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National 5 Expressions and Formulae

1. Simplify, giving your answer in surd form: 32

2. (a) Simplify (i) 3

64

x

xx (ii) 2

5

45 4 xx

(b) The number of people attending a football match was 3·12 × 104. If each person paid £27,

how much was collected? Give you answer in Scientific Notation.

3. Expand and simplify where appropriate:

(a) d(4d – e) (b) (g + 4)(g + 9)

4. Factorise: (a) y² – 6y (b) t² – 49 (c) x² + 7x + 12

5. Express x² + 6x + 7 in the form (x + p)² + q.

6. Write )4()4(

)4)(34(2

x

x

xx in its simplest form.

7. Write each of the following as a single fraction:

(a) )0,(53

baba

(b) )0(5

gg

ef

8. Points P and Q have coordinates (–5, –4) and (6, 3) respectively. Calculate the gradient of PQ.

9. Calculate the volume of a sphere with radius 2·3 cm, giving your answer correct to 2 significant

figures.

2·3 cm

10. The logo for Cyril's Cars is shown below. The logo is a sector of a circle of radius 6∙2 cm. The

reflex angle at the centre is 240o.

(a) Calculate the length of the arc AB.

(b) Cyril wants to jazz up the logo by outlining it with coloured rope. He buys 20 metres of

rope. How many logos would he be able to makeup?

11. Sherbet in a sweet shop is stored in a cylindrical container like the one shown in diagram 1.

The sherbet is sold in conical containers with diameter 5 cm and height 6 cm as shown in

diagram 2.

The shop owner thinks he can fill 260 cones from the cylinder. Is he correct?

240o

A

B

Diagram 1

32cm

20cm

6 cm

Diagram 2

5 cm

End of Question Paper

Expressions and Formulae: Marking Scheme

Points of reasoning are marked # in the table.

Question Main points of expected responses

1 1 start of process

2 simplified surd 1 √16√2 (or equivalent)

2 4√2

2 (a) (i)

(ii)

(b)

1 simplify numerator

2 correct answer

3 correct coefficient

4 simplify indices

5 calculation of amount

6 express in standard form

1 x10

2 x7

3 20

4 3

2x in answer 20 3

2x

5 27 × 3·12 × 104

=84·24 × 104

6 £8·424 × 105

3 (a)

(b) 1 multiply out brackets

2 multiply out the brackets

3 collect like terms

1 4d2 – de

2 g2 + 4g + 9g + 36

3 g2 + 13g + 36

4 (a)

(b)

(c)

1 factorise expression

2 factorise difference of two

squares

3 start to factorise trinomial

expression

4 complete factorisation

1 y(y – 6)

2 (t + 7)(t – 57)

3 (x 3)(x 4) ie evidence of

brackets, x, 3 and 4

4 (x + 3)(x + 4)


2 complete process

1 (x + 3)2

2 (x + 3)2 – 2

6 1 reduce to simplest form 1

4

34

x

x

7 (a)

(b)

1 denominator correct

2 numerator correct

3 multiply by inversion of

fraction

4 correct answer

1

ab

///

2

ab

ab 53

3

e

g

4

e

fg

5

8 1 evidence of gradient

calculation

2 correct gradient

1 Uses2 1

2 1

y y

x x

or equivalent

2 11

7

9 1 substitute and start

calculation

2 complete calculation

3 round calculation to 2

significant figures

1 3323

4

167123

4 or

equivalent

2 50·939 cm³ or equivalent

3 51 cm3

10 (a)

(b)

1 correct ratio and substitution

2 calculate arc length

#2.1 valid strategy

#2.2 interpretation of answer

1 412360

240

2 25·957 cm or equivalent

#2.1 eg 2 000 ÷ 38

#2.2 (for 52∙63) 52 logos can

be made.

11 #2.1 uses valid strategy to find

volumes of cone and

cylinder

1 calculate volume of cylinder

2 calculate volume of cone

# 2.2 states conclusion

# 2.1 Substitutes relevant values

into correct formulae

1 10 048 cm3 or equivalent

2 39∙25 cm3 or equivalent

# 2.2 Shop owner is wrong

because only 256 cones

can be filled




37

x

xx (ii) 332 2

1 xx

(b) The number of people attending a musical was 2·64 × 103. If each person paid £34, how

much was collected. Give you answer in Scientific Notation.


