s y s a g e P
m o r f
s e c r u o s e r
5
l a n o i t a N
-
e c n e l l e c x E M A T H E M A T I C S r o f
m u l u c i r r u C
CFE Mathematics
National 5
Pegasys 2013
Practice Unit Tests CFE Mathematics
National 5
Practice Unit Tests Practice Unit Assessment (1) for National 5 Expressions and Formulae
1. Simplify, giving your answer in surd form: 32
Contents & Information 5 2. (a) Simplify (i) (ii) 5x4 4x 2
4 (b) The number of people attending 9 a Practice football match Unit was Tests...... 3·12 × 10 . If(3 each for person each paid unit) £27, how much was collected? Give you answer in Scientific Notation.
Answers & marking schemes
3. Expand and simplify where appropriate:
(a) d(4d – e) (b) (g + 4)(g + 9) Pegasys Educational Publishing
4. Factorise: (a) y² – 6y (b) t² – 49 (c) x² + 7x + 12 Detailed marking schemes
5. Express x² + 6x + 7 in the form (x + p)² + q.
6. Write in its simplest form.
7. Write each of the following as a single fraction:
(a) (b)
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
A 240o 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.
Pegasys 2013 B (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.
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
Practice Unit Assessment (1) for Expressions and Formulae: Marking Scheme
Points of reasoning are marked # in the table.
Question Main points of expected responses
Pegasys 2013 1 1 start of process 1 √16√2 (or equivalent) 2 simplified surd 2 4√2 2 (a) (i) 1 simplify numerator 1 x10 2 correct answer 2 x7 (ii) 3 correct coefficient 3 20 4 3 3 simplify indices 4 x 2 x 2 5 in answer 20 (b) calculation of amount 5 27 × 3·12 × 104 4 6 =84·24 × 10 express in standard form 6 £8·424 × 105 3 (a) 1 multiply out brackets 1 4d2 – de (b) 2 multiply out the brackets 2 g2 + 4g + 9g + 36 3 collect like terms 3 g2 + 13g + 36 4 (a) 1 factorise expression 1 y(y – 6) (b) 2 factorise difference of two 2 (t + 7)(t – 57) squares (c) 3 start to factorise trinomial 3 (x 3)(x 4) ie evidence of expression 4 complete factorisation brackets, x, 3 and 4 4 (x + 3)(x + 4)
5 1 start of process 1 (x + 3)2 2 complete process 2 (x + 3)2 – 2
6 1 reduce to simplest form 4x 3 1 x 4
7 (a) 1 denominator correct /// 1 ab 3b 5a (b) 2 numerator correct 2 ab g 3 multiply by inversion of 3 fraction e 4 fg correct answer 4 5e
1 8 evidence of gradient y2- y 1 1 Uses or equivalent calculation x2- x 1 7 2 correct gradient 2 11
9 1 substitute and start 4 1 2 33 calculation 3 4 12 167 or 3 equivalent 2 complete calculation 2 50·939 cm³ or equivalent Pegasys 2013 3 round calculation to 2 3 51 cm3 significant figures
10 (a) 1 correct ratio and substitution 240 1 12 4 360 2 calculate arc length 2 25·957 cm or equivalent (b) #2.1 valid strategy #2.2 interpretation of answer #2.1 eg 2 000 ÷ 38 #2.2 (for 52∙63) 52 logos can be made.
11 #2.1 uses valid strategy to find # 2.1 Substitutes relevant values volumes of cone and into correct formulae cylinder
1 calculate volume of cylinder 1 10 048 cm3 or equivalent 2 calculate volume of cone 2 39∙25 cm3 or equivalent
# 2.2 states conclusion # 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: 54
1 2. (a) Simplify (i) (ii) 2x 2 3x 3 Pegasys 2013 (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 in its simplest form.
