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Contents & Information
9 Practice Unit Tests...... (3 for each unit)
Answers & marking schemes
Detailed marking schemes
CFE Mathematics
National 5
Practice Unit Tests
Pegasys Educational Publishing
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2
Practice Unit Assessment (1) for 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
3
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? (#2.1 and #2.2)
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? (#2.1 and #2.2)
240o
A
B
Diagram 1
32cm
20cm
6 cm
Diagram 2
5 cm
End of Question Paper
4
Practice Unit Assessment (1) for 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 – 7)
3 (x 3)(x 4) ie evidence of
brackets, x, 3 and 4
4 (x + 3)(x + 4)
5 1 start of process
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
5
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
6
Practice Unit Assessment (2) for National 5 Expressions and Formulae
1. Simplify, giving your answer in surd form: 54
2. (a) Simplify (i) 2
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.
3. Expand and simplify where appropriate:
(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
xx in its simplest form.
7. Write each of the following as a single fraction:
(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
7
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? (#2.1 and #2.2)
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? (#2.1 and #2.2)
230o
A
B
25cm
15cm
End of Question Paper
Diagram 2
Diagram 1
8 cm
6 cm
8
Practice Unit Assessment (2) for 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 √9√6
2 3√6
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 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)
(b) 1 multiply out brackets
2 multiply out the brackets
3 collect like terms
1 6g2 – gh
2 d2 – 7d + 3d – 21
3 d2 – 4d – 21
4 (a)
(b)
(c)
1 factorise expression
2 factorise difference of two
squares
3 start to factorise trinomial
expression
4 complete factorisation
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)
5 1 start of process
2 complete process
1 (x – 4)2
2 (x – 4)2 – 15
6 1 reduce to simplest form 1
3
13
x
x
7 (a)
(b)
1 denominator correct
2 numerator correct
3 multiply by inversion of
fraction
4 correct answer
1
cd
///
2
cd
cd 75
3
k
h
4
7
h
8 1 evidence of gradient
calculation
2 correct gradient
1 Uses2 1
2 1
y y
x x
or equivalent
2 9
1
9
9 1 substitute and start
calculation
2 complete calculation
3 round calculation to 2
significant figures
1 3733
4
653503
4 or
equivalent
2 212·067 cm³ or equivalent
3 210 cm3
10 (a)
(b)
1 correct ratio and substitution
2 calculate arc length
#2.1 valid strategy
#2.2 interpretation of answer
1 211360
230
2 22·468 cm or equivalent
#2.1 eg 4 000 ÷ 22·468
#2.2 (for 178∙02) 178 cones can
be trimmed.
11 #2.1 uses valid strategy to find
volumes of cone and
cylinder
1 calculate volume of cone
2 calculate volume of cylinder
# 2.2 states conclusion
# 2.1 Substitutes relevant values
into correct formulae
1 75·36 cm3 or equivalent
2 4415∙625 cm3 or equivalent
# 2.2 cylinder is not big enough
since 75·36 × 60 >
volume of cylinder
10
Practice Unit Assessment (3) for National 5 Expressions and Formulae
1. Simplify, giving your answer in surd form: 147
2. (a) Simplify (i) 3
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.
3. Expand and simplify where appropriate:
(a) m(3m – n) (b) (p + 5)(p + 8)
4. Factorise: (a) h² – 11h (b) q² – 144 (c) a² – 12a + 32
5. Express x² + 7x + 9 in the form (x + p)² + q.
6. Write )52()52(
)7)(52(2
x
x
xx in its simplest form.
7. Write each of the following as a single fraction:
(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
11
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?
