Practice Unit Assessment (1) for National 5 Expressions and Formulae
1.
Simplify, giving your answer in surd form:
2.
(a)
Simplify
(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.
(i)
32
x4  x6
x3
(ii)
5x 4  4 x
 52
Expand and simplify where appropriate:
(a)
d(4d – e)
y² – 6y
(g + 4)(g + 9)
t² – 49
4.
Factorise:
5.
Express x² + 6x + 7 in the form (x + p)² + q.
6.
Write
7.
Write each of the following as a single fraction:
(a)
(a)
(b)
(b)
(c)
x² + 7x + 12
(4 x  3)( x  4)
( x  4) in its simplest form.
( x  4) 2
3 5

a b
(a, b  0)
(b)
f e

5 g
( g  0)
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
 Pegasys 2013
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
240o
B
11.
(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?
Sherbet in a sweet shop is stored in a cylindrical container like the one shown in diagram 1.
20cm
32cm
Diagram 1
The sherbet is sold in conical containers with diameter 5 cm and height 6 cm as shown in
diagram 2.
5 cm
6 cm
Diagram 2
The shop owner thinks he can fill 260 cones from the cylinder. Is he correct?
End of Question Paper
 Pegasys 2013
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
2
1
2
3
4
5
start of process
simplified surd
simplify numerator
correct answer
correct coefficient
simplify indices
calculation of amount
1
2
1
2
3
6
express in standard form
4 (a)
(b)

2
3
1
2
(c)
3
4
multiply out brackets
multiply out the brackets
collect like terms
factorise expression
factorise difference of two
squares
start to factorise trinomial
expression
complete factorisation
6
1
2
3
1
2
5
1
2
6
1
2 (a) (i)
(ii)
(b)
3 (a)
(b)
1
4
5
3
√16√2 (or equivalent)
4√2
x10
x7
20
3
3
x 2 in answer 20 x 2
27 × 3·12 × 104
=84·24 × 104
£8·424 × 105
4d2 – de
g2 + 4g + 9g + 36
g2 + 13g + 36
y(y – 6)
(t + 7)(t – 57)
4
(x 3)(x 4) ie evidence of
brackets, x, 3 and 4
(x + 3)(x + 4)
start of process
complete process
1
2
(x + 3)2
(x + 3)2 – 2
1
reduce to simplest form
1
4x  3
x4
7 (a)
1
denominator correct
1
(b)
2
numerator correct
3
multiply by inversion of
fraction
correct answer
3
1
evidence of gradient
calculation
1
Uses
2
correct gradient
2
7
11
4
8
 Pegasys 2013
2
4
///
ab
3b  5a
ab
g

e
fg
5e
y 2  y1
or equivalent
x 2  x1
9
10 (a)
(b)
11
1
substitute and start
calculation
1
2
complete calculation
2
4
   2  33
3
4
   12  167 or
3
equivalent
50·939 cm³ or equivalent
3
round calculation to 2
significant figures
3
51 cm3
1
correct ratio and substitution
1
2
240
   12  4
360
calculate arc length
2
25·957 cm or equivalent
#2.1
#2.2
valid strategy
interpretation of answer
#2.1
uses valid strategy to find
volumes of cone and
cylinder
1
2
calculate volume of cylinder
calculate volume of cone
# 2.2 states conclusion
 Pegasys 2013
#2.1 eg 2 000 ÷ 38
#2.2 (for 52∙63) 52 logos can
be made.
# 2.1 Substitutes relevant values
into correct formulae
1
2
10 048 cm3 or equivalent
39∙25 cm3 or equivalent
# 2.2 Shop owner is wrong
because only 256 cones
can be filled
Practice Unit Assessment (2) for National 5 Expressions and Formulae
1.
Simplify, giving your answer in surd form:
2.
(a)
Simplify
(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.
(i)
54
x 7  x 3
x2
1
(ii)
2 x 2  3 x 3
Expand and simplify where appropriate:
(a)
g(6g – h)
k² – 7k
(d + 3)(d – 7)
x² – 81
4.
Factorise:
5.
Express x² – 8x + 1 in the form (x + p)² + q.
6.
Write
7.
Write each of the following as a single fraction:
(a)
(a)
(b)
(b)
(c)
z² + 10z + 21
(3x  1)( x  3)
( x  3) in its simplest form.
( x  3) 2
5 7

c d
(c, d  0)
(b)
k k

7 h
(h  0)
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
 Pegasys 2013
10.
The diagram shows a sector of a circle with radius 5·6 cm and angle at the centre 230o.
A
230o
B
11.
(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?
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.
6 cm
8 cm
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.
15cm
25cm
Diagram 2
Will this container be large enough to hold the juice?
End of Question Paper
 Pegasys 2013
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
simplify surd
1
3√6
2 (a) (i)

2
3
4
5
simplify numerator
correct answer
correct coefficient
simplify indices
calculation of amount