7. Write each of the following as a single fraction:
(a) (b)
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 230o
B
Pegasys 2013 (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. 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
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
Pegasys 2013 2 (a) (i) 1 simplify numerator 1 x4 2 correct answer 2 x2 (ii) 3 correct coefficient 3 6 4 4 5 5 simplify indices x 2 in answer 6x 2 5 (b) calculation of amount 5 34 × 2·64 × 103 =89·76 × 103 6 express in standard form 6 8·976 × 104 3 (a) 1 multiply out brackets 1 6g2 – gh (b) 2 multiply out the brackets 2 d2 – 7d + 3d – 21 3 collect like terms 3 d2 – 4d – 21 4 (a) 1 factorise expression 1 k(k – 7) (b) 2 factorise difference of two 2 (x + 9)(x – 9) squares (c) 3 start to factorise trinomial 3 (z 3)(z 7) ie evidence of expression 4 complete factorisation brackets, z, 3 and 7 4 (z + 3)(z + 7)
5 1 start of process 1 (x – 4)2 2 complete process 2 (x – 4)2 – 15
6 1 reduce to simplest form 3x 1 1 x 3
7 (a) 1 denominator correct /// 1 cd 5d 7c (b) 2 numerator correct 2 cd h 3 multiply by inversion of 3 fraction k 4 k correct answer 4 7
1 8 evidence of gradient y2- y 1 1 Uses or equivalent calculation x2- x 1 1 2 correct gradient 2 9
9 1 substitute and start 4 1 3 73 calculation 3 4 50 653 or 3 equivalent 2 complete calculation 2 212·067 cm³ or equivalent 3 round calculation to 2 3 210 cm3 significant figures
Pegasys 2013 10 (a) 1 correct ratio and substitution 230 1 11 2 360
2 calculate arc length 2 22·468 cm or equivalent (b) #2.1 valid strategy #2.1 eg 4 000 ÷ 22·468 #2.2 interpretation of answer #2.2 (for 178∙02) 178 cones can be trimmed.
11 #2.1 uses valid strategy to find # 2.1 Substitutes relevant values volumes of cone and into correct formulae cylinder
1 calculate volume of cone 1 75·36 cm3 or equivalent 2 calculate volume of cylinder 2 4415∙625 cm3 or equivalent
# 2.2 states conclusion # 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: 147
1 2. (a) Simplify (i) (ii) 6x 3 3x 2
(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.
Pegasys 2013 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 in its simplest form.
7. Write each of the following as a single fraction:
(a) (b)
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
10. (a) Calculate the area of the sector of a circle in the diagram which has radius 6∙8cm.
o O135
Pegasys 2013 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
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
Pegasys 2013 2 (a) (i) 1 simplify numerator 1 x10 2 correct answer 2 x13 (ii) 3 correct coefficient 3 18 4 4 5 5 simplify indices 3 in answer 3 5 x 18x (b) calculation of distance 5 30 × 2·4 × 104 4 6 =72 × 10 express in standard form 6 7·2 × 105 3 (a) 1 multiply out brackets 1 3m2 – mn (b) 2 multiply out the brackets 2 p2 + 8p + 5p + 40 3 collect like terms 3 p2 + 13p + 40 4 (a) 1 factorise expression 1 h(h – 11) (b) 2 factorise difference of two 2 (q + 12)(q – 12) squares (c) 3 start to factorise trinomial 3 (a 4)(a 8) ie evidence of expression 4 complete factorisation brackets, a, 4 and 8 4 (a – 4)(a – 8)
5 1 start of process 1 (x + 3·5)2 2 complete process 2 (x + 3·5)2 – 3·25
6 1 reduce to simplest form x 7 1 2x 5
7 (a) 1 denominator correct /// 1 mn 4n 9m (b) 2 numerator correct 2 mn l 3 multiply by inversion of 3 fraction k 4 4l correct answer 4 k 2
1 8 evidence of gradient y2- y 1 1 Uses or equivalent calculation x2- x 1 1 2 correct gradient 2 7
9 1 substitute and start 1 1 232 7 calculation 3 1 37 03 or 3 equivalent 2 complete calculation 2 37·75806 cm³ or equivalent 3 round calculation to 2 3 38 cm3 significant figures
Pegasys 2013 10 (a) 1 correct ratio and substitution 135 1 682 360
2 calculate sector area 2 54·4476 cm or equivalent (b) #2.1 valid strategy #2.1 eg 6 500 ÷ 54·4476 #2.2 interpretation of answer #2.2 (for 119∙38) 119 sectors can be cut.
