International A Level (IAL)MathsEdexcelPure 3Exam QuestionsAlgebra & FunctionsRational Expressions

Rational Expressions (Edexcel International A Level (IAL) Maths: Pure 3): Exam Questions

Exam code: YMA01

4 hours43 questions
Easy
Medium
Hard
Very Hard
1
Sme Calculator3 marks

Simplify

(i) x squared over x 

(ii) fraction numerator x open parentheses x minus 1 close parentheses over denominator x end fraction 

(iii) fraction numerator 6 x plus 2 over denominator 2 end fraction 

How did you do?

2
Sme Calculator3 marks

(i) Factorise x squared plus 7 x plus 12

(ii) Hence simplify fraction numerator x squared plus 7 x plus 12 over denominator 2 left parenthesis x space plus space 3 right parenthesis end fraction   

How did you do?

3a
Sme Calculator2 marks

Simplify fully fraction numerator 2 x squared plus 10 x over denominator 2 left parenthesis x space plus space 5 right parenthesis end fraction

How did you do?

3b
Sme Calculator3 marks

Simplify fully fraction numerator 3 x squared over denominator x space plus space 4 end fraction cross times fraction numerator x squared plus 5 x plus 4 over denominator x end fraction

How did you do?

4
Sme Calculator4 marks

(i) Fully factorise x cubed minus 9 x squared plus 20 x  

(ii) Hence simplify fraction numerator x cubed minus 9 x squared plus 20 x over denominator x squared minus 5 x end fraction 

How did you do?

5a
Sme Calculator2 marks

The function  is given by straight f left parenthesis x right parenthesis equals 2 x cubed plus 7 x squared minus 4 x.

Show that straight f left parenthesis x right parenthesis equals x left parenthesis 2 x minus 1 right parenthesis left parenthesis x plus 4 right parenthesis.

How did you do?

5b
Sme Calculator2 marks

Hence, or otherwise, write down the real solutions to the equation

fraction numerator straight f left parenthesis x right parenthesis over denominator x plus 1 end fraction equals 0.

How did you do?

6a
Sme Calculator2 marks

The function straight f left parenthesis x right parenthesis is given by

straight f left parenthesis x right parenthesis equals x cubed minus 4 x squared minus 7 x plus 10

Work out straight f left parenthesis 1 right parenthesis and hence write down a factor of straight f left parenthesis x right parenthesis.

How did you do?

6b
Sme Calculator2 marks

Work out straight f left parenthesis x right parenthesis divided by left parenthesis x plus 2 right parenthesis.

How did you do?

6c
Sme Calculator3 marks

Write straight f left parenthesis x right parenthesis space in the form left parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis left parenthesis x plus c right parenthesis where a comma space b space and space c are integers to be found.

How did you do?

7
Sme Calculator2 marks

Which one of the following algebraic fractions is improper?  Explain your answer.

fraction numerator x squared plus 5 x minus 1 over denominator x cubed minus 2 end fraction

fraction numerator x squared plus 3 x plus 2 over denominator x squared minus 3 x plus 2 end fraction

fraction numerator x space plus space 1 over denominator left parenthesis x minus 1 right parenthesis squared end fraction

How did you do?

8
Sme Calculator3 marks

Find the remainder when x cubed plus 2 x squared minus 5 x plus 8 is divided by left parenthesis x minus 3 right parenthesis.

How did you do?

9a
Sme Calculator3 marks

Given that left parenthesis x squared minus 8 x minus 20 right parenthesis divided by left parenthesis x minus 2 right parenthesis equals A x plus B plus fraction numerator C over denominator x minus 2 end fraction

where A comma space B space and space C are integer constants.

In terms of A comma space B and/or C as appropriate

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

How did you do?

9b
Sme Calculator4 marks

Find the values of A comma space B space and space C.

How did you do?

10
Sme Calculator4 marks

The function straight f left parenthesis x right parenthesis is given by

straight f left parenthesis x right parenthesis equals x squared plus a x plus b

where a space and space b are integer constants.

It is also given that  space straight f left parenthesis 3 right parenthesis equals straight f left parenthesis negative 8 right parenthesis equals 0.

Find the values of a space and space b.

How did you do?

1
Sme Calculator4 marks

Simplify

(i) fraction numerator x over denominator 2 x end fraction  

(ii) fraction numerator x plus 1 over denominator x left parenthesis x plus 1 right parenthesis end fraction  

(iii) fraction numerator 6 x plus 12 over denominator x squared plus 2 x end fraction 

How did you do?

2a
Sme Calculator3 marks

Simplify fully fraction numerator 2 x squared plus 6 x over denominator x cubed plus 3 x squared end fraction

How did you do?

