Printables Exam Builder

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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:

√2

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 $5\sqrt2$

1 mark for combining like terms to reach 7√2 $7\sqrt2$

Simplify the following expression:

$ \frac{3x^3y^2}{9xy^{-1}} \times \frac{2x^{-2}y^3}{4y} $

$ \frac{2x}{3y} $

1 mark for correctly applying the rules of indices to simplify powers of x.

1 mark for correctly applying the rules of indices to simplify powers of y.

1 mark for combining the fractions and simplifying the entire expression correctly.

Simplify the following expression:

$ \frac{3x^2y}{4z} \times \frac{8z^2}{6xy} $

$ \frac{4xz}{y} $

1 mark for correctly multiplying the numerators and denominators.

1 mark for correctly simplifying the expression.

Simplify the following expression by rationalising the denominator:

$ \frac{3\sqrt{5}}{\sqrt{2} - 1} $

$ 3\sqrt{10} + 3\sqrt{5} $

1 mark for multiplying numerator and denominator by the conjugate of the denominator.

1 mark for correctly applying the difference of two squares in the denominator.

1 mark for correctly simplifying the numerator.

1 mark for the final simplified expression.

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