National 5 Maths...11 Practice Unit Assessment (1) for National 5 Expressions and Formulae 1....
Transcript of National 5 Maths...11 Practice Unit Assessment (1) for National 5 Expressions and Formulae 1....
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Dalkeith High School
National 5 Maths
Expressions and Formulae
Revision Booklet
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Revision Questions
Assessment Standard 1.1 Page 3
Assessment Standard 1.2 Page 4
Assessment Standard 1.3 Page 5
Assessment Standard 1.4 Page 6
Practice Assessments
Practice A Page 10
Practice B Page 14
Practice C Page 18
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National 5 Expressions and Formulae Revision Questions
Assessment Standard 1.1
Answers
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Assessment Standard 1.2
Answers
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Assessment Standard 1.3
Answers
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Assessment Standard 1.4
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Answers
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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.
cm2·3
cm
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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
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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 x
10
2 x
7
3
20
4
32x in answer 20
32x
5 27 × 3·12 × 10
4
=84·24 × 104
6 £8·424 × 10
5
3 (a)
(b)
1 multiply out brackets
2 multiply out the brackets
3 collect like terms
1 4d
2 – de
2
g2 + 4g + 9g + 36
3 g
2 + 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)
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 Uses
2 1
2 1
y y
x x
or equivalent
2
11
7
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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 cm
3
10 (a)
(b)
1 correct ratio and substitution
2 calculate arc length
#2.1 valid strategy
#2.2 interpretation of answer
1 412
360
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 cm
3 or equivalent
2
39∙25 cm3
or equivalent
# 2.2 Shop owner is wrong
because only 256 cones
can be filled
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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 21
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
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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
End of Question Paper
Diagram 2
Diagram 1
8 cm
6 cm
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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)
(ii)
(b)
1 simplify numerator
2 correct answer
3 correct coefficient
4 simplify indices
5 calculation of amount
6 express in standard form
1 x
4
2 x
2
3
6
4 2
5x in answer 2
5
6
x
5 34 × 2·64 × 10
3
=89·76 × 103
6 8·976 × 10
4
3 (a)
(b)
1 multiply out brackets
2 multiply out the brackets
3 collect like terms
1 6g
2 – gh
2
d2 – 7d + 3d – 21
3 d
2 – 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
k
8 1 evidence of gradient
calculation
2 correct gradient
1 Uses
2 1
2 1
y y
x x
or equivalent
2
9
1
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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 cm
3
10 (a)
(b)
1 correct ratio and substitution
2 calculate arc length
#2.1 valid strategy
#2.2 interpretation of answer
1 211
360
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 cm
3 or equivalent
2
4415∙625 cm3
or equivalent
# 2.2 cylinder is not big enough
since 75·36 × 60 >
volume of cylinder
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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 31
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 – 12z + 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
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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]
End of Question Paper
O
42o
135o
11 cm
18 cm
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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)
(ii)
(b)
1 simplify numerator
2 correct answer
3 correct coefficient
4 simplify indices
5 calculation of distance
6 express in standard form
1 x
10
2 x
13
3
18
4 3
5x in answer 3
5
18
x
5 30 × 2·4 × 10
4
=72 × 104
6 7·2 × 10
5
3 (a)
(b)
1 multiply out brackets
2 multiply out the brackets
3 collect like terms
1 3m
2 – 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 Uses
2 1
2 1
y y
x x
or equivalent
2
7
1
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9 1 substitute and start
calculation
2 complete calculation
3 round calculation to 2
significant figures
1 732
3
1 2
03373
1 or
equivalent
2 37·75806 cm³ or equivalent
3 38 cm
3
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 cm
3 or equivalent
2
569∙91 cm3
or equivalent
# 2.2 cone is big enough
since 523·33 < 569∙91