(a) g(6g – h) (b) (d + 3)(d – 7)

4. Factorise: (a) k² – 7k (b) x² – 81 (c) z² + 10z + 21

5. Express x² – 8x + 1 in the form (x + p)² + q.

6. Write )3()3(

)3)(13(2

x

x



(a) )0,(75

dcdc

(b) )0(7

hh

kk

8. Points R and S have coordinates (3, –2) and (–6, –3) respectively. Calculate the gradient of RS.

9. Calculate the volume of a sphere with radius 3·7 cm, giving your answer correct to 2 significant

figures.

3·7 cm

10. The diagram shows a sector of a circle with radius 5·6 cm and angle at the centre 230o.

(a) Calculate the length of the arc AB.

(b) The sector has to be made up into a cone with a fur trim round its base. How many cones

could be trimmed from 40 metres of fur?

11. During a cross country race, juice is distributed to the runners in conical containers with diameter

6 cm and height 8 cm as shown in diagram 1.

At the end of the race juice from 60 cones is poured into a cylinderical container with dimensions

as shown in Diagram 2.

Will this container be large enough to hold the juice?

230o

A

B

25cm

15cm


Diagram 2

Diagram 1

8 cm

6 cm




1 1 simplify surd 1 3√6

2 (a) (i)

(ii)

(b)


2 correct answer


4 simplify indices

5 calculation of amount


1 x4

2 x2

3 6

4 25

x in answer 25

6

x

5 34 × 2·64 × 103

=89·76 × 103

6 8·976 × 104

3 (a)




1 6g2 – gh

2 d2 – 7d + 3d – 21

3 d2 – 4d – 21

4 (a)

(b)

(c)



squares


expression


1 k(k – 7)

2 (x + 9)(x – 9)

3 (z 3)(z 7) ie evidence of

brackets, z, 3 and 7

4 (z + 3)(z + 7)


2 complete process

1 (x – 4)2

2 (x – 4)2 – 15


3

13

x

x

7 (a)

(b)


2 numerator correct


fraction

4 correct answer

1

cd

///

2

cd

cd 75

3

k

h

4

7

k


calculation

2 correct gradient

1 Uses2 1

2 1

y y

x x

or equivalent

2 9

1


calculation



significant figures

1 3733

4

653503

4 or

equivalent


3 210 cm3

10 (a)

(b)


2 calculate arc length

#2.1 valid strategy


1 211360

230


#2.1 eg 4 000 ÷ 22·468

#2.2 (for 178∙02) 178 cones can

be trimmed.


volumes of cone and

cylinder


2 calculate volume of cylinder




1 75·36 cm3 or equivalent


# 2.2 cylinder is not big enough

since 75·36 × 60 >

volume of cylinder




82

x

xx (ii) 236 3

1 xx

(b) A factory produces 2·4 × 104 cakes every day. How many cakes will it produce in the

month of April? Give you answer in Scientific Notation.


(a) m(3m – n) (b) (p + 5)(p + 8)

4. Factorise: (a) h² – 11h (b) q² – 144 (c) a² – 12z + 32

5. Express x² + 7x + 9 in the form (x + p)² + q.

6. Write )52()52(

)7)(52(2

x

x



(a) )0,(94

nmnm

(b) )0(4

hl

k

k

8. Points C and D have coordinates (–8, –2) and (6, –4) respectively. Calculate the gradient of CD.

9. Calculate the volume of a cone with diameter 4·6 cm and height 7 cm giving your answer correct

to 2 significant figures.

7 cm

4·6 cm

10. (a) Calculate the area of the sector of a circle in the diagram which has radius 6∙8cm.

(b) These sectors have to be cut from a piece of card with an area of 6500 cm².

Assuming there is not waste, how many sectors can be cut from the card?

11. A candle is in the shape of a sphere with a diameter of 10 cm.

(a) Calculate the volume of the candle.

The candle was melted down and poured into a conical container like the one shown in this

diagram.

