aqa level 2 certificate in further mathematics worksheets teacher booklet pdf

M Method marks are awarded for a correct method which could lead to a correct answer.. A Accuracy marks are awarded when following on from a correct method.. B Marks awarded independent

Further Mathematics

Worksheets - Teacher Booklet

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2 Geometric Problems and Proof 17

Glossary for Mark Schemes

These examinations are marked in such a way as to award positive achievement wherever possible.Thus,for these papers,marks are awarded under various categories

M Method marks are awarded for a correct method which could lead

to a correct answer

A Accuracy marks are awarded when following on from a correct

method It is not necessary to always see the method This can beimplied

B Marks awarded independent of method

M Dep A method mark dependent on a previous method mark being

awarded

B Dep A mark that can only be awarded if a previous independent mark

has been awarded

ft Follow through marks Marks awarded following a mistake in an

earlier step

SC Special case Marks awarded within the scheme for a common

misinterpretation which has some mathematical worth

oe Or equivalent Accept answers that are equivalent

eg, accept 0.5 as well as

2 1

Write down the equation of each of these circles.

(d) Centre (8, 15) radius 17

Does this circle pass through the origin?

912

PQ is a diameter of a circle, centre C.

Question 5 (non-calculator)

A (12, 6) and B (14, 4) are two points on a circle, centre C (20, 12).

(a) Work out the coordinates of the midpoint M, of AB (2 marks)

Not drawnaccurately

(0,2), (0, 12) and (4, 12) are three points on a circle, centre C.

Question 7

AB is a diameter of the circle ABC.

Mark Scheme

Gradient AC =

24

36

B (6, k)

C (4, 6)

A (2, 3)

A circle has equation (x 5)2

Question 10

A circle passes through the points (0, 3) and (0, 11) and has centre (6, k)

(a) Work out the value of k.

The equation of this circle, centre C, is (x 3)2

+(y 5)2

=17

P (4, 1) is a point on the circle.

(a) Show working to explain why OP is a tangent to the circle (5 marks)

(b) Show that the length OP is equal to the radius of the circle (3 marks)

Mark Scheme

(a) C is (3, 5) B1

Gradient CP =

43

15

O

C

P (4, 1)

x y

Question 12 (non-calculator)

The equation of this circle is x2+ y2= 20

P (4, 2) is a point on the circle.

Work out the equation of the tangent to the circle at P.

x y

O

SQ is a tangent to the circle at Q.

PR=QR

Mark Scheme

Let angle SQR = x

 angle RPQ = x alternate segment

M1 Any order of angles

 angle RQP = x isosceles triangle M1

Not drawnaccurately

P R

Q S

PQRS is a cyclic quadrilateral (converse of)

opposite angles add up to 180

A1 SC2 ‘Correct’ solution without reasons

Q

Not drawnaccurately

P

Question 4

O is the centre of the circle.

AOBC and EDC are straight lines.

D E

y

x

QS bisects both of the angles PSR and PQR.

-2 base angles equal

A1 SC2 ‘Correct’ solution without reasons

Not drawnaccurately

T

S X

R

O is the centre of the circle.

AB bisects angle OBC.

Angle at centre = 2 angle at circumference

A1 SC3 ‘Correct’ solution without reasons

Not drawnaccurately

O

B

A

y O x

Ox O

C

 TQP = x =  RST exterior angle of cyclic

quadrilateral = opposite interior angle

  VTR =  RST

PVT is tangent

(converse of) alternate segment theorem

A2 SC3 ‘Correct’ solution without reasons

Not drawnaccurately

P

T

S R

x

ABF is a common tangent to the two circles at A and B.

CDE is a straight line.

AC is parallel to BD.

Mark Scheme

  DCA = x corresponding angles

equal

M1

ie, DAB =  EBF

 AD is parallel to BE

(converse of) corresponding angles equal

A2 SC3 ‘Correct’ solution without reasons

Not drawnaccurately

8y + 24 + 6  3y or 5y + 30 M2 M1 4 terms with 3 correct

5y + 30 and 5(y + 6) A1 oe eg, 5y + 30 and states both terms

8a3+ 4a2 4a2 or 8a3 M2 M1 3 terms with 2 correct

number

a and b are positive integers.

a < b

Prove that

b bx

a ax

and explains that as numerator is smaller

than denominator value will be < 1

Question 5

(a) Express x2+ 6x + 11 in the form (x + a)2+ b where a and b are integers (2 marks)

(b) Hence, prove that x2+ 6x + 11 is always positive (2 marks)

f(x) = (2x + 3)2+ 8(x + 2) for all values of x.

Prove that there is exactly one value of x for which f(x) = 0 (4 marks)

The nth term of a sequence is

2

1

n(n + 1)

(a) Work out an expression for the (n 1)th term of the sequence

(b) Hence, or otherwise, prove that the sum of any consecutive pair of terms of the

Question 9

Prove that

105

A1 oe eg, Explains that 10 > 0 and

Work out the exact value of sin 60+sin 120+sin 270.

Mark Scheme

Question 3 (non-calculator)

Work out the area of triangle ABC.