11 #2.1 uses valid strategy to find # 2.1 Substitutes relevant values volumes of cone and into correct formulae sphere
1 calculate volume of sphere 1 523·33 cm3 or equivalent 2 calculate volume of cone 2 569∙91 cm3 or equivalent
# 2.2 states conclusion # 2.2 cone is big enough since 523·33 < 569∙91
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.
Pegasys 2013 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 2x S 6 3
Change the subject of the formula to x.
y 6. The diagram shows the parabola with equation y kx 2 . 40
What is the value of k? 35 30
25
20
15
10
y 5
– 3 – 2 –8 1 0 1 2 3 x
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. 6
Write down the values of a and b. 4
2
Pegasys 2013
– 2 – 1 0 1 2 3 4 x 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.
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.
6·3m 3·3m
5·4m
Pegasys 2013 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. B 126o A C
D
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.
16. Here is a regular, 5 – sided polygon.
Calculate the size of the shaded angle.
Pegasys 2013 17. Sketch the graph of y = 4sin xo for 0° 吵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° 吵x 360° .
End of Question Paper
Pegasys 2013 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 1 3p 2 simplify numbers 2 18 3 solve 3 p < 6
3 #2.1 uses correct strategy and sets #2.1 3a + 2c = 32·25 up equation 4 1 multiply by appropriate 1 3a + 5b = 39 Factor 5a – 5b = –15 or equivalent 2 solve for a 2 a = 3 3 solve for b 3 b = 6
5 1 subtract 6 1 S – 6 2 multiply by 3 2 (S – 6) × 3 (or equivalent) 3 divide by 2 3(S 6) 3 2 (or equivalent)
6 1 correct value of k 1 k = 5
7 1 find value of ‘a’ 1 a = –1 2 find value of ‘b’ 2 b = 2
8 1 identify and annotate roots 1 –3, 1 and (0, –3) and y-intercept 2 identify and annotate turning 2 (–1, –4) point 3 draw correct shape of graph 3 correctly annotated graph
9 (a) 1 axis of symmetry 1 x = 3
(b) 2 turning point 2 (–3, 4) 3 nature 3 minimum turning point
10 1 solve equation 1 x = –7, x = 3
11 1 correct substitution 1 2 22 41 7 2
2 32 2 evaluation discriminant 3 x = 1·8 3 solve for 1 root 4 x = –3·8 Pegasys 2013 4 complete solution (rounding not required)
12 1 correct substitution 1 (2)2 – 4 × 3 × –1 2 evaluate discriminant 2 16
#2.2 interpret result #2.2 real and unequal roots Since b2 – 4ac > 0
13 1 calculates and adds squares 1 3·32 +5·42 = 40·05 of two short sides
2 squares longest side 2 6·32 = 39·69
#2.2 interprets result #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 1 either angle BAD or angle BCD = 90° 2 subtract 2 360 – (90 + 90 + 126) 3 correct answer 3 54°
15 1 use volume scale factor 1 (15/150)3 × 12000
2 correct answer 2 12 cm3
16 #2.1 use a valid strategy #2.1 eg centre angles 360/5 = 72° each
1 correct answer 1 108°
17 1 correct amplitude and period 1 4 / – 4 and 360° 2 correctly annotated graph 2 Correct graph complete with roots and amplitude.
18 1 correct period 1 120°
19 1 solve for sin x° 1 sin x° = 0·25 2 solve for x 2 14·5° 3 complete solution 3 165·5°
Practice Unit Assessment (2) for National 5 Relationships
Pegasys 2013 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 4B A 2 5
Change the subject of the formula to B.
y 6. The diagram shows the parabola with equation y kx 2 . 16
What is the value of k? 14 12
10
8
6
4
2
– 3 – 2 – 1 0 1 2 3 x
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. y Pegasys 2013 8
6
4
2
– 2 – 1 0 1 2 3 4x
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.
