Secondary 4 Elementary Mathematics Vectors Matrices Quiz

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices

Name: __________________________
Class: __________________________
Date: __________________________
Score: ________ / 50

Duration: 45 minutes
Total Marks: 50

Instructions:

1. Answer all questions.
2. Write your answers in the spaces provided.
3. Show all necessary working clearly. No marks will be given for correct answers without working.
4. Give non-exact numerical answers correct to 3 significant figures, unless otherwise specified.
5. An electronic calculator is expected to be used for this quiz.

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Section A: Matrix Operations and Properties (Questions 1–5)

[15 Marks]

1. Given matrices $A = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ -2 & 0 \end{pmatrix}$.

Calculate the matrix $2A - B$.

<br> <br> <br> **Answer:** $\begin{pmatrix} \_\_\_ & \_\_\_ \\ \_\_\_ & \_\_\_ \end{pmatrix}$ [2]

2. Given $P = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 1 & 0 \\ -1 & 2 \end{pmatrix}$.

Find the product $PQ$.

<br> <br> <br> **Answer:** $\begin{pmatrix} \_\_\_ & \_\_\_ \\ \_\_\_ & \_\_\_ \end{pmatrix}$ [2]

3. Find the values of $x$ and $y$ if: $\begin{pmatrix} x & 3 \\ 2y & 1 \end{pmatrix} + \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix} = \begin{pmatrix} 5 & 2 \\ 10 & 4 \end{pmatrix}$

<br> <br> <br> $x = $ _______________ $y = $ _______________ [2]

4. Matrix $M = \begin{pmatrix} k & 2 \\ 3 & k \end{pmatrix}$. Given that the determinant of $M$ is 7, find the possible values of $k$.

<br> <br> <br> $k = $ _______________ or _______________ [3]

5. A shop sells apples and oranges. The price of an apple is \a$and an orange is$$b$. On Monday, 10 apples and 5 oranges were sold for$$15$. On Tuesday, 6 apples and 8 oranges were sold for$$14.40$.

Write down a matrix equation of the form $\mathbf{AX} = \mathbf{B}$ to represent this information, where $\mathbf{X} = \begin{pmatrix} a \\ b \end{pmatrix}$.

<br> <br> <br> **Answer:** $\begin{pmatrix} \_\_\_ & \_\_\_ \\ \_\_\_ & \_\_\_ \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} \_\_\_ \\ \_\_\_ \end{pmatrix}$ [2]

(Note: Do not solve for $a$ and $b$ in this question.)

6. Given $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

Verify whether $AB = BA$ by calculating both products. State your conclusion.

<br> <br> <br> <br> <br> **Conclusion:** _________________________________________________________ [2]

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Section B: Vectors in Two Dimensions (Questions 7–12)

[18 Marks]

7. Given vectors $\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$.

Calculate the vector $3\mathbf{a} - 2\mathbf{b}$.

<br> <br> <br> **Answer:** $\begin{pmatrix} \_\_\_ \\ \_\_\_ \end{pmatrix}$ [2]

8. Points $A$ and $B$ have coordinates $(2, 5)$ and $(8, -1)$ respectively.

Find the column vector $\vec{AB}$.

<br> <br> <br> **Answer:** $\begin{pmatrix} \_\_\_ \\ \_\_\_ \end{pmatrix}$ [1]

9. Using the coordinates from Question 8, calculate the magnitude of vector $\vec{AB}$, denoted as $|\vec{AB}|$. Give your answer in the form $\sqrt{n}$ where $n$ is an integer.

<br> <br> <br> $|\vec{AB}| = $ _______________ [2]

10. In triangle $OAB$, $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$. $M$ is the midpoint of $AB$.

Express $\vec{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ in its simplest form.

<br> <br> <br> $\vec{OM} = $ _________________________ [2]

11. Given $\mathbf{u} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ k \end{pmatrix}$.

If $\mathbf{u}$ and $\mathbf{v}$ are parallel, find the value of $k$.

<br> <br> <br> $k = $ _______________ [2]

12. The position vectors of points $P$ and $Q$ are $\mathbf{p} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 4 \\ 7 \end{pmatrix}$. Point $R$ lies on the line segment $PQ$ such that $PR : RQ = 1 : 2$.

Find the position vector of $R$, $\vec{OR}$.

<br> <br> <br> <br> **Answer:** $\begin{pmatrix} \_\_\_ \\ \_\_\_ \end{pmatrix}$ [3]

13. Vector $\mathbf{w} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}$.

Find the unit vector in the direction of $\mathbf{w}$.

<br> <br> <br> **Answer:** $\begin{pmatrix} \_\_\_ \\ \_\_\_ \end{pmatrix}$ [2]

14. Points $A(1, 2)$, $B(4, 6)$ and $C(7, 10)$ are given.

By using vectors, show that $A, B$ and $C$ are collinear.