2
3
4
5
x4
x2
6
6
express in standard form
4 (a)
(b)
1
2
3
1
2
(c)
3
4
multiply out brackets
multiply out the brackets
collect like terms
factorise expression
factorise difference of two
squares
start to factorise trinomial
expression
complete factorisation
6
1
2
3
1
2
5
1
2
6
(ii)
(b)
3 (a)
(b)
1
1
3
5
 52
x 2 in answer 6 x
34 × 2·64 × 103
=89·76 × 103
8·976 × 104
6g2 – gh
d2 – 7d + 3d – 21
d2 – 4d – 21
k(k – 7)
(x + 9)(x – 9)
4
(z 3)(z 7) ie evidence of
brackets, z, 3 and 7
(z + 3)(z + 7)
start of process
complete process
1
2
(x – 4)2
(x – 4)2 – 15
1
reduce to simplest form
1
3x  1
x3
7 (a)
1
denominator correct
1
(b)
2
numerator correct
3
multiply by inversion of
fraction
correct answer
3
1
evidence of gradient
calculation
1
Uses
2
correct gradient
2
1
9
4
8
 Pegasys 2013
2
4
///
cd
5d  7c
cd
h

k
k
7
y 2  y1
or equivalent
x 2  x1
9
10 (a)
1
substitute and start
calculation
1
2
complete calculation
2
4
   3  73
3
4
   50  653 or
3
equivalent
212·067 cm³ or equivalent
3
round calculation to 2
significant figures
3
210 cm3
1
correct ratio and substitution
1
230
   11  2
360
2
calculate arc length
2
22·468 cm or equivalent
(b)
11
#2.1
#2.2
valid strategy
interpretation of answer
#2.1 eg 4 000 ÷ 22·468
#2.2 (for 178∙02) 178 cones can
be trimmed.
#2.1
uses valid strategy to find
volumes of cone and
cylinder
# 2.1 Substitutes relevant values
into correct formulae
1
2
calculate volume of cone
calculate volume of cylinder
# 2.2 states conclusion
 Pegasys 2013
1
2
75·36 cm3 or equivalent
4415∙625 cm3 or equivalent
# 2.2 cylinder is not big enough
since 75·36 × 60 >
volume of cylinder
Practice Unit Assessment (3) for National 5 Expressions and Formulae
1.
Simplify, giving your answer in surd form:
2.
(a)
Simplify
(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.
(i)
147
x 2  x8
x 3
1
(ii)
6 x 3  3x  2
Expand and simplify where appropriate:
(a)
m(3m – n)
(p + 5)(p + 8)
h² – 11h
(b)
q² – 144
4.
Factorise:
5.
Express x² + 7x + 9 in the form (x + p)² + q.
6.
Write
7.
Write each of the following as a single fraction:
(a)
(a)
(b)
(c)
a² – 12z + 32
(2 x  5)( x  7)
( x  2  5) in its simplest form.
(2 x  5) 2
4 9

m n
(m, n  0)
(b)
4 k

k l
(h  0)
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.
4·6 cm
7 cm
 Pegasys 2013
10.
(a)
Calculate the area of the sector of a circle in the diagram which has radius 6∙8cm.
o
O135
42o
(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.
11 cm
18 cm
(b)
Will the cone be big enough to hold the wax? [assume there is no wax lost during the
melting process]
End of Question Paper
 Pegasys 2013
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
simplify surd
1
7√3
2 (a) (i)

2
3
4
5
simplify numerator
correct answer
correct coefficient
simplify indices
calculation of distance

2
3
x10
x13
18
4
5
6
express in standard form
4 (a)
(b)
1
2
3
1
2
(c)
3
4
multiply out brackets
multiply out the brackets
collect like terms
factorise expression
factorise difference of two
squares
start to factorise trinomial
expression
complete factorisation
6
1
2
3
1
2
x 3 in answer 18 x
30 × 2·4 × 104
=72 × 104
7·2 × 105
3m2 – mn
p2 + 8p + 5p + 40
p2 + 13p + 40
h(h – 11)
(q + 12)(q – 12)
5
1
2
6
(ii)
(b)
3 (a)
(b)
1
1
3
5
 53
4
(a 4)(a 8) ie evidence of
brackets, a, 4 and 8
(a – 4)(a – 8)
start of process
complete process
1
2
(x + 3·5)2
(x + 3·5)2 – 3·25
1
reduce to simplest form
1
x7
2x  5
7 (a)
1
denominator correct
1
(b)
2
numerator correct
3
multiply by inversion of
fraction
correct answer
3
1
evidence of gradient
calculation
1
Uses
2
correct gradient
2

4
8
 Pegasys 2013
2
4
///
mn
4n  9m
mn
l

k
4l
k2
1
7
y 2  y1
or equivalent
x 2  x1
9
10 (a)
1
substitute and start
calculation
1
2
complete calculation
2
1
   2  32  7
3
1
   37  03 or
3
equivalent
37·75806 cm³ or equivalent
3
round calculation to 2
significant figures
3
38 cm3
1
correct ratio and substitution
1
135
   6  82
360
2
calculate sector area
2
54·4476 cm or equivalent
(b)
11
#2.1
#2.2
valid strategy
interpretation of answer
#2.1 eg 6 500 ÷ 54·4476
#2.2 (for 119∙38) 119 sectors
can be cut.
#2.1
uses valid strategy to find
volumes of cone and
sphere
# 2.1 Substitutes relevant values
into correct formulae
1
2
calculate volume of sphere
calculate volume of cone
# 2.2 states conclusion
 Pegasys 2013
1
2
523·33 cm3 or equivalent
569∙91 cm3 or equivalent
# 2.2 cone is big enough
since 523·33 < 569∙91