Section 6

Assignment 6

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 6.

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:

1.1, 2.6

2.7, 2.8, 2.9

5.1, 5.2, 5.3, 5.4, 5.5

4.1, 4.2, 4.3

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

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 Using a suitable approximation, find the value of cos?(10-5) correct

to 11 decimal places, where the angle is in radians. (4)

2 A curve has equation y = 1 - 2cos x, where 0 = x = 2p.

Find the coordinates of the points where the curve crosses the x-axis. (2)

3 Use the binomial expansion to find (1.0006)13 correct to 8 d.p. (5)

4 (a) The sketch shows the graph of y = sin-1 x.

Write down the coordinates of the points P and Q, the end-points

of the graph. (2)

(b) Sketch the graph of y = -sin-1(x - 1). (3)

GCE A level (Part 2) Mathematics ? Tutor notes ? Assignment 6

5

The figure above shows the triangle ABC, with AB = 8 cm,

AC = 11 cm and Ð BAC = 0.7 radians. The arc BD, where D lies on AC, is an arc of a circle with centre A and radius 8 cm. The region R, shown shaded in Figure 1, is bounded by the straight lines BC and CD and the arc BD.

Find:

(a) the length of the arc BD (2)

(b) the perimeter of R, giving your answer to 3 significant figures (4)

(c) the area of R, giving your answer to 3 significant figures.

6 A sequence of terms u1, u2, u3, … is defined by (5)

un = 24-n

(a) Write down the exact values of u1, u2, u3. (2)

(b) Find the value of k such that uk = 0.

7 The function f is defined by (2)

5x+1 3

f :x ! 2+x-2- x+2, x 1

x

(a) Show that f (x)= x-2 1, x 1

(b) Find f?-1(x)

The function g is defined by g : x ? x2 +5, x ?? (4)

(3)

(b) Solve fg (3)

8 f (x)=( 3x2)+( 16 )2 =(1-A3x)+(2+B x)+(2+Cx)2 , x 13.

1-3x 2+x

(a) Find the values of A and C and show that B = 0. (4)

(b) Hence, or otherwise, find the series expansion of f(x), in ascending powers of x, up to and including the term in x3.

Simplify each term. (7)

9 The figure below shows part of the graph of y = f?(x), where f?(x) = |x + a| + b and a, b are constants.

It has a minimum point at the point P with coordinates (1, –2) and passes through the x-axis at the point Q with coordinates (3, 0).

(a) Write down the values of a and b (2)

(b) Giving the coordinates of the points corresponding to P and Q in each case, sketch the graphs of

(i) y = 2f?(x) (3)

(ii) y = –f?(2x) (3)

(c) Find the two solutions to the equation f?(x) = ½ x (4)

1

10 (a) If P is irrational, show by the method of contradiction that

P

is also irrational. (4)

(b) Give an example of two irrational numbers, a and b, where

a ¹ b, such that ab is rational. (1)

GCE A level (Part 2) Mathematics ? Tutor notes ? Assignment 6

11

The figure above shows part of the curve with equation y = f (x), x ?? , where f is an increasing function of x. The curve passes through the points P(0, -2) and Q(3, 0) as shown.

In separate diagrams, sketch the curve with equation

(a) y = f?-1?(x) (3)

(b) y = f (3x). (3)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

(Total marks for Assignment 6 = 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 6

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 6.

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:

1.1, 2.6

2.7, 2.8, 2.9

5.1, 5.2, 5.3, 5.4, 5.5

4.1, 4.2, 4.3

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

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 Using a suitable approximation, find the value of cos?(10-5) correct

to 11 decimal places, where the angle is in radians. (4)

2 A curve has equation y = 1 - 2cos x, where 0 = x = 2p.

Find the coordinates of the points where the curve crosses the x-axis. (2)

3 Use the binomial expansion to find (1.0006)13 correct to 8 d.p. (5)

4 (a) The sketch shows the graph of y = sin-1 x.

Write down the coordinates of the points P and Q, the end-points

of the graph. (2)

(b) Sketch the graph of y = -sin-1(x - 1). (3)

GCE A level (Part 2) Mathematics ? Tutor notes ? Assignment 6

5

The figure above shows the triangle ABC, with AB = 8 cm,

AC = 11 cm and Ð BAC = 0.7 radians. The arc BD, where D lies on AC, is an arc of a circle with centre A and radius 8 cm. The region R, shown shaded in Figure 1, is bounded by the straight lines BC and CD and the arc BD.

Find:

(a) the length of the arc BD (2)

(b) the perimeter of R, giving your answer to 3 significant figures (4)

(c) the area of R, giving your answer to 3 significant figures.

6 A sequence of terms u1, u2, u3, … is defined by (5)

un = 24-n

(a) Write down the exact values of u1, u2, u3. (2)

(b) Find the value of k such that uk = 0.

7 The function f is defined by (2)

5x+1 3

f :x ! 2+x-2- x+2, x 1

x

(a) Show that f (x)= x-2 1, x 1

(b) Find f?-1(x)

The function g is defined by g : x ? x2 +5, x ?? (4)

(3)

(b) Solve fg (3)

8 f (x)=( 3x2)+( 16 )2 =(1-A3x)+(2+B x)+(2+Cx)2 , x 13.

1-3x 2+x

(a) Find the values of A and C and show that B = 0. (4)

(b) Hence, or otherwise, find the series expansion of f(x), in ascending powers of x, up to and including the term in x3.

Simplify each term. (7)

9 The figure below shows part of the graph of y = f?(x), where f?(x) = |x + a| + b and a, b are constants.

It has a minimum point at the point P with coordinates (1, –2) and passes through the x-axis at the point Q with coordinates (3, 0).

(a) Write down the values of a and b (2)

(b) Giving the coordinates of the points corresponding to P and Q in each case, sketch the graphs of

(i) y = 2f?(x) (3)

(ii) y = –f?(2x) (3)

(c) Find the two solutions to the equation f?(x) = ½ x (4)

1

10 (a) If P is irrational, show by the method of contradiction that

P

is also irrational. (4)

(b) Give an example of two irrational numbers, a and b, where

a ¹ b, such that ab is rational. (1)

GCE A level (Part 2) Mathematics ? Tutor notes ? Assignment 6

11

The figure above shows part of the curve with equation y = f (x), x ?? , where f is an increasing function of x. The curve passes through the points P(0, -2) and Q(3, 0) as shown.

In separate diagrams, sketch the curve with equation

(a) y = f?-1?(x) (3)

(b) y = f (3x). (3)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

(Total marks for Assignment 6 = 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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