Write your answer in its simplest form

C

A

B

26

5 cm

45

Show that tan2 

θcos

θsin

θcos1

2 2

tan 

θcos

1

required for all 3 marks

Alt

θcos

θcos

required for all 3 marks

ACD = [53.1, 53.13010235] A1 Allow 53 with method seen

Angle ABD = [53.1, 53.13010235] B1 ft ft From 3rd mark their angle ACD

Not drawnaccurately

B

D

A hanging basket is made from a hemisphere and three chains.

The radius of the hemisphere is 10 cm

Each chain is 30 cm long

The chains are equally spaced around the rim of the hemisphere

Work out angle AOB.

(5 marks)

Mark Scheme

A triangle formed with A, B and the centre of

the hemisphere with 2 sides of 10 cm and an

O

ft For other two solutions

Question 8 (calculator)

Solve the following equation for 0   360

Mark Scheme

(3cos  1)( cos  + 1) M2 M2 Fully correct use of quadratic formula

M1 (acos  + b)(ccos  + d) where ac = 3 and bd = 1 or

quadratic formula with one sign error

25

48

410

23

23

25

21

32

21

(12 marks)

Mark Scheme

Each question 2 marks M1 for a correct row by column multiplication A1 for the correct answer

2 3 2 1

42

2 1

57

23

43

1 3 1

3

2

37

22

3 marks per question 1 mark for multiplication of row by column, 1 mark for 2 simplified elements,

1 for other 2 elements correct Part (c) 2 marks.

10

22

1623

12 19 6

727

322216

Work out, giving your answers as simply as possible.

03

10

04

01

01

(13 marks)

Mark Scheme

Each question 2 marks M1 for a correct row by column multiplication A1 for the correct answer

(f) 3 marks 2 for 1 pair correctly multiplied, 1 for final answer.

01

a a

a a

1

a a

a

a (f) 2

19

33

(14 marks)

Mark Scheme

(a) to (d) 2 marks each

(e) and (f) 3 marks each, 1 for a correct multiplication, 1 for two elements correct, 1 for all correct.

x x

1517

69

2

0

x x x

y y y

96

36

31227

9

2 2

x x x

x x

47

32

41

36

013

Question 2

P= 52 10 Q= 34 12 C=  23 Work out

(2 marks)

The point A(m,n) is transformed to the point A (2, 0) by the matrix 

32

Work out the values ofmandn

The matrix A represents a reflection in the liney = x

Write down the matrix A

The unit square is transformed by the matrix A and then by rotation through90 aboutO

Work out the matrix representing the combined transformation

(2 marks)

Mark Scheme

Reflection, in the line y = x B1, B1

Question 9 (non-calculator)

The unit square OABC is transformed by the matrix 0h 0h to the square OABC.

The area of OABC is 27.

Work out the exact value ofh

The point P (2, 7) is transformed by matrix BA to P.

Show that P lies on the line 7x +2y =0

or 4 correct with one incorrect

A rhombus and a rectangle are shown.

The perimeter of the rhombus is greater than the perimeter of the rectangle

Mark Scheme

4(2y + 6) > 2y + 10 + 2y + 10 + y + 4 + y + 4 M2 oe eg, 8y + 24 > 6y + 28

M1 4(2y + 6) or 2y + 10 + 2y + 10 + y + 4 + y + 4

Question 5

p < – 1 and q > 1

Tick the correct box for each statement

Always true Sometimes true Never true

(a) Write down the coordinates of points A and B (2 marks)

Question 7

(b) Sketch y = x2+ 3x

Label the x values of the points of intersection with the x-axis (2 marks)

Mark Scheme

(a) x(x + 3) B1

0 and 3 labelled on x-axis A1 ft ft Their factors in (a)

(c) x <  3 and x > 0 B2 ft ft Their factors in (a)

1 < x <

2

A triangle and a square are shown.

Work out the range of values of n for which

area of triangle < area of square

Question 4

Work out the range for each of these functions

Give a reason whyx 0 is not a suitable domain for f(x) (1 mark)

Mark Scheme

(a) Not defined when x = 3

or cannot divide by 0 when x = 3

Question 8

(b) Sketch the function f(x)= x2 5x 14 for allx

Label the points of intersection with thexandyaxes (3 marks)

(3, 0) and (7, 0) marked or used M1

(1, 2) and (4,1) marked or used M1

Either of their triangular areas calculated

For each of these straight lines, work out

(ii) The gradient of the line that is perpendicular to the given line (1 mark for each part)

(iii) They-intercept of the line (1 mark for each part)

(a) y =5x 4 (b) 3y =9 6x (c) 3y 12=2x

(d) 5x 2y +15=0 (e)

4

x 3

1



Question 2

For each of these straight line segments, AB, work out

(iii) The length of AB, giving your answer as an integer or a surd (2 marks for each part)

In each of these line segments, B lies between A and C.

Work out the coordinates of C in each case (2 marks for each part)

Question 4

Work out the coordinates of the points of intersection of the curve y = x2+7 and

Question 5

Line L has equation y + 3x = 7

Line N is perpendicular to line L and passes through (3, 1)

Work out the equation of line N.

