Secondary 4 Elementary Mathematics Vectors Matrices Quiz

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices

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

Duration: 60 Minutes
Total Marks: 50
Instructions: Answer all questions. Show all necessary working. Calculators are permitted.

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Section A: Matrices (Questions 1–10)

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

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

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

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

5. Given $R = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $S = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}$, find $RS$.
   
   [3 marks]

6. Determine if $RS = SR$ for the matrices in Question 5. Show your working.
   
   [3 marks]

7. Find the value of $x$ and $y$ if $\begin{pmatrix} x & 2 \\ -3 & y \end{pmatrix} + \begin{pmatrix} 4 & 1 \\ 2 & 5 \end{pmatrix} = \begin{pmatrix} 7 & 3 \\ -1 & 8 \end{pmatrix}$.
   
   [3 marks]

8. Given $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, find a matrix $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that $AB = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.
   
   [4 marks]

9. A matrix $C = \begin{pmatrix} 10 & 20 \\ 30 & 40 \end{pmatrix}$ represents the number of apples and oranges sold by two shops. If the price of an apple is \0.50$and an orange is$$0.80$, represent the prices as a matrix$P$and find$PC$ to determine the total revenue for each shop.
   
   [4 marks]

10. Solve for the matrix $X$ in the equation $2X + \begin{pmatrix} 1 & 4 \\ -2 & 3 \end{pmatrix} = \begin{pmatrix} 5 & 10 \\ 0 & 7 \end{pmatrix}$.
    
    [4 marks]

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Section B: Vectors (Questions 11–20)

11. Given vector $\mathbf{a} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$, find the magnitude $|\mathbf{a}|$.
    
    [2 marks]

12. If $\vec{AB} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$ and $\vec{BC} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}$, find $\vec{AC}$.
    
    [2 marks]

13. Given $\mathbf{u} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} -2 \\ 6 \end{pmatrix}$, find $2\mathbf{u} - 3\mathbf{v}$.
    
    
    [3 marks]

14. Point $P$ has position vector $\vec{OP} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$ and point $Q$ has position vector $\vec{OQ} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$. Find the vector $\vec{PQ}$.
    
    [2 marks]

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

16. 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 $\mathbf{b}$.
    
    [3 marks]

17. Given $\vec{XY} = 3\mathbf{a} - 2\mathbf{b}$ and $\vec{YZ} = \mathbf{a} + 4\mathbf{b}$, find $\vec{XZ}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.
    
    [3 marks]

18. In a quadrilateral $ABCD$, $\vec{AB} = \mathbf{u}$ and $\vec{AD} = \mathbf{v}$. If $\vec{DC} = 2\mathbf{u}$, express $\vec{AC}$ in terms of $\mathbf{u}$ and $\mathbf{v}$.
    
    [4 marks]

19. Point $R$ lies on the line segment $PQ$ such that $PR:RQ = 1:3$. Given $\vec{OP} = \mathbf{p}$ and $\vec{OQ} = \mathbf{q}$, express $\vec{OR}$ in terms of $\mathbf{p}$ and $\mathbf{q}$.
    
    [4 marks]

20. Given $\mathbf{a} = \begin{pmatrix} k \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$. If $\mathbf{a}$ and $\mathbf{b}$ are parallel, find the value of $k$.
    
    [4 marks]

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

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

2. $\begin{pmatrix} 3(4) & 3(7) \\ 3(-1) & 3(3) \end{pmatrix} = \begin{pmatrix} 12 & 21 \\ -3 & 9 \end{pmatrix}$ [2 marks]

3. $\begin{pmatrix} 2-3 & 5-(-2) \\ 0-4 & -1-1 \end{pmatrix} = \begin{pmatrix} -1 & 7 \\ -4 & -2 \end{pmatrix}$ [2 marks]

4. $(2 \times 4) + (3 \times -1) = 8 - 3 = 5$ [2 marks]

5. $RS = \begin{pmatrix} (1\cdot 0 + 2\cdot -1) & (1\cdot 1 + 2\cdot 2) \\ (3\cdot 0 + 4\cdot -1) & (3\cdot 1 + 4\cdot 2) \end{pmatrix} = \begin{pmatrix} -2 & 5 \\ -4 & 11 \end{pmatrix}$ [3 marks]

6. $SR = \begin{pmatrix} (0\cdot 1 + 1\cdot 3) & (0\cdot 2 + 1\cdot 4) \\ (-1\cdot 1 + 2\cdot 3) & (-1\cdot 2 + 2\cdot 4) \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 5 & 6 \end{pmatrix}$. Since $RS \neq SR$, they are not equal. [3 marks]

