O LevelMathsCambridgeExam QuestionsCoordinate Geometry & GraphsFurther Graphs & Tangents

Further Graphs & Tangents (Cambridge O Level Maths)

Exam Questions

5 hours32 questions

Select a question set to download for
Further Graphs & Tangents

Select an answer set to view for
Further Graphs & Tangents

Medium
Hard
Very Hard
1
Sme Calculator3 marks

cie-igcse-2020-may-jun-p4-tz3-q10b

The diagram shows the graph of another function.
 
By drawing a suitable tangent, find an estimate for the gradient of the function at the point P.

How did you do?

Did this page help you?

2a
Sme Calculator2 marks

Write down the equation of your tangent in the form y equals m x plus c.


y equals .................................................

How did you do?

2b
Sme Calculator3 marks

cie-igcse-2020-march-p4-tz2-q2b

i)
Draw the tangent to the curve at x equals 1.


[1]

ii)
Use your tangent to estimate the gradient of the curve at x equals 1.


[2]

How did you do?

Did this page help you?

3
Sme Calculator3 marks

cie-igcse-2019-march-p2-q16

By drawing a suitable tangent, estimate the gradient of the curve at the point P.

How did you do?

Did this page help you?

4a
Sme Calculator2 marks

q18-specimen-paper1-2025-4024-01-cambridge-o-level-maths-d

The diagram shows a sketch of the graph of y space equals space 3 space cross times space 2 to the power of x space plus space 5 space.

The graph crosses the y-axis at point A.
Find the coordinates of point A.

How did you do?

4b
Sme Calculator1 mark

Write down the equation of the asymptote to the graph of  y space equals space 3 space cross times space 2 to the power of x space plus space 5 space.

How did you do?

Did this page help you?

5a
Sme Calculator2 marks

The speed–time graph shows the first 10 seconds of the journey of a car.

q13-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

By drawing a tangent, estimate the gradient of the curve at t space equals space 2.

How did you do?

5b
Sme Calculator1 mark

Write down what the gradient of the curve represents in terms of the car’s motion for the first 10 seconds of its journey.

How did you do?

Did this page help you?

6
Sme Calculator2 marks

Sketch the graph of the function y equals 1 over x

q16b-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

How did you do?

Did this page help you?

1a
Sme Calculator2 marks

space space space space space y space equals space x to the power of 2 space end exponent plus 1 over x comma space space space x space not equal to space 0

Complete the table.

x 0.2 0.3 0.5 1 1.5 2 2.5
y 5.0 3.4 2.3   2.9   6.7

How did you do?

1b
Sme Calculator4 marks

On the grid, draw the graph of y space equals space x to the power of 2 space end exponent space plus space 1 over x for 0.2 space less or equal than space x space less or equal than space 2.5.
The graph of y space equals space x to the power of 2 space end exponent space plus space 1 over x for negative 2.5 space less or equal than space x space less or equal than space minus 0.2 has been drawn for you.

cie-igcse-2020-oct-nov-p4-tz2-q7b

How did you do?

1c
Sme Calculator3 marks

By drawing suitable straight lines on the grid, solve the following equations.

i)blank over blank
x to the power of 2 space end exponent plus space 1 over x space equals space minus 2
 
x space equals ................................................ [1]
ii)blank over blank
x to the power of 2 space end exponent plus space 1 over x space plus space x space minus space 1 equals space 0
 
x space equals ................................................ [2]

How did you do?

1d
Sme Calculator1 mark

k is an integer and the equation x to the power of 2 space end exponent plus space 1 over x space equals space k has three solutions.

Write down a possible value of k.


k space equals ................................................

How did you do?

Did this page help you?

2a
Sme Calculator3 marks

The table shows some values of  y equals x squared over 2 plus 1 over x squared minus 2 over x, x not equal to 0.


x negative 3 negative 2 negative 1 negative 0.5 negative 0.3   0.2 0.3 0.5 1 2 3
y 5.3 3.3   8.1 17.8     4.5 0.1 negative 0.5 1.3  
 
Complete the table.

How did you do?

2b
Sme Calculator5 marks

On the grid, draw the graph of  y equals x squared over 2 plus 1 over x squared minus 2 over x  for  negative 3 less or equal than x less or equal than negative 0.3  and  0.2 less or equal than x less or equal than 3.

56XoM-7K_cie-igcse-2019-oct-nov-p4-tz2-q5b

How did you do?

2c
Sme Calculator2 marks

Use your graph to solve  x squared over 2 plus 1 over x squared minus 2 over x less or equal than 0.

