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A LevelMaths: PureOCRExam Questions2. Algebra & Functions2.6 Rational Expressions (A Level only)

Rational Expressions (A Level only) (OCR A Level Maths: Pure)

Exam Questions

4 hours43 questions
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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 

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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 

 

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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

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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

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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 

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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.

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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.

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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.

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6b
Sme Calculator2 marks

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

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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.

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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

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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.

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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.

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9b
Sme Calculator4 marks

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

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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.

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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 

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2a
Sme Calculator3 marks

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

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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

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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

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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.

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3b
Sme Calculator1 mark

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

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3c
Sme Calculator3 marks

Fully factorise straight f left parenthesis x right parenthesis.

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3d
Sme Calculator2 marks

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

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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.

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4b
Sme Calculator2 marks

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

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4c
Sme Calculator2 marks

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

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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.

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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.

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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.

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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.

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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. 

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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.

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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

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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.

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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.

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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.

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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 

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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

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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

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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

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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.

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3b
Sme Calculator4 marks

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

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3c
Sme Calculator2 marks

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

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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.

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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.

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5b
Sme Calculator2 marks

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

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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.

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6b
Sme Calculator2 marks

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

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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).

 

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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.

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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

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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.

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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?

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10b
Sme Calculator2 marks

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

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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.

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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.

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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 

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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

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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

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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

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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.

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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.

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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.

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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.

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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.

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6b
Sme Calculator3 marks

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

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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.

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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.

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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.

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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

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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.

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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.

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