Secondary 3 Additional Mathematics Vectors Matrices Quiz

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Secondary 3 Additional Mathematics Quiz - Vectors Matrices

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

Duration: 60 minutes
Total Marks: 50

Instructions:

• Answer ALL questions.
• Show your working clearly. Marks will be awarded for correct method even if the final answer is wrong.
• Non-programmable scientific calculators may be used.
• Give answers as exact values unless otherwise stated.
• Vectors may be written in column form $\begin{pmatrix} x \\ y \end{pmatrix}$ or component form $x\mathbf{i} + y\mathbf{j}$.

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Section A: Vector Basics and Operations (Questions 1–5)

1. Given $\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$, find $2\mathbf{a} - 3\mathbf{b}$.
[2 marks]

 

 

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2. The position vector of point $P$ is $\begin{pmatrix} 4 \\ 7 \end{pmatrix}$ and the position vector of point $Q$ is $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$. Find the vector $\overrightarrow{PQ}$ and hence find the distance $PQ$.
[3 marks]

 

 

 

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3. Find the magnitude of the vector $\mathbf{v} = 5\mathbf{i} - 12\mathbf{j}$.
[2 marks]

 

 

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4. Given that $\mathbf{p} = \begin{pmatrix} 2 \\ k \end{pmatrix}$ and $|\mathbf{p}| = 5$, find the possible values of $k$.
[3 marks]

 

 

 

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5. Vectors $\mathbf{m} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$ and $\mathbf{n} = \begin{pmatrix} -9 \\ t \end{pmatrix}$ are parallel. Find the value of $t$.
[2 marks]

 

 

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Section B: Scalar Product and Applications (Questions 6–10)

6. Given $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$, find the scalar product $\mathbf{a} \cdot \mathbf{b}$.
[2 marks]

 

 

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7. Find the angle between the vectors $\mathbf{u} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 5 \\ -12 \end{pmatrix}$, giving your answer correct to the nearest degree.
[4 marks]

 

 

 

 

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8. Given $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$, determine whether $\mathbf{a}$ and $\mathbf{b}$ are perpendicular. Justify your answer.
[2 marks]

 

 

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9. The vectors $\mathbf{p} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 2 \\ k \end{pmatrix}$ are perpendicular. Find the value of $k$.
[2 marks]

 

 

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10. Given $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$, find the projection of $\mathbf{a}$ onto $\mathbf{b}$.
[3 marks]

 

 

 

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Section C: Matrices – Operations and Properties (Questions 11–15)

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

 

 

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12. Given $M = \begin{pmatrix} 3 & 2 \\ -1 & 4 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix}$, find $MN$.
[3 marks]

 

 

 

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13. Find the determinant of the matrix $P = \begin{pmatrix} 5 & 3 \\ -2 & 4 \end{pmatrix}$.
[2 marks]

 

 

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14. Find the inverse of the matrix $Q = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$, or explain why it does not exist.
[3 marks]

 

 

 

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15. Given $R = \begin{pmatrix} 4 & -2 \\ 6 & -3 \end{pmatrix}$, show that $R$ does not have an inverse.
[2 marks]

 

 

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Section D: Matrices – Simultaneous Equations and Applications (Questions 16–20)

16. Write the following simultaneous equations as a matrix equation $A\mathbf{x} = \mathbf{b}$:
$3x + 2y = 12$
$5x - y = 7$
[2 marks]

 

 

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17. Use a matrix method to solve the simultaneous equations:
$2x + 3y = 13$
$5x - 2y = 4$
[5 marks]

 

 

 

 

 

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18. The matrix $M = \begin{pmatrix} 1 & 2 \\ 3 & 2 \end{pmatrix}$ has an eigenvalue $\lambda = 4$. Find a corresponding eigenvector.
[3 marks]

 

 

 

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19. A transformation is represented by the matrix $T = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. The point $A(3, 5)$ is transformed by $T$ to point $A'$. Find the coordinates of $A'$. Describe the geometric effect of this transformation.
[3 marks]

 

 

 

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20. A shop sells two types of items. On Monday, 4 units of Item P and 3 units of Item Q are sold for $47. On Tuesday, 6 units of Item P and 5 units of Item Q are sold for $73.
(a) Write two equations and express them in matrix form.
(b) Use a matrix method to find the price of each item.
[5 marks]

 

 

 

 

 

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Secondary 3 Additional Mathematics Quiz - Vectors Matrices

Answer Key

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1.
$2\mathbf{a} = 2\begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$
$3\mathbf{b} = 3\begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} -3 \\ 15 \end{pmatrix}$
$2\mathbf{a} - 3\mathbf{b} = \begin{pmatrix} 6 \\ -4 \end{pmatrix} - \begin{pmatrix} -3 \\ 15 \end{pmatrix} = \begin{pmatrix} 9 \\ -19 \end{pmatrix}$

Answer: $\begin{pmatrix} 9 \\ -19 \end{pmatrix}$
[2 marks] — 1 mark for each of $2\mathbf{a}$ and $3\mathbf{b}$ correctly computed; 1 mark for correct final subtraction.