..................... less or equal than x less or equal than ....................

How did you do?

2d
Sme Calculator1 mark

Find the smallest positive integer value of k for which  x squared over 2 plus 1 over x squared minus 2 over x equals k has two solutions for  negative 3 less or equal than x less or equal than negative 0.3  and  0.2 less or equal than x less or equal than 3.

How did you do?

2e
Sme Calculator3 marks

By drawing a suitable straight line, solve  x squared over 2 plus 1 over x squared minus 2 over x equals 3 x plus 1  for  negative 3 less or equal than x less or equal than negative 0.3  and  0.2 less or equal than x less or equal than 3.

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

How did you do?

Did this page help you?

3a
Sme Calculator3 marks

The table shows some values for y equals x cubed plus x squared minus 5 x.

x negative 3 negative 2 negative 1.5 negative 1 0 1 1.5 2 2.5 3
y negative 3 6 6.4   0   negative 1.9 2 9.4  

 

Complete the table.

How did you do?

3b
Sme Calculator4 marks

On the grid, draw the graph of  y equals x cubed plus x squared minus 5 x  for  negative 3 less or equal than x less or equal than 3.

l8xiBxEy_cie-igcse-2019-oct-nov-p4-tz3-q3b

How did you do?

3c
Sme Calculator2 marks

Use your graph to solve the equation  x cubed plus x squared minus 5 x equals 0.

x = ..................... or x = ..................... or x = .....................

How did you do?

3d
Sme Calculator3 marks

By drawing a suitable tangent, find an estimate of the gradient of the curve at x equals 2.

How did you do?

3e
Sme Calculator1 mark

Write down the largest value of the integer, k, so that the equation  x cubed plus x squared minus 5 x equals k  has three solutions for  negative 3 less or equal than x less or equal than 3.

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

How did you do?

Did this page help you?

4a
Sme Calculator3 marks

The table shows some values for space y equals 3 over 10 x cubed minus 2 x for space minus 3 less or equal than x less or equal than 3.

x -3 -2 -1.5 -1 0 1 1.5 2 3
y     2.0 1.7 0   -2.0 -1.6  

Complete the table.

How did you do?

4b
Sme Calculator4 marks

On the grid, draw the graph of space y equals 3 over 10 x cubed minus 2 x for negative 3 less or equal than x less or equal than 3.

cie-igcse-2019-march-p4-q5

How did you do?

4c
Sme Calculator4 marks

On the grid, draw a suitable straight line to solve the equation space 3 over 10 x cubed minus 2 x equals 1 half open parentheses 1 minus x close parentheses for negative 3 less or equal than x less or equal than 3.
 

x equals....................... or x equals....................... or x equals......................

How did you do?

4d
Sme Calculator1 mark

For negative 3 less or equal than x less or equal than 3, the equation space 3 over 10 x cubed minus 2 x equals 1 space has n solutions.
 
Write down the value of n.
 

n equals..................................................

How did you do?

Did this page help you?

5a
Sme Calculator3 marks

The table shows some values of  y equals x cubed minus 3 x squared plus x.

x -0.75 -0.5 -0.25 0 0.5 1 1.5 2 2.5 2.75
y -2.9 -1.4 -0.5   -0.1 -1 -1.9   -0.6  

 

Complete the table.

How did you do?

5b
Sme Calculator3 marks

On the grid, draw the graph of  y equals x cubed minus 3 x squared plus x  for  negative 0.75 less or equal than x less or equal than 2.75.

6TC-Enhq_cie-igcse-2018-oct-nov-p4-tz1-q3b

How did you do?

5c
Sme Calculator3 marks

Use your graph to complete the inequalities in x for which y greater than negative 1.
 

.................... less than x less than .................... and  x greater than ....................

How did you do?

5d
Sme Calculator5 marks

The equation  x cubed minus 3 x squared plus 2 x minus 1 equals 0 can be solved by drawing a straight line on the grid.

i)
Write down the equation of this line.
 

[2]

ii)
On the grid, draw this line and use it to solve the equation  x cubed minus 3 x squared plus 2 x minus 1 equals 0. 
  
 

x = ............................................... [3]

How did you do?

5e
Sme Calculator3 marks

By drawing a suitable tangent, find an estimate for the gradient of the graph of  y equals x cubed minus 3 x squared plus x  at  x equals negative 0.25.

How did you do?

Did this page help you?

