A LevelMathsEdexcelPureExam QuestionsAlgebra & FunctionsRational Expressions

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 

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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 

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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 

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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

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5a2 marks

Simplify

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

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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

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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.

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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

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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.

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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.

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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

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10a2 marks

Simplify 

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

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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

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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

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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 

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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 

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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.

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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.

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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.

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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.

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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.

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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.

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7a2 marks

Simplify 

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

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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

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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

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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

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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

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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

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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.

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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.

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3a3 marks

Simplify 

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

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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

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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.

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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.

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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

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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

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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

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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.

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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.

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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.

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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.

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