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2.
$\overrightarrow{PQ} = \vec{q} - \vec{p} = \begin{pmatrix} -2 \\ 3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} = \begin{pmatrix} -6 \\ -4 \end{pmatrix}$
$PQ = |\overrightarrow{PQ}| = \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}$

Answer: $\overrightarrow{PQ} = \begin{pmatrix} -6 \\ -4 \end{pmatrix}$, $PQ = 2\sqrt{13}$ (or $\sqrt{52}$)
[3 marks] — 1 mark for correct $\overrightarrow{PQ}$; 1 mark for correct magnitude calculation; 1 mark for simplified exact answer.

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3.
$|\mathbf{v}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

Answer: $13$
[2 marks] — 1 mark for correct substitution; 1 mark for correct answer.

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4.
$|\mathbf{p}| = \sqrt{2^2 + k^2} = 5$
$4 + k^2 = 25$
$k^2 = 21$
$k = \pm\sqrt{21}$

Answer: $k = \sqrt{21}$ or $k = -\sqrt{21}$
[3 marks] — 1 mark for setting up the magnitude equation; 1 mark for solving $k^2 = 21$; 1 mark for both values.

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5.
If $\mathbf{m}$ and $\mathbf{n}$ are parallel, then $\mathbf{n} = \lambda\mathbf{m}$ for some scalar $\lambda$.
From the first component: $-6\lambda = -9 \Rightarrow \lambda = \frac{3}{2}$
From the second component: $-4\lambda = t \Rightarrow t = -4 \times \frac{3}{2} = -6$

Answer: $t = -6$
[2 marks] — 1 mark for setting up proportionality; 1 mark for correct value.

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6.
$\mathbf{a} \cdot \mathbf{b} = (2)(-1) + (3)(4) = -2 + 12 = 10$

Answer: $10$
[2 marks] — 1 mark for correct formula; 1 mark for correct answer.

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7.
$\mathbf{u} \cdot \mathbf{v} = (3)(5) + (4)(-12) = 15 - 48 = -33$
$|\mathbf{u}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
$|\mathbf{v}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
$\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|} = \frac{-33}{5 \times 13} = \frac{-33}{65}$
$\theta = \cos^{-1}\left(\frac{-33}{65}\right) \approx 120.51°$

Answer: $121°$ (to nearest degree)
[4 marks] — 1 mark each for: scalar product, $|\mathbf{u}|$, $|\mathbf{v}|$, and correct angle.

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8.
$\mathbf{a} \cdot \mathbf{b} = (1)(-3) + (2)(1) = -3 + 2 = -1$
Since $\mathbf{a} \cdot \mathbf{b} \neq 0$, the vectors are not perpendicular.

Answer: Not perpendicular, because $\mathbf{a} \cdot \mathbf{b} = -1 \neq 0$.
[2 marks] — 1 mark for computing scalar product; 1 mark for correct conclusion with justification.

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9.
If $\mathbf{p}$ and $\mathbf{q}$ are perpendicular, then $\mathbf{p} \cdot \mathbf{q} = 0$.
$(4)(2) + (1)(k) = 0$
$8 + k = 0$
$k = -8$

Answer: $k = -8$
[2 marks] — 1 mark for setting scalar product to zero; 1 mark for correct value.

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10.
Projection of $\mathbf{a}$ onto $\mathbf{b}$ is given by $\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2}\,\mathbf{b}$.
$\mathbf{a} \cdot \mathbf{b} = (3)(2) + (1)(-2) = 6 - 2 = 4$
$|\mathbf{b}|^2 = 2^2 + (-2)^2 = 4 + 4 = 8$
Projection $= \frac{4}{8}\begin{pmatrix} 2 \\ -2 \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 2 \\ -2 \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$

Answer: $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$
[3 marks] — 1 mark for $\mathbf{a} \cdot \mathbf{b}$; 1 mark for $|\mathbf{b}|^2$; 1 mark for correct final vector.

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11.
$A + B = \begin{pmatrix} 2+0 & -1+5 \\ 3+(-2) & 4+1 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 1 & 5 \end{pmatrix}$

Answer: $\begin{pmatrix} 2 & 4 \\ 1 & 5 \end{pmatrix}$
[2 marks] — 1 mark for correct addition; 1 mark for correct final matrix.

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12.
$MN = \begin{pmatrix} 3 & 2 \\ -1 & 4 \end{pmatrix}\begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix}$
$= \begin{pmatrix} 3(1)+2(3) & 3(-2)+2(0) \\ -1(1)+4(3) & -1(-2)+4(0) \end{pmatrix}$
$= \begin{pmatrix} 3+6 & -6+0 \\ -1+12 & 2+0 \end{pmatrix}$
$= \begin{pmatrix} 9 & -6 \\ 11 & 2 \end{pmatrix}$

Answer: $\begin{pmatrix} 9 & -6 \\ 11 & 2 \end{pmatrix}$
[3 marks] — 1 mark for correct method; 1 mark for correct first row; 1 mark for correct second row.

