Section 3

Assignment 3

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 3.

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:

5.7, 5.8, 2.3

7.3

6.1, 6.2, 6.3, 6.4, 6.6, 6.7, 2.9

8.1, 8.2, 8.3

10.1, 10.2, 10.3, 10.4

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

assignment.

GCE A level (AS/Part 1) Mathematics ? Section 3 ? Assignment 3

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 y = x2 – kvx, where k is a constant. dy

(a) Find (2)

dx

(b) Given that y is decreasing at x = 4, find the set of possible values of k. (2)

2 (a) Given that log2 (3p – 1) – 2log2 q = 3

express p in terms of q (3)

(b) Solve the equation e2x – 5ex + 4 = 0, giving your answers

in exact form. (4)

3 (a) Sketch on the same axes the curve C with equation y = 4x – x2 and the line L with equation y = x, showing

the maximum point of C, the points where C crosses the

x-axis and the points where L crosses C. (4)

(b) Calculate the area enclosed by the line L, the curve C and

the positive x-axis. (5)

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

4 (a) The figure below shows the right-angled triangle PQR with angle PQR = ?°

P

Show that sin2 ? + cos2 ? = 1 (3)

(b) Show that ç -tanxöæ÷ç 1 + =1ö÷ cosx æ 1 (4)

ècosx øèsinx ø sinx

5 The figure below shows a sketch of part of the curve C with equation y = x3 – 10x2 + kx, where k is constant.

The point P on C is the maximum turning point.

Given that the x-coordinate of P is 2.

(a) Show that k = 28. (3)

The line through P parallel to the x-axis cuts the y-axis at the point N.

The region R is bounded by C, the y-axis and PN, as shown shaded in the figure above.

(b) Use calculus to find the exact area of R. (6)

6 (a) Solve, for –180° ? 180°

1 – 2cos (? – 25°) = 0 (3)

(b) Solve, for 0 ? 360°

3cos2 ? = 1 + sin ?

GCE A level (AS/Part 1) Mathematics ? Section 3 ? Assignment 3

giving your answers correct to 1 decimal place where

appropriate. (6)

7

A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, x cm, as shown in the figure above.

The volume of the cuboid is 81 cubic centimetres.

(a) Show that the total length, L cm, of the twelve edges of the cuboid is given by

L=12x+162 2 (3) x

(b) Use calculus to find the minimum value of L. (6)

(c) Justify, by further differentiation, that the value of L that

you have found is a minimum. (2)

8 (a) The vectors p, q and r are given as

p ??? -32 ? ,q =?? m6 ??? and r =??? -46 ???

= ?

? ?

Show that for any value of m, p + q is parallel to 2q – r (4)

???? ????

(b) In the triangle ABC, AB=12i +5jand AC=7i +17j

????

(i) Find ? BC . (2)

(ii) Show that triangle ABC is isosceles. (3)

(iii) Find the length of AC and hence show that the angle at B

is 90° (3)

4 © 2017 The Open School Trust – National Extension College GCE A level (AS/Part 1) Mathematics ? Section 3 ? Assignment 3

9

The figure above shows part of the straight-line graph.

(a) Find the equation of the line. (2)

The graph is drawn to represent the equation y = Axn, where A and n are constant.

(b) Hence find the value of A and of n. (5)

(Total marks for Assignment 3 = 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.

Assignment 3

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 3.

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:

5.7, 5.8, 2.3

7.3

6.1, 6.2, 6.3, 6.4, 6.6, 6.7, 2.9

8.1, 8.2, 8.3

10.1, 10.2, 10.3, 10.4

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

assignment.

GCE A level (AS/Part 1) Mathematics ? Section 3 ? Assignment 3

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 y = x2 – kvx, where k is a constant. dy

(a) Find (2)

dx

(b) Given that y is decreasing at x = 4, find the set of possible values of k. (2)

2 (a) Given that log2 (3p – 1) – 2log2 q = 3

express p in terms of q (3)

(b) Solve the equation e2x – 5ex + 4 = 0, giving your answers

in exact form. (4)

3 (a) Sketch on the same axes the curve C with equation y = 4x – x2 and the line L with equation y = x, showing

the maximum point of C, the points where C crosses the

x-axis and the points where L crosses C. (4)

(b) Calculate the area enclosed by the line L, the curve C and

the positive x-axis. (5)

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

4 (a) The figure below shows the right-angled triangle PQR with angle PQR = ?°

P

Show that sin2 ? + cos2 ? = 1 (3)

(b) Show that ç -tanxöæ÷ç 1 + =1ö÷ cosx æ 1 (4)

ècosx øèsinx ø sinx

5 The figure below shows a sketch of part of the curve C with equation y = x3 – 10x2 + kx, where k is constant.

The point P on C is the maximum turning point.

Given that the x-coordinate of P is 2.

(a) Show that k = 28. (3)

The line through P parallel to the x-axis cuts the y-axis at the point N.

The region R is bounded by C, the y-axis and PN, as shown shaded in the figure above.

(b) Use calculus to find the exact area of R. (6)

6 (a) Solve, for –180° ? 180°

1 – 2cos (? – 25°) = 0 (3)

(b) Solve, for 0 ? 360°

3cos2 ? = 1 + sin ?

GCE A level (AS/Part 1) Mathematics ? Section 3 ? Assignment 3

giving your answers correct to 1 decimal place where

appropriate. (6)

7

A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, x cm, as shown in the figure above.

The volume of the cuboid is 81 cubic centimetres.

(a) Show that the total length, L cm, of the twelve edges of the cuboid is given by

L=12x+162 2 (3) x

(b) Use calculus to find the minimum value of L. (6)

(c) Justify, by further differentiation, that the value of L that

you have found is a minimum. (2)

8 (a) The vectors p, q and r are given as

p ??? -32 ? ,q =?? m6 ??? and r =??? -46 ???

= ?

? ?

Show that for any value of m, p + q is parallel to 2q – r (4)

???? ????

(b) In the triangle ABC, AB=12i +5jand AC=7i +17j

????

(i) Find ? BC . (2)

(ii) Show that triangle ABC is isosceles. (3)

(iii) Find the length of AC and hence show that the angle at B

is 90° (3)

4 © 2017 The Open School Trust – National Extension College GCE A level (AS/Part 1) Mathematics ? Section 3 ? Assignment 3

9

The figure above shows part of the straight-line graph.

(a) Find the equation of the line. (2)

The graph is drawn to represent the equation y = Axn, where A and n are constant.

(b) Hence find the value of A and of n. (5)

(Total marks for Assignment 3 = 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.

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