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Further Graphs & Tangents (Cambridge (CIE) IGCSE Maths: Extended): Exam Questions

Exam code: 0580 & 0980

5 hours32 questions

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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 a function.  

By drawing a suitable tangent, find an estimate for the gradient of the function at the point P.

How did you do?

2a
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?

2b
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?

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?

4a1 mark

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

Write down the equation of the line of symmetry of the graph.

How did you do?

4b3 marks

On the grid, draw the tangent to the curve at the point where x equals 0.5.

Find the gradient of this tangent.

How did you do?

5a2 marks

The diagram shows part of the graph of  y space equals space x squared space minus space 2 x space plus space 3

q3-2-14-medium-cie-igcse-maths-extended

By drawing a suitable straight line, use your graph to find estimates for the solutions of x squared space minus space 3 x space minus space 1 space equals space 0

How did you do?

5b
Sme Calculator3 marks

P is the point on the graph of y space equals space x squared space minus space 2 x space plus space 3 where x space equals space 2

Calculate an estimate for the gradient of the graph at the point P.

How did you do?

6a
Sme Calculator2 marks

Complete the table of values for y space equals space x squared space minus space x space minus space 6

x

-3

-2

-1

0

1

2

3

y

6

 

 

-6

 

 

 

How did you do?

6b2 marks

On the grid, draw the graph of y space equals space x squared space minus space x space minus space 6 for values of x from negative 3 to 3

q4b-2-14-medium-cie-igcse-maths-extended

How did you do?

6c2 marks

Use your graph to find estimates of the solutions to the equation x squared space minus space x space minus space 6 space equals space minus 2

How did you do?

7a
Sme Calculator2 marks

Complete the table of values for y space equals space x squared space minus space 2 x

x

-2

-1

0

1

2

3

4

y

 

3

0

 

 

3

 

How did you do?

7b2 marks

On the grid, draw the graph of y = x2 – 2x for values of x from –2 to 4

q3b-2-14-very-hard-cie-igcse-maths-extended

How did you do?

7c2 marks

Solve x squared space minus space 2 x space minus space 2 space equals space 1

How did you do?

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?

1b4 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?

1c3 marks

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

i) x to the power of 2 space end exponent plus space 1 over x space equals space minus 2

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

ii)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?

1d1 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?

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?

2b5 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?

2c2 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?

2d1 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?

2e3 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?

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?

3b4 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?

3c2 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?

3e1 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?

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?

4b4 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?

4c4 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?

4d1 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?

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?

5b3 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?

5c3 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?

5d5 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?

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?

6b5 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?

6d4 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]

How did you do?

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?

1b4 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?

1c1 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?

1d4 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?

2a4 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?

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?

3b1 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?

3c3 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?

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?

4b4 marks

On the grid, draw the graph of 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.

cie-igcse-2018-march-p4-q3b

How did you do?

4c3 marks

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

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

How did you do?

4d5 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]

How did you do?