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Further Graphs & Tangents (Cambridge (CIE) O Level Maths): Exam Questions

Exam code: 4024

5 hours32 questions

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Further Graphs & Tangents

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1
Sme Calculator4 marks

On the diagram, sketch the graph of y space equals space open parentheses x space plus space 1 close parentheses open parentheses x space minus space 3 close parentheses squared.
Label the values where the graph meets the x-axis and the y-axis.

cie-igcse-2020-oct-nov-p4-tz3-q7b

Ā 

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2
Sme Calculator2 marks

Sketch the graph of the function.

y equals 1 over x

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3
Sme Calculator3 marks

The diagrams A, B, C, D, E and F are six graphs of different functions.

cie-igcse-q21-graphs-diagrams

Complete the table to identify the correct graph for each function. One has been done for you.Ā 

Function

y space equals space x space plus space 1

y space equals space 1 space minus space x over 3

y space equals space 2 x squared

y space equals space minus 4 over x

Diagram

E

Ā 

Ā 

Ā 

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1a
Sme Calculator3 marks
q26-paper2-spec2025-cie-igcse-further-maths

straight f left parenthesis x right parenthesis space equals space x left parenthesis x space plus space 2 right parenthesis left parenthesis x space ā€“ space 3 right parenthesis space

On the diagram, sketch the graph ofĀ y space equals space straight f left parenthesis x right parenthesis space space space space space space for space ā€“ 3 space less-than or slanted equal to space x space less-than or slanted equal to space 4 . Show the values of the intersections with the axes.Ā 

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1b
Sme Calculator3 marks

Expand and simplify.

x left parenthesis x space plus space 2 right parenthesis left parenthesis x space ā€“ space 3 right parenthesis

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2a
Sme Calculator4 marks

The diagram shows the graph ofĀ y space equals space straight f open parentheses x close parentheses forĀ  negative 3 space less or equal than space x space less or equal than space 3.

cie-igcse-2020-oct-nov-p4-tz1-q5a

i) SolveĀ space straight f left parenthesis space x right parenthesis space equals space 14. space

space x space equals space................................................ [1]

ii) By drawing a suitable tangent, find an estimate of the gradient of the graph at the point left parenthesis negative 2 comma space 4 right parenthesis.

Ā 

[3]

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2b
Sme Calculator3 marks

By drawing a suitable straight line on the grid, solvespace space straight f left parenthesis space x right parenthesis space equals space 2 x space minus space 2 space forĀ negative 3 space less or equal than space x space less or equal than space 3.

space x space equals space................................................

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3a
Sme Calculator3 marks

The table shows some values for y equals 2 x cubed minus 4 x squared plus 3.

x

-1

-0.5

0

0.5

1

1.5

2

y

-3

1.75

Ā 

Ā 

Ā 

0.75

3

Complete the table.

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3b
Sme Calculator4 marks

Draw the graph of y equals 2 x cubed minus 4 x squared plus 3Ā forĀ negative 1 less or equal than x less or equal than 2.

cie-igcse-2020-march-p4-tz2-q2a

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3c
Sme Calculator4 marks

i) Use your graph to solve the equation 2 x cubed minus 4 x squared plus 3 equals 1.5.

x equalsĀ ........................ or x equalsĀ ........................ or x equalsĀ ........................ [3]

ii) The equation 2 x cubed minus 4 x squared plus 3 equals kĀ has only one solution for negative 1 less or equal than x less or equal than 2.
Write down a possible integer value of k.

[1]

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4a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals x squared over 4 minus 4 over x,Ā x not equal to 0 Ā 
Complete the table forĀ straight f left parenthesis x right parenthesis.

x

0.5

1

2

3

4

5

6

straight f left parenthesis x right parenthesis

-7.9

-3.8

Ā 

0.9

Ā 

5.5

8.3

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4b
Sme Calculator3 marks

The graph of y equals straight f left parenthesis x right parenthesis forĀ negative 6 less or equal than x less or equal than negative 0.5 is drawn on the grid.

cie-igcse-2018-oct-nov-p4-tz3-q4b

On the same grid, draw the graph ofĀ y equals straight f left parenthesis x right parenthesis forĀ 0.5 less or equal than x less or equal than 6.

