A LevelMathsEdexcelPureExam QuestionsDifferentiationApplications of Differentiation

Applications of Differentiation (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

4 hours45 questions
Easy
Medium
Hard
Very Hard
1
2 marks

straight f open parentheses x close parentheses equals x cubed plus 2 x squared minus 8 x plus 5

Find straight f apostrophe apostrophe open parentheses x close parentheses

How did you do?

2
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2 marks

The curve C has equation y equals straight f open parentheses x close parentheses

The curve

  • passes through the point P open parentheses 3 comma space minus 10 close parentheses

  • has a turning point at P

Given that

fraction numerator straight d y over denominator straight d x end fraction equals 2 x cubed minus 9 x squared plus 5 x plus k

where k is a constant,

show that k equals 12.

How did you do?

3
1 mark
Graph of a positive cubic curve C which intersects the x axis at the origin and a minimum point touches the x axis at the point x=6. The maximum point with coordinates (2, 8) is marked on the graph.
Figure 1

Figure 1 shows a sketch of a curve C with equation y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis is a cubic expression in x.

The curve

  • passes through the origin

  • has a maximum turning point at left parenthesis 2 comma space 8 right parenthesis

  • has a minimum turning point at left parenthesis 6 comma space 0 right parenthesis

Write down the set of values of x for which

straight f apostrophe left parenthesis x right parenthesis less than 0

How did you do?

4
2 marks

The curve C has equation y equals straight f left parenthesis x right parenthesis where x element of straight real numbers

Given that

  • straight f prime left parenthesis x right parenthesis equals 2 x plus 1 half cos space x

  • the point P left parenthesis 0 comma space 3 right parenthesis lies on C

find the equation of the tangent to the curve at P, giving your answer in the form y equals m x plus c, where m and c are constants to be found.

How did you do?

5a
2 marks

A curve C has equation

y = 3x^{2} - 2x \qquad x \in \mathbb{R}

Find \frac{\text{d}y}{\text{d}x}.

How did you do?

5b
2 marks

The points P and Q lie on C and have x-coordinates 3 and -2 respectively.

Find the gradient of C at P and the gradient of C at Q.

How did you do?

6a
3 marks

A curve C has equation

y = 2x^{3} - 3x^{2} - 1 \qquad x \in \mathbb{R}

The point P(2, 3) lies on C.

Find the gradient of C at the point P.

How did you do?

6b
2 marks

Hence find the equation of the tangent to C at P, giving your answer in the form y = mx + c, where m and c are constants to be found.

How did you do?

7a
2 marks

The function \text{f} is defined by

\text{f}(x) = x^{3} + x^{2} - 5x \qquad x \in \mathbb{R}

Find \text{f}'(x).

How did you do?

7b
2 marks

Using algebra, solve the equation 3x^{2} + 2x - 5 = 0.

How did you do?

7c
2 marks

Hence find the set of values of x for which \text{f} is a decreasing function.

How did you do?

8a
3 marks

A curve C has equation

y = 3x^{3} + 6x^{2} - 5x + 1 \qquad x \in \mathbb{R}

Find \frac{\text{d}y}{\text{d}x} and \frac{\text{d}^{2}y}{\text{d}x^{2}}.

How did you do?

8b
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4 marks

Verify that C has a stationary point at x = \frac{1}{3} and determine its nature, giving a reason for your answer.

How did you do?

9a
2 marks

A curve C has equation

y = 3x - \frac{1}{2}x^{2} \qquad x \in \mathbb{R}

The point P lies on C and has x-coordinate 5.

Find the gradient of C at P.

How did you do?

9b
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3 marks

Find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers to be found.

How did you do?

10
3 marks

The function \text{f} is defined by

\text{f}(x) = 2x^{2} - 16x \qquad x \in \mathbb{R}

Find the set of values of x for which \text{f} is an increasing function.

How did you do?

11
4 marks

A curve C has equation

y = \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - 6x + 2 \qquad x \in \mathbb{R}

Find the x-coordinates of the stationary points on C.

How did you do?

12
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5 marks

A curve C has equation

y = 2x^{2} - \frac{2}{3}x^{3} - \frac{5}{3} \qquad x \in \mathbb{R}

Show that the point (2, 1) is a local maximum point on C.

How did you do?

1a
2 marks

A curve has equation

y equals 2 over 3 x cubed minus 7 over 2 x squared minus 4 x plus 5

Find fraction numerator straight d y over denominator straight d x end fraction writing your answer in simplest form.

How did you do?

1b
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4 marks

Hence find the range of values of x for which y is decreasing.

How did you do?

