Dalkeith High School. National 5 Maths Relationships Revision Booklet

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1 Dalkeith High School National 5 Maths Relationships Revision Booklet 1

2 Revision Questions Assessment Standard 1.1 Page 3 Assessment Standard 1. Page 5 Assessment Standard 1.3 Page 8 Assessment Standard 1.4 Page 9 Assessment Standard 1.5 Page 1 Practice Assessments Practice A Page 14 Practice B Page 0 Practice C Page 6

3 National 5 Relationships Revision Questions Assessment Standard 1.1 3

4 Answers 4

5 Assessment Standard 1. 5

6 6

7 Answers 7

8 Assessment Standard 1.3 Answers 8

9 Assessment Standard 1.4 9

10 c) Kite OABC and a circle with centre O are shown. AB is a tangent to the circle at A. BC is a tangent to the circle at C. Given that angle AOC = 140 o, calculate angle ABC. 10

11 Exercise 16 Calculate the size of the shaded angle in these regular polygons: Answers 11

12 Assessment Standard 1.5 Exercise 17 Exercise 18 1

13 Exercise 19 Answers Exercise 17 (Ask your teacher) Exercise 18 a) 10 o b) 10 o c) 90 o d) 60 o e) 10 o f) 180 o g) 36 o h) 90 o Exercise 19 a)5.7 o and o b) 66.4 o and 93.6 o c) 9.6 o and o d) 74.1 o and 54.1 o e) 78.5 o and 81.5 o f) 6.7 o and o g) 48. o and o 13

14 Practice Unit Assessment (1) for National 5 Relationships 1. A straight line with gradient 3 passes through the point (, 5). Determine the equation of this straight line.. Solve the inequation 4p 1 < p The Stuart family visit a new attraction in Edinburgh. They paid 3.5 for 3 adult tickets and 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 S x 6 3 Change the subject of the formula to x. 6. The diagram shows the parabola with equation y kx. What is the value of k? y x 14

15 7. The equation of the quadratic function whose graph is shown below is of the form y = (x + a) + b, where a and b are integers. Write down the values of a and b. y 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) + 4. (a) (b) Write down the equation of its axis of symmetry. 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) = Solve the equation x + x 7 = 0 using the quadratic formula. 1. Determine the nature of the roots of the equation 3x + x 1 = 0 using the discriminant. 15

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. 6 3m 3 3m 5 4m 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 16, calculate angle ADC. A B 16 o 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 cm 3. A similar miniature version is 15cm long. Calculate how much water the miniature version would hold. 16

17 16. Here is a regular, 5 sided polygon. Calculate the size of the shaded angle. 17. Sketch the graph of y = 4sin x o for 0 x Write down the period of the graph of the equation y = cos 3x o. 19. Solve the equation 4sin x 1 = 0, 0 x 360. End of Question Paper 17

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 ( )) (or equivalent) 1 simplify for p simplify numbers 3 solve 3 #.1 uses correct strategy and sets up equation 4 1 multiply by appropriate Factor 3 solve for a solve for b 5 1 subtract 6 multiply by 3 3 divide by 3p 18 3 p < 6 1 #.1 3a + c = a + 5b = 39 5a 5b = 15 or equivalent a = 3 3 b = 6 1 S 6 (S 6) 3 (or equivalent) 3 3( S 6) 6 1 correct value of k 1 k = 5 (or equivalent) 7 1 find value of a find value of b 8 1 identify and annotate roots and y-intercept identify and annotate turning point 3 draw correct shape of graph 1 a = 1 b = 1 3, 1 and (0, 3) ( 1, 4) 3 correctly annotated graph 9 (a) 1 axis of symmetry 1 x = 3 (b) 3 turning point nature ( 3, 4) 3 minimum turning point 10 1 solve equation 1 x = 7, x = correct substitution

19 3 4 evaluation discriminant solve for 1 root complete solution 3 x = x = 3 8 (rounding not required) 1 1 correct substitution evaluate discriminant #. interpret result 13 1 calculates and adds squares of two short sides 1 () #. real and unequal roots Since b 4ac > = squares longest side 6 3 = #. interprets result 14 1 radius and tangent 3 subtract correct answer 15 1 use volume scale factor #. so and hence triangle is not rightangled using converse of Pythagoras. The corners of the room are not right angled. either angle BAD or angle BCD = ( ) (15/150) correct answer 1 cm 3 16 #.1 use a valid strategy #.1 eg centre angles 360/5 = 7 each 1 correct answer correct amplitude and period correctly annotated graph complete with roots and amplitude. 1 4 / 4 and 360 Correct graph 18 1 correct period solve for sin x solve for x 3 complete solution 1 sin x =

20 Practice Unit Assessment () for National 5 Relationships 1. A straight line with gradient 4 passes through the point (, 4). Determine the equation of this straight line.. Solve the inequation 7m + 5 < m The Clelland family visit a new attraction in Inverness. They paid 9.40 for adult tickets and 4 child tickets. Write an equation to represent this information. 4. Solve the following system of equations algebraically: 7x + y = 3 x y = 6 5. Here is a formula Change the subject of the formula to B. A 4B 5 6. The diagram shows the parabola with equation y kx. What is the value of k? y x 0

21 7. The equation of the quadratic function whose graph is shown below is of the form y = (x + a) + b, where a and b are integers. Write down the values of a and b. y x 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) 3. (a) (b) Write down the equation of its axis of symmetry. 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) = Solve the equation x 3x = 0 using the quadratic formula. 1. Determine the nature of the roots of the equation 4x + 3x + 5 = 0 using the discriminant. 1

22 13. A shape has dimensions as shown in the diagram. 1 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. Given that angle LMN is 14, calculate angle LPN. L P 14 o M 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.