2b
Sme Calculator3 marks

Simplify fully fraction numerator x plus 4 over denominator x cubed end fraction cross times fraction numerator x squared plus 2 x over denominator x plus 4 end fraction

How did you do?

2c
Sme Calculator3 marks

Simplify fully fraction numerator x squared plus 4 x over denominator 3 x plus 6 end fraction divided by fraction numerator 2 x plus 8 over denominator x plus 2 end fraction

How did you do?

3a
Sme Calculator2 marks

The function straight f left parenthesis x right parenthesis is given by

straight f left parenthesis x right parenthesis equals 3 x cubed minus 5 x squared minus 4 x plus 4

Show that straight f open parentheses 2 over 3 close parentheses space equals 0.

How did you do?

3b
Sme Calculator1 mark

Hence write down a factor of straight f left parenthesis x right parenthesis.

How did you do?

3c
Sme Calculator3 marks

Fully factorise straight f left parenthesis x right parenthesis.

How did you do?

3d
Sme Calculator2 marks

Write down the solutions to the equation straight f left parenthesis x right parenthesis space equals 0.

How did you do?

4a
Sme Calculator2 marks

Show that left parenthesis 2 x minus 3 right parenthesis spaceis a factor of 2 x cubed minus 13 x squared plus 23 x minus 12.

How did you do?

4b
Sme Calculator2 marks

Fully factorise 2 x cubed minus 13 x squared plus 23 x minus 12.

How did you do?

4c
Sme Calculator2 marks

Find all the real solutions to 2 x cubed minus 13 x squared plus 23 x minus 12 equals 0.

How did you do?

5
Sme Calculator2 marks

Given that left parenthesis 2 x minus 1 right parenthesis space is a factor of 2 x cubed plus x squared minus 25 x plus a find the value of a.

How did you do?

6a
Sme Calculator3 marks

Work out left parenthesis x cubed plus 3 x squared minus 2 x plus 4 right parenthesis divided by left parenthesis x plus 1 right parenthesis.

How did you do?

6b
Sme Calculator3 marks

Work out  fraction numerator 2 x cubed minus 4 x plus 3 over denominator x space minus space 2 end fraction.

How did you do?

7a
Sme Calculator3 marks

Given  left parenthesis x squared plus 8 x minus 4 right parenthesis divided by left parenthesis x minus 3 right parenthesis equals x plus 11 plus fraction numerator 29 over denominator x minus 3 end fraction

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

How did you do?

7b
Sme Calculator3 marks

(i) Write down the degree of x squared plus 8 x minus 4. 

(ii) Write down the degree of x minus 3. 

(iii) Explain why you would expect the quotient to be of degree 1 in this case. 

How did you do?

8
Sme Calculator3 marks

One of the three algebraic fractions below is improper (‘top-heavy’).

fraction numerator x plus 2 over denominator x squared plus 2 end fraction

fraction numerator x over denominator x plus 2 end fraction

fraction numerator space 1 over denominator x plus 2 end fraction

Identify which fraction is improper and write it in the form  A plus fraction numerator B over denominator x plus 2 end fraction, where A and B are integers to be found.

How did you do?

9a
Sme Calculator2 marks

Simplify fully fraction numerator x cubed minus 2 x squared minus 8 x over denominator x space minus space 4 end fraction

How did you do?

9b
Sme Calculator3 marks

Hence solve the equation fraction numerator x cubed minus 2 x squared minus 8 x over denominator x space minus space 4 end fraction equals x squared plus 10 x plus 16.

How did you do?

10
Sme Calculator2 marks

It is given that

fraction numerator straight f left parenthesis x right parenthesis over denominator x plus 2 end fraction equals 3 x plus 4 minus fraction numerator 2 over denominator x plus 2 end fraction

Find straight f left parenthesis x right parenthesis.

How did you do?

11
Sme Calculator2 marks

The result of dividing x squared plus a x minus 5 by left parenthesis x plus 1 right parenthesis is x plus 3 plus space fraction numerator d over denominator x plus 1 end fraction.

Find the values of a and d.

How did you do?

1
Sme Calculator6 marks

Simplify fully

(i) fraction numerator x space plus space 3 over denominator x squared plus 3 x end fraction  

(ii) fraction numerator x cubed plus x over denominator x to the power of 4 end fraction  

(iii) fraction numerator x cubed plus 3 x squared minus 4 x over denominator x to the power of 4 minus x cubed end fraction 

How did you do?

2a
Sme Calculator3 marks

Simplify fully fraction numerator x squared plus x minus 2 over denominator x cubed plus 4 x squared minus 4 x minus 1 end fraction

How did you do?

2b
Sme Calculator3 marks

Simplify fully fraction numerator 3 x plus 9 over denominator x plus 2 end fraction cross times fraction numerator x squared plus 6 x plus 8 over denominator x plus 3 end fraction

How did you do?