2a
3 marks

A curve C has equation

y equals x squared minus 2 x minus 24 square root of x comma space space space space space space space space x greater than 0

Find

(i) fraction numerator straight d y over denominator straight d x end fraction

(ii) fraction numerator straight d squared y over denominator straight d x squared end fraction

How did you do?

2b
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2 marks

Verify that C has a stationary point when x equals 4.

How did you do?

2c
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2 marks

Determine the nature of this stationary point, giving a reason for your answer.

How did you do?

3a
2 marks
A graph showing a parabola opening upwards, with points P(2, 10) and Q on the curve labelled. On the parabola, point Q is above and to the right of point P. Axes are marked with origin O, x and y directions.
Figure 1

Figure 1 shows part of the curve with equation y equals 3 x squared minus 2

The point P open parentheses 2 comma space 10 close parentheses lies on the curve.

Find the gradient of the tangent to the curve at P.

How did you do?

3b
3 marks

The point Q with x coordinate 2 plus h also lies on the curve.

Find the gradient of the line P Q, giving your answer in terms of h in simplest form.

How did you do?

3c
1 mark

Explain briefly the relationship between part (b) and the answer to part (a).

How did you do?

4
5 marks

In this question your must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve C has equation

table row cell y equals 1 third x squared minus 2 square root of x plus 3 end cell blank cell x greater or equal than 0 end cell end table

The point P lies on C and has x coordinate 4

The line l is tangent to C at P.

Show that l has equation

13 x minus 6 y minus 26 equals 0

How did you do?

5
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3 marks

The function \text{f} is defined by

\text{f}(x) = -9x^2 + 5x - 3 \qquad x \in \mathbb{R}

Find the set of values of x for which \text{f} is an increasing function.

How did you do?

6
3 marks

The function \text{f} is defined by

\text{f}(x) = x^3 - 3x^2 + 6x - 7 \qquad x \in \mathbb{R}

Show that \text{f} is increasing for all x \in \mathbb{R}.

How did you do?

7a
1 mark

The curve C has equation

y = 2x^3 - 3x^2 + 4x - 3

Show that the point P(2, 9) lies on C.

How did you do?

7b
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3 marks

Show that the value of \frac{\text{d}y}{\text{d}x} at P is 16.

How did you do?

7c
2 marks

Find an equation of the tangent to C at P.

How did you do?

8a
2 marks

The curve C has equation

y = 3x^2 - 6 + \frac{4}{x}

The point P(1, 1) lies on C.

Find an expression for \frac{\text{d}y}{\text{d}x}.

How did you do?

8b
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3 marks

Show that an equation of the normal to C at P is

x + 2y = 3

How did you do?

8c
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2 marks

The normal cuts the x-axis at the point Q.

Find the length of PQ, giving your answer as an exact value.

How did you do?

9a
2 marks

Given that

y = 2x^3 - 8\sqrt{x}

find \frac{\text{d}y}{\text{d}x}.

How did you do?

9b
2 marks

Find \frac{\text{d}^2y}{\text{d}x^2}.

How did you do?

10a
3 marks

A curve has the equation

y = x^3 - 12x + 7

Find expressions for \frac{\text{d}y}{\text{d}x} and \frac{\text{d}^2y}{\text{d}x^2}.

How did you do?

10b
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3 marks

Determine the coordinates of the local minimum of the curve.

How did you do?

11a
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5 marks

The diagram below shows part of the curve with equation

y = x^3 + 11x^2 + 35x + 25

The curve touches the x-axis at A and cuts the x-axis at C. The points A and B are stationary points on the curve.

q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

Using calculus, and showing all your working, find the coordinates of A and B.

How did you do?

11b
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2 marks

Show that (-1, 0) is a point on the curve and explain why this must be the point C.

How did you do?

12a
2 marks

A company manufactures food tins in the shape of cylinders which must have a constant volume of 150\pi \text{ cm}^3. To lessen material costs the company would like to minimise the surface area of the tins.

By first expressing the height h of the tin in terms of its radius r, show that the surface area of the cylinder is given by

S = 2\pi r^2 + \frac{300\pi}{r}

How did you do?

12b
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4 marks

Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.

How did you do?

1a
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4 marks

The curve C has equation y equals straight f open parentheses x close parentheses where

straight f open parentheses x close parentheses equals a x cubed plus 15 x squared minus 39 x plus b

and a and b are constants.

Given

  • the point open parentheses 2 comma space 10 close parentheses lies on C

  • the gradient of the curve at open parentheses 2 comma space 10 close parentheses is negative 3

(i) show that the value of a is negative 2

(ii) find the value of b.

How did you do?

1b
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3 marks

Hence show that C has no stationary points.

How did you do?

