National 5 Maths...11 Practice Unit Assessment (1) for National 5 Expressions and Formulae 1. Simplify, giving your answer in surd form: 32 2. (a) Simplify (i) 3 4 6 x ux (ii) 2 5

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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




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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Answers

5


Answers

6


9

Answers

11

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

12

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

13

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)


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

14

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

15




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.


(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

16

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

17




1 1 simplify surd

1 3√6

2 (a) (i)

(ii)

(b)


2 correct answer


4 simplify indices

5 calculation of amount


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 6g

2 – gh

2

d2 – 7d + 3d – 21

3 d

2 – 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


1

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 Uses

2 1

2 1

y y

x x

or equivalent

2

9

1

18


calculation



significant figures

1

3733

4

653503

4 or

equivalent


3 210 cm

3

10 (a)

(b)


2 calculate arc length

#2.1 valid strategy


1 211

360

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 cm

3 or equivalent

2

4415∙625 cm3

or equivalent

# 2.2 cylinder is not big enough

since 75·36 × 60 >

volume of cylinder

19




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.


(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

20

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

21




1 1 simplify surd

1 7√3

2 (a) (i)

(ii)

(b)


2 correct answer


4 simplify indices

5 calculation of distance


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 3m

2 – 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


1

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 Uses

2 1

2 1

y y

x x

or equivalent

2

7

1

22


calculation



significant figures

1 732

3

1 2

03373

1 or

equivalent


3 38 cm

3

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 cm

3 or equivalent

2

569∙91 cm3

or equivalent

# 2.2 cone is big enough

since 523·33 < 569∙91

Documents

National 5 Maths...11 Practice Unit Assessment (1) for National 5 Expressions and Formulae 1. Simplify, giving your answer in surd form: 32 2. (a) Simplify (i) 3 4 6 x ux (ii) 2 5