6a
Sme Calculator3 marks

Complete the table of values for y equals x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction comma space x not equal to 0.

x negative 3 negative 2 negative 1 negative 0.5 negative 0.3   0.3 0.5 1 2 3
y negative 9.1 negative 2.8 negative 0.8   negative 5.6   negative 5.5 negative 2.0     8.9

How did you do?

6b
Sme Calculator5 marks

On the grid, draw the graph of y equals x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction for negative 3 less or equal than x less or equal than negative 0.3 and 0.3 less or equal than x less or equal than 3.

cie-igcse-2018-may-jun-2-p4-q6b

How did you do?

6c
Sme Calculator5 marks
i)
By drawing a suitable tangent, find an estimate of the gradient of the curve at x equals negative 2.

[3]

ii)
Write down the equation of the tangent to the curve at x equals negative 2.
Give your answer in the form y equals m x plus c.

y equals ................................................ [2]

How did you do?

6d
Sme Calculator4 marks

Use your graph to solve the equations.

i)table row blank row blank end table
x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction equals 0

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

ii)begin mathsize 16px style table row blank row blank end table end style
x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction plus 4 equals 0

x equals .................... or x equals .................... or x equals .................... [3]

How did you do?

Did this page help you?

7a
Sme Calculator2 marks

The table shows some values for space y space equals space x cubed over 3 – space 2 x space plus space 5 spacecorrect to 1 decimal place.

x –3 –2.5 –2 –1 0 1 2 2.5 3
y   4.8 6.3 6.7 5.0 3.3 3.7 5.2  

Complete the table.

How did you do?

7b
Sme Calculator4 marks

Draw the graph of y space equals space x cubed over 3 space – space 2 x space plus space 5 space for space – 3 space less-than or slanted equal to space x space less-than or slanted equal to space 3.

q21b-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

How did you do?

7c
Sme Calculator4 marks

By drawing a suitable line on the grid, find the roots of the equation x cubed over 3 space – space 1.5 x space plus space 1 space equals space 0.

How did you do?

Did this page help you?

8
Sme Calculator6 marks

q7c-june-2021-4024-21-cambridge-o-level-maths-d

The grid shows the graph ofy equals x cubed minus 2 x squared minus 5 x plus 6.

i)
x cubed minus 2 x squared minus 5 x plus 6 equals k has exactly 2 solutions.

Use the graph to find the possible values of k.

[2]

ii)
By drawing a suitable line on the grid, find the solutions of  x cubed minus 2 x squared minus 7 x plus 5 equals 0
x = .................. , x = .................. , x = .................. [4]

How did you do?

Did this page help you?

1a
Sme Calculator3 marks

The table shows some values of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 for 0.15 less or equal than x less or equal than 3.5 space.

x 0.15 0.2 0.5 1 1.5 2 2.5 3 3.5
y 3.30   0.88   -0.04   -0.43 -0.58 -0.73

Complete the table. 

How did you do?

1b
Sme Calculator4 marks

On the grid, draw the graph of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 for 0.15 less or equal than x less or equal than 3.5 space.
The last two points have been plotted for you.

cie-igcse-2019-may-jun-p4-tz2-q5b

How did you do?

1c
Sme Calculator1 mark

Use your graph to solve the equation fraction numerator 1 over denominator 2 x end fraction minus x over 4 equals 1 half for 0.15 less or equal than x less or equal than 3.5 space. 
 

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

How did you do?

1d
Sme Calculator4 marks
i)
On the grid, draw the line y equals 2 minus x space. 
 
[2]
ii)
Write down the x co-ordinates of the points where the line y equals 2 minus x crosses the graph
of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 for 0.15 less or equal than x less or equal than 3.5.
 
x = .................... and x = .................... [2]

How did you do?

1e
Sme Calculator2 marks

Show that the graph of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 can be used to find the value of square root of 2 for 0.15 less or equal than x less or equal than 3.5 space.

How did you do?

Did this page help you?

2a
Sme Calculator4 marks

The diagram shows the graph of y equals f open parentheses x close parentheses where f open parentheses x close parentheses equals x squared minus 2 over x minus 2 comma space x not equal to 0.

cie-igcse-2019-may-jun-p4-tz3-q5a

On the grid, draw a suitable straight line to solve the equation
x squared minus 2 over x minus 7 equals negative 3 x for negative 3 less or equal than x less or equal than 3.
 

x = ...................... or x = ..........................

How did you do?

2b
Sme Calculator3 marks

By drawing a suitable tangent, find an estimate of the gradient of the curve at x equals negative 2.

How did you do?

2c
Sme Calculator7 marks
i)
Complete the table for y equals text g end text left parenthesis x right parenthesis where text g end text left parenthesis x right parenthesis equals 2 to the power of negative x end exponent for negative 3 less or equal than x less or equal than 3.
x -3 -2 -1 0 1 2 3
y     2 1 0.5   0.125

 

[3]

ii)
On the grid, draw the graph of y equals text g end text left parenthesis x right parenthesis.
 
[3]
iii)
Use your graph to find the positive solution to the equation text f end text left parenthesis x right parenthesis equals text g end text left parenthesis x right parenthesis.
 
x = .............................................. [1]

How did you do?

Did this page help you?

3a
Sme Calculator7 marks

The graph of y equals 10 minus 8 x squared for  negative 1.5 less or equal than x less or equal than 1.5  is drawn on the grid.

cie-igcse-2018-may-jun-1-7

And the table shows some values for y equals x cubed plus 3 x plus 4.

x negative 1.5 negative 1 negative 0.5 0 0.5 1 1.5
y negative 3.9       5.6 8 11.9

i)
Complete the table.


[3]

ii)
On the grid, draw the graph of y equals x cubed plus 3 x plus 4 for negative 1.5 less or equal than x less or equal than x 1.5.


[4]

How did you do?

3b
Sme Calculator1 mark

Show that the values of x where the two curves intersect are the solutions to the equation x cubed plus 8 x squared plus 3 x minus 6 equals 0.

How did you do?

3c
Sme Calculator3 marks

By drawing a suitable straight line, solve the equation x cubed plus 5 x plus 2 equals 0  for  negative 1.5 less or equal than x less or equal than 1.5.

x equals..............................................

How did you do?

Did this page help you?

4a
Sme Calculator3 marks

The table shows some values for space y equals 2 x plus 1 over x minus 3 space for  0.125 less or equal than x less or equal than 3.

x 0.125 0.25 0.375 0.5 0.75 1 1.5 2 2.5 3
space y space 5.25 1.5 0.42     0 0.67 1.5   3.33

Complete the table.

How did you do?

4b
Sme Calculator4 marks

On the grid, draw the graph of space y equals 1 half x plus 1 over x minus 3 space for 0.125 less or equal than x less or equal than 3.

cie-igcse-2018-march-p4-q3b

How did you do?

4c
Sme Calculator3 marks

Use your graph to solve space 2 x plus 1 over x minus 3 greater or equal than 2.
 

................................................. 
 ................................................. 

How did you do?

4d
Sme Calculator5 marks

The equation space 1 over x equals 7 minus 3 x space can be solved using your graph in part (b) and a straight line.

i)

Write down the equation of this straight line.
 

[2]

ii)table row blank row blank end table

Draw this straight line and solve the equation space 1 over x equals 7 minus 3 x.
 

x equals.................... or x equals.................... [3]

How did you do?

Did this page help you?

5
Sme Calculator8 marks

The table shows some values of x and the corresponding values of y for space y space equals space 2 x cubed space – space 3 x squared space plus space 5.

x negative 1.5 negative 1 negative 0.5 0 0.5 1 1.5 2
y   0 4 5 4.5 4 5 9

i)
Complete the table. 

[1]

ii)
Using a scale of 4 cm to represent 1 unit, draw a horizontal x-axis for – 1.5 space less or equal than space x space less or equal than space 2.
Using a scale of 2 cm to represent 5 units, draw a vertical y-axis for – 10 space less or equal than space y space less or equal than space 10.

Draw the graph of y space equals space 2 x cubed space – space 3 x squared space plus space 5 for – 1.5 space less or equal than space x space less or equal than space 2.
tT_F1C0y_q6aii-june-2018-4024-2-cambridge-o-level-maths-d

[3]

iii)
Use your graph to estimate the gradient of the curve when x space equals space 1.5.

[2]

iv)
By drawing a suitable line on your graph, find the solution of the equation 2 x cubed space – space 3 x squared space plus space 4 space equals space 0.

x = ...................[2]

How did you do?

Did this page help you?

6
Sme Calculator1 mark

q6b-june-2018-4024-2-cambridge-o-level-maths-d

The graph shows a sketch of the curve y space equals space p over x
Two points on the curve are left parenthesis 3 comma space 0.4 right parenthesis and left parenthesis q comma space 2.4 right parenthesis

i)
Find p and q.

p = ...................
q = ...............[2]

ii)
Calculate the gradient of the straight line joining the points left parenthesis 3 comma space 0.4 right parenthesis and left parenthesis q comma space 2.4 right parenthesis.

[2]

How did you do?

Did this page help you?