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

The matrix B is defined by

$\mathbf{B}=\left(\begin{array}{cc}p& 0\\ 0& q\end{array}\right)$

where $p$ and $q$ are integers.

State the value of $p$ and the value of $q$ when B represents

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

[1]

(b) a reflection in the $y$-axis.

[1]

Prove by induction that for $n\in {\mathrm{\mathbb{Z}}}^{+}$

${\left(\begin{array}{cc}1& r\\ 0& 2\end{array}\right)}^n=\left(\begin{array}{cc}1& (2^n-1)r\\ 0& 2^n\end{array}\right)$

where $r$ is a constant.

$\mathbf{M}=\left(\begin{array}{cc}4& 0\\ 0& 5\end{array}\right)$ $\mathbf{N}={\left(\begin{array}{cc}1& -2\\ 0& 2\end{array}\right)}^4$

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 $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ where $a$, $b$, $c$ and $d$ are integers.

[1]

(ii) Determine B

[2]

Hexagon $S$ is transformed onto hexagon $S\text{'}$ by matrix B

Given that the area of $S\text{'}$ is 720 square units, determine the area of $S$

$\mathbf{A}=\left(\begin{array}{cc}1& 0\\ 0& 3\end{array}\right)$

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 $H$ is transformed onto the hexagon $H\text{'}$ by the matrix C.

Given that the area of hexagon $H$ is 5 square units, determine the area of hexagon $H\text{'}$

$\mathbf{A}=\left(\begin{array}{cc}-3& 8\\ -3& k\end{array}\right)$ where $k$ is a constant.

The transformation represented by $\mathbf{A}$ transforms triangle $T$ to triangle $T\text{'}$.

The area of triangle $T\text{'}$ is three times the area of triangle $T$.

Determine the possible values of $k$.

(i)

$\mathbf{P}=\left(\begin{array}{cc}0& -1\\ -1& 0\end{array}\right)$

The matrix P represents a geometrical transformation $U$

(a) Describe $U$ fully as a single geometrical transformation.

[2]

The transformation $V$, represented by the 2 $\times$ 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 $U$ followed by $V$ is the transformation $T$, which is represented by the matrix R

(c) determine the matrix R

[2]

The transformation $W$ is represented by the matrix

$\left(\begin{array}{cc}-2& 2\sqrt{3}\\ 2\sqrt{3}& 2\end{array}\right)$

Show that there is a real number $\lambda$ for which $W$ maps the point $(\lambda ,1)$ onto the point $(4\lambda ,4)$, giving the exact value of $\lambda$

$\mathbf{A}=\left(\begin{array}{cc}-\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ \frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right)$

Determine the matrix ${\mathbf{A}}^2$

Describe fully the single geometrical transformation represented by the matrix ${\mathbf{A}}^2$

Hence determine the smallest positive integer value of $n$ for which ${\mathbf{A}}^n=\mathbf{I}$

The matrix B represents a stretch scale factor 4 parallel to the $x$-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 $P$ is transformed onto the parallelogram $P\text{'}$ by the matrix C

Given that the area of parallelogram $P\text{'}$ is 20 square units, determine the area of parallelogram $P$

$\mathbf{P}=\left(\begin{array}{cc}\frac{1}{2}& -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right)$

The matrix $\mathbf{P}$ represents the transformation $U$

Give a full description of $U$ as a single geometrical transformation.

The transformation $V$ , represented by the 2×2 matrix $\mathbf{Q}$ , is a reflection in the line $y=-x$

Write down the matrix $\mathbf{Q}$

The transformation $U$ followed by the transformation $V$ is represented by the matrix $\mathbf{R}$

Determine the matrix $\mathbf{R}$

The transformation $W$ is represented by the matrix $3\mathbf{R}$

The transformation $W$ maps a triangle $T$ to a triangle $T\text{'}$

The transformation $W\text{'}$ maps the triangle $T\text{'}$ back to the original triangle $T$

Determine the matrix that represents $W\text{'}$

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

(a) Write down the $2\times 2$ matrix that represents a rotation of $210°$ anticlockwise about the origin.

[1]

(b) Write down the $2\times 2$ matrix that represents a stretch parallel to the $y$-axis with scale factor 5.

[1]

The transformation $T$ is a rotation of $210°$ anticlockwise about the origin followed by a stretch parallel to the $y$-axis with scale factor 5.

(c) Determine the $2\times 2$ matrix that represents $T$.

[2]

$\mathit{M}=\left(\begin{array}{cc}k& k+3\\ -5& 1-k\end{array}\right)\text{where }k\text{ is a constant}$

(a) Find det $\mathit{M}$, giving your answer in simplest form in terms of $k$.

[2]

A closed shape $R$ is transformed to a closed shape $R\text{'}$ by the transformation represented by the matrix $\mathit{M}$.

Given that the area of $R$ is 2 square units and that the area of $R\text{'}$ is $16k$ square units,

(b) determine the possible values of $k$.

[3]

The triangle $T$ has vertices $A(2,1)$, $B(2,3)$ and $C(0,1)$.

The triangle $T\text{'}$ is the image of $T$ under the transformation represented by the matrix

$\mathbf{P}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$

Find the coordinates of the vertices of $T\text{'}$.

Describe fully the transformation represented by $\mathit{P}$.

The $2\times 2$ matrix Q represents a reflection in the $x$-axis and the $2\times 2$ matrix R represents a rotation through $90°$ 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

$\mathbf{A}=\left(\begin{array}{cc}4& -5\\ -3& 2\end{array}\right)$

The transformation represented by A maps triangle $T$onto triangle $T\text{'}$.

Given that the area of triangle $T$ is 23 cm²,

determine the area of triangle $T\text{'}$.

The point $P$ has coordinates $(3p+2,2p-1)$where $p$ is a constant. The transformation represented by A maps $P$ onto the point $P\text{'}$ with coordinates $(17,-18)$

Determine the value of $p$.

Given that

$\mathbf{B}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$

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.

$\mathbf{A}=\left(\begin{array}{cc}1& 0\\ 0& 3\end{array}\right)$

(a) Describe fully the single transformation represented by the matrix $\mathbf{A}$.

The matrix $\mathbf{B}$ represents a rotation of $45°$ clockwise about the origin.

[2]

(b) Write down the matrix $\mathbf{B}$ , giving each element of the matrix in exact form.

[1]

The transformation represented by matrix $\mathbf{A}$ followed by the transformation represented by matrix $\mathbf{B}$ is represented by the matrix $\mathbf{C}$.

(c) Determine $\mathbf{C}$.

[2]

The trapezium $T$ has vertices at the points $(-2, 0)$, $(-2, k)$, $(5, 8)$ and $(5, 0)$, where $k$ is a positive constant. Trapezium $T$ is transformed onto the trapezium $T\text{'}$ by the matrix

$\left(\begin{array}{cc}5& 1\\ -2& 3\end{array}\right)$

Given that the area of trapezium $T\text{'}$ is 510 square units, calculate the exact value of $k$.