Transformations using Matrices (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

The matrix B is defined by

where and are integers.

State the value of and the value of when B represents

(a) an enlargement about the origin with scale factor −2

[1]

(b) a reflection in the -axis.

[1]

Prove by induction that for

where is a constant.

The transformation represented by matrix M followed by the transformation represented by matrix N is represented by the matrix B

(i) Determine N in the form where , , and are integers.

[1]

(ii) Determine B

[2]

Hexagon is transformed onto hexagon by matrix B

Given that the area of is 720 square units, determine the area of

Describe the single geometrical transformation represented by the matrix A.

The matrix B represents a rotation of 210° anticlockwise about centre (0, 0).

Write down the matrix B, giving each element in exact form.

The transformation represented by matrix A followed by the transformation represented by matrix B is represented by the matrix C.

Find C.

The hexagon is transformed onto the hexagon by the matrix C.

Given that the area of hexagon is 5 square units, determine the area of hexagon

where is a constant.

The transformation represented by transforms triangle to triangle .

The area of triangle is three times the area of triangle .

Determine the possible values of .

(i)

The matrix P represents a geometrical transformation

(a) Describe fully as a single geometrical transformation.

[2]

The transformation , represented by the 2 2 matrix Q, is a rotation through 240° anticlockwise about the origin followed by an enlargement about (0, 0) with scale factor 6

(b) Determine the matrix Q, giving each entry in exact numerical form.

[2]

Given that followed by is the transformation , which is represented by the matrix R

(c) determine the matrix R

[2]

The transformation is represented by the matrix

Show that there is a real number for which maps the point onto the point , giving the exact value of

Determine the matrix

Describe fully the single geometrical transformation represented by the matrix

Hence determine the smallest positive integer value of for which

The matrix B represents a stretch scale factor 4 parallel to the -axis.

Write down the matrix B

The transformation represented by matrix A followed by the transformation represented by matrix B is represented by the matrix C

Determine the matrix C

The parallelogram is transformed onto the parallelogram by the matrix C

Given that the area of parallelogram is 20 square units, determine the area of parallelogram

The matrix represents the transformation

Give a full description of as a single geometrical transformation.

The transformation , represented by the 2×2 matrix , is a reflection in the line

Write down the matrix

The transformation followed by the transformation is represented by the matrix

Determine the matrix

The transformation is represented by the matrix

The transformation maps a triangle to a triangle

The transformation maps the triangle back to the original triangle

Determine the matrix that represents

The elements of each matrix should be expressed in exact numerical form.

(a) Write down the matrix that represents a rotation of anticlockwise about the origin.

[1]

(b) Write down the matrix that represents a stretch parallel to the -axis with scale factor 5.

[1]

The transformation is a rotation of anticlockwise about the origin followed by a stretch parallel to the -axis with scale factor 5.

(c) Determine the matrix that represents .

[2]

(a) Find det , giving your answer in simplest form in terms of .

[2]

A closed shape is transformed to a closed shape by the transformation represented by the matrix .

Given that the area of is 2 square units and that the area of is square units,

(b) determine the possible values of .

[3]

The triangle has vertices , and .

The triangle is the image of under the transformation represented by the matrix

Find the coordinates of the vertices of .

Describe fully the transformation represented by .

The matrix Q represents a reflection in the -axis and the matrix R represents a rotation through anticlockwise about the origin.

Write down the matrix Q and the matrix R.

Find the matrix RQ.

Give a full geometrical description of the single transformation represented by the answer to part (d).

The matrix A is defined by

The transformation represented by A maps triangle onto triangle .

Given that the area of triangle is 23 cm²,

determine the area of triangle .

The point has coordinates where is a constant. The transformation represented by A maps onto the point with coordinates

Determine the value of .

Given that

describe fully the single geometrical transformation represented by matrix B.

The transformation represented by matrix A followed by the transformation represented by matrix C is equivalent to the transformation represented by matrix B.

Determine C.

(a) Describe fully the single transformation represented by the matrix .

The matrix represents a rotation of clockwise about the origin.

[2]

(b) Write down the matrix , giving each element of the matrix in exact form.

[1]

The transformation represented by matrix followed by the transformation represented by matrix is represented by the matrix .

(c) Determine .

[2]

The trapezium has vertices at the points , , and , where is a positive constant. Trapezium is transformed onto the trapezium by the matrix

Given that the area of trapezium is 510 square units, calculate the exact value of .