13. A shape has dimensions as shown in the diagram.
Pegasys 2013 12·5m 10m
7·5m
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. L Given that angle LMN is 142°, calculate angle LPN.
1
4
2 M P o
N
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. .
16. Here is a regular, 12 – sided polygon.
Pegasys 2013 Calculate the size of the shaded angle.
17. Sketch the graph of y = 7cos xo for 0° 吵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° 吵x 360° .
End of Question Paper
Pegasys 2013 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 1 5m 2 simplify numbers 2 25 3 solve 3 m < 25
3 #2.1 uses correct strategy and sets #2.1 2a + 4c = 29·4 up equation 4 1 multiply by appropriate 1 7x + 2y = 32 factor 4x – 2y = 12 or equivalent 2 solve for x 2 x = 4 3 solve for y 3 y = 2
5 1 add 2 1 A + 2 2 multiply by 5 2 (A + 2) × 5 (or equivalent) 3 divide by 4 5(A 2) 3 4 (or equivalent)
6 1 correct value of k 1 k = 2
7 1 find value of ‘a’ 1 a = –1 2 find value of ‘b’ 2 b = 4
8 1 identify and annotate roots 1 5, 7 and (0, 35) and y-intercept 2 identify and annotate turning 2 (6, –1) point 3 draw correct shape of graph 3 correctly annotated graph
9 (a) 1 axis of symmetry 1 x = –4
(b) 2 turning point 2 (–4, –3) 3 nature 3 minimum turning point
10 1 solve equation 1 x = –5, x = 10
11 1 correct substitution 1 3 32 41 2 2
2 17 2 evaluation discriminant 3 x = 3·6 3 solve for 1 root 4 x = –0·6 Pegasys 2013 4 complete solution (rounding not required)
12 1 correct substitution 1 (3)2 – 4 × 4 × 5 2 evaluate discriminant 2 –71
#2.2 interpret result #2.2 roots are not real since b2 – 4ac < 0
13 1 calculates and adds squares 1 7·52 +102 = 156·25 of two short sides
2 squares longest side 2 12·52 = 156·25
#2.2 interprets result #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 1 either angle PLM or angle MNP = 90° 2 subtract 2 360 – (90 + 90 + 142) 3 correct answer 3 38°
15 1 use volume scale factor 1 (10/30)3 × 15000
2 correct answer 2 55·6 cm3
16 #2.1 use a valid strategy #2.1 eg centre angles 360/12 = 30° each
1 correct answer 1 150°
17 1 correct amplitude and period 1 7 / – 7 and 360° 2 correctly annotated graph 2 Correct graph complete with roots and amplitude.
18 1 correct period 1 72°
19 1 solve for cos x° 1 cos x° = 2/7 2 solve for x 2 73·4° 3 complete solution 3 286·6°
Practice Unit Assessment (3) for National 5 Relationships
Pegasys 2013 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 5m k 7 4
Change the subject of the formula to m.
y 6. The diagram shows the parabola with 2 24 equation y= kx 21 What is the value of k? 18
15
12 y
9 8 6
3 6
– 3 – 2 – 1 0 1 2 3 x
4 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. 2 Write down the values of a and b.
Pegasys 2013
– 4 – 3 –2 –1 0 1 2 x
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.
13. A shape has dimensions as shown.
A 6·7m B
Pegasys 2013 8m 4·6m
D C
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. X 139o W Y
Z
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.
16. Here is a regular, 10 – sided polygon.
Pegasys 2013 Calculate the size of the shaded angle.