(b) Will the cone be big enough to hold the wax? [assume there is no wax lost during the

melting process]


O

42o

135o

11 cm

18 cm




1 1 simplify surd 1 7√3

2 (a) (i)

(ii)

(b)


2 correct answer


4 simplify indices

5 calculation of distance


1 x10

2 x13

3 18

4 35

x in answer 35

18

x

5 30 × 2·4 × 104

=72 × 104

6 7·2 × 105

3 (a)




1 3m2 – mn

2 p2 + 8p + 5p + 40

3 p2 + 13p + 40

4 (a)

(b)

(c)



squares


expression


1 h(h – 11)

2 (q + 12)(q – 12)

3 (a 4)(a 8) ie evidence of

brackets, a, 4 and 8

4 (a – 4)(a – 8)


2 complete process

1 (x + 3·5)2

2 (x + 3·5)2 – 3·25


52

7

x

x

7 (a)

(b)


2 numerator correct


fraction

4 correct answer

1

mn

///

2

mn

mn 94

3

k

l

4 2

4

k

l


calculation

2 correct gradient

1 Uses2 1

2 1

y y

x x

or equivalent

2 7

1


calculation



significant figures

1 7323

1 2

03373

1 or

equivalent


3 38 cm3

10 (a)

(b)


2 calculate sector area

#2.1 valid strategy


1 286360

135


#2.1 eg 6 500 ÷ 54·4476

#2.2 (for 119∙38) 119 sectors

can be cut.


volumes of cone and

sphere

1 calculate volume of sphere





1 523·33 cm3 or equivalent


# 2.2 cone is big enough

since 523·33 < 569∙91

National 5 Relationships

1. A straight line with gradient – 3 passes through the point (– 2, 5).

Determine the equation of this straight line.

2. Solve the inequation 4p – 12 < p + 6.

3. The Stuart family visit a new attraction in Edinburgh. They paid £32.25 for 3 adult tickets and 2

child tickets.

Write an equation to represent this information.

4. Solve the following system of equations algebraically:

3a + 5b = 39

a – b = – 3

5. Here is a formula

63

2

xS

Change the subject of the formula to x.

6. The diagram shows the parabola with

equation 2kxy .

What is the value of k?

y

x 0 1 2 3 – 1 – 2 – 3

5

10

15

20

25

30

35

40

7. The equation of the quadratic function whose graph is shown below is of the form y = (x + a)2 + b,

where a and b are integers.

Write down the values of a and b.

8. Sketch the graph y = (x – 1)(x + 3) on plain paper.

Mark clearly where the graph crosses the axes and state the coordinates of the turning point.

9. A parabola has equation y = (x – 3)2 + 4.

(a) Write down the equation of its axis of symmetry.

(b) Write down the coordinates of the turning point on the parabola and state whether it is a

maximum or minimum.

10. Solve the equation (x – 3)(x + 7) = 0

11. Solve the equation x2 + 2x – 7 = 0 using the quadratic formula.

12. Determine the nature of the roots of the equation 3x2 + 2x – 1 = 0 using the discriminant.

y

x 1 2 3 4 0 – 1 – 2

2

4

6

8

13. To check that a room has perfect right angles, a builder measures two sides of the room and its

diagonal. The measurements are shown in this diagram.

Are the corners of the room right – angled?

14. The diagram shows kite ABCD and a circle with centre B.

AD is the tangent to the circle at A and CD is the tangent

to the circle at C.

Given that angle ABC is 126°, calculate angle ADC.

15. A water container is in the shape of a cylinder which is 150 cm long. The volume of water in the

container is 12 000 cm3.

A similar miniature version is 15cm long.

Calculate how much water the miniature version would hold.

3·3m

5·4m

6·3m

A

B

C

D

126o

16. Here is a regular, 5 – sided polygon.

Calculate the size of the shaded angle.

17. Sketch the graph of y = 4sin xo for 0° 360x .

18. Write down the period of the graph of the equation y = cos 3xo.

19. Solve the equation 4sin x° – 1 = 0, 0° 360x .


Relationships: Marking Scheme



1 1 correct substitution 1 y – 5 = –3(x –(–2))

(or equivalent)

2 1 simplify for p

2 simplify numbers

3 solve

1 3p

2 18

3 p < 6

3 #2.1 uses correct strategy and sets

up equation

#2.1 3a + 2c = 32·25

4 1 multiply by appropriate

Factor

2 solve for a

3 solve for b

1 3a + 5b = 39

5a – 5b = –15

or equivalent

2 a = 3

3 b = 6

5 1 subtract 6

2 multiply by 3

3 divide by 2

1 S – 6

2 (S – 6) × 3

(or equivalent)

3 2

)6(3 S

(or equivalent)

6 1 correct value of k

1 k = 5

7 1 find value of ‘a’

2 find value of ‘b’

1 a = –1

2 b = 2

8 1 identify and annotate roots

and y-intercept

2 identify and annotate turning

point

3 draw correct shape of graph

1 –3, 1 and (0, –3)