(#2.1 and #2.2)
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] (#2.1 and #2.2)
End of Question Paper
O
42o
135o
11 cm
18 cm
12
Practice Unit Assessment (3) for 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 √49√3 (or equivalent)
2 7√3
2 (a) (i)
(ii)
(b)
1 simplify numerator
2 correct answer
3 correct coefficient
4 simplify indices
5 calculation of distance
6 express in standard form
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)
(b) 1 multiply out brackets
2 multiply out the brackets
3 collect like terms
1 3m2 – mn
2 p2 + 8p + 5p + 40
3 p2 + 13p + 40
4 (a)
(b)
(c)
1 factorise expression
2 factorise difference of two
squares
3 start to factorise trinomial
expression
4 complete factorisation
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)
5 1 start of process
2 complete process
1 (x + 3·5)2
2 (x + 3·5)2 – 3·25
6 1 reduce to simplest form 1
52
7
x
x
7 (a)
(b)
1 denominator correct
2 numerator correct
3 multiply by inversion of
fraction
4 correct answer
1
mn
///
2
mn
mn 94
3
k
l
4 2
4
k
l
8 1 evidence of gradient
calculation
2 correct gradient
1 Uses2 1
2 1
y y
x x
or equivalent
2 7
1
13
9 1 substitute and start
calculation
2 complete calculation
3 round calculation to 2
significant figures
1 7323
1 2
03373
1 or
equivalent
2 38·75806 cm³ or equivalent
3 39 cm3
10 (a)
(b)
1 correct ratio and substitution
2 calculate sector area
#2.1 valid strategy
#2.2 interpretation of answer
1 286360
135
2 54·4476 cm or equivalent
#2.1 eg 6 500 ÷ 54·4476
#2.2 (for 119∙38) 119 sectors
can be cut.
11 #2.1 uses valid strategy to find
volumes of cone and
sphere
1 calculate volume of sphere
2 calculate volume of cone
# 2.2 states conclusion
# 2.1 Substitutes relevant values
into correct formulae
1 523·33 cm3 or equivalent
2 569∙91 cm3 or equivalent
# 2.2 cone is big enough
since 523·33 < 569∙91
14
Practice Unit Assessment (1) for 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
15
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
16
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
17
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° ≤ x ≤ 360°.
18. Write down the period of the graph of the equation y = cos 3xo.
19. Solve the equation 4sin x° – 1 = 0, 0° ≤ x ≤ 360°.
End of Question Paper
18
Practice Unit Assessment (1) for Relationships: Marking Scheme
Points of reasoning are marked # in the table.
Question Main points of expected responses
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
19
2 evaluation discriminant
3 solve for 1 root
4 complete solution
2 32
3 x = 1·8
4 x = –3·8
(rounding not required)
12 1 correct substitution
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
2 correctly annotated graph
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°
20
Practice Unit Assessment (2) for National 5 Relationships
1. A straight line with gradient 4 passes through the point (2, –4).
Determine the equation of this straight line.
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.
Write an equation to represent this information.
4. Solve the following system of equations algebraically:
7x + 2y = 32
2x – y = 6
5. Here is a formula
25
4
BA
Change the subject of the formula to B.
6. The diagram shows the parabola with
equation 2kxy .
What is the value of k?
y
x 0 1 2 3 – 1 – 2 – 3
2
4
6
8
10
12
14
16
21
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 – 5)(x – 7) 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 + 4)2 – 3.
(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 – 10)(x + 5) = 0
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
22
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
23
16. Here is a regular, 12 – sided polygon.
Calculate the size of the shaded angle.
17. Sketch the graph of y = 7cos xo for 0° ≤ x ≤ 360°.
18. Write down the period of the graph of the equation y = sin 5xo.
19. Solve the equation 7cos x° – 2 = 0, 0° ≤ x ≤ 360°.
End of Question Paper
24
Practice Unit Assessment (2) for Relationships: Marking Scheme
Points of reasoning are marked # in the table.
Question Main points of expected responses
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 < 5
3 #2.1 uses correct strategy and sets
up equation
#2.1 2a + 4c = 29·4
4 1 multiply by appropriate
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)
6 1 correct value of k
1 k = 2
7 1 find value of ‘a’
2 find value of ‘b’
1 a = –1
2 b = 4
8 1 identify and annotate roots
and y-intercept
2 identify and annotate turning
point
3 draw correct shape of graph
1 5, 7 and (0, 35)
2 (6, –1)
3 correctly annotated graph
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
11 1 correct substitution
1
2
21433 2
25
2 evaluation discriminant
3 solve for 1 root
4 complete solution
2 17
3 x = 3·6
4 x = –0·6
(rounding not required)
12 1 correct substitution
2 evaluate discriminant
#2.2 interpret result
1 (3)2 – 4 × 4 × 5
2 –71
#2.2 roots are not real
since b2 – 4ac < 0
13 1 calculates and adds squares
of two short sides
2 squares longest side
#2.2 interprets result
1 7·52 +102 = 156·25
2 12·52 = 156·25
#2.2 so 7·52 +102 = 12·52
and hence triangle is right-
angled using converse of
Pythagoras. The shape is a rectangle
14 1 radius and tangent
2 subtract
3 correct answer
1 either angle PLM or
angle MNP = 90°
2 360 – (90 + 90 + 142)
3 38°
15 1 use volume scale factor
2 correct answer
1 (10/30)3 × 15000
2 55·6 cm3
16 #2.1 use a valid strategy
1 correct answer
#2.1 eg centre angles
360/12 = 30° each
1 150°
17 1 correct amplitude and period
2 correctly annotated graph
complete with roots and
amplitude.
1 7 / – 7 and 360°
2 Correct graph
18 1 correct period 1 72°
19 1 solve for cos x°
2 solve for x
3 complete solution
1 cos x° = 2/7
2 73·4°
3 286·6°
26
Practice Unit Assessment (3) for National 5 Relationships
1. A straight line with gradient ½ passes through the point (1, 5).
Determine the equation of this straight line.
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.
Write an equation to represent this information.
4. Solve the following system of equations algebraically:
5c – 2d = 36
c + d = 17
5. Here is a formula
4
57
mk
Change the subject of the formula to m.
6. The diagram shows the parabola with
equation 2y kx
What is the value of k?
y
x 0 1 2 3 – 1 – 2 – 3
3
6
9
12
15
18
21
24
27
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 – 4)(x + 2) 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 = 5 – (x + 3)2 .
(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 – 7)(x + 1) = 0
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
28
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
29
16. Here is a regular, 10 – sided polygon.
Calculate the size of the shaded angle.
17. Sketch the graph of y = –3sin xo for 0° ≤ x ≤ 360°.
18. Write down the period of the graph of the equation y = sin ½ xo.
19. Solve the equation 5tan x° – 7 = 0, 0° ≤ x ≤ 360°.
End of Question Paper
30
Practice Unit Assessment (3) for Relationships: Marking Scheme
Points of reasoning are marked # in the table.
Question Main points of expected responses
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
3 #2.1 uses correct strategy and sets
up equation
#2.1 4c + 2l = 9·4
4 1 multiply by appropriate
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)
6 1 correct value of k
1 k = 3
7 1 find value of ‘a’
2 find value of ‘b’
1 a = 1
2 b = 3
8 1 identify and annotate roots
and y-intercept
2 identify and annotate turning
point
3 draw correct shape of graph
1 –2, 4 and (0, –8)
2 (1, –9)
3 correctly annotated graph
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
11 1 correct substitution
1
2
71455 2
31
2 evaluation discriminant
3 solve for 1 root
4 complete solution
2 53
3 x = 1·1
4 x = –6·1
(rounding not required)
12 1 correct substitution
2 evaluate discriminant
#2.2 interpret result
1 (6)2 – 4 × 9 × 1
2
#2.2 equal roots
since b2 – 4ac = 0
13 1 calculates and adds squares
of two short sides
2 squares longest side
#2.2 interprets result
1 4·62 +6·72 = 66·05
2 82 = 64
#2.2 so 4·62 +6·72 ≠ 82
and hence triangle is not right-
angled using converse of
Pythagoras. Angle DAB is not a right
angle.
14 1 radius and tangent
2 subtract
3 correct answer
1 either angle ZWX or
angle ZYX = 90°
2 360 – (90 + 90 + 139)
3 41°
15 1 use volume scale factor
2 correct answer
1 (9/21)3 × 50
2 4 cm3
16 #2.1 use a valid strategy
1 correct answer
#2.1 eg centre angles
360/10 = 36° each
1 144°
17 1 correct amplitude and period
2 correctly annotated graph
complete with roots and
amplitude.
