International A Level (IAL)Further MathsEdexcelFurther Pure 1Exam QuestionsMatricesMatrix Algebra

Matrix Algebra (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

Exam code: YFM01

1 hour12 questions
1a
3 marks

Given that

bold A equals open parentheses table row k 3 row cell negative 1 end cell cell k plus 2 end cell end table close parentheses comma    text where  end text k text  is a constant end text

show that det left parenthesis bold A right parenthesis greater than 0 for all real values of k,

How did you do?

1b
2 marks

find bold A to the power of negative 1 end exponent in terms of k.

How did you do?

2
4 marks

(i) The matrix A is defined by

bold A equals open parentheses table row cell 3 k end cell cell 4 k minus 1 end cell row 2 6 end table close parentheses

where k is a constant.

(a) Determine the value of k for which A is singular.

Given that A is non-singular,

[2]

(b) determine bold A to the power of negative 1 end exponent in terms of k, giving your answer in simplest form.

[2]

How did you do?

3a
4 marks

Prove by induction that for n element of straight integer numbers to the power of plus

open parentheses table row 1 r row 0 2 end table close parentheses to the power of n equals open parentheses table row 1 cell left parenthesis 2 to the power of n minus 1 right parenthesis r end cell row 0 cell 2 to the power of n end cell end table close parentheses

where r is a constant.

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3b
3 marks

bold M equals open parentheses table row 4 0 row 0 5 end table close parentheses bold N equals open parentheses table row 1 cell negative 2 end cell row 0 2 end table close parentheses to the power of 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 open parentheses table row a b row c d end table close parentheses where a, b, c and d are integers.

[1]

(ii) Determine B

[2]

How did you do?

3c
Sme Calculator
2 marks

Hexagon S is transformed onto hexagon S apostrophe by matrix B

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

How did you do?

4a
3 marks

bold M equals open parentheses table row cell 2 k plus 1 end cell k row cell k plus 7 end cell cell k plus 4 end cell end table close parentheses where k is a constant

Show that M is non-singular for all real values of k.

How did you do?

4b
2 marks

Determine bold M to the power of negative 1 end exponent in terms of k.

How did you do?

5
8 marks

bold B equals open parentheses table row a cell negative 4 end cell row 2 3 end table close parentheses bold BC equals open parentheses table row 2 5 1 row 1 4 2 end table close parentheses where a is a constant

Determine, in terms of a, the matrix bold C

How did you do?

6a
2 marks

Given that

bold A equals open parentheses table row 2 cell negative 1 end cell 3 row cell negative 2 end cell 3 0 end table close parentheses and bold B equals open parentheses table row 1 k row 0 cell negative 3 end cell row cell 2 k end cell 2 end table close parentheses

where k is a non-zero constant,

determine the matrix AB

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6b
3 marks

determine the value of k for which det(AB) = 0

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7a
2 marks

bold M equals open parentheses table row k k row 3 5 end table close parentheses where k is a non-zero constant

Determine bold M to the power of negative 1 end exponent , giving your answer in simplest form in terms of k .

How did you do?

7b
2 marks

Hence, given that bold N to the power of negative 1 end exponent equals open parentheses table row k k row 4 cell negative 1 end cell end table close parentheses

determine left parenthesis bold MN right parenthesis to the power of negative 1 end exponent, giving your answer in simplest form in terms of k.

How did you do?

8a
1 mark

bold A equals open parentheses table row cell negative fraction numerator square root of 3 over denominator 2 end fraction end cell cell negative 1 half end cell row cell 1 half end cell cell negative fraction numerator square root of 3 over denominator 2 end fraction end cell end table close parentheses

Determine the matrix bold A squared

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8b
2 marks

Describe fully the single geometrical transformation represented by the matrix bold A squared

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8c
1 mark

Hence determine the smallest positive integer value of n for which bold A to the power of n equals bold I

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8d
1 mark

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

Write down the matrix B

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8e
2 marks

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

Determine the matrix C

How did you do?

8f
2 marks

The parallelogram P is transformed onto the parallelogram P apostrophe by the matrix C

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

How did you do?

9
5 marks

bold M equals open parentheses table row cell 3 x end cell 7 row cell 4 x plus 1 end cell cell 2 minus x end cell end table close parentheses

Find the range of values of x for which the determinant of the matrix bold M is positive.

How did you do?

10a
2 marks

A equals open parentheses table row 3 a row cell negative 2 end cell cell negative 2 end cell end table close parentheses

where a is a non-zero constant and a not equal to 3

Determine A to the power of negative 1 end exponent giving your answer in terms of a

How did you do?

10b
3 marks

Given that A plus A to the power of negative 1 end exponent equals I where I is the 2 cross times 2 identity matrix,

determine the value of a

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11a
4 marks

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

(a) Write down the 2 cross times 2 matrix that represents a rotation of 210 degree anticlockwise about the origin.

[1]

(b) Write down the 2 cross 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 degree anticlockwise about the origin followed by a stretch parallel to the y-axis with scale factor 5.

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

[2]

How did you do?

11b
5 marks

bold italic M equals open parentheses table row k cell k plus 3 end cell row cell negative 5 end cell cell 1 minus k end cell end table close parentheses    text where  end text k text  is a constant end text

(a) Find det bold italic M, giving your answer in simplest form in terms of k.

[2]

A closed shape R is transformed to a closed shape R apostrophe by the transformation represented by the matrix bold italic M.

Given that the area of R is 2 square units and that the area of R apostrophe is 16 k square units,

(b) determine the possible values of k.

[3]

How did you do?

12a
2 marks

The matrix M is defined by

bold M equals open parentheses table row cell k plus 5 end cell cell negative 2 end cell row cell negative 3 end cell k end table close parentheses

Determine the values of k for which M is singular.

How did you do?

12b
2 marks

Given that M is non-singular,

find bold M to the power of negative 1 end exponent in terms of k.

How did you do?