2 (–1, –4)

3 correctly annotated graph

9 (a)

(b)

1 axis of symmetry

2 turning point

3 nature

1 x = 3

2 (–3, 4)

3 minimum turning point

10 1 solve equation

1 x = –7, x = 3

11 1 correct substitution

1

2

71422 2

2 evaluation discriminant

3 solve for 1 root

4 complete solution

2 32

3 x = 1·8

4 x = –3·8

(rounding not required)


2 evaluate discriminant

#2.2 interpret result

1 (2)2 – 4 × 3 × –1

2 16

#2.2 real and unequal roots

Since b2 – 4ac > 0

13 1 calculates and adds squares

of two short sides

2 squares longest side

#2.2 interprets result

1 3·32 +5·42 = 40·05

2 6·32 = 39·69

#2.2 so 3·32 + 5·42 ≠ 6·32

and hence triangle is not right-

angled using converse of

Pythagoras. The corners of the room are not

right angled.

14 1 radius and tangent

2 subtract

3 correct answer

1 either angle BAD or

angle BCD = 90°

2 360 – (90 + 90 + 126)

3 54°

15 1 use volume scale factor

2 correct answer

1 (15/150)3 × 12000

2 12 cm3

16 #2.1 use a valid strategy

1 correct answer

#2.1 eg centre angles

360/5 = 72° each

1 108°

17 1 correct amplitude and period


complete with roots and

amplitude.

1 4 / – 4 and 360°

2 Correct graph

18 1 correct period 1 120°

19 1 solve for sin x°

2 solve for x

3 complete solution

1 sin x° = 0·25

2 14·5°

3 165·5°


1. A straight line with gradient 4 passes through the point (2, –4).


2. Solve the inequation 7m + 5 < 2m + 30.

3. The Clelland family visit a new attraction in Inverness. They paid £29.40 for 2 adult tickets and 4

child tickets.



7x + 2y = 32

2x – y = 6


25

4

BA

Change the subject of the formula to B.


equation 2kxy .


y

x 0 1 2 3 – 1 – 2 – 3

2

4

6

8

10

12

14

16




8. Sketch the graph y = (x – 5)(x – 7) on plain paper.


9. A parabola has equation y = (x + 4)2 – 3.



maximum or minimum.


11. Solve the equation x2 – 3x – 2 = 0 using the quadratic formula.

12. Determine the nature of the roots of the equation 4x2 + 3x + 5 = 0 using the discriminant.

y

x 1 2 3 4 0 – 1 – 2

2

4

6

8

13. A shape has dimensions as shown in the diagram.

Kalen thinks it is a rectangle. Is he correct?

14. The diagram shows kite PNML and a circle with centre M.

PL is the tangent to the circle at L and PN is the tangent

to the circle at N.

Given that angle LMN is 142°, calculate angle LPN.

15. A cuboid has length 30 cm and a volume of 1500 cm³

A similar miniature version is 10 cm long.

Calculate the volume of the miniature cuboid.

.

10m

7·5m

12·5m

L

M

N

P

14

2o



17. Sketch the graph of y = 7cos xo for 0° 360x .

18. Write down the period of the graph of the equation y = sin 5xo.

19. Solve the equation 7cos x° – 2 = 0, 0° 360x .





1 1 correct substitution 1 y + 4 = 4(x ––2)

(or equivalent)

2 1 simplify for m

2 simplify numbers

3 solve

1 5m

2 25

3 m < 25


up equation

#2.1 2a + 4c = 29·4


factor

2 solve for x

3 solve for y

1 7x + 2y = 32

4x – 2y = 12

or equivalent

2 x = 4

3 y = 2

5 1 add 2

2 multiply by 5

3 divide by 4

1 A + 2

2 (A + 2) × 5

(or equivalent)

3 4

)2(5 A

(or equivalent)


1 k = 2



1 a = –1

2 b = 4


and y-intercept


point


1 5, 7 and (0, 35)

2 (6, –1)


9 (a)

(b)

1 axis of symmetry

2 turning point

3 nature

1 x = –4

2 (–4, –3)

3 minimum turning point

10 1 solve equation

1 x = –5, x = 10


1

2

21433 2


3 solve for 1 root

4 complete solution

2 17

3 x = 3·6

4 x = –0·6





1 (3)2 – 4 × 4 × 5

2 –71

#2.2 roots are not real

since b2 – 4ac < 0


of two short sides



1 7·52 +102 = 156·25

2 12·52 = 156·25

#2.2 so 7·52 +102 = 12·52

and hence triangle is right-


Pythagoras. The shape is a rectangle


2 subtract

3 correct answer

1 either angle PLM or

angle MNP = 90°

2 360 – (90 + 90 + 142)

3 38°


2 correct answer

1 (10/30)3 × 15000

2 55·6 cm3


1 correct answer


360/12 = 30° each

1 150°




amplitude.