23 16. Here is a regular, 1 sided polygon. Calculate the size of the shaded angle. 17. Sketch the graph of y = 7cos x o for 0 x Write down the period of the graph of the equation y = sin 5x o. 19. Solve the equation 7cos x = 0, 0 x 360. End of Question Paper 3

24 Practice Unit Assessment () 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 ) (or equivalent) 1 simplify for m simplify numbers 3 solve 3 #.1 uses correct strategy and sets up equation 4 1 multiply by appropriate factor 3 solve for x solve for y 5 1 add multiply by 5 3 divide by 4 5m 5 3 m < 5 1 #.1 a + 4c = x + y = 3 4x y = 1 or equivalent x = 4 3 y = 1 A + (A + ) 5 (or equivalent) 3 5( A ) correct value of k 1 k = (or equivalent) 7 1 find value of a find value of b 8 1 identify and annotate roots and y-intercept identify and annotate turning point 3 draw correct shape of graph 1 a = 1 b = 4 1 5, 7 and (0, 35) (6, 1) 3 correctly annotated graph 9 (a) 1 axis of symmetry 1 x = 4 (b) 3 turning point nature ( 4, 3) 3 minimum turning point 10 1 solve equation 1 x = 5, x = correct substitution

25 3 4 evaluation discriminant solve for 1 root complete solution 3 x = x = 0 6 (rounding not required) 1 1 correct substitution evaluate discriminant #. interpret result 13 1 calculates and adds squares of two short sides 1 (3) #. roots are not real since b 4ac < = squares longest side 1 5 = #. interprets result 14 1 radius and tangent 3 subtract correct answer 15 1 use volume scale factor #. so = 1 5 and hence triangle is rightangled using converse of Pythagoras. The shape is a rectangle either angle PLM or angle MNP = ( ) (10/30) correct answer 55 6 cm 3 16 #.1 use a valid strategy #.1 eg centre angles 360/1 = 30 each 1 correct answer correct amplitude and period correctly annotated graph complete with roots and amplitude. 1 7 / 7 and 360 Correct graph 18 1 correct period solve for cos x solve for x 3 complete solution 1 cos x = /

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.. Solve the inequation 5k 3 < k A group of friends met in a coffee bar. They paid 9.40 for 4 cappuccinos and lattes. Write an equation to represent this information. 4. Solve the following system of equations algebraically: 5c d = 36 c + d = Here is a formula 5m k 7 4 Change the subject of the formula to m. 6. The diagram shows the parabola with equation y kx What is the value of k? y x 6

27 7. The equation of the quadratic function whose graph is shown below is of the form y = (x + a) + b, where a and b are integers. Write down the values of a and b. y x 8. Sketch the graph y = (x 4)(x + ) 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). (a) (b) Write down the equation of its axis of symmetry. 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) = Solve the equation x + 5x 7 = 0 using the quadratic formula. 1. Determine the nature of the roots of the equation 9x + 6x + 1 = 0 using the discriminant. 7

28 13. A shape has dimensions as shown. A 6 7m B 8m 4 6m D C Is angle DAB = 90 o 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. W X 139 o Y Z 15. A tube of toothpaste is 1 cm long and has a volume of 50cm³ A similar miniature version is 9cm long. Calculate how much toothpaste the miniature version would hold. 8

29 16. Here is a regular, 10 sided polygon. Calculate the size of the shaded angle. 17. Sketch the graph of y = 3sin x o for 0 x Write down the period of the graph of the equation y = sin ½ x o. 19. Solve the equation 5tan x 7 = 0, 0 x 360. End of Question Paper 9

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) 1 simplify for k simplify numbers 3 solve 3 #.1 uses correct strategy and sets up equation 4 1 multiply by appropriate Factor 3 solve for c solve for d 5 1 subtract 7 multiply by 4 3 divide by 5 3k 1 3 k < 4 1 #.1 4c + l = c d = 36 5c + d = 34 or equivalent c = 10 3 d = 7 1 k 7 (k 7) 4 (or equivalent) 3 4( k 7) correct value of k 1 k = 3 (or equivalent) 7 1 find value of a find value of b 8 1 identify and annotate roots and y-intercept identify and annotate turning point 3 draw correct shape of graph 1 a = 1 b = 3 1, 4 and (0, 8) (1, 9) 3 correctly annotated graph 9 (a) 1 axis of symmetry 1 x = 3 (b) