Section 1

Assignment 1

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 1.

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: 2.1, 2.2, 2.3, 2.6, 2.7, 2.8

3.1

5.1, 5.3, 5.5

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 1, 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 1 ¦ Assignment 1

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 Express each of the following in the form 3x

(a) 27 (1)

1

(b)

3 3

2 The triangle ABC has sides AB = 9 cm and BC = 10 cm.

The angle BAC = 40°. (2)

Find the value of the angle BCA, correct to 1 decimal place.

3 A line l passes through the points A(2,3) and B(5,–1) (4)

(a) Find the gradient of the line l (2)

(b) Find the equation of l in the form ax + by + c = 0

4 f (x) = 4x3 – 4x2 – 11x + 6 (3)

(a) Show that (x – 2) is a factor of f(x)

(b) Find the values of the constants a, b and c such that (2)

f(x) = (x – 2)(ax2 + bx + c) (2)

(c) Express f(x) as the product of three linear factors. (2)

5 The graph of f(x) = (x + a)2 + b has a minimum point at P(3, –2).

(a) State the values of a and of b. (2)

(b) Find the value of y where the graph of y = f (x) crosses

the y-axis. (1)

(c) Find the values of x at which the graph of y = f (x)

crosses the x-axis, giving your answers in exact form. (2)

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

(d) Sketch the graph of y = f(x), showing clearly the axis of

symmetry and stating its equation. (3)

6 Find the value(s) of the constant a in each of the following cases

(a) The equation 3x2 – 2ax + 12 = 0 has repeated roots (3)

(b) The polynomial x3 – 5x2 + ax – 2 has (x + 2) as a factor. (3)

7 Simplify

(a) (3 2) (1) 2

48x3

(b) (4x-5/2 )2 (3)

11 5 + where

(c) giving your answer in the form a b c

3 5 2 3-

a and c are integers and b is rational. (4)

8 The figure below shows an equilateral triangle ABC with a square BCDE drawn with BC as base:

9 f(x) = (x – 1)2(x – 2)(x + 3)

(a) Calculate the value of y at the point where the graph of

y = f (x) crosses the y-axis (1)

(b) Sketch the curve of y = f (x), showing clearly the points at which the curve touches or crosses the x-axis. (4)

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

10 (a) Express (i) 4 and (ii) 16 in the form 2x (1)

(b) Hence solve the equation 2(4x+1) = 16x – 2 (4)

11 On separate diagrams and for values of x such that 0!x!360°, sketch the graphs of

(a) y = sin x (1)

(b) y = sin (x + 60°), giving the coordinates of the points where

the graph crosses the x-axis (3)

(c) y = 1 + sin 2x, giving the coordinates of its maximum points (4)

12 The figure below shows the points A(–2, –3), B(4, 2) and C(9, p) with angle ABC = 90°

(a) Find the equation of the line l1 passing through the points A

and B. (2)

(b) Find the equation of the line l2 passing through B and

perpendicular to AB. (3)

(c) Given that the point C(9, p) lies on l2, find the value of p. (3)

(d) D is the point such that ABCD is a square. Find the

coordinates of D. (3)

Total marks for Assignment 1 = 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 © 2017 The Open School Trust – National Extension College

Assignment 1

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 1.

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: 2.1, 2.2, 2.3, 2.6, 2.7, 2.8

3.1

5.1, 5.3, 5.5

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 1, 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 1 ¦ Assignment 1

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 Express each of the following in the form 3x

(a) 27 (1)

1

(b)

3 3

2 The triangle ABC has sides AB = 9 cm and BC = 10 cm.

The angle BAC = 40°. (2)

Find the value of the angle BCA, correct to 1 decimal place.

3 A line l passes through the points A(2,3) and B(5,–1) (4)

(a) Find the gradient of the line l (2)

(b) Find the equation of l in the form ax + by + c = 0

4 f (x) = 4x3 – 4x2 – 11x + 6 (3)

(a) Show that (x – 2) is a factor of f(x)

(b) Find the values of the constants a, b and c such that (2)

f(x) = (x – 2)(ax2 + bx + c) (2)

(c) Express f(x) as the product of three linear factors. (2)

5 The graph of f(x) = (x + a)2 + b has a minimum point at P(3, –2).

(a) State the values of a and of b. (2)

(b) Find the value of y where the graph of y = f (x) crosses

the y-axis. (1)

(c) Find the values of x at which the graph of y = f (x)

crosses the x-axis, giving your answers in exact form. (2)

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

(d) Sketch the graph of y = f(x), showing clearly the axis of

symmetry and stating its equation. (3)

6 Find the value(s) of the constant a in each of the following cases

(a) The equation 3x2 – 2ax + 12 = 0 has repeated roots (3)

(b) The polynomial x3 – 5x2 + ax – 2 has (x + 2) as a factor. (3)

7 Simplify

(a) (3 2) (1) 2

48x3

(b) (4x-5/2 )2 (3)

11 5 + where

(c) giving your answer in the form a b c

3 5 2 3-

a and c are integers and b is rational. (4)

8 The figure below shows an equilateral triangle ABC with a square BCDE drawn with BC as base:

9 f(x) = (x – 1)2(x – 2)(x + 3)

(a) Calculate the value of y at the point where the graph of

y = f (x) crosses the y-axis (1)

(b) Sketch the curve of y = f (x), showing clearly the points at which the curve touches or crosses the x-axis. (4)

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

10 (a) Express (i) 4 and (ii) 16 in the form 2x (1)

(b) Hence solve the equation 2(4x+1) = 16x – 2 (4)

11 On separate diagrams and for values of x such that 0!x!360°, sketch the graphs of

(a) y = sin x (1)

(b) y = sin (x + 60°), giving the coordinates of the points where

the graph crosses the x-axis (3)

(c) y = 1 + sin 2x, giving the coordinates of its maximum points (4)

12 The figure below shows the points A(–2, –3), B(4, 2) and C(9, p) with angle ABC = 90°

(a) Find the equation of the line l1 passing through the points A

and B. (2)

(b) Find the equation of the line l2 passing through B and

perpendicular to AB. (3)

(c) Given that the point C(9, p) lies on l2, find the value of p. (3)

(d) D is the point such that ABCD is a square. Find the

coordinates of D. (3)

Total marks for Assignment 1 = 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 © 2017 The Open School Trust – National Extension College

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