1 7 / – 7 and 360°

2 Correct graph


19 1 solve for cos x°

2 solve for x

3 complete solution

1 cos x° = 2/7

2 73·4°

3 286·6°


1. A straight line with gradient ½ passes through the point (1, 5).


2. Solve the inequation 5k – 3 < 2k + 9.

3. A group of friends met in a coffee bar. They paid £9.40 for 4 cappuccinos and 2 lattes.



5c – 2d = 36

c + d = 17


4

57

mk

Change the subject of the formula to m.


equation 2y kx


y

x 0 1 2 3 – 1 – 2 – 3

3

6

9

12

15

18

21

24




8. Sketch the graph y = (x – 4)(x + 2) on plain paper.


9. A parabola has equation y = 5 – (x + 3)2 .



maximum or minimum.


11. Solve the equation x2 + 5x – 7 = 0 using the quadratic formula.

12. Determine the nature of the roots of the equation 9x2 + 6x + 1 = 0 using the discriminant.

x

y

–1 0 1 2 –2 – 3 – 4

2

4

6

8

13. A shape has dimensions as shown.

Is angle DAB = 90o in this shape?

14. The diagram shows kite WXYZ and a circle with centre X.

WZ is the tangent to the circle at W and YZ is the tangent

to the circle at Y.

Given that angle WXY is 139°, calculate angle WZY.

15. A tube of toothpaste is 21 cm long and has a volume of 50cm³

A similar miniature version is 9cm long.

Calculate how much toothpaste the miniature version would hold.

4·6m

6·7m

8m

W

X

Y

Z

139o

A B

C D



17. Sketch the graph of y = –3sin xo for 0° 360x .

18. Write down the period of the graph of the equation y = sin ½ xo.

19. Solve the equation 5tan x° – 7 = 0, 0° 360x .





1 1 correct substitution 1 y – 5 = ½ (x –1)

(or equivalent)

2 1 simplify for k

2 simplify numbers

3 solve

1 3k

2 12

3 k < 4


up equation

#2.1 4c + 2l = 9·4


Factor

2 solve for c

3 solve for d

1 5c – 2d = 36

5c + 2d = 34

or equivalent

2 c = 10

3 d = 7

5 1 subtract 7

2 multiply by 4

3 divide by 5

1 k – 7

2 (k – 7) × 4

(or equivalent)

3 5

)7(4 k

(or equivalent)


1 k = 3



1 a = 1

2 b = 3


and y-intercept


point


1 –2, 4 and (0, –8)

2 (1, –9)


9 (a)

(b)

1 axis of symmetry

2 turning point

3 nature

1 x = –3

2 (–3, 5)

3 maximum turning point

10 1 solve equation

1 x = –1, x = 7


1

2

71455 2


3 solve for 1 root

4 complete solution

2 53

3 x = 1·1

4 x = –6·1





1 (6)2 – 4 × 9 × 1

2

#2.2 equal roots

since b2 – 4ac = 0


of two short sides



1 4·62 +6·72 = 66·05

2 82 = 64

#2.2 so 4·62 +6·72 ≠ 82

and hence triangle is not right-


Pythagoras. Angle DAB is not a right

angle.


2 subtract

3 correct answer

1 either angle ZWX or

angle ZYX = 90°

2 360 – (90 + 90 + 139)

3 41°


2 correct answer

1 (9/21)3 × 50

2 4 cm3


1 correct answer


360/10 = 36° each

1 144°




amplitude.

1 – 3 / 3 and 360°

2 Correct graph


19 1 solve for tan x°

2 solve for x

3 complete solution

1 tan x° = 1·4

2 54·5°

3 234·5°

National 5 Applications

1. A farmer wishes to spread fertiliser on a triangular plot of ground.

The diagram gives the dimensions of the plot.

Calculate the area of this plot to the nearest square metre.

2. The diagram shows the paths taken by two runners, Barry and Charlie. Barry runs 350 metres from

point S to position R. Charlie runs 300 metres to position T.

` What is the shortest distance between the two runners? [i.e. the distance TR on the diagram]

35 m

44 m

58o

S R

T

350 m

300 m

12o

3. On an orienteering course there are three checkpoints at points U, V and W as shown in the diagram

below.

W is 220 kilometres from V and 400 kilometres from U.

W is on a bearing of 125° from V.

Calculate the bearing of W from U. i.e. the size of angle NUW in the diagram.

Give your answer to the nearest degree.

U

V

W

125o

220 km

400 km

N

4. The diagrams below show 2 directed line segments u and v.

Draw the resultant of 3u+ v.

5. The diagram below shows a square based model of a glass pyramid of height 8 cm. Square OPQR

has a side length of 6 cm.

The coordinates of Q are (6, 6, 0). R lies on the y-axis.

Write down the coordinates of S.

6. The forces acting on a body are represented by three vectors a, b and c as given below.

52

2

5

a

55

7

3

b

2

6

51

c

Find the resultant force.

O

R Q (6, 6, 0)

P

S

x

y

z

u v

7. Vector

3

5p and vector

3

1q .

Calculate qp 2

8. Kashef bought a new car for £24 000. Its value decreased by 12% each year. Find the value of the

car after 5 years.

9. A desk top has measurements as shown in the diagram.

Calculate the exact area of the desk top (in m2).

10. A man invested some money in a Building Society last year.

It has increased in value by 15% and is now worth £2760.

Calculate how much the man invested.

11. The cost of a set menu meal in 7 different café style restaurants were as follows:

£14 £17 £13 £14 £11 £19 £17

(a) Calculate the mean and standard deviation of these costs.

(b) In 7 up market restaurants the mean cost of a meal was £22 with a standard deviation of 2·2.

Using these statistics, compare the profits of the two companies and make two valid comparisons.

m7

32

m3

21

12. A primary teacher took a note of the results in a spelling test and the number of hours of TV that

some of her pupils watched in a week. She then drew the following graph.

(a) Determine the gradient and the y-intercept of the line of best fit shown.

(b) Using these values for the gradient and the y-intercept, write down the equation of the line.

(c) Estimate the mark in the spelling test if the pupil spent 25 hours watching television.


Spel

ling T

est

(S)

Res

ult

, s

5

10

15

20

25

0

0 10 Hours spent watching TV, (h)

20 30 40 50

Applications: Marking Scheme



1

1 substitute into

formula

2 correct answer

1 58sin44352

1

2 653 m2

2 1 use correct formula

2 substitute correctly

3 process to s2

4 take square root

1 selects cosine rule

2

12cos3503002350300 222s

3 7 089

4 84·1 metres (rounding not

required)

3 #2.1 uses correct strategy

1 finds angle U

2 states bearing from

U

#2.1 400

125sin220sin

U then valid

steps below

1 26·8

2 153·2o (rounding not required)

4 1 draws 3u

2 applies head-to-tail

method when adding

v

3 draws resultant from

tail of 3u to head of

v.

5

1 correct point 1 (3, 3, 8)

3u

v

3u + v

6 1 add to get resultant

2 correct answer 1

2

6

51

55

7

3

52

2

5

2

6

15

53

7 1 correct scalar

multiplication then

addition

2 calculate magnitude

3 correct answer

1

3

11

3

1

6

10

2 22 311

3 130

8 1 start calculation

2 process calculation

3 correct answer

Note: repeated subtraction

method can be used

1 0·88

2 24 000 × 0·885

3 £12 665·57

equivalent

9

1 area calculation

2 correct answer

1 3

5

7

17

2 21

14

21

85 m²

10 #2.1 appropriate strategy

1 correct answer

#2.1 eg 1 + 0·15 x = £2760

1 £2 400

11 (a)

(b)

1 mean for A

2 calculates 2)( xx

3 substitute into

formula

4 correct standard

deviation

#2.2 Compares mean and

standard deviation in a

valid way for data

1 105 ÷ 7 = 15

2 1, 4, 4, 1, 16, 16, 4

3 6

46

4 2·77 (rounding not required)

(Equivalent calculations can be used)

#2.2 On average up market prices more

expensive

There is less of a spread in up market

restaurants

12 (a)

(b)

(c)

1 chooses 2 distinct

points and

substitutes into

gradient formula

2 calculates gradient

3 finds intercept

4 writes down

equation

# 2.2 estimate mark

1 4010

57522

m

2 2

1m (or based on gradient

line of best fit

3 c = 27·5 (approximately or by

calculation or from

graph)

4 S = 2

1 h + 27·5

(or equivalent)

#2.2 Approximately 15 hours

Pegasys 2013


1. A children’s play park, which is triangular in shape, has to be covered with a protective matting.


Calculate the area, to the nearest square metre, of protective matting needed.

2. The diagram shows the courses followed by two ships, the Westminster and the Bogota, after they

leave Port A. The Westminster sails 520 metres to position W and the Bogota 580 metres to

position B.

` How far apart are the ships?[i.e. the distance WB on the diagram]

A B

W

580 m

520 m

14o

22 m 24 m

56o

Pegasys 2013

3. On a radar screen, three planes, P, Q and R are at the positions shown in the diagram.

R is 300 kilometres from Q and 450 kilometres from P.

R is on a bearing of 132° from Q.

Calculate the bearing of R from P. i.e. the size of angle NPR in the diagram.


P

Q

R

132o

300 km

450 km

N

Pegasys 2013

4. The diagrams below show 2 directed line segments a and b.

Draw the resultant of 2a + 2b.

5. The diagram below shows a square based model of a glass pyramid of height 10 cm. Square OPQR

has a side length of 8 cm.

The coordinates of R are (0, 8, 0). P lies on the x-axis.

Write down the coordinates of S.

6. The forces acting on a body are represented by three vectors k, l and m as given below.

4

52

3

k

51

4

2

l

4

0

53

m


O

Q

P

S

x

y

z

a b

R (0, 8, 0)

Pegasys 2013

7. Vector

6

3a and vector

5

2b .

Calculate ba 2

8. Due to inflation, house prices are expected to rise by 3∙6% each year.

What will the average house price be in 3 years if it is £142,000 today?

9. A room has dimensions as shown in the diagram.

Calculate the exact amount of carpet that would have to be bought for the room.

10. A woman bought an antique painting last year.

It has increased in value by 35% and is now worth £3 510.

Calculate how much the woman paid for the painting.

11. A quality control examiner on a production line measures the weight, in grams, of cakes coming off

the line. In a sample of eight cakes the weights were

150 147 148 153 149 143 145 149

(a) Find the mean and standard deviation of the above weights.

(b) On a second production line, a sample of 8 cakes gives a mean of 148 and a standard

deviation of 6·1.

Using these statistics, compare the profits of the two companies and make two valid

comparisons.

m8

34

m4

32

Pegasys 2013

12. The diagram below shows the connection between the thickness of insulation in a roof and the

heat lost through the roof. The line of best fit has been drawn.



(c) Estimate the thickness of insulation for a heat loss of 2·5 kilowatts.


Thic

knes

s of

insu

lati

on i

n c

enti

met

res

(T)

5

10

15

20

25

0

0 1 2 3 4 5 Heat loss from roof in kilowatts (H)

Pegasys 2013




1

1 substitute into

formula

2 correct answer

1 56sin24222

1

2 219 m2



3 process to a2

4 take square root


2

14cos5805202580520 222a

3 21517


required)


1 finds angle P


P

#2.1 450

132sin300sin

P then valid

steps below

1 29·7

2 150·3o (rounding not required)

4 1 draws 2a


method when adding

2b


tail of 2a to head of

2b

5


v

2a

2a + 2b

2b

Pegasys 2013


2 correct answer 1

4

0

53

51

4

2

4

52

3

2

56

56

51

7 1 correct scalar

multiplication then

addition


3 correct answer

1

4

1

10

4

6

3

2 22 )4()1(

3 17



3 correct answer

Note: repeated addition

method can be used

1 1·036

2 142 000 × 1·036³

3 £157 894

equivalent

9

1 area calculation

2 correct answer

1 4

11

8

35

2 32

112

32

385 m²


1 correct answer

#2.1 eg 1 + 0·35 x = £3510

1 £2 600

11 (a)

(b)

1 mean for A

2 calculates

3 substitute into

formula

4 correct standard

Deviation



valid way for data

1 1184 ÷ 8 = 148

2 4, 1, 0, 25, 1, 25, 9, 1

3 7

66

4 3·07 (rounding not required)


#2.2 On average weights the same

Wider spread on second line.