17. Sketch the graph of y = –3sin xo for 0° 吵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° 吵x 360° .
End of Question Paper
Pegasys 2013 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 1 3k 2 simplify numbers 2 12 3 solve 3 k < 4
3 #2.1 uses correct strategy and sets #2.1 4c + 2l = 9·4 up equation 4 1 multiply by appropriate 1 5c – 2d = 36 Factor 5c + 2d = 34 or equivalent 2 solve for c 2 c = 10 3 solve for d 3 d = 7
5 1 subtract 7 1 k – 7 2 multiply by 4 2 (k – 7) × 4 (or equivalent) 3 divide by 5 4(k 7) 3 5 (or equivalent)
6 1 correct value of k 1 k = 3
7 1 find value of ‘a’ 1 a = 1 2 find value of ‘b’ 2 b = 3
8 1 identify and annotate roots 1 –2, 4 and (0, –8) and y-intercept 2 identify and annotate turning 2 (1, –9) point 3 draw correct shape of graph 3 correctly annotated graph
9 (a) 1 axis of symmetry 1 x = –3
(b) 2 turning point 2 (–3, 5) 3 nature 3 maximum turning point
10 1 solve equation 1 x = –1, x = 7
11 1 correct substitution 1 5 52 4 1 7 2
2 53 2 evaluation discriminant 3 x = 1·1 3 solve for 1 root 4 x = –6·1 Pegasys 2013 4 complete solution (rounding not required)
12 1 correct substitution 1 (6)2 – 4 × 9 × 1 2 evaluate discriminant 2
#2.2 interpret result #2.2 equal roots since b2 – 4ac = 0
13 1 calculates and adds squares 1 4·62 +6·72 = 66·05 of two short sides
2 squares longest side 2 82 = 64
#2.2 interprets result #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 1 either angle ZWX or angle ZYX = 90° 2 subtract 2 360 – (90 + 90 + 139) 3 correct answer 3 41°
15 1 use volume scale factor 1 (9/21)3 × 50
2 correct answer 2 4 cm3
16 #2.1 use a valid strategy #2.1 eg centre angles 360/10 = 36° each
1 correct answer 1 144°
17 1 correct amplitude and period 1 – 3 / 3 and 360° 2 correctly annotated graph 2 Correct graph complete with roots and amplitude.
18 1 correct period 1 720°
19 1 solve for tan x° 1 tan x° = 1·4 2 solve for x 2 54·5° 3 complete solution 3 234·5°
Practice Unit Assessment (1) for National 5 Applications
1. A farmer wishes to spread fertiliser on a triangular plot of ground.
Pegasys 2013 The diagram gives the dimensions of the plot.
35 m 58o
44 m
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.
T
300 m
12o S R 350 m
N
` What is the shortest distance between the two runners? [i.e. the distance TR on the diagram]
U
400 km
125o 3. On an orienteering course thereV are three checkpoints at points U, V and W as shown in the diagram below. 220 km Pegasys 2013 W
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.
Pegasys 2013 4. The diagrams below show 2 directed line segments u and v.
u 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.
z
S y
R Q (6, 6, 0) x O P
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.
5 3 15 a 2 b 7 c 6 2 5 55 2
Find the resultant force.
Pegasys 2013 5 1 7. Vector p and vector q . 3 3
Calculate 2 p q
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. 3 2 m 7
2 1 m 3
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.
Pegasys 2013 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.
25
20 ) S (
t s
e 15 T
s g
, n t i l l l 10 u s e e p S R 5
0 0 10 20 30 40 50 Hours spent watching TV, (h)
(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
Pegasys 2013 Practice Unit Assessment (1) for Applications: Marking Scheme
Points of reasoning are marked # in the table.
Main points of expected responses Question 1 1 substitute into 1 1 35 44 sin 58 formula 2
2 correct answer 2 653 m2
2 1 use correct formula 1 selects cosine rule
2 substitute correctly 2 s 2 3002 3502 2 300 350 cos12
3 7 089 3 process to s2 4 84·1 metres (rounding not 4 take square root required)
3 #2.1 uses correct strategy 220sin125 #2.1 sinU then valid 400 steps below
1 finds angle U 1 26·8
2 states bearing from 2 153·2o (rounding not required) U 4 1 draws 3u v 2 applies head-to-tail method when adding v 3u 3u + v 3 draws resultant from tail of 3u to head of v.
