A LevelFurther Maths: Core PureEdexcelTopic Questions2. Matrices2.2 Transformations using Matrices

Transformations using Matrices (Edexcel A Level Further Maths: Core Pure)

Topic Questions

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1a
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bold M equals open parentheses table row 1 cell negative square root of 3 end cell row cell square root of 3 end cell 1 end table close parentheses

a)
Show that bold M is non-singular.

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1b
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The hexagon R is transformed to the hexagon S by the transformation represented by the matrix bold M.

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

(b)
find the area of hexagon S.

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1c
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The matrix bold M represents an enlargement, with centre (0, 0) and scale factor k, where k greater than 0, followed by a rotation anti clockwise through an angle theta about (0, 0).

(c)
Find the value of k.

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1d
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(d)
Find the value of theta.

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2a
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bold A equals open parentheses table row 2 a row cell a minus 4 end cell b end table close parentheses

where a and b spaceare non-zero constants.

Given that the matrix bold A is self-inverse,

(a)
determine the value of b and the possible values for a.

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2b
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The matrix bold A represents a linear transformation M.

Using the smaller value of a from part (a),

(b) show that the invariant points of the linear transformation M form a line, stating the equation of this line.

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3a
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bold P equals open parentheses table row p cell 2 p end cell row cell negative 1 end cell cell 3 p end cell end table close parentheses

wherespace p is a positive constant.

The matrix bold P represents a linear transformation U.

The triangle T has vertices at the points with coordinates (1, 2), (3, 2) and (2, 5).

The area of the image of T under the linear transformation U is 15

(a)
Determine the value ofspace p.

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3b
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The transformation V consists of a stretch scale factor 3 parallel to the x-axis with the y-axis invariant followed by a stretch scale factor –2 parallel to the y-axis with the x-axis invariant. The transformation V is represented by the matrix bold Q.

(b)
Write down the matrix bold Q.

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3c
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Given that U followed by V is the transformation W, which is represented by the matrix bold R,

(c)
find the matrix bold R.

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4a
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bold M equals open parentheses table row 4 cell negative 5 end cell row 2 cell negative 7 end cell end table close parentheses

(a)
Show that the matrix bold M is non-singular.

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4b
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The transformation T of the plane is represented by the matrix bold M.

The triangle R is transformed to the triangle S by the transformation T.

Given that the area of S is 63 square units,

(b)
find the area of R.

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4c
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(c)
Show that the line y equals 2 x is invariant under the transformation T.

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