2c
Sme Calculator3 marks

Simplify fully fraction numerator x squared plus 8 x minus 9 over denominator x squared plus 7 x plus 12 end fraction divided by fraction numerator x squared plus 11 x plus 18 over denominator 2 x squared plus 7 x minus 4 end fraction

How did you do?

3a
Sme Calculator2 marks

The function straight f left parenthesis x right parenthesis is given by

straight f left parenthesis x right parenthesis equals 4 x cubed minus 7 x squared minus 21 x plus 18

Show that left parenthesis 4 x minus 3 right parenthesis is a factor of straight f left parenthesis x right parenthesis.

How did you do?

3b
Sme Calculator4 marks

Hence, or otherwise, fully factorise straight f left parenthesis x right parenthesis.

How did you do?

3c
Sme Calculator2 marks

Write down the roots of straight f left parenthesis x right parenthesis.

How did you do?

4
Sme Calculator5 marks

Show that left parenthesis 5 x minus 2 right parenthesis is a factor of 25 x cubed plus 55 x squared minus 56 x plus 12.

Hence find all the real solutions to the equation 25 x cubed plus 55 x squared minus 56 x plus 12 equals 0.

How did you do?

5a
Sme Calculator2 marks

Given that left parenthesis 4 x minus 5 right parenthesis is a factor of 4 x cubed minus 9 x squared plus a x plus 30 find the value of a.

How did you do?

5b
Sme Calculator2 marks

Hence, or otherwise, fully factorise 4 x cubed minus 9 x squared plus a x plus 30.

How did you do?

6a
Sme Calculator2 marks

Work out left parenthesis x cubed plus 5 x squared minus 4 right parenthesis divided by left parenthesis x minus 5 right parenthesis.

How did you do?

6b
Sme Calculator2 marks

Work out  fraction numerator 3 x cubed plus 2 x minus 5 over denominator x squared plus 1 end fraction.

How did you do?

7
Sme Calculator4 marks

(i) Find the remainder when x cubed minus 2 x squared plus 4 x minus 3 is divided by x minus 2.

(ii) Find the value of straight f left parenthesis 2 right parenthesis when straight f left parenthesis x right parenthesis equals x cubed minus 2 x squared plus 4 x minus 3.

(iii) Comment on your answers to parts (i) and (ii).  

How did you do?

8
Sme Calculator3 marks

One of the three algebraic fractions below is improper (‘top-heavy’):

fraction numerator x squared minus 5 x plus 1 over denominator x plus 1 end fraction

fraction numerator x plus 2 over denominator left parenthesis x plus 1 right parenthesis squared end fraction

fraction numerator space x squared minus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis cubed end fraction

Identify which fraction is improper and write it in the form  A x plus B plus fraction numerator C over denominator x plus 1 end fraction, where A comma space B and C are integers to be found.

How did you do?

9a
Sme Calculator3 marks

Simplify fraction numerator x cubed minus 7 x squared plus 14 x minus 8 over denominator x space minus space 1 end fraction

How did you do?

9b
Sme Calculator3 marks

Hence solve fraction numerator x cubed minus 7 x squared plus 14 x minus 8 over denominator x space minus space 1 end fraction space equals 2 x squared minus 5 x plus 2.

How did you do?

10a
Sme Calculator1 mark

It is given that

fraction numerator straight f left parenthesis x right parenthesis over denominator straight g left parenthesis x right parenthesis end fraction space equals 2 x plus 3 minus fraction numerator 4 over denominator x space plus space 1 end fraction

Why would assuming that straight g left parenthesis x right parenthesis equals x plus 1 be a logical first step in attempting to determine the precise forms of straight f left parenthesis x right parenthesis and straight g left parenthesis x right parenthesis?

How did you do?

10b
Sme Calculator2 marks

By first making the assumption from part (a), find straight f left parenthesis x right parenthesis.

How did you do?

10c
Sme Calculator2 marks

Explain, with an example, why the forms of straight f left parenthesis x right parenthesis and straight g open parentheses x close parentheses determined in parts (a) and (b) are not the only possible forms for those functions.

How did you do?

11
Sme Calculator3 marks

When x cubed plus a x squared plus 4 x minus 1 is divided by x plus 2 the quotient is x squared minus 4 x plus 12 and the remainder is b.

Find the values of a and b.

How did you do?

1
Sme Calculator8 marks

Simplify fully

(i) fraction numerator x squared plus 5 x over denominator x cubed plus 3 x squared minus 10 x end fraction  

(ii) fraction numerator x squared minus 4 over denominator x to the power of 4 minus 16 end fraction  

(iii) fraction numerator left parenthesis x plus 2 right parenthesis squared plus left parenthesis x plus 2 right parenthesis left parenthesis x plus 4 right parenthesis over denominator x squared plus 5 x plus 6 end fraction 

How did you do?

