Simplify the following algebraic fraction to its lowest terms:

$$\frac{2x^2 - 5x - 3}{x - 3}$$

2x + 1

2 marks for correct factorisation of the numerator (2x + 1)(x - 3).

1 mark for correct simplification of the fraction.

A point P is located at (2, 3) in the 2D plane. The following vector is applied to point P.

$$\begin{bmatrix} 3 \\ -2 \\ \end{bmatrix}$$

What is the new position of point P after the vector translation?

(5, 1)

1 mark for correctly adding the x-coordinates.

1 mark for correctly adding the y-coordinates.

If a point P(2, 5) is translated by the vector **v** to (5, 3), what is the column vector **v**?

$$\begin{bmatrix} 3 \\ -2 \\ \end{bmatrix}$$

1 marks for the correct x translation value.

1 mark for the correct y translation value.

In vector notation, the vector A to B is represented as **\(\vec{AB} = 3\vec{i} - 2\vec{j}\)** and the vector B to C is represented as **\(\vec{BC} = -\vec{i} + 4\vec{j}\)**.

What is the vector C to A?

\(\vec{CA} = -2\vec{i} - 2\vec{j}\)

1 mark for correctly finding the vector B to A.

1 mark for correctly adding the vectors B to A and B to C to find the vector C to A.

Consider a rectangle with vertices at the points (0,0), (0,3), (4,0) and (4,3) on a coordinate plane.

The rectangle undergoes a translatation and the 4 vertices move to (5,2), (5,5), (9,2) and (9,5).

Describe the translation using a column vector.

$$\begin{bmatrix} 5 \\ 2 \\ \end{bmatrix}$$

1 marks for the correct x translation value.

1 mark for the correct y translation value.

A point is located at coordinates (2, 3).

The point undergoes a transformation and moves to (6, 0).

Describe the transformation using a column vector.

Translation and $$\begin{bmatrix} 5 \\ 2 \\ \end{bmatrix}$$

1 mark for identifying translation.

1 mark for the correct column vector.

A rectangle has vertices at points A (2, 1), B (5, 1), C (5, 3) and D (2, 3) on a coordinate plane.

The rectangle is translated according to the vector **v**:

$$\begin{bmatrix} 4 \\ -2 \\ \end{bmatrix}$$

What are the coordinates of the translated rectangle?

E (6, -1), F (9, -1), G (9, 1) and H (6, 1)

3 marks for all correct.

2 marks for any 3 correct.

1 mark for any 1 correct.

In a coordinate plane, point A is at (2, 3) and point B is at (4, 7).

Translate point A by the vector:

$$\begin{bmatrix} 3 \\ -1 \\ \end{bmatrix}$$

Translate point B by the vector:

$$\begin{bmatrix} -2 \\ 2 \\ \end{bmatrix}$$

What are the new coordinates of points A and B?

Point A is now at (3, 4) and point B is now at (5, 8).

1 mark for correctly translating point A's x-coordinate.

1 mark for correctly translating point A's y-coordinate.

1 mark for correctly translating point B's x-coordinate.

1 mark for correctly translating point B's y-coordinate.

If a point A(1, 4) is translated by the vector **v** to (4, 2), what is the column vector **v**?

$$\begin{bmatrix} 3 \\ -2 \\ \end{bmatrix}$$

1 marks for the correct x translation value.

1 mark for the correct y translation value.

Given the vectors of A to B as **\(\vec{a}\)** and B to C as **\(\vec{b}\)** , find the vector of C to A?

**-a - b**

1 mark for understanding that the vector from C to A is the negative of the vector from A to C. 1 mark for correctly applying this to find the vector -a - b.

Given the vectors A to B as **\(3\vec{i} + 2\vec{j}\)** and B to C as **\(2\vec{i} - 4\vec{j}\)**, calculate the vector C to A.

**\(-5\vec{i} + 2\vec{j}\)**

1 mark for correctly identifying that both vector A to B and B to C need to be reversed, giving -3i - 2j or -2i + 4j.

1 mark for correctly calculating the resultant vector C to A.

Express the following fraction with a rationalised denominator:

3√2/2

1 mark for multiplying both the numerator and denominator by √2.

1 mark for the correct answer.

Simplify the following surd: √50 + 2√2

7√2

1 mark for simplifying √50 = 5√2

1 mark for combining like terms to reach 7√2