix}$.
[2]

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10. Find the determinant of $A = \begin{pmatrix} 5 & 2 \\ 1 & 3 \end{pmatrix}$.
[2]

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11. Given $T = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, describe the transformation represented by $T$ on a 2D vector.
[2]

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12. Solve for $x$ and $y$:
$\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}$.
[2]

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13. The position vectors of $X$ and $Y$ are $\mathbf{x} = 2\mathbf{i} + \mathbf{j}$ and $\mathbf{y} = 5\mathbf{i} - 3\mathbf{j}$. Find the vector $\overrightarrow{XY}$.
[2]

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Section C: Applied Vectors and Matrices (Questions 14–20)

14. A translation matrix $S = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is applied to $\mathbf{w} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$. State the resulting vector.
[2]

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15. Triangle $ABC$ has vertices $A(0,0)$, $B(2,0)$, $C(0,3)$. A matrix $K = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ is applied to each position vector. State the new coordinates of $B'$.
[2]

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16. Vector $\mathbf{d} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}$ represents a displacement. A scale factor of $\frac{1}{2}$ is applied. Write the new vector.
[2]

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17. Given $\mathbf{m} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{n} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, find the matrix $P$ such that $P\mathbf{m} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $P\mathbf{n} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$.
[2]

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18. A line has direction vector $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$. Find a unit vector in the same direction.
[2]

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19. Matrix $L = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$. State whether $L$ has an inverse. Give a reason using determinant.
[2]

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20. Two forces $\mathbf{F_1} = 4\mathbf{i} + 2\mathbf{j}$ N and $\mathbf{F_2} = -3\mathbf{i} + 5\mathbf{j}$ N act on a point. Find the resultant force vector.
[2]

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Answer Key: Secondary 3 Additional Mathematics Quiz - Vectors Matrices

Total Marks: 40 (20 questions × 2 marks)

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Section A: Vector Basics

Q1. $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2+4 \\ -3+1 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}$
Marks: 2 (1 for correct x, 1 for correct y)
Teaching: Add corresponding components.

Q2. $2\mathbf{p} = 6\mathbf{i} - 4\mathbf{j}$; $2\mathbf{p} - \mathbf{q} = (6 - (-1))\mathbf{i} + (-4 - 5)\mathbf{j} = 7\mathbf{i} - 9\mathbf{j}$
Marks: 2
Teaching: Scalar multiply then subtract components.

Q3. $|\mathbf{v}| = \sqrt{(-5)^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13$
Marks: 2 (1 formula, 1 answer)
Teaching: Magnitude = $\sqrt{x^2+y^2}$.

Q4. $\overrightarrow{AB} = \begin{pmatrix} 4-1 \\ 6-2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$
Marks: 2
Teaching: Terminal minus initial point.

Q5. $x = 10\cos30^\circ = 8.7$, $y = 10\sin30^\circ = 5.0$; vector = $8.7\mathbf{i} + 5.0\mathbf{j}$
Marks: 2
Teaching: Component = magnitude × trig ratio.

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Section B: Vector Geometry and Matrices

Q6. $\overrightarrow{PQ} = (3,1)$, $\overrightarrow{PR} = (1,4)$. Ratios $3/1 \neq 1/4$ so not parallel.
Marks: 2
Teaching: Parallel if one is scalar multiple of other.

Q7. $\mathbf{v} = 2\mathbf{u}$ so parallel (same direction, scaled by 2).
Marks: 2
Teaching: Check scalar multiple.

Q8. $M+N = \begin{pmatrix} 1+0 & 2+1 \\ 3+1 & 4+0 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 4 & 4 \end{pmatrix}$
Marks: 2

Q9. $MN = \begin{pmatrix} 2(1)+0(2) & 2(4)+0(1) \\ 1(1)+3(2) & 1(4)+3(1) \end{pmatrix} = \begin{pmatrix} 2 & 8 \\ 7 & 7 \end{pmatrix}$
Marks: 2

Q10. $\det A = 5(3) - 2(1) = 15 - 2 = 13$
Marks: 2

Q11. $T$ rotates vectors $90^\circ$ anticlockwise about origin.
Marks: 2
Teaching: Standard rotation matrix.

Q12. From equations: $2x+y=5$, $x+y=3$ → subtract: $x=2$, then $y=1$.
Marks: 2

Q13. $\overrightarrow{XY} = \mathbf{y} - \mathbf{x} = (5-2)\mathbf{i} + (-3-1)\mathbf{j} = 3\mathbf{i} - 4\mathbf{j}$
Marks: 2

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Section C: Applied Vectors and Matrices

Q14. $S\mathbf{w} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ (identity matrix).
Marks: 2

Q15. $KB = \begin{pmatrix} 2&0\\0&2 \end{pmatrix}\begin{pmatrix}2\\0\end{pmatrix} = \begin{pmatrix}4\\0\end{pmatrix}$ → $B'(4,0)$.
Marks: 2

Q16. New vector = $\frac{1}{2}\begin{pmatrix}6\\8\end{pmatrix} = \begin{pmatrix}3\\4\end{pmatrix}$.
Marks: 2

Q17. $P = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ (columns are images of basis vectors).
Marks: 2

Q18. Magnitude = 5; unit = $\frac{1}{5}\begin{pmatrix}3\\-4\end{pmatrix} = \begin{pmatrix}0.6\\-0.8\end{pmatrix}$.
Marks: 2

Q19. $\det L = 1(4)-2(2)=0$; no inverse.
Marks: 2

Q20. $\mathbf{F_R} = (4-3)\mathbf{i} + (2+5)\mathbf{j} = \mathbf{i} + 7\mathbf{j}$ N.
Marks: 2