Section 8

Assignment 8

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 8.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in the topics listed above. These address the following in Pure Mathematics part of the specification:

9.1, 9.2, 9.3, 9.4

7.4, 7.5

3.3, 7.5

8.5

7.6, 8.7, 8.8

How your tutor will mark your work

Your tutor will:

??check that you have answered each question ??check your workings as well as your answer ??give you feedback ??make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 8, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

© 2019 The Open School Trust – National Extension College 1

.

GCE A level (Part 2) Mathematics ? Section 8 ? Assignment 8

What to do

Answer as many questions as you can.

It is always important to show your working in your responses.

Guide time

The guide time for this assignment is 1 hour 30 minutes.

1 Show that the area enclosed by the curve defined parametrically by equations x = t y, =t3 for t =0 and the x-axis between x = 0

8

and x = 2 is given by 2 (4)

7

2 The radius of a circular disc is increasing at the rate of 0.75 cms-1.

Find the rate at which the area is increasing when the radius is 40 cm,

leaving your answer in terms of p . (5)

3 Find, by using the substitution x = sin ?, the exact value of the integral

(

??dx (7)

)0

4 A curve has equation x2 + 2xy - 3y2 + 16 = 0.

dy

Find the coordinates of the points on the curve where = 0. (7) dx

18

5 f (x)= 3 x + -20 x

(a) Show that the equation f?(x) = 0 has a root a in the interval

[1.1,1.2]. (2)

(b) Find f '?(x). (3)

(c) Using x0 = 1.1 as a first approximation to a, apply the NewtonRaphson procedure once to f?(x) to find a second approximation

to a, giving your answer to 3 significant figures. (2)

2 © 2019 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 8 ? Assignment 8

6

The diagram above shows the graph of the curve with equation y = xe2x, x = 0.

The finite region R bounded by the lines x = 1, the x-axis and the curve is shown shaded in the figure.

(a) Use integration to find the exact value for the area of R. (5) (b) Complete a copy of the table below with the values of y

corresponding to x = 0.4 and 0.8. (1)

Table

x 0 0.2 0.4 0.6 0.8 1

y = xe2x 0 0.29836 1.99207 7.38906

(c) Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4

significant figures. (4)

7 A curve has parametric equations x = 2cot t, y = 2sin2 t, 0 t .

dy

(a) Find an expression for in terms of the parameter t. (4) dx

(b) Find an equation of the tangent to the curve at the point where

t = . (4)

(c) Find a Cartesian equation of the curve in the form y = f?(x).

State the domain on which the curve is defined. (4)

8 f?(x) = 2x3 - x - 4

(a) Show that the equation f?(x) = 0 can be written as

2 1

x = + (3) x 2

© 2019 The Open School Trust – National Extension College 3

GCE A level (Part 2) Mathematics ? Section 8 ? Assignment 8

The equation 2x3 - x - 4 = 0 has a root between 1.35 and 1.4.

(b) Use the iteration formula

2 1

xn+1 = + ,

xn 2

with x0 = 1.35, to find (to 2 d.p.), the values of x1, x2 and x3.

The only real root of f?(x) = 0 is a.

(c) By choosing a suitable interval, prove that a = 1.392, to 3 (3)

decimal places. (3)

9 Liquid is pouring into a container at a constant rate of 20 cm3 s-1 and is leaking out at a rate proportional to the volume of liquid aleady in the container.

(a) Explain why, at time t seconds, the volume, V cm3, of liquid in the container satisfies the differential equation dV

= 20-kV ,

dt

where k is a positive constant. (3)

The container is initially empty.

(b) By solving the differential equation, show that V = A + Be-kt,

giving values of A and B in terms of k. (6) dV

Given also that =10 when t = 5,

dt

(c) find the volume of liquid in the container at 10 s after the start. (5)

Total marks for Assignment 8 = 75

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your tutor will send you helpful feedback and advice to help you progress through the course.