1 – 3 / 3 and 360°
2 Correct graph
18 1 correct period 1 720°
19 1 solve for tan x°
2 solve for x
3 complete solution
1 tan x° = 1·4
2 54·5°
3 234·5°
32
Practice Unit Assessment (1) for 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
33
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
34
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
35
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
36
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.
End of Question Paper
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
37
Practice Unit Assessment (1) for Applications: Marking Scheme
Points of reasoning are marked # in the table.
Question Main points of expected responses
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
38
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
39
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
Practice Unit Assessment (2) for National 5 Applications
1. A children’s play park, which is triangular in shape, has to be covered with a protective matting.
The diagram gives the dimensions of the plot.
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.
Give your answer to the nearest degree.
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
Find the resultant force.
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.
(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 thickness of insulation for a heat loss of 2·5 kilowatts.
End of Question Paper
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
Practice Unit Assessment (2) for Applications: Marking Scheme
Points of reasoning are marked # in the table.
Question Main points of expected responses
1
1 substitute into
formula
2 correct answer
1 56sin24222
1
2 219 m2
2 1 use correct formula
2 substitute correctly
3 process to a2
4 take square root
1 selects cosine rule
2
14cos5805202580520 222a
3 21517
4 146·7 metres (rounding not
required)
3 #2.1 uses correct strategy
1 finds angle P
2 states bearing from
P
#2.1 450
132sin300sin
P then valid
steps below
1 29·7
2 150·3o (rounding not required)
4 1 draws 2a
2 applies head-to-tail
method when adding
2b
3 draws resultant from
tail of 2a to head of
2b
5
1 correct point 1 (4, 4, 10)
v
2a
2a + 2b
2b
Pegasys 2013
6 1 add to get resultant
2 correct answer 1
4
0
53
51
4
2
4
52
3
2
56
56
51
7 1 correct scalar
multiplication then
addition
2 calculate magnitude
3 correct answer
1
4
1
10
4
6
3
2 22 )4()1(
3 17
8 1 start calculation
2 process calculation
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
10 #2.1 appropriate strategy
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
#2.2 Compares mean and
standard deviation in a
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)
(Equivalent calculations can be used)
#2.2 On average weights the same
Wider spread on second line.
Pegasys 2013
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 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
Practice Unit Assessment (3) for National 5 Applications
1. Turf has to be laid on a triangular plot of garden
.
The diagram gives the dimensions of the plot.
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.
Give your answer to the nearest degree.
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
Find the resultant force.
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.
(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) Use your equation to estimate the number of points gained by a team who win 27 games.
End of Question Paper
Wins
W
P
4 8 12 16 20
Poin
ts
10
20
30
40
50
60
70
80
Pegasys 2013
Practice Unit Assessment (3) for Applications: Marking Scheme
Points of reasoning are marked # in the table.
Question Main points of expected responses
1
1 substitute into
formula
2 correct answer
1 102sin25272
1
2 330 m2
2 1 use correct formula
2 substitute correctly
3 process to t2
4 take square root
1 selects cosine rule
2
17cos262422624 222t
3 58·53
4 7·7 metres (rounding not
required)
3 #2.1 uses correct strategy
1 finds angle K
2 states bearing from
K
#2.1 40
125sin30sin
K then valid
steps below
1 38
2 142o (rounding not required)
4 1 draws k
2 applies head-to-tail
method when adding
2l
3 draws resultant from
tail of k to head of
2l
5
1 correct point 1 (4, 4, 0)
k
2l
Pegasys 2013
6 1 add to get resultant
2 correct answer 1
2
1
50
72
2
1
32
4
2
52
6
0
7 1 correct scalar
multiplication then
addition
2 calculate magnitude
3 correct answer
1
8
5
10
4
18
9
2 22 85
3 89
8 1 start calculation
2 process calculation
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
10 #2.1 appropriate strategy
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
#2.2 Compares mean and
standard deviation in a
valid way for data
1 198 ÷ 6 = 33
2 4, 36, 100, 225, 9, 36
3 5
410
4 9 (rounding not required)
(Equivalent calculations can be used)
#2.2 On average second group had
higher marks
Second group’s marks less spread out
Pegasys 2013
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
1 612
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