Question 7

A curve has equation y = x3+ x2+ 2x 4

Work out the equation of the tangent to this curve where x =2

A curve has equation y = x3+ 2x2 9x + 3

Work out the equation of the normal to this curve at the point (1,3)

Give your answer in the form ax + by + c = 0, where a, b and c are integers (5 marks)

A curve has equation y = x3 6x2

x either side of 0 and 4

M1

Maximum at (0, 20) Minimum at (4,12) A1 If previous M1 earned

B1 ft For labelling the stationary points

0, 20

y

O y

x

Question 10

A curve has equation y = x3x2+ k x 2

(a) Write down an expression for

(b) The curve has a minimum point at the point where x =2

(c) Work out thexcoordinate of the maximum point on the curve (3 marks)

(a) Show that the line y =

2

1

x4

(b) The point B on the curve is such that the tangent at B is perpendicular to the tangent at A,

as shown in the diagram

9answergiven

Not drawnaccurately

y

10 Factor Theorem

Question 1

(b) Write down the x values of the three points where the graph of y = x3 5x2 36x

(a) Show that (x + 5) is a factor of x3+ 7x2+ 2x 40 (2 marks)

(b) Work out the other two linear factors of x3+ 7x2+ 2x 40 (3 marks)

Substitutes another value into the

expression and tests for ‘= 0’

Question 4

A sketch of y = x3+ 5x2+ 9x + k where k is an integer, is shown.

2

(a) (x + 3) is a factor of f(x) = x3+ x2+ ax  72 where a is an integer.

Substitutes another value into the

expression and tests for ‘= 0’

Question 6

(x − 3) and (x + 4) are factors of f(x) = x3+ ax2+ bx + 24 where a and b are integers.

(a) (x − 5) is a factor of f(x) = x3+ kx2+ 9x − 20 where k is an integer.

(b) Express f(x) as a product of (x − 5) and a quadratic factor (2 marks)

(c) Show that (x − 5) is the only linear factor of f(x) (2 marks)

(c) Tests ‘b2− 4ac’ for the quadratic M1 ft Their quadratic

or attempts to solve their quadratic = 0

Shows ‘b2− 4ac’ = −15 (or < 0) and states

no more linear factors

A1 States 'no solutions' to their quadratic = 0

Question 8

Mark Scheme

Substitutes a value of x into the

expression and tests for ‘= 0’

M1 Attempts to work out the quadratic factor

Sight of x2and−18 in a quadratic factor

or sight of x2and−9 in a quadratic factor

or sight of x2and 2 in a quadratic factor

Alt 1 Substitutes a value of x into the

expression and tests for ‘= 0’

M1

Works out first linear factor

(x + 1), (x + 2) or (x − 9)

A1

Substitutes another value into the

expression and tests for ‘= 0’

M1

Alt 2 Substitutes a value of x into the

expression and tests for ‘= 0’

M1

Works out first linear factor

(x + 1), (x + 2) or (x − 9)

A1

A linear sequence starts

Mark Scheme

For the nth terms of quadratic sequences two methods are shown (see example 2).

Other valid methods may be used

(a) Show that the nth term of the quadratic sequence

Question 4 (non calculator)

(a) Write down the nth term of the linear sequence

(c) For the sequence in part 4(b), show that the 30th term is equal to the product

This pattern of rectangles continues.

Show that the sequence of numbers formed by the areas of these rectangles has nth term

Question 6

A linear sequence starts

a + b a + 3b a + 5b a + 7b …………

The 5th and 8th terms have values 35 and 59

1



n

n (3 marks)

(b) Which are the first two consecutive terms with a difference less than 0.01? (2 marks)

(c) Write down the limiting value of the sequence as n  (1 mark)

n n n

)1(

4313

n n n

n

n

=

)1(

=

)1(

The nth term of a sequence is

23

12

2 2





n n

(a) Show that the difference between the first two terms is

(b)

3

12 Algebraic problems – including ratio

 If, in a problem, two numbers are in the ratio 4 : 7, use 4xand 7xas the numbers

(usually leading to a linear equation); otherwise, use xandyas the numbers(which will lead to simultaneous equations)

12

x : y = 6 : 5

)(

Question 3

A point P divides XY in the ratio 3 : 7

Y (6a, 11b)

X (a, b)

P

Here is a linear sequence

Question 5

You are given that ab + a = 5 and a : b = 4 : 3

a = 2

The sum of the ages of two people is 90 years.

Six years ago, their ages were in the ratio 8 : 5

How old are they now?

Do not use trial and improvement.

Question 7

O is the centre of the circle.

Given that x : y = 4 : 5

Work out the value of y.

Do not use trial and improvement.

A rectangular picture is surrounded by a frame of constant width.

All measurements are in centimetres

3

(

their

)18

7x

3x

99

a

b

A cuboid has dimensions 2n, n and n 1 cm.

A diagonal has length 2n + 1 cm.

2n + 1

2n

n

n 1

AQA Level 2 Certificate in Further Mathematics

from 2011 onwards

Qualification Accreditation Number: 600/2123/8

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