Differentiation (Cambridge O Level Additional Maths)

Exam Questions

3 hours28 questions
1
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3 marks

Given that y space equals space tan space x, use calculus to find the approximate change in y as x increases from negative pi over 4 to h minus pi over 4, where h is small.

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2
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3 marks

Find the x-coordinate of the stationary point on the curve y space equals open parentheses 2 minus space square root of 3 close parentheses x squared space plus space x minus 1, giving your answer in the form a plus b square root of space 3 end root,where a and b are rational numbers.

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3
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4 marks

The radius, r space cm, of a circle is increasing at the rate of 5 space cms to the power of – 1 end exponent. Find, in terms of pi, the rate at which the area of the circle is increasing when r space equals space 3.

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4
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4 marks

The volume, V, of a sphere of radius r is given by V equals 4 over 3 straight pi straight r cubed.

The radius, r space cm, of a sphere is increasing at the rate of 0.5 space cms to the power of negative 1 end exponent. Find, in terms of straight pi, the rate of change of the volume of the sphere when space r space equals space 0.25.

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5a
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5 marks

Given that y equals open parentheses x squared minus 1 close parentheses square root of 5 x plus 2 end root, show that fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator A x squared plus B x plus C over denominator 2 square root of 5 x plus 2 end root end fraction, where A comma space B and C are integers.

5b
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3 marks

Find the coordinates of the stationary point of the curve y equals open parentheses x squared minus 1 close parentheses square root of 5 x plus 2 end root space for space x greater than 0. Give each coordinate correct to 2 significant figures.

5c
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2 marks

Determine the nature of this stationary point.

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6
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4 marks

Variables x and y are such that y space equals space sin space x plus straight e to the power of negative x end exponent. Use differentiation to find the approximate change in y as xincreases from straight pi over 4 space to space straight pi over 4 plus h, where h is small.

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7a
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6 marks

The equation of a curve is y space equals space x square root of 16 minus x squared end root space for space 0 less or equal than x less or equal than 4.

Find the exact coordinates of the stationary point of the curve.

7b
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5 marks

Find fraction numerator straight d over denominator straight d x end fraction open parentheses 16 minus x squared close parentheses to the power of 3 over 2 end exponentand hence evaluate the area enclosed by the curve y equals x square root of 16 minus x squared end root and the lines y space equals space 0 comma space x space equals 1 space and space x space equals space 3.

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8a
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5 marks

A curve has equation y space equals space left parenthesis 2 x minus 1 right parenthesis space square root of 4 x plus 3 end root.

Show that fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 4 open parentheses A x plus B close parentheses over denominator square root of 4 x plus 3 end root end fraction, where A and B are constants.

8b
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1 mark

Hence write down the x-coordinate of the stationary point of the curve.

8c
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2 marks

Determine the nature of this stationary point.

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9a
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4 marks

Find the equation of the tangent to the curve y space equals space x cubed space minus 6 x squared space plus 3 x plus 10 at the point where x space equals space 1.

9b
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5 marks

Find the coordinates of the point where this tangent meets the curve again.

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10a
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2 marks

It is given that y space equals space ln left parenthesis 1 plus sin space x right parenthesis for 0 space less than space x space less than space straight pi .

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

10b
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2 marks

Find the value of fraction numerator straight d y over denominator straight d x end fraction when x equals straight pi over 6, giving your answer in the form fraction numerator 1 over denominator square root of a end fraction, where a is an integer.

10c
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5 marks

Find the values of x for which fraction numerator straight d y over denominator straight d x end fraction equals tan space x.

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11a
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2 marks

A curve has equation y space equals space x space cos space x.

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

11b
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4 marks

Find the equation of the normal to the curve at the point where x space equals space straight pi, giving your answer in the form y space equals space m x plus c.

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12
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6 marks

Find the equation of the tangent to the curve y equals fraction numerator ln open parentheses 3 x squared minus 1 close parentheses over denominator x plus 2 end fraction at the point where x equals 1. Give your answer in the formspace y equals m x plus c, where m and c are constants correct to 3 decimal places.

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1
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7 marks

A curve has equation y space equals space ln left parenthesis 5 space – space 3 x right parenthesis where x space less than space 5 over 3. The normal to the curve at the point where x space equals space – 5 , cuts the x-axis, at the point P.
Find the equation of the normal and the x-coordinate of P.

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2
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6 marks

Variables x and y are such that y space equals space e to the power of x over 2 end exponent space plus space x cos space 2 x , where x is in radians. Use differentiation to find the approximate change in y as x increases from 1 to 1 plus h, where h is small.

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3
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6 marks

The tangent to the curve y equals ln open parentheses 3 x squared minus 4 close parentheses minus x cubed over 6, at the point where x space equals space 2, meets the y-axis at the point P. Find the exact coordinates of P.

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4
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8 marks

q11-0606-m20-qp-22-additional-maths

A container is a circular cylinder, open at one end, with a base radius of r cm and a height of h cm. The volume of the container is 1000 cm3. Given that r and h can vary and that the total outer surface area of the container has a minimum value, find this value.

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5a
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6 marks

Find the x-coordinates of the stationary points of the curve y space equals space e to the power of 3 x end exponent space left parenthesis 2 x plus 3 right parenthesis to the power of 6.

5b
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2 marks

A curve has equation y space equals space straight f left parenthesis x right parenthesis and has exactly two stationary points. Given that straight f to the power of " left parenthesis x right parenthesis space equals space 4 x minus 7, straight f space apostrophe left parenthesis 0.5 right parenthesis space equals space 0 space and space straight f apostrophe left parenthesis 3 right parenthesis space equals space 0, use the second derivative test to determine the nature of each of the stationary points of this curve.