<br> <br> <br> <br> <br> [2]

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Section C: Combined Applications (Questions 15–20)

[17 Marks]

15. A transformation is represented by the matrix $T = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

Describe fully the geometric transformation represented by $T$.

<br> <br> <br> **Answer:** _________________________________________________________ [2]

16. Triangle $ABC$ has vertices $A(1,1)$, $B(3,1)$ and $C(1,4)$. The triangle is transformed by the matrix $M = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$.

(a) Find the coordinates of the image $A'B'C'$. (b) Calculate the ratio of the Area of $\triangle A'B'C'$ to the Area of $\triangle ABC$.

<br> <br> <br> <br> (a) $A'($___,___$)$, $B'($___,___$)$, $C'($___,___$)$ (b) Ratio = _______________ [3]

17. In parallelogram $ABCD$, $\vec{AB} = \mathbf{a}$ and $\vec{AD} = \mathbf{b}$. The diagonals $AC$ and $BD$ intersect at $X$.

Express $\vec{AX}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

<br> <br> <br> $\vec{AX} = $ _________________________ [2]

18. Given that $\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 8 \end{pmatrix}$.

Find the values of $x$ and $y$ by forming and solving simultaneous equations, or by using the inverse matrix method.

<br> <br> <br> <br> <br> $x = $ _______________ $y = $ _______________ [4]

19. Points $P, Q, R$ have position vectors $\mathbf{p} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$, $\mathbf{q} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 8 \\ 5 \end{pmatrix}$.

Show that $\vec{PQ} = \vec{QR}$. What does this imply about the points $P, Q, R$?

<br> <br> <br> <br> **Implication:** _________________________________________________________ [3]

20. A matrix $A = \begin{pmatrix} 3 & k \\ 2 & 4 \end{pmatrix}$ is singular.

Find the value of $k$.

<br> <br> <br> $k = $ _______________ [2]

*** End of Quiz ***

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices (Answer Key)

1. Calculate $2A - B$. $2A = \begin{pmatrix} 4 & -2 \\ 0 & 6 \end{pmatrix}$ $2A - B = \begin{pmatrix} 4 & -2 \\ 0 & 6 \end{pmatrix} - \begin{pmatrix} 1 & 4 \\ -2 & 0 \end{pmatrix} = \begin{pmatrix} 4-1 & -2-4 \\ 0-(-2) & 6-0 \end{pmatrix} = \begin{pmatrix} 3 & -6 \\ 2 & 6 \end{pmatrix}$ Answer: $\begin{pmatrix} 3 & -6 \\ 2 & 6 \end{pmatrix}$ [2]

2. Find $PQ$. $PQ = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 2 \end{pmatrix}$ Row 1, Col 1: $(3)(1) + (1)(-1) = 3 - 1 = 2$ Row 1, Col 2: $(3)(0) + (1)(2) = 0 + 2 = 2$ Row 2, Col 1: $(2)(1) + (4)(-1) = 2 - 4 = -2$ Row 2, Col 2: $(2)(0) + (4)(2) = 0 + 8 = 8$ Answer: $\begin{pmatrix} 2 & 2 \\ -2 & 8 \end{pmatrix}$ [2]

3. Solve for $x$ and $y$. $\begin{pmatrix} x+2 & 3-1 \\ 2y+4 & 1+3 \end{pmatrix} = \begin{pmatrix} 5 & 2 \\ 10 & 4 \end{pmatrix}$ $x + 2 = 5 \Rightarrow x = 3$ $2y + 4 = 10 \Rightarrow 2y = 6 \Rightarrow y = 3$ Answer: $x = 3, y = 3$ [2]

4. Determinant of $M$ is 7. $\det(M) = (k)(k) - (2)(3) = k^2 - 6$ $k^2 - 6 = 7 \Rightarrow k^2 = 13 \Rightarrow k = \pm\sqrt{13}$ Answer: $k = \sqrt{13}$ or $-\sqrt{13}$ [3]

5. Matrix Equation. Equations: $10a + 5b = 15$ and $6a + 8b = 14.4$. Answer: $\begin{pmatrix} 10 & 5 \\ 6 & 8 \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 15 \\ 14.4 \end{pmatrix}$ [2]

6. Verify $AB = BA$. $AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$ $BA = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$ Since $\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} \neq \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$, $AB \neq BA$. Conclusion: Matrix multiplication is not commutative. [2]

7. Calculate $3\mathbf{a} - 2\mathbf{b}$. $3\begin{pmatrix} 3 \\ -2 \end{pmatrix} - 2\begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 9 \\ -6 \end{pmatrix} - \begin{pmatrix} -2 \\ 8 \end{pmatrix} = \begin{pmatrix} 9 - (-2) \\ -6 - 8 \end{pmatrix} = \begin{pmatrix} 11 \\ -14 \end{pmatrix}$ Answer: $\begin{pmatrix} 11 \\ -14 \end{pmatrix}$ [2]