5 1 correct point 1 (3, 3, 8)
Pegasys 2013 6 1 add to get resultant 5 3 15 1 2 2 7 6 correct answer 2 5 5 5 2 35 2 15 6
7 1 correct scalar 10 1 11 1 multiplication then 6 3 3 addition
2 2 calculate magnitude
3 3 correct answer 130
8 1 start calculation 1 0·88 2 process calculation 2 24 000 × 0·885 3 correct answer 3 £12 665·57
Note: repeated subtraction equivalent method can be used
9 1 area calculation 17 5 1 7 3 2 correct answer 85 1 2 4 m² 21 21 10 #2.1 appropriate strategy #2.1 eg 1 + 0·15 x = £2760
1 correct answer 1 £2 400
11 (a) 1 mean for A 1 105 ÷ 7 = 15
2 calculates 2 1, 4, 4, 1, 16, 16, 4
3 substitute into 3 formula 4 2·77 (rounding not required) 4 correct standard (Equivalent calculations can be used) deviation
#2.2 On average up market prices more (b) #2.2 Compares mean and expensive standard deviation in a valid way for data There is less of a spread in up market restaurants
12 (a) 1 chooses 2 distinct 22 5 7 5 1 m points and 10 40 substitutes into gradient formula
Pegasys 2013 2 1 2 m (or based on gradient calculates gradient 2 line of best fit 3 finds intercept 3 c = 27·5 (approximately or by calculation or from graph) (b) 4 writes down equation 1 4 S = h + 27·5 2 (c) (or equivalent) # 2.2 estimate mark #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.
22 m 24 m
56o
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.
W
520 m
14o A B 580 m
` How far apart are the ships?[i.e. the distance WB on the diagram]
Pegasys 2013 3. On a radar screen, three planes, P, Q and R are at the positions shown in the diagram.
N
P
450 km
o Q 132
300 km
R
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.
Pegasys 2013 4. The diagrams below show 2 directed line segments a and b.
b a
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.
z
S y
R (0, 8, 0) Q x O P
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.
3 2 35 k 2 5 l 4 m 0 4 15 4
Find the resultant force.
Pegasys 2013 3 2 7. Vector a and vector b . 6 5
Calculate a 2b
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. 3 4 m 8
3 2 m 4
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. 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. ) T (
s e r t e m i t n e c
n i
n o i t a l u s n i
f o
s s e n k c i h T 25
20 (a) Determine the gradient and the y-intercept of the line of best fit shown. 15 (b) Using these values for the gradient and the y-intercept, write down the equation of the line. 10 (c) Estimate the thickness of insulation for a heat loss of 2·5 kilowatts.
5
0 End of Question Paper 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.
Main points of expected responses Question 1 1 substitute into 1 1 22 24 sin 56 formula 2
2 correct answer 2 219 m2
2 1 use correct formula 1 selects cosine rule
2 substitute correctly 2 a 2 520 2 5802 2 520 580 cos14
3 21517 3 process to a2 4 146·7 metres (rounding not 4 take square root required)
3 #2.1 uses correct strategy 300sin132 #2.1 sin P then valid 450 steps below v
1 finds angle P 1 29·7
2 states bearing from 2 150·3o (rounding not required) P 4 1 draws 2a
2 applies head-to-tail method when adding 2b 2b 2a 3 draws resultant from 2a + 2b tail of 2a to head of 2b
5 1 correct point 1 (4, 4, 10)
Pegasys 2013 6 1 add to get resultant 3 2 35 1 2 25 4 0 correct answer 4 15 4 15 2 65 65
7 1 correct scalar 3 4 1 1 multiplication then 6 10 4 addition
2 2 calculate magnitude
3 3 correct answer 17
8 1 start calculation 1 1·036 2 process calculation 2 142 000 × 1·036³ 3 correct answer 3 £157 894
Note: repeated addition equivalent method can be used
9 1 area calculation 35 11 1 8 4 2 correct answer 385 1 2 12 m² 32 32 10 #2.1 appropriate strategy #2.1 eg 1 + 0·35 x = £3510
1 correct answer 1 £2 600
11 (a) 1 mean for A 1 1184 ÷ 8 = 148
2 calculates 2 4, 1, 0, 25, 1, 25, 9, 1
3 substitute into 3 formula 4 3·07 (rounding not required) 4 correct standard (Equivalent calculations can be used) Deviation
#2.2 On average weights the same (b) #2.2 Compares mean and standard deviation in a Wider spread on second line. valid way for data
12 (a) 1 chooses 2 distinct 20 10 1 m points and 15 35 substitutes into
Pegasys 2013 gradient formula 2 m 5 (or based on gradient line of best fit 2 calculates gradient 3 c = 27·5 (approximately or by 3 finds intercept calculation or from graph)
4 writes down 4 T = 5 H + 27·5 (b) equation (or equivalent)
(c) # 2.2 estimate mark #2.2 Approximately 15 cm
Practice Unit Assessment (3) for National 5 Applications
Pegasys 2013 1. Turf has to be laid on a triangular plot of garden .