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4c
Sme Calculator3 marks

By drawing a suitable tangent, estimate the gradient of the graph ofĀ y equals straight f left parenthesis x right parenthesis at the point (ā€“4, 5).

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4d
Sme Calculator1 mark

straight g left parenthesis x right parenthesis equals 9 over x,Ā  x not equal to 0 Ā 
Complete the table forĀ straight g left parenthesis x right parenthesis. Ā 

x

-4

-3

-2

-1

Ā 

1

2

3

4

straight g left parenthesis x right parenthesis

-2.3

Ā 

-4.5

-9

Ā 

9

4.5

Ā 

2.3

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4e
Sme Calculator4 marks

On the same grid, draw the graph ofĀ Ā y equals straight g left parenthesis x right parenthesis forĀ Ā negative 4 less or equal than x less or equal than negative 1Ā  andĀ Ā 1 less or equal than x less or equal than 4.

How did you do?

4f
Sme Calculator2 marks

i) Use your graphs to find the value ofĀ x whenĀ Ā straight f left parenthesis x right parenthesis equals straight g left parenthesis x right parenthesis.

Ā Ā 

xĀ = ................................................. [1]

ii) Write down an inequality to show the positive values ofĀ x for whichĀ Ā straight f left parenthesis x right parenthesis greater than straight g left parenthesis x right parenthesis.

Ā [1]

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5a
Sme Calculator4 marks

i) y equals 2 to the power of x Complete the table

x

0

1

2

3

4

y

Ā 

2

4

8

Ā 

Ā  [2]

ii) y equals 14 minus x squared Complete the table

x

0

1

2

3

4

y

Ā 

13

10

5

Ā 

Ā  [2]

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5b
Sme Calculator6 marks

On the grid, draw the graphs ofĀ  y equals 2 to the power of x andĀ  y equals 14 minus x squaredĀ  forĀ  0 less or equal than x less or equal than 4.

cie-igcse-2018-may-jun-3-p4-q2b

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5c
Sme Calculator2 marks

Use your graphs to solve the equations.

i) 2 to the power of x equals 12

x equals............................................... [1]

ii) 2 to the power of x equals 14 minus x squared

x equals............................................... [1]

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5d
Sme Calculator3 marks

i) On the grid, draw the line from the point left parenthesis 4 comma space 2 right parenthesis that has a gradient ofĀ negative 4.

[1]

ii) Complete the statement.

This straight line is a .................................. to the graph of y equals 14 minus x squared at the point ( ....... , ....... ). [2]

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1a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis space equals space 20 over x space plus space x comma space space space space x space not equal to space 0 Complete the table.

x

-10

-8

-5

-2

-1.6

Ā 

1.6

2

5

8

10

straight f left parenthesis x right parenthesis

-12

-10.5

-9

-12

-14.1

Ā 

14.1

12

Ā 

Ā 

12

How did you do?

1b
Sme Calculator5 marks

On the grid, draw the graph ofĀ  y space equals space straight f left parenthesis x right parenthesis for space ā€“ 10 space less-than or slanted equal to space x space less-than or slanted equal to space ā€“ 1.6 space andĀ  1.6 space less-than or slanted equal to space x space less-than or slanted equal to space 10.

cie-igcse-spec-p4-q3b-graph

How did you do?

1c
Sme Calculator2 marks

Using your graph, solve the equation straight f left parenthesis x right parenthesis space equals space 11.

x = ..................Ā  Ā orĀ  Ā x = .................

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1d
Sme Calculator2 marks

k is a prime number and space straight f left parenthesis x right parenthesis space equals space k spacehas no solutions.

Find the possible values of k.

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1e
Sme Calculator1 mark

The gradient of the graph ofĀ y space equals space straight f left parenthesis x right parenthesis at the pointspace left parenthesis 2 comma space 12 right parenthesis spaceis negative 4.

Write down the coordinates of the other point on the graph of y space equals space straight f left parenthesis x right parenthesis where the gradient is negative 4.

Ā (....................... , .......................)

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1f
Sme Calculator6 marks

i) The equationĀ straight f left parenthesis x right parenthesis space equals space x squared can be written asĀ space x cubed space plus space p x squared space plus space q space equals space 0. Show that space p space equals space minus 1 spaceand q space equals space minus 20.

[2]

ii) On the grid, draw the graph ofĀ  y space equals space x squared space forĀ  ā€“ 4 space less-than or slanted equal to space x space less-than or slanted equal to space 4.Ā  Ā 

[2]

iii) Using your graphs, solve the equationĀ space x cubed space space ā€“ space x squared space space ā€“ space 20 space equals space 0.

Ā 

x = .............................................. [1]

iv)

qJ9LLAKn_cie-igcse-spec-p4-q3fiv-graph

The diagram shows a sketch of the graph ofĀ y space equals space x cubed space space ā€“ space x squared space space ā€“ space 20.
P is the point left parenthesis n comma space 0 right parenthesis.

Write down the value of n.

n space equals .............................................. [1]

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2a
Sme Calculator3 marks

The table shows some values forĀ y equals x cubed plus 3 x squared plus 2.

x

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

y

-4.1

Ā 

5.1

6

5.4

4

2.6

Ā 

2.9

Ā 

12.1

Complete the table.

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2b
Sme Calculator4 marks

On the grid, draw the graph ofĀ y equals x cubed plus 3 x squared plus 2 forĀ negative 3.5 less or equal than x less or equal than 1.5 space.

cie-igcse-2019-may-jun-1-q2-2

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2c
Sme Calculator1 mark

Use your graph to solve the equationĀ x cubed plus 3 x squared plus 2 equals 0 forĀ negative 3.5 less or equal than x less or equal than 1.5 space. Ā 

x = ....................................................

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2d
Sme Calculator2 marks

By drawing a suitable straight line, solve the equationĀ x cubed plus 3 x squared plus 2 x plus 2 equals 0 forĀ negative 3.5 less or equal than x less or equal than 1.5 space. Ā 

x = ....................................................

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2e
Sme Calculator1 mark

ForĀ negative 3.5 less or equal than x less or equal than 1.5 space comma the equationĀ x cubed plus 3 x squared plus 2 equals k has three solutions andĀ k is an integer.

Write down a possible value of k. Ā 

k = ....................................................

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3a
Sme Calculator2 marks

The table shows some values ofĀ Ā y equals x cubed minus 3 x minus 1.

x

-3

-2.5

-2

-1.5

-1

0

1

1.5

2

2.5

3

y

-19

-9.1

Ā 

0.1

1

-1

-3

-2.1

1

7.1

Ā 

Ā  Complete the table of values.

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3b
Sme Calculator4 marks

Draw the graph ofĀ Ā y equals x cubed minus 3 x minus 1Ā  forĀ Ā negative 3 less or equal than x less or equal than 3.

cie-igcse-2018-oct-nov-p4-tz2-q5b

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3c
Sme Calculator5 marks

A straight line through (0, ā€“17) is a tangent to the graph ofĀ Ā y equals x cubed minus 3 x minus 1.

i) On the grid, draw this tangent.Ā  Ā Ā 

Ā 

[1]

ii) Find the co-ordinates of the point where the tangent meets your graph.Ā  Ā 

Ā 

(................ , ................) [1]

iii) Find the equation of the tangent.
Give your answer in the formĀ Ā y equals m x plus c. Ā 

Ā 

yĀ = ................................................ [3]

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3d
Sme Calculator4 marks

By drawing a suitable straight line on the grid, solve the equationĀ Ā x cubed minus 6 x minus 3 equals 0. Ā 

x = ................... orĀ x = ................... orĀ x = ...................

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4
Sme Calculator4 marks
q25-june-2021-4024-11-cambridge-o-level-maths-d

The diagram shows a circle centre C space left parenthesis 0 comma space 1 right parenthesis. space P space left parenthesis 3 comma space 5 right parenthesis spaceis a point on the circumference of the circle.
Find the equation of the tangent at P.

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