Pegasys 2013

12 (a)

(b)

(c)


points and

substitutes into

gradient formula


3 finds intercept

4 writes down

equation

# 2.2 estimate mark

1 5351

1020

m

2 5m (or based on gradient line

of best fit

3 c = 27·5 (approximately or by

calculation or from

graph)

4 T = 5 H + 27·5

(or equivalent)

#2.2 Approximately 15 cm

Pegasys 2013


1. Turf has to be laid on a triangular plot of garden

.


Calculate the area, to the nearest square metre, of turf that is required.

2. Billy and Peter are bowlers. They are playing a game and after they each throw their first bowl they

are in the positions shown in the diagram.

` How far apart are the bowls after this first throw?[i.e. the distance PB on the diagram]

P T

B

26 m

24 m

17o

27 m

25 m 102o

Pegasys 2013

3. The positions of three players, K, L and M are shown in this diagram.

Player M is 30 metres from player L and 40 metres from player K.

M is on a bearing of 125° from L.

Calculate the bearing of player M from player K. i.e. the size of angle NKM in the diagram.


K

L

M

125o

30 m

40 m

N

Pegasys 2013

4. The diagrams below show 2 directed line segments k and l.

Draw the resultant of k + 2l.

5. The diagram below shows a square based model of a glass pyramid of height 5 cm. The base OPQR

is a square.

The coordinates of S are (2, 2, 5). P lies on the x-axis and R lies on the y – axis.

Write down the coordinates of Q.

6. The forces acting on a body are represented by three vectors x, y and z as given below.

1

32

4

x

50

72

2

y

2

1

2

z


k

l

O

Q

P

S

x

y

z

R

(2, 2, 5)

Pegasys 2013

7. Vector

6

3x and vector

5

2y .

Calculate yx 23

8. Chocolate fountains have become very popular at parties.

At one party 23% of the remaining chocolate was used every 20 minutes.

If 2kg of melted chocolate was added to the fountain at the start of the night,

how much would be left after 1 hour?

9. Calculate the area of this piece of ground which has dimensions as shown in the diagram.

10. I bought a car three years ago.

Since then it has decreased in value by 45% and is now worth £6875.

How much did I pay for the car?

11. A set of Maths test marks for a group of students are shown below.

35 27 43 18 36 39

(a) Find the mean and standard deviation.

(b) Another group had a mean of 37 and a standard deviation of 8∙6.

Compare the test marks of the two classes.

m5

110

m4

16

Pegasys 2013

12. A selection of the number of games won and the total points gained by teams in the Scottish

Premier League were plotted on this scattergraph and the line of best fit was drawn.



(c) Use your equation to estimate the number of points gained by a team who win 27 games.


Wins

W

P

4 8 12 16 20

Poin

ts

10

20

30

40

50

60

70

80

Pegasys 2013




1

1 substitute into

formula

2 correct answer

1 102sin25272

1

2 330 m2



3 process to t2

4 take square root


2

17cos262422624 222t

3 58·53


required)


1 finds angle K


K

#2.1 40

125sin30sin

K then valid

steps below

1 38

2 142o (rounding not required)

4 1 draws k


method when adding

2l


tail of k to head of

2l

5


k

2l

Pegasys 2013


2 correct answer 1

2

1

50

72

2

1

32

4

2

52

6

0

7 1 correct scalar

multiplication then

addition


3 correct answer

1

8

5

10

4

18

9

2 22 85

3 89



3 correct answer

Note: repeated addition

method can be used

1 0·77

2 2 000 × 0·77³

3 913g

equivalent – 3

9

1 area calculation

2 correct answer

1 5

51

4

25

2 4

363

4

255 m²


1 correct answer

#2.1 eg (1 – 0·45) x = £6 875

1 £12 500

11 (a)

(b)

1 mean

2 calculates

3 substitute into

formula

4 correct standard

deviation



valid way for data

1 198 ÷ 6 = 33

2 4, 36, 100, 225, 9, 36

3 5

410

4 9 (rounding not required)


#2.2 On average second group had

higher marks

Second group’s marks less spread out

Pegasys 2013

12 (a)

(b)

(c)


points and

substitutes into

gradient formula


3 finds intercept

4 writes down

equation

# 2.2 estimate mark

1 612

2040

m

2 3

10m (or based on gradient line

of best fit)

3 c = 0 (approximately or by

calculation or from

graph)

4 P = 3

10W

(or equivalent)

#2.2 90 points

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