CIE IGCSE Maths: Extended

Topic Questions

IGCSEMaths: ExtendedCIETopic Questions2. Algebra & Graphs2.18 Differentiation

Syllabus Edition

First teaching 2021

Last exams 2024

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2.18 Differentiation

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1a
Sme Calculator9 marks

A curve has the equation  y equals x cubed plus 8 x squared plus 5 x.

Work out the coordinates of the two turning points.


(.................... , ....................) and (.................... , ....................)

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1b
Sme Calculator3 marks

Determine whether each of the turning points is a maximum or a minimum.
Give reasons for your answers.

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2
Sme Calculator6 marks

space y equals x to the power of 4 minus 4 x cubed space

Find the two stationary points on the graph of space y equals x to the power of 4 minus 4 x cubed.
 

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

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3
Sme Calculator8 marks

A curve has equation space y equals 4 x cubed minus 3 x plus 3.

i)

Find the coordinates of the two stationary points.
 

( .................... , .................... ) and ( .................... , .................... ) [5]

ii)

Determine whether each of the stationary points is a maximum or a minimum.
Give reasons for your answers.
 

[3]

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4a
Sme Calculator5 marks

A curve has equation y equals x cubed minus 3 x plus 4.

Work out the coordinates of the two stationary points.

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

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4b
Sme Calculator3 marks

Determine whether each stationary point is a maximum or a minimum.
Give reasons for your answers.

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5a
Sme Calculator6 marks

A curve has equation y space equals space x cubed space – space 6 x to the power of 2 space end exponent space plus space 16.

Find the coordinates of the two turning points.
 

(............ , ............) and (............ , ............)

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5b
Sme Calculator3 marks

Determine whether each of the turning points is a maximum or a minimum.
Give reasons for your answers.

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6a
Sme Calculator2 marks

The curve bold C has equation space y space equals space 5 x cubed – space x to the power of 2 space end exponent – space 6 x space plus space 4.

Find  fraction numerator straight d y over denominator straight d x end fraction.

   fraction numerator straight d y over denominator straight d x end fraction = ..............................................

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6b
Sme Calculator4 marks

There are two points on the curve bold C at which the gradient of the curve is 2.

Find the x coordinate of each of these two points.
Show clear algebraic working.

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7a
Sme Calculator2 marks

y space equals space x cubed space – space 6 x squared space – space 15 x.

Find fraction numerator straight d y over denominator straight d x end fraction.

fraction numerator straight d y over denominator straight d x end fraction space equals....................................

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7b
Sme Calculator4 marks

The curve with equation y space equals space x cubed space – space 6 x squared space – space 15 x has two stationary points.

Work out the coordinates of these two stationary points.

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1a
Sme Calculator2 marks

Differentiate space 6 space plus space 4 x space minus space x squared.

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1b
Sme Calculator2 marks

Find the coordinates of the turning point of the graph of y space equals space 6 space plus space 4 x space minus space x squared.

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

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2
Sme Calculator3 marks

Calculate the gradient of  y space equals space 24 space plus space 5 x space minus space x to the power of 2 space end exponent at space x space equals negative 1.5.

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3a
Sme Calculator1 mark

Use differentiation to find fraction numerator d y over denominator d x end fraction for the following:

y space equals space x to the power of 4

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3b
Sme Calculator1 mark
y space equals space 2 x cubed

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3c
Sme Calculator1 mark

y space equals space 4.

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4a
Sme Calculator2 marks

For the curve with equation y space equals space 2 x squared space minus space 6 x space minus space 11 colon

find fraction numerator d y over denominator d x end fraction

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4b
Sme Calculator2 marks

find the coordinates of the point on the curve where the gradient is 2.

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5a
Sme Calculator2 marks

A curve has equation y space equals space x cubed space plus space 7 over 2 x squared space minus space 2 x plus space 9
Find  fraction numerator d y over denominator d x end fraction

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5b
Sme Calculator4 marks

Find the gradient of the curve at the point where:

(i)     x space equals space minus 3

[2]

(ii)    x space equals space 2 over 3

[2]

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5c
Sme Calculator1 mark

What can you say about the tangents to the curves at these two points?

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1
Sme Calculator5 marks

cie-igcse-2020-oct-nov-p4-tz1-q10a

 The diagram shows a sketch of the curve  y space equals space x to the power of 2 space end exponent space plus space 3 x space minus space 4.

i)
Differentiate  y space equals space x to the power of 2 space end exponent space plus space 3 x space minus space 4
  
 [2]
ii)
Find the equation of the tangent to the curve at the point (2, 6). 
 
[3]

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2
Sme Calculator2 marks

space y equals x to the power of p plus 2 x to the power of q space

fraction numerator d y over denominator d x end fraction equals 11 x to the power of 10 plus 10 x to the power of 4, where fraction numerator d y over denominator d x end fraction is the derived function.
 
Find the value of p and the value of q.
 

p equals................................................
q equals................................................

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3a
Sme Calculator2 marks

A curve, C, has equation y space equals space 2 x squared space plus space 8 k squared x space minus space 3 where k is a constant.

Show that when k = 0, the turning point on C has coordinates (0, -3).

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3b
Sme Calculator4 marks

Show that when k space not equal to space 0, the turning point on C must have a negative x-coordinate.

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3c
Sme Calculator2 marks

When k space not equal to space 0 determine whether or not the y-coordinate of the turning point is negative.

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4
Sme Calculator6 marks

The curve bold C has equation y space equals space a x cubed space plus space b x squared space – space 12 x space plus space 6 where a and b are constants.

The point A with coordinates (2, –6) lies on bold C.
The gradient of the curve at A is 16.

Find the y coordinate of the point on the curve whose x coordinate is 3.
Show clear algebraic working.

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5
Sme Calculator7 marks

Part of the graph with equation y space equals space 2 x to the power of 4 space minus space 16 x squared space plus space 3 is shown below.

q5-2-18-very-hard-cie-igcse-maths-extended


The graph has three stationary points, indicated on the graph by points P , Q and R.
Find the area of the triangle PQR.

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