2a
2 marks

Factorise completely 9 x minus x cubed

How did you do?

2b
2 marks

The curve C has equation

y equals 9 x minus x cubed

Sketch C showing the coordinates of the points at which the curve cuts the x-axis.

How did you do?

2c
3 marks

The line l has equation y equals k where k is a constant.

Given that C and l intersect at 3 distinct points, find the range of values for k, writing your answer in set notation.

Solutions relying on calculator technology are not acceptable.

How did you do?

3a
3 marks

The curve C has equation

table row cell y equals 5 x to the power of 4 minus 24 x cubed plus 42 x squared minus 32 x plus 11 end cell blank cell x element of straight real numbers end cell end table

Find

(i) fraction numerator straight d y over denominator straight d x end fraction

(ii) fraction numerator straight d squared y over denominator straight d x squared end fraction

How did you do?

3b
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4 marks

(i) Verify that C has a stationary point at x equals 1

(ii) Show that this stationary point is a point of inflection, giving reasons for your answer.

How did you do?

4a
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4 marks

A company makes drinks containers out of metal.

The containers are modelled as closed cylinders with base radius r cm and height h cm and the capacity of each container is 355 cm3

The metal used

  • for the circular base and the curved sides costs 0.04 pence/cm2

  • for the circular top costs 0.09 pence/cm2

Both metals used are of negligible thickness.

Show that the total cost, C pence, of the metal for one container is given by

C equals 0.13 pi r squared plus fraction numerator 28.4 over denominator r end fraction

How did you do?

4b
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4 marks

Use calculus to find the value of r for which C is a minimum, giving your answer to 3 significant figures.

How did you do?

4c
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2 marks

Using fraction numerator straight d squared C over denominator straight d r squared end fraction prove that the cost is minimised for the value of r found in part (b).

How did you do?

4d
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2 marks

Hence find the minimum value of C, giving your answer to the nearest integer.

How did you do?

5a
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4 marks
Prism with a top face in the shape of a sector. The angle of the sector BAC is 0.8 radians, height h and radius r
Figure 5

A company makes toys for children.

Figure 5 shows the design for a solid toy that looks like a piece of cheese.

The toy is modelled so that

  • face A B C is a sector of a circle with radius r cm and centre A

  • angle B A C equals 0.8 radians

  • faces A B C and D E F are congruent

  • edges A D comma space C F and B E are perpendicular to faces A B C and D E F

  • edges A D comma space C F and B E have length h cm

Given that the volume of the toy is 240 space cm cubed show that the surface area of the toy, S space cm squared, is given by

S equals 0.8 r squared plus 1680 over r

making your method clear.

How did you do?

5b
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4 marks

Using algebraic differentiation, find the value of r for which S has a stationary point.

How did you do?

5c
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2 marks

Prove, by further differentiation, that this value of r gives the minimum surface area of the toy.

How did you do?

6
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5 marks

The function \text{f} is defined by

\text{f}(x) = x^3 - 5x^2 + 3x - 2 \qquad x \in \mathbb{R}

Find the set of values of x for which \text{f} is a decreasing function.

How did you do?

7
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3 marks

The function \text{f} is defined by

\text{f}(x) = 7x^2 - 2x(x^2 + 5) \qquad x \in \mathbb{R}

Show that \text{f} is a decreasing function for all x \in \mathbb{R}.

How did you do?

8
Sme Calculator
5 marks

The curve C has equation

y = 3x^2 - 6x + \sqrt{2x}

The point P(2, 2) lies on C.

Find an equation of the tangent to C at P, giving your answer in the form y = mx + c, where m and c are constants to be found.

How did you do?

9
Sme Calculator
6 marks

The curve C has equation

y = \frac{9}{\sqrt{3x}} - \frac{3}{x}

The point P(3, 2) lies on C.

The normal to C at P intersects the x-axis at the point Q.

Find the coordinates of Q.

How did you do?

10a
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3 marks

Given that

y = \frac{4}{x} - \sqrt[3]{\frac{27}{x}}

find \frac{\text{d}y}{\text{d}x}.

How did you do?

10b
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2 marks

Find \frac{\text{d}^2y}{\text{d}x^2}.

How did you do?

11
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5 marks

The curve C has equation

y = x(x + 6)^2 + 4(3x + 11)

Find the coordinates of the stationary point of C and determine its nature.

How did you do?

12a
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3 marks

The diagram below shows a part of the curve with equation y = \text{f}(x), where

\text{f}(x) = 460 - \frac{x^3}{300} - \frac{8100}{x}, \quad x > 0

The point A is the maximum point of the curve.

KTI0dIN4_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

Find \text{f}'(x).

