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

Exam code: 9MA0

2 hours30 questions
Easy
Medium
Hard
13 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?

22 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 plus 3 right parenthesis end fraction 

How did you do?

33 marks

(i) 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?

42 marks

Which one, out of the following expressions, is an improper algebraic fraction? 

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?

5a2 marks

Simplify

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

How did you do?

5b2 marks

Simplify 

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

How did you do?

6a3 marks

Given 

fraction numerator x squared plus 8 x minus 4 over denominator x minus 3 end fraction identical to 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?

6b3 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 how you know that the degree of the quotient must be 1

How did you do?

73 marks

Given

fraction numerator x squared minus 8 x minus 20 over denominator x minus 2 end fraction identical to x minus 6 minus fraction numerator 32 over denominator x minus 2 end fraction

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

How did you do?

82 marks

Use polynomial division to 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?

93 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?

10a2 marks

Simplify 

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

How did you do?

10b2 marks

Simplify

 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?

10c3 marks

Simplify

 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?

114 marks

Simplify

(i) fraction numerator x plus 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?

126 marks

Simplify

(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?

12 marks

The function straight f is defined by

straight f open parentheses x close parentheses equals fraction numerator 8 x plus 5 over denominator 2 x plus 3 end fraction space space space space space space space space space space space space space x greater than negative 3 over 2

Show that

straight f open parentheses x close parentheses equals A plus fraction numerator B over denominator 2 x plus 3 end fraction

where A and B are constants to be found.

How did you do?

23 marks

Given that

fraction numerator x cubed plus 3 x squared minus 2 x plus 4 over denominator x plus 1 end fraction identical to A x squared plus B x plus C plus fraction numerator D over denominator x plus 1 end fraction space space space space space space space space space space space space space x element of straight real numbers space space space space x not equal to negative 1

find the values of the constants A, B, C and D.

How did you do?

33 marks

Given that

fraction numerator 2 x cubed minus 4 x plus 3 over denominator x minus 2 end fraction identical to P x squared plus Q x plus R plus fraction numerator S over denominator x minus 2 end fraction space space space space space space space space space space space space space x element of straight real numbers space space space space x not equal to 2

find the values of the constants P, Q, R and S.

How did you do?

42 marks

Given

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 in the form a x squared plus b x plus c, where a, b and c are constants to be found.

How did you do?

53 marks

One of the three expressions below is an improper algebraic fraction.

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 the improper algebraic fraction 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?

63 marks

One of the three expressions below is an improper algebraic fraction.

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 the improper algebraic fraction 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?

7a2 marks

Simplify 

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

How did you do?

7b2 marks

Solve the equation

fraction numerator x cubed minus 2 x squared minus 8 x over denominator x minus 4 end fraction equals x squared plus 10 x plus 16

How did you do?

8a4 marks

Simplify

 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?

8b2 marks

Simplify

 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?

8c3 marks

Simplify

 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?

1a3 marks

Given that

fraction numerator x squared plus 8 x minus 3 over denominator x plus 2 end fraction identical to A x plus B plus fraction numerator C over denominator x plus 2 end fraction space space space space space space space space space space space space space x element of straight real numbers space space space space x not equal to negative 2

find the values of the constants A, B and C

How did you do?

1b
Sme Calculator4 marks

Hence, using algebraic integration, find the exact value of

integral subscript 0 superscript 6 fraction numerator x squared plus 8 x minus 3 over denominator x plus 2 end fraction space straight d x

giving your answer in the form a plus b ln 2 where a and b are integers to be found.

How did you do?

23 marks

Given that

fraction numerator x cubed plus 5 x squared minus 4 over denominator x minus 5 end fraction identical to A x squared plus B x plus C plus fraction numerator D over denominator x minus 5 end fraction space space space space space space space space space space space space space x element of straight real numbers space space space space x not equal to 5

find the values of the constants A, B, C and D.

How did you do?

3a3 marks

Simplify 

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

How did you do?

3b3 marks

Solve the equation

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

How did you do?

43 marks

Given that

fraction numerator x squared plus a x minus 5 over denominator x plus 1 end fraction identical to x plus 3 plus fraction numerator d over denominator x plus 1 end fraction space space space space space space space space space space space space space x element of straight real numbers space space space space x not equal to negative 1

find the values of the constants a and d.

How did you do?

54 marks

When 

x cubed plus a x squared plus 4 x minus 1

is divided by open parentheses x plus 2 close parentheses

  • the quotient is open parentheses x squared minus 4 x plus 12 close parentheses

  • the remainder is b

Find the values of a and b.

How did you do?

6a4 marks

Simplify 

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?

6b3 marks

Simplify 

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?

6c5 marks

Simplify

 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?

73 marks

One of the three expressions below is an improper algebraic fraction.

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 the improper algebraic fraction and write it in the form 

  A x squared plus B x plus C plus fraction numerator D over denominator x minus 4 end fraction

where A comma space B, C and D are integers to be found.

How did you do?

85 marks

Given that

fraction numerator 3 x to the power of 4 plus 2 x cubed minus 5 x plus 2 over denominator x minus 3 end fraction identical to A x cubed plus B x squared plus C x plus D plus fraction numerator E over denominator x minus 3 end fraction space space space space space space space space space space space space space x element of straight real numbers space space space space x not equal to 3

find the values of the constants A, B, C, D and E.

How did you do?

95 marks

Given

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

find the values of the constants a and b.

How did you do?

104 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 open parentheses x plus 4 close parentheses

  • 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

  • the remainder is c

Find the values of a comma space b and c.

How did you do?