Section 2

Assignment 2

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 2.

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 elements in the specification:

1.1

2.4, 2.5

3.2

4.1

6.1, 6.3, 6.4, 6.5

7.1, 7.2, 7.3

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

assignment.

© 2017 The Open School Trust – National Extension College 1

GCE A level (AS/Part 1) Mathematics ¦ Section 2 ¦ Assignment 2

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. This is the sort of time you would be expected to spend on these questions in an exam situation. At this stage of your studies do not worry if you spend more time than this.

1 6 – 4x – x2 = q – (x + p)2 where p and q are integers.

(a) Find the value of p and the value of q. (3)

(b) State the coordinates of the maximum point. (1)

2 Show that the product of two odd numbers is an odd number. (4)

3 (a) Show that the line y = 4 is a tangent to the curve with

equation y x= + 4 (3)

x

(b) Give the coordinates of the point at which the line touches

the curve. (2)

4 Find the values of x such that 2 log2 x – log2(5x – 8) = 1 (5)

5 Find the set of values of x for which

(a) 3(2 – x) 4 – x (2)

(b) 3x2 + 5x – 2 0 (4)

2 © 2017 The Open School Trust – National Extension College

GCE A level (AS/Part 1) Mathematics ¦ Section 2 ¦ Assignment 2

6

The circle C with centre T and radius r has equation x2+ y2 - 20x-16y+139 = 0

(a) Find the coordinates of the centre of C. (3)

(b) Show that r = 5. (2)

The line L has equation x = 13 and crosses C at points P and Q as shown in the figure above.

(c) Find the y-coordinate of P and the y-coordinate of Q. (3)

7 A curve has equation y = 3x + 2

x

dy

(a) Find dx . (4)

(b) Find the coordinates of the two points on the curve at

which the gradient is 1. (4)

(c) Find the equation of the normal to the curve at the point where x = 2, giving your answer in the form

ax + by + c = 0, where a, b and c are integers. (4)

8 (a) If y = 2x, express (i) 22x+2 (1)

(ii) 22x+2 - 21(2x)+5 (1) in terms of y.

(b) Solve the equation 22x+2 - 21(2x)+5= 0 , giving your answers to 3 significant figures where appropriate. (6)

9 (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of (3 – 2x)6 giving each

term in its simplest form. (4)

© 2017 The Open School Trust – National Extension College 3 GCE A level (AS/Part 1) Mathematics ¦ Section 2 ¦ Assignment 2

(b) Hence find the term in x2 in the expansion of

(1 – x) (3 – 2x)6 (3)

An approximate value of 2.99986 is needed.

(c) Determine a suitable value of x to be substituted into the

expansion in part (a). (2)

(d) Use the first 2 terms of your expansion to find the value of

2.99986 correct to 4 decimal places. (2)

10 The circle C has equation x2 – 2x + y2 + 4y = 20

The line l with equation x + 7y = 12 intersects the circle at the points A and B. The x-coordinate of A is positive.

(a) Find the coordinates of the points A and B. (4)

(b) Find the equation of the tangents to the circle at the points

A and B. (5)

(c) Show that the tangents cross at the point P(2, 5) (4)

(d) Find the length AP. (3)

(Total marks for Assignment 2 = 79)

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 © 2017 The Open School Trust – National Extension College

Assignment 2

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 2.

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 elements in the specification:

1.1

2.4, 2.5

3.2

4.1

6.1, 6.3, 6.4, 6.5

7.1, 7.2, 7.3

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

assignment.

© 2017 The Open School Trust – National Extension College 1

GCE A level (AS/Part 1) Mathematics ¦ Section 2 ¦ Assignment 2

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. This is the sort of time you would be expected to spend on these questions in an exam situation. At this stage of your studies do not worry if you spend more time than this.

1 6 – 4x – x2 = q – (x + p)2 where p and q are integers.

(a) Find the value of p and the value of q. (3)

(b) State the coordinates of the maximum point. (1)

2 Show that the product of two odd numbers is an odd number. (4)

3 (a) Show that the line y = 4 is a tangent to the curve with

equation y x= + 4 (3)

x

(b) Give the coordinates of the point at which the line touches

the curve. (2)

4 Find the values of x such that 2 log2 x – log2(5x – 8) = 1 (5)

5 Find the set of values of x for which

(a) 3(2 – x) 4 – x (2)

(b) 3x2 + 5x – 2 0 (4)

2 © 2017 The Open School Trust – National Extension College

GCE A level (AS/Part 1) Mathematics ¦ Section 2 ¦ Assignment 2

6

The circle C with centre T and radius r has equation x2+ y2 - 20x-16y+139 = 0

(a) Find the coordinates of the centre of C. (3)

(b) Show that r = 5. (2)

The line L has equation x = 13 and crosses C at points P and Q as shown in the figure above.

(c) Find the y-coordinate of P and the y-coordinate of Q. (3)

7 A curve has equation y = 3x + 2

x

dy

(a) Find dx . (4)

(b) Find the coordinates of the two points on the curve at

which the gradient is 1. (4)

(c) Find the equation of the normal to the curve at the point where x = 2, giving your answer in the form

ax + by + c = 0, where a, b and c are integers. (4)

8 (a) If y = 2x, express (i) 22x+2 (1)

(ii) 22x+2 - 21(2x)+5 (1) in terms of y.

(b) Solve the equation 22x+2 - 21(2x)+5= 0 , giving your answers to 3 significant figures where appropriate. (6)

9 (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of (3 – 2x)6 giving each

term in its simplest form. (4)

© 2017 The Open School Trust – National Extension College 3 GCE A level (AS/Part 1) Mathematics ¦ Section 2 ¦ Assignment 2

(b) Hence find the term in x2 in the expansion of

(1 – x) (3 – 2x)6 (3)

An approximate value of 2.99986 is needed.

(c) Determine a suitable value of x to be substituted into the

expansion in part (a). (2)

(d) Use the first 2 terms of your expansion to find the value of

2.99986 correct to 4 decimal places. (2)

10 The circle C has equation x2 – 2x + y2 + 4y = 20

The line l with equation x + 7y = 12 intersects the circle at the points A and B. The x-coordinate of A is positive.

(a) Find the coordinates of the points A and B. (4)

(b) Find the equation of the tangents to the circle at the points

A and B. (5)

(c) Show that the tangents cross at the point P(2, 5) (4)

(d) Find the length AP. (3)

(Total marks for Assignment 2 = 79)

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 © 2017 The Open School Trust – National Extension College

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