How did you do?

12b
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4 marks

Use your answer to part (a) to find the coordinates of point A.

How did you do?

13a
1 mark

A garden bed is to be divided by fencing into four identical isosceles triangles, arranged as shown in the diagram below:

dVG~C3Lv_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

The base of each triangle is 2x metres, and the equal sides are each y metres in length.

Although x and y can vary, the total amount of fencing to be used is fixed at P metres.

Explain why 0 < x < \frac{P}{6}.

How did you do?

13b
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4 marks

Show that

A^2 = \frac{4}{9} P^2 x^2 - \frac{16}{3} P x^3

where A is the total area of the garden bed.

How did you do?

13c
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4 marks

Using your answer to (b) find, in terms of P, the maximum possible area of the garden bed.

How did you do?

13d
1 mark

Describe the shape of the bed when the area has its maximum value.

How did you do?

1a
4 marks
Diagram of a cylindrical shape with a hemispherical top, showing radius as "r m" and height as "h m" with arrows for dimensions.
Figure 9

[A sphere of radius r has volume 4 over 3 pi r cubed and surface area 4 pi r squared]

A manufacturer produces a storage tank.

The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9.

The walls of the tank are assumed to have negligible thickness.

The cylinder has radius r metres and height h metres and the hemisphere has radius r metres.

The volume of the tank is 6 m3.

Show that, according to the model, the surface area of the tank, in m2, is given by

12 over r plus 5 over 3 pi r squared

How did you do?

1b
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4 marks

The manufacturer needs to minimise the surface area of the tank.

Use calculus to find the radius of the tank for which the surface area is a minimum.

How did you do?

1c
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2 marks

Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

How did you do?

2
Sme Calculator
4 marks

The function \text{f} is defined by

\text{f}(x) = 4x + \frac{3}{x} \qquad x \in \mathbb{R}, \, x \neq 0

Find the set of values of x for which \text{f} is a decreasing function.

How did you do?

3
Sme Calculator
4 marks

The function \text{f} is defined by

\text{f}(x) = \sqrt{x} - \frac{7}{\sqrt{x}} \qquad x > 0

Show that \text{f} is an increasing function for all x > 0.

How did you do?

4a
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7 marks

The curve C has equation

y = 5 - (x - 3)^2

The points A and B lie on C and have x-coordinates 0 and 6 respectively.

The tangents to C at A and B intersect at the point D.

Find the coordinates of D.

How did you do?

4b
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2 marks

Find the exact area of triangle ABD.

How did you do?

5a
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6 marks

A curve C has equation y = \text{f}(x), where

\text{f}(x) = \frac{1}{\sqrt{x}}, \quad x > 0

The point P lies on C such that the normal to C at P passes through the origin O.

Find the coordinates of P, giving your answer in the form \left(2^a, 2^b\right), where a and b are rational constants to be found.

How did you do?

5b
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1 mark

Write down the equation of the normal to C at P.

How did you do?

5c
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4 marks

Show that an equation of the tangent to C at P is

\left(2^{\frac{1}{3}}\right) x + \left(2^{\frac{5}{6}}\right) y = 3

How did you do?

6a
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3 marks

The curve C has equation

y = 7 - 2x^2 + \sqrt{x}, \quad x \geq 0

Find \dfrac{\text{d}y}{\text{d}x} and \dfrac{\text{d}^2 y}{\text{d}x^2}.

How did you do?

6b
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4 marks

The curve C has a stationary point at P.

Find the coordinates of P and determine its nature, justifying your answer.

How did you do?

7a
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3 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve C with equation

y = 3 - \frac{1}{4} x^2, \quad y > 0

The point P(x, y) lies on C and O is the origin.

mao-shtQ_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

Show that the square of the distance from O to P is given by

OP^2 = \frac{1}{16} x^4 - \frac{1}{2} x^2 + 9

How did you do?

7b
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8 marks

Using calculus, find the exact minimum distance from O to C. You must justify that your answer is a minimum.

How did you do?

8a
2 marks

Figure 2 shows the design for a patio table top.

q7a-7-2-applications-of-differentiation-very-hard-a-level-maths-pure

The table top is modelled as a sector of a circle with radius r metres and central angle \theta radians, where 0 < \theta < 2\pi.

The area of the table top is fixed at A m2.

Explain why r > \sqrt{\dfrac{A}{\pi}}.

How did you do?

8b
2 marks

Show that the perimeter P metres of the table top is given by

P = 2r + \frac{2A}{r}

How did you do?

8c
5 marks

Using calculus, show that the minimum possible value for P is equal to the perimeter of a square of area A. Justify that your value is a minimum.

How did you do?