2a
Sme Calculator3 marks

Simplify fully fraction numerator x squared minus 5 x plus 4 over denominator x cubed minus 2 x squared minus 11 x plus 12 end fraction

How did you do?

2b
Sme Calculator3 marks

Simplify fully fraction numerator 2 x squared minus x minus 6 over denominator x squared minus 4 end fraction cross times fraction numerator 5 x space plus space 10 over denominator 4 x squared minus 9 end fraction

How did you do?

2c
Sme Calculator3 marks

Simplify fully fraction numerator 45 x cubed plus 90 x squared minus 5 x minus 10 over denominator x squared minus 25 end fraction divided by fraction numerator 3 x squared plus 7 x plus 2 over denominator 2 x squared plus 9 x minus 5 end fraction

How did you do?

3
Sme Calculator4 marks

Given that left parenthesis 2 x minus 5 right parenthesis is a factor of the function

straight f left parenthesis x right parenthesis equals 2 x cubed plus k x squared minus 11 x minus 60

find the value of k and fully factorise straight f left parenthesis x right parenthesis.

How did you do?

4
Sme Calculator5 marks

Show that left parenthesis 9 x squared minus 4 right parenthesis is a factor of 9 x to the power of 4 minus 40 x squared plus 16 and hence find all the real solutions to the equation 9 x to the power of 4 minus 40 x squared plus 16 equals 0.

How did you do?

5a
Sme Calculator2 marks

Show that left parenthesis a x minus 2 right parenthesis is a factor of 3 a x squared plus left parenthesis a minus 6 right parenthesis x minus 2.

How did you do?

5b
Sme Calculator3 marks

Given that x equals fraction numerator negative space 1 over denominator a space minus space 4 end fraction is a root of  3 a x squared plus left parenthesis a minus 6 right parenthesis x minus 2, find the value of a.

How did you do?

6a
Sme Calculator3 marks

Work out left parenthesis 3 x to the power of 4 plus 2 x cubed minus 5 x plus 2 right parenthesis divided by left parenthesis x minus 3 right parenthesis.

How did you do?

6b
Sme Calculator3 marks

Work out fraction numerator x to the power of 5 over denominator 3 x squared minus 2 end fraction

How did you do?

7a
Sme Calculator2 marks

For a polynomial straight f left parenthesis x right parenthesis, the Remainder Theorem states that

When straight f left parenthesis x right parenthesis is divided by left parenthesis a x minus b right parenthesis space the remainder is straight f open parentheses b over a close parentheses.

Use the Remainder Theorem to find the remainder when 8 x cubed plus 6 x squared minus x minus 2 is divided by left parenthesis 2 x plus 1 right parenthesis.

How did you do?

7b
Sme Calculator2 marks

Work out the remainder when 6 x squared minus x minus 2 is divided by left parenthesis 2 x plus 1 right parenthesis.

How did you do?

8
Sme Calculator3 marks

One of the three algebraic fractions below is improper (‘top-heavy’):

fraction numerator x plus 4 over denominator left parenthesis x minus 4 right parenthesis squared end fraction

fraction numerator 5 x minus 25 over denominator left parenthesis x minus 4 right parenthesis left parenthesis x plus 4 right parenthesis end fraction

fraction numerator space x cubed minus 2 x squared plus 6 x minus 1 over denominator left parenthesis x minus 4 right parenthesis end fraction

Identify which fraction is improper and rewrite it as a quotient and a remainder term.

How did you do?

9
Sme Calculator5 marks

Solve fraction numerator 8 x to the power of 4 plus 25 x cubed plus 3 x squared minus 32 x minus 4 over denominator x squared plus x minus 2 end fraction equals 2 x squared plus 16 x plus 3

How did you do?

10
Sme Calculator5 marks

It is given that

fraction numerator left parenthesis x plus a right parenthesis cubed over denominator x space plus space b end fraction equals left parenthesis x squared plus 7 x plus 13 right parenthesis plus fraction numerator 1 over denominator x space plus space b end fraction

where a and b are integers.

Find the values of a space and space b.

How did you do?

11
Sme Calculator4 marks

When 2 x cubed plus left parenthesis a plus b right parenthesis x squared plus left parenthesis a minus b right parenthesis x minus 3 is divided by x plus 4 space space the quotient is 2 x squared plus left parenthesis 2 a plus 3 right parenthesis x plus left parenthesis 2 b minus 5 right parenthesis and the remainder is c.

Find the values of a comma space b and c.

How did you do?