Section 7

Assignment 7

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 7.

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 Maths part of the specification:

5.6, 5.8

4.4, 4.5, 4.6

7.1, 7.2, 7.4

8.2, 8.3, 8.4, 8.6

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 7, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

© 2018 The Open School Trust – National Extension College 1 GCE A level (Part 2) Mathematics ? Section 7 ? Assignment 7

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 Find the value of xdx (3)

2 The adult population of a town is 25 000 at the end of Year 1.

A model predicts that the adult population of the town will increase by 3 per cent each year, forming a geometric sequence.

(a) Show that the predicted adult population at the end of Year 2

is 25 750 (1)

(b) Write down the common ratio of the geometric sequence. (1)

The model predicts that Year N will be the first year in which the adult population of the town exceeds 40?000

(c) Show that

(N – 1) log1.03 log1.6 (3)

(d) Find the value of N.

3 (a) Differentiate with respect to x (2)

(i) e3x + ln 2x (2)

(ii) 5 + x2 (3)

(b) (i) Differentiate e3x cos x with respect to x. (2)

(ii) The curve C has equation y = e3x cos x

Find an equation of the tangent to C at the point where x = 0

(3)

4 (a) Using the identity cos2 ? + sin2 ? = 1, show that

tan2 ? = sec2 ? – 1 (2)

(b) Solve, for 2tan 0°???360°2 , the equation ? = 2 (6)

2 ? + 4 sec ? + sec

5 A farmer has a pay scheme to keep fruit pickers working throughout

2 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 7 ? Assignment 7

the 30-day season.

He pays £a for their first day, £(a + d) for their second day, £(a + 2d) for their third day, and so on, thus increasing the daily payment by £d for each extra day they work.

A picker who works for all 30 days will earn £103.50 on the final day. (a) Use this information to form an equation in a and d. (2) A picker who works all 30 days will earn a total of £2452.50.

(b) Show that 15(a + 103.5) = 2452.5 (2)

(c) Hence find the value of the first day’s pay and the daily increase in payment. (4)

6 (a) Find the x-coordinates of the turning points of the curve

y = x3 – 3x2 – 9x + 11. (3)

(b) Find the nature of each of these turning points. (3)

(c) Find the coordinates of the point of inflection of the curve (3)

7 (a) Use the identity cos(A + B) = cosAcosB – sinAsinB to show that

cos 2A = 1 – 2sin2A (2)

The curves C1 and C2 have equations

C1: y = 3sin 2x

C2: y = 4sin2x – 2cos 2x

(b) Show that the x-coordinate of the points where C1 and C2 intersect

satisfy the equation 4cos 2x + 3sin 2x = 2 (3)

(c) Express 4cos 2x + 3sin 2x in the form R cos(2x – a), where R 0 and 0 a 90°, giving the value of a to 2 decimal places. (3)

(d) Hence find, for 0 x 180°, all the solutions of the equation

4cos 2x + 3sin 2x = 2 giving your answers to 1 decimal place. (4)

© 2018 The Open School Trust – National Extension College 3 GCE A level (Part 2) Mathematics ? Section 7 ? Assignment 7

8 g(x)=

A B

(a) Express g(x) in the form + , where A and B are 1+4x 2-x

constants to be found. (3)

The finite region R is bounded by the curve with equation y = g(x), the coordinate axes and the line x = ½

(b) Find the area of R, giving your answer in the form a ln 2 + b ln 3 (7)

9 (a) Prove that

2 x (3)

(i) 1+cosx = 2cos

2 sinx x

(ii) = tan (2)

1+cosx 2

(b) Hence show that tan 22.5°= 2-1 (3)

Total marks for Assignment 7 = 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 © 2018 The Open School Trust – National Extension College

Assignment 7

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 7.

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 Maths part of the specification:

5.6, 5.8

4.4, 4.5, 4.6

7.1, 7.2, 7.4

8.2, 8.3, 8.4, 8.6

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 7, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

© 2018 The Open School Trust – National Extension College 1 GCE A level (Part 2) Mathematics ? Section 7 ? Assignment 7

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 Find the value of xdx (3)

2 The adult population of a town is 25 000 at the end of Year 1.

A model predicts that the adult population of the town will increase by 3 per cent each year, forming a geometric sequence.

(a) Show that the predicted adult population at the end of Year 2

is 25 750 (1)

(b) Write down the common ratio of the geometric sequence. (1)

The model predicts that Year N will be the first year in which the adult population of the town exceeds 40?000

(c) Show that

(N – 1) log1.03 log1.6 (3)

(d) Find the value of N.

3 (a) Differentiate with respect to x (2)

(i) e3x + ln 2x (2)

(ii) 5 + x2 (3)

(b) (i) Differentiate e3x cos x with respect to x. (2)

(ii) The curve C has equation y = e3x cos x

Find an equation of the tangent to C at the point where x = 0

(3)

4 (a) Using the identity cos2 ? + sin2 ? = 1, show that

tan2 ? = sec2 ? – 1 (2)

(b) Solve, for 2tan 0°???360°2 , the equation ? = 2 (6)

2 ? + 4 sec ? + sec

5 A farmer has a pay scheme to keep fruit pickers working throughout

2 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 7 ? Assignment 7

the 30-day season.

He pays £a for their first day, £(a + d) for their second day, £(a + 2d) for their third day, and so on, thus increasing the daily payment by £d for each extra day they work.

A picker who works for all 30 days will earn £103.50 on the final day. (a) Use this information to form an equation in a and d. (2) A picker who works all 30 days will earn a total of £2452.50.

(b) Show that 15(a + 103.5) = 2452.5 (2)

(c) Hence find the value of the first day’s pay and the daily increase in payment. (4)

6 (a) Find the x-coordinates of the turning points of the curve

y = x3 – 3x2 – 9x + 11. (3)

(b) Find the nature of each of these turning points. (3)

(c) Find the coordinates of the point of inflection of the curve (3)

7 (a) Use the identity cos(A + B) = cosAcosB – sinAsinB to show that

cos 2A = 1 – 2sin2A (2)

The curves C1 and C2 have equations

C1: y = 3sin 2x

C2: y = 4sin2x – 2cos 2x

(b) Show that the x-coordinate of the points where C1 and C2 intersect

satisfy the equation 4cos 2x + 3sin 2x = 2 (3)

(c) Express 4cos 2x + 3sin 2x in the form R cos(2x – a), where R 0 and 0 a 90°, giving the value of a to 2 decimal places. (3)

(d) Hence find, for 0 x 180°, all the solutions of the equation

4cos 2x + 3sin 2x = 2 giving your answers to 1 decimal place. (4)

© 2018 The Open School Trust – National Extension College 3 GCE A level (Part 2) Mathematics ? Section 7 ? Assignment 7

8 g(x)=

A B

(a) Express g(x) in the form + , where A and B are 1+4x 2-x

constants to be found. (3)

The finite region R is bounded by the curve with equation y = g(x), the coordinate axes and the line x = ½

(b) Find the area of R, giving your answer in the form a ln 2 + b ln 3 (7)

9 (a) Prove that

2 x (3)

(i) 1+cosx = 2cos

2 sinx x

(ii) = tan (2)

1+cosx 2

(b) Hence show that tan 22.5°= 2-1 (3)

Total marks for Assignment 7 = 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 © 2018 The Open School Trust – National Extension College

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