The diagram gives the dimensions of the plot.
27 m
102o 25 m
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.
B
24 m
17o P T 26 m
` How far apart are the bowls after this first throw?[i.e. the distance PB on the diagram]
Pegasys 2013 3. The positions of three players, K, L and M are shown in this diagram.
N
K
40 m
o L 125
30 m
M
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.
Pegasys 2013 4. The diagrams below show 2 directed line segments k and l.
k 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.
z
S(2, 2, 5) y
R Q x O P
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.
4 2 2 x 2 3 y 2 7 z 1 1 0 5 2
Find the resultant force.
Pegasys 2013 3 2 7. Vector x and vector y . 6 5
Calculate 3x 2 y
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.
1 6 m 4 1 10 m 5 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.
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.
P 80
70
60 s t n i o
P 50
40
30
20
10
4 8 12 16 20 W Wins
(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
Practice Unit Assessment (3) for Applications: Marking Scheme
Points of reasoning are marked # in the table. Pegasys 2013 Main points of expected responses Question 1 1 substitute into 1 1 27 25 sin102 formula 2
2 correct answer 2 330 m2
2 1 use correct formula 1 selects cosine rule
2 substitute correctly 2 t 2 242 262 2 24 26 cos17
3 process to t2 3 58·53
4 take square root 4 7·7 metres (rounding not required)
3 #2.1 uses correct strategy 30sin125 #2.1 sin K then valid 40 steps below
1 finds angle K 1 38
2 states bearing from 2 142o (rounding not required) K 4 1 draws k
2 applies head-to-tail method when adding 2l 2l k 3 draws resultant from tail of k to head of 2l
5 1 correct point 1 (4, 4, 0)
6 1 add to get resultant 4 2 1 2 2 3 2 7 1 correct answer 1 0 5 2
Pegasys 2013 0 2 6 2 5
7 1 correct scalar 9 4 5 1 multiplication then 18 10 8 addition
2 2 calculate magnitude
3 3 correct answer 89
8 1 start calculation 1 0·77 2 process calculation 2 2 000 × 0·77³ 3 correct answer 3 913g
Note: repeated addition equivalent – 3 method can be used
9 1 area calculation 25 51 1 4 5 2 correct answer 255 3 2 63 m² 4 4 10 #2.1 appropriate strategy #2.1 eg (1 – 0·45) x = £6 875
1 correct answer 1 £12 500
11 (a) 1 mean 1 198 ÷ 6 = 33
2 calculates 2 4, 36, 100, 225, 9, 36
3 substitute into 3 formula 4 9 (rounding not required) 4 correct standard (Equivalent calculations can be used) deviation
#2.2 On average second group had (b) #2.2 Compares mean and higher marks standard deviation in a valid way for data Second group’s marks less spread out
12 (a) 1 chooses 2 distinct 40 20 1 m points and 12 6 substitutes into gradient formula 10 2 m (or based on gradient line 3 2 calculates gradient of best fit)
Pegasys 2013 3 finds intercept 3 c = 0 (approximately or by calculation or from graph) 4 writes down (b) equation 10 4 P = W 3 (or equivalent)
#2.2 90 points (c) # 2.2 estimate mark
Pegasys 2013