4 © 2019 The Open School Trust – National Extension College

Assignment 8

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 8.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in the topics listed above. These address the following in Pure Mathematics part of the specification:

9.1, 9.2, 9.3, 9.4

7.4, 7.5

3.3, 7.5

8.5

7.6, 8.7, 8.8

How your tutor will mark your work

Your tutor will:

??check that you have answered each question ??check your workings as well as your answer ??give you feedback ??make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 8, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

© 2019 The Open School Trust – National Extension College 1

.

GCE A level (Part 2) Mathematics ? Section 8 ? Assignment 8

What to do

Answer as many questions as you can.

It is always important to show your working in your responses.

Guide time

The guide time for this assignment is 1 hour 30 minutes.

1 Show that the area enclosed by the curve defined parametrically by equations x = t y, =t3 for t =0 and the x-axis between x = 0

8

and x = 2 is given by 2 (4)

7

2 The radius of a circular disc is increasing at the rate of 0.75 cms-1.

Find the rate at which the area is increasing when the radius is 40 cm,

leaving your answer in terms of p . (5)

3 Find, by using the substitution x = sin ?, the exact value of the integral

(

??dx (7)

)0

4 A curve has equation x2 + 2xy - 3y2 + 16 = 0.

dy

Find the coordinates of the points on the curve where = 0. (7) dx

18

5 f (x)= 3 x + -20 x

(a) Show that the equation f?(x) = 0 has a root a in the interval

[1.1,1.2]. (2)

(b) Find f '?(x). (3)

(c) Using x0 = 1.1 as a first approximation to a, apply the NewtonRaphson procedure once to f?(x) to find a second approximation

to a, giving your answer to 3 significant figures. (2)

2 © 2019 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 8 ? Assignment 8

6

The diagram above shows the graph of the curve with equation y = xe2x, x = 0.

The finite region R bounded by the lines x = 1, the x-axis and the curve is shown shaded in the figure.

(a) Use integration to find the exact value for the area of R. (5) (b) Complete a copy of the table below with the values of y

corresponding to x = 0.4 and 0.8. (1)

Table

x 0 0.2 0.4 0.6 0.8 1

y = xe2x 0 0.29836 1.99207 7.38906

(c) Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4

significant figures. (4)

7 A curve has parametric equations x = 2cot t, y = 2sin2 t, 0 t .

dy

(a) Find an expression for in terms of the parameter t. (4) dx

(b) Find an equation of the tangent to the curve at the point where

t = . (4)

(c) Find a Cartesian equation of the curve in the form y = f?(x).

State the domain on which the curve is defined. (4)

8 f?(x) = 2x3 - x - 4

(a) Show that the equation f?(x) = 0 can be written as

2 1

x = + (3) x 2

© 2019 The Open School Trust – National Extension College 3

GCE A level (Part 2) Mathematics ? Section 8 ? Assignment 8

The equation 2x3 - x - 4 = 0 has a root between 1.35 and 1.4.

(b) Use the iteration formula

2 1

xn+1 = + ,

xn 2

with x0 = 1.35, to find (to 2 d.p.), the values of x1, x2 and x3.

The only real root of f?(x) = 0 is a.

(c) By choosing a suitable interval, prove that a = 1.392, to 3 (3)

decimal places. (3)

9 Liquid is pouring into a container at a constant rate of 20 cm3 s-1 and is leaking out at a rate proportional to the volume of liquid aleady in the container.

(a) Explain why, at time t seconds, the volume, V cm3, of liquid in the container satisfies the differential equation dV

= 20-kV ,

dt

where k is a positive constant. (3)

The container is initially empty.

(b) By solving the differential equation, show that V = A + Be-kt,

giving values of A and B in terms of k. (6) dV

Given also that =10 when t = 5,

dt

(c) find the volume of liquid in the container at 10 s after the start. (5)

Total marks for Assignment 8 = 75

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your tutor will send you helpful feedback and advice to help you progress through the course.

4 © 2019 The Open School Trust – National Extension College

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