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13.
$\det(P) = (5)(4) - (3)(-2) = 20 + 6 = 26$

Answer: $26$
[2 marks] — 1 mark for correct formula; 1 mark for correct answer.

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14.
$\det(Q) = (2)(3) - (1)(5) = 6 - 5 = 1 \neq 0$, so the inverse exists.
$Q^{-1} = \frac{1}{\det(Q)}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \frac{1}{1}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$

Answer: $\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$
[3 marks] — 1 mark for determinant; 1 mark for correct formula application; 1 mark for correct final answer.

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15.
$\det(R) = (4)(-3) - (-2)(6) = -12 + 12 = 0$
Since $\det(R) = 0$, the matrix $R$ is singular and does not have an inverse.

Answer: $\det(R) = 0$, so $R$ has no inverse.
[2 marks] — 1 mark for computing determinant; 1 mark for correct conclusion.

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16.
$\begin{pmatrix} 3 & 2 \\ 5 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 12 \\ 7 \end{pmatrix}$

Answer: $A = \begin{pmatrix} 3 & 2 \\ 5 & -1 \end{pmatrix}$, $\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 12 \\ 7 \end{pmatrix}$
[2 marks] — 1 mark for correct coefficient matrix; 1 mark for correct column vector form.

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17.
Matrix form: $\begin{pmatrix} 2 & 3 \\ 5 & -2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 13 \\ 4 \end{pmatrix}$
$\det = (2)(-2) - (3)(5) = -4 - 15 = -19$
$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{-19}\begin{pmatrix} -2 & -3 \\ -5 & 2 \end{pmatrix}\begin{pmatrix} 13 \\ 4 \end{pmatrix}$
$= \frac{1}{-19}\begin{pmatrix} (-2)(13)+(-3)(4) \\ (-5)(13)+(2)(4) \end{pmatrix}$
$= \frac{1}{-19}\begin{pmatrix} -26-12 \\ -65+8 \end{pmatrix}$
$= \frac{1}{-19}\begin{pmatrix} -38 \\ -57 \end{pmatrix}$
$= \begin{pmatrix} 2 \\ 3 \end{pmatrix}$

Answer: $x = 2$, $y = 3$
[5 marks] — 1 mark for matrix form; 1 mark for determinant; 1 mark for inverse matrix; 1 mark for multiplication; 1 mark for correct final answer.

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18.
We need $(M - 4I)\mathbf{v} = \mathbf{0}$:
$M - 4I = \begin{pmatrix} 1-4 & 2 \\ 3 & 2-4 \end{pmatrix} = \begin{pmatrix} -3 & 2 \\ 3 & -2 \end{pmatrix}$
Row 2 = −(Row 1), so we solve $-3x + 2y = 0 \Rightarrow y = \frac{3}{2}x$.
Let $x = 2$, then $y = 3$.

Answer: Any non-zero scalar multiple of $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ (e.g., $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$)
[3 marks] — 1 mark for $M - \lambda I$; 1 mark for solving the system; 1 mark for correct eigenvector.

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19.
$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 3 \\ 5 \end{pmatrix} = \begin{pmatrix} 0(3)+(-1)(5) \\ 1(3)+0(5) \end{pmatrix} = \begin{pmatrix} -5 \\ 3 \end{pmatrix}$
The transformation is a rotation of 90° anticlockwise about the origin.

Answer: $A' = (-5, 3)$; rotation of 90° anticlockwise about the origin.
[3 marks] — 1 mark for matrix multiplication; 1 mark for correct coordinates; 1 mark for correct geometric description.

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20.
(a) Let the price of Item P be $p$ dollars and Item Q be $q$ dollars.
$4p + 3q = 47$
$6p + 5q = 73$
Matrix form: $\begin{pmatrix} 4 & 3 \\ 6 & 5 \end{pmatrix}\begin{pmatrix} p \\ q \end{pmatrix} = \begin{pmatrix} 47 \\ 73 \end{pmatrix}$

(b) $\det = (4)(5) - (3)(6) = 20 - 18 = 2$
$\begin{pmatrix} p \\ q \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 5 & -3 \\ -6 & 4 \end{pmatrix}\begin{pmatrix} 47 \\ 73 \end{pmatrix}$
$= \frac{1}{2}\begin{pmatrix} 5(47)+(-3)(73) \\ -6(47)+4(73) \end{pmatrix}$
$= \frac{1}{2}\begin{pmatrix} 235-219 \\ -282+292 \end{pmatrix}$
$= \frac{1}{2}\begin{pmatrix} 16 \\ 10 \end{pmatrix}$
$= \begin{pmatrix} 8 \\ 5 \end{pmatrix}$

Answer: Item P = $8, Item Q = $5
[5 marks] — Part (a): 2 marks (1 for equations, 1 for matrix form). Part (b): 3 marks (1 for determinant/inverse, 1 for multiplication, 1 for correct prices).

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