8. Column vector $\vec{AB}$. $\vec{AB} = \begin{pmatrix} 8 - 2 \\ -1 - 5 \end{pmatrix} = \begin{pmatrix} 6 \\ -6 \end{pmatrix}$ Answer: $\begin{pmatrix} 6 \\ -6 \end{pmatrix}$ [1]

9. Magnitude $|\vec{AB}|$. $|\vec{AB}| = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72}$ Answer: $\sqrt{72}$ [2]

10. Express $\vec{OM}$. $\vec{OM} = \vec{OA} + \vec{AM} = \mathbf{a} + \frac{1}{2}\vec{AB}$ $\vec{AB} = \mathbf{b} - \mathbf{a}$ $\vec{OM} = \mathbf{a} + \frac{1}{2}(\mathbf{b} - \mathbf{a}) = \mathbf{a} + \frac{1}{2}\mathbf{b} - \frac{1}{2}\mathbf{a} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ Answer: $\frac{1}{2}(\mathbf{a} + \mathbf{b})$ or $\frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ [2]

11. Parallel vectors. If parallel, $\mathbf{u} = k\mathbf{v}$ or ratios of components are equal. $\frac{4}{2} = \frac{6}{k} \Rightarrow 2 = \frac{6}{k} \Rightarrow 2k = 6 \Rightarrow k = 3$ Answer: $k = 3$ [2]

12. Position vector of $R$. $\vec{OR} = \mathbf{p} + \frac{1}{1+2}\vec{PQ} = \mathbf{p} + \frac{1}{3}(\mathbf{q} - \mathbf{p})$ $\mathbf{q} - \mathbf{p} = \begin{pmatrix} 4-1 \\ 7-3 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ $\vec{OR} = \begin{pmatrix} 1 \\ 3 \end{pmatrix} + \frac{1}{3}\begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ 4/3 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 + 1.33... \end{pmatrix} = \begin{pmatrix} 2 \\ 13/3 \end{pmatrix}$ Answer: $\begin{pmatrix} 2 \\ 13/3 \end{pmatrix}$ [3]

13. Unit vector. $|\mathbf{w}| = \sqrt{(-3)^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$ Unit vector = $\frac{1}{5}\begin{pmatrix} -3 \\ 4 \end{pmatrix} = \begin{pmatrix} -3/5 \\ 4/5 \end{pmatrix}$ Answer: $\begin{pmatrix} -0.6 \\ 0.8 \end{pmatrix}$ [2]

14. Collinearity. $\vec{AB} = \begin{pmatrix} 4-1 \\ 6-2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ $\vec{BC} = \begin{pmatrix} 7-4 \\ 10-6 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ Since $\vec{AB} = \vec{BC}$ and they share point $B$, $A, B, C$ are collinear. [2]

15. Geometric Transformation. Matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ maps $(1,0) \to (0,1)$ and $(0,1) \to (-1,0)$. Answer: Rotation $90^\circ$ anti-clockwise about the origin $(0,0)$. [2]

16. Transformation $M = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$. (a) Multiply coordinates by 2. $A'(2,2), B'(6,2), C'(2,8)$. (b) Scale factor $k=2$. Area scale factor $k^2 = 4$. Answer: (a) $A'(2,2), B'(6,2), C'(2,8)$; (b) 4 [3]

17. Vector $\vec{AX}$. Diagonals of a parallelogram bisect each other. $\vec{AC} = \vec{AB} + \vec{BC} = \mathbf{a} + \mathbf{b}$ $\vec{AX} = \frac{1}{2}\vec{AC} = \frac{1}{2}(\mathbf{a} + \mathbf{b})$ Answer: $\frac{1}{2}(\mathbf{a} + \mathbf{b})$ [2]

18. Solve simultaneous equations. $2x + y = 7$ (1) $x + 3y = 8$ (2) From (2), $x = 8 - 3y$. Substitute into (1): $2(8 - 3y) + y = 7 \Rightarrow 16 - 6y + y = 7 \Rightarrow -5y = -9 \Rightarrow y = 1.8$ $x = 8 - 3(1.8) = 8 - 5.4 = 2.6$ Answer: $x = 2.6, y = 1.8$ [4]

19. Show $\vec{PQ} = \vec{QR}$. $\vec{PQ} = \mathbf{q} - \mathbf{p} = \begin{pmatrix} 5-2 \\ 3-1 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ $\vec{QR} = \mathbf{r} - \mathbf{q} = \begin{pmatrix} 8-5 \\ 5-3 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ Vectors are equal. Implication: Points $P, Q, R$ are collinear and $Q$ is the midpoint of $PR$. [3]

20. Singular Matrix. Determinant is 0. $(3)(4) - (2)(k) = 0 \Rightarrow 12 - 2k = 0 \Rightarrow 2k = 12 \Rightarrow k = 6$ Answer: $k = 6$ [2]