5c
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5 marks

In this question all lengths are in centimetres.

q12c-0606-s20-qp-21-additional-maths

The diagram shows a solid cuboid with height h and a rectangular base measuring 4 x by x. The volume of the cuboid is 40 space cm cubed. Given that x and h can vary and that the surface area of the cuboid has a minimum value, find this value.

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6a
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5 marks

Find the equation of the tangent to the curve 2 y equals tan space 2 x plus 7 at the point where x equals straight pi over 8.
Give your answer in the form a x minus y equals straight pi over b plus c, where a comma space b and c are integers.

6b
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2 marks

This tangent intersects the x-axis at P and the y-axis at Q. Find the length of P Q.

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7a
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5 marks

y space equals space x square root of x plus 2 end root Given that , show that fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator A x plus B over denominator 2 square root of x plus 2 end root end fraction, where A and B are constants.

7b
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3 marks

Find the exact coordinates of the stationary point of the curve y space equals space x square root of x plus 2 end root.

7c
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2 marks

Determine the nature of this stationary point.

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8a
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2 marks

Differentiate y space equals space tan left parenthesis x plus 4 right parenthesis space minus 3 space sin space x with respect to x.

8b
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6 marks

Variables x and y are such that y equals fraction numerator ln open parentheses 2 x plus 5 close parentheses over denominator 2 e to the power of 3 x end exponent end fraction. Use differentiation to find the approximate change in y as x increases from 1 space to space 1 space plus space h, where h is small.

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9a
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4 marks

It is given that y equals fraction numerator tan space 3 x over denominator sin space x end fraction.

Find the exact value of fraction numerator straight d y over denominator straight d x end fraction when x equals straight pi over 3.

9b
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1 mark

Hence find the approximate change in y as x increases from straight pi over 3 space to space straight pi over 3 plus h comma where h is small.

9c
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2 marks

Given that x is increasing at the rate of 3 units per second, find the corresponding rate of change in y when x equals straight pi over 3 , giving your answer in its simplest surd form.

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10a
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3 marks

It is given that y space equals space ln left parenthesis sin space x plus 3 space cos space x right parenthesis space for space 0 less than space x space less than straight pi over 2.

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

10b
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3 marks

Find the value of x for which fraction numerator straight d y over denominator straight d x end fraction equals negative 1 half.

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11a
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3 marks

Given that y equals fraction numerator straight e to the power of 2 x minus 3 end exponent over denominator x squared plus 1 end fraction, find fraction numerator straight d y over denominator straight d x end fraction.

11b
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3 marks

Hence, given that y is increasing at the rate of 2 units per second, find the exact rate of change of x when x space equals space 2.

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1a
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3 marks

In this question, all lengths are in centimetres.

q11-2025-specimen-paper-2-cie-igcse-additional-maths

The diagram shows a cone of base radius x, height y and sloping edge square root of x squared plus y squared end root. The volume of the cone is 10 straight pi space cm cubed.

Show that the curved surface area, S, of the cone is given by S space equals fraction numerator pi square root of x to the power of 6 plus 900 end root over denominator x end fraction.

1b
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5 marks

Given that x can vary and that S has a minimum value, find the value of x for which S is a minimum.

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2
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6 marks

Variables x and y are such that y equals fraction numerator e to the power of 3 x end exponent sin space x over denominator x squared end fraction .Use differentiation to find the approximate change in yas x increases from 0.5 space to space 0.5 plus h, where h spaceis small.

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3a
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3 marks

In this question all lengths are in centimetres.
The volume, V, of a cone of height h and base radius r is given by V equals 1 third pi r squared h

q11-0606-s20-qp-23-additional-maths

The diagram shows a large hollow cone from which a smaller cone of height 180 and base radius 90 has been removed. The remainder has been fitted with a circular base of radius 90 to form a container for water. The depth of water in the container is w and the surface of the water is a circle of radius R.

Find an expression for R in terms of w and show that the volume V of the water in the container is given by V equals space straight pi over 12 open parentheses w plus 180 close parentheses cubed space minus 486000 straight pi.

3b
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4 marks

Water is poured into the container at a rate of 10 space 000 space cm cubed straight s to the power of negative 1 end exponent. Find the rate at which the depth of the water is increasing when w space equals space 10

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4a
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6 marks

A curve has equation y equals fraction numerator ln open parentheses 3 x squared minus 5 close parentheses over denominator 2 x plus 1 end fraction space for space 3 x squared greater than 5

Find the equation of the normal to the curve at the point where x space equals space square root of 2.

4b
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1 mark

Find the approximate change in y as x increases from square root of 2 space to space square root of 2 space plus h, where h is small.

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5a
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3 marks

q9-0606-w20-qp-23-additional-maths

The rectangle A B C D E represents a ploughed field where A B space equals space 300 space straight m and A E space equals space 400 space straight m. Joseph needs to walk from A to D in the least possible time. He can walk at 0.9 space ms to the power of negative 1 end exponent on the ploughed field and at 1.5 space ms to the power of negative 1 end exponent on any part of the path B C D along the edge of the field. He walks from A to C and then from C to D. The distance B C space equals space x space straight m.

Find, in terms of x, the total time, T space straight s, Joseph takes for the journey.

5b
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6 marks

Given that x can vary, find the value of x for which T is a minimum and hence find the minimum value of T.

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