7. $x + 4 = 7 \Rightarrow x = 3$; $y + 5 = 8 \Rightarrow y = 3$. [3 marks]

8. Let $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. $2a + c = 1$, $2b + 3d = 0$, $0a + 3c = 0$, $0b + 3d = 1$. From $3c = 0, c = 0$. Then $2a = 1 \Rightarrow a = 0.5$. From $3d = 1, d = 1/3$. Then $2b + 3(1/3) = 0 \Rightarrow 2b = -1 \Rightarrow b = -0.5$. $B = \begin{pmatrix} 0.5 & -0.5 \\ 0 & 1/3 \end{pmatrix}$ [4 marks]

9. $P = \begin{pmatrix} 0.5 & 0.8 \end{pmatrix}$. $PC = \begin{pmatrix} 0.5 & 0.8 \end{pmatrix} \begin{pmatrix} 10 & 20 \\ 30 & 40 \end{pmatrix} = \begin{pmatrix} (5+24) & (10+32) \end{pmatrix} = \begin{pmatrix} 29 & 42 \end{pmatrix}$. Revenue: Shop 1 = \29$, Shop 2 =$$42$. [4 marks]

10. $2X = \begin{pmatrix} 5-1 & 10-4 \\ 0-(-2) & 7-3 \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 2 & 4 \end{pmatrix} \Rightarrow X = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ [4 marks]

11. $|\mathbf{a}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ [2 marks]

12. $\vec{AC} = \vec{AB} + \vec{BC} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 1 \\ 8 \end{pmatrix}$ [2 marks]

13. $2\begin{pmatrix} 4 \\ 2 \end{pmatrix} - 3\begin{pmatrix} -2 \\ 6 \end{pmatrix} = \begin{pmatrix} 8 \\ 4 \end{pmatrix} - \begin{pmatrix} -6 \\ 18 \end{pmatrix} = \begin{pmatrix} 14 \\ -14 \end{pmatrix}$ [3 marks]

14. $\vec{PQ} = \vec{OQ} - \vec{OP} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} - \begin{pmatrix} 5 \\ 2 \end{pmatrix} = \begin{pmatrix} -6 \\ 2 \end{pmatrix}$ [2 marks]

15. $|\mathbf{w}| = \sqrt{3^2 + 4^2} = 5$. Unit vector = $\frac{1}{5} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 0.6 \\ 0.8 \end{pmatrix}$ [3 marks]

16. $\vec{AB} = \mathbf{b} - \mathbf{a}$. $\vec{AM} = \frac{1}{2}(\mathbf{b} - \mathbf{a})$. $\vec{OM} = \vec{OA} + \vec{AM} = \mathbf{a} + \frac{1}{2}\mathbf{b} - \frac{1}{2}\mathbf{a} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ [3 marks]

17. $\vec{XZ} = \vec{XY} + \vec{YZ} = (3\mathbf{a} - 2\mathbf{b}) + (\mathbf{a} + 4\mathbf{b}) = 4\mathbf{a} + 2\mathbf{b}$ [3 marks]

18. $\vec{AC} = \vec{AB} + \vec{BC}$. Since $\vec{DC} = 2\mathbf{u}$ and $ABCD$ is a quad, $\vec{BC} = \vec{BA} + \vec{AD} + \vec{DC}$ is not necessarily true unless it's a parallelogram. Correct path: $\vec{AC} = \vec{AB} + \vec{BC}$. If $\vec{BC}$ is not given, we use $\vec{AC} = \vec{AD} + \vec{DC} = \mathbf{v} + 2\mathbf{u}$ (assuming $D$ is the vertex). Wait, $\vec{AC} = \vec{AB} + \vec{BC}$. If $\vec{DC} = 2\mathbf{u}$, then $\vec{CD} = -2\mathbf{u}$. $\vec{AC} = \vec{AD} + \vec{DC} = \mathbf{v} + 2\mathbf{u}$ [4 marks]

19. $\vec{PR} = \frac{1}{4}\vec{PQ} = \frac{1}{4}(\mathbf{q} - \mathbf{p})$. $\vec{OR} = \vec{OP} + \vec{PR} = \mathbf{p} + \frac{1}{4}\mathbf{q} - \frac{1}{4}\mathbf{p} = \frac{3}{4}\mathbf{p} + \frac{1}{4}\mathbf{q}$ [4 marks]

20. Parallel vectors: $\mathbf{a} = k\mathbf{b}$. $\begin{pmatrix} k \\ 3 \end{pmatrix} = m \begin{pmatrix} 2 \\ -1 \end{pmatrix} \Rightarrow 3 = m(-1) \Rightarrow m = -3$. Then $k = (-3)(2) = -6$. [4 marks]