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

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Further Graphs & Tangents

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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.

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2a
Sme Calculator2 marks

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

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

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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]

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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.

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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.

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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.

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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.

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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.

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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

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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

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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

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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]

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

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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.

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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

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

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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.

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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 = ...................................................

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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.

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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

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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 = .....................

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3d
Sme Calculator3 marks

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

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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 = ..................................................

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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.

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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

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

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

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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.

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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

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

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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]

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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.

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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

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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]

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6d
Sme Calculator4 marks

Use your graph to solve the equations.

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

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

ii) 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]

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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.

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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

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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.

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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]

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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

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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= ..............................................

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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]

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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.

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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 = ..........................

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2b
Sme Calculator3 marks

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

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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]

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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.

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

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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

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4c
Sme Calculator3 marks

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

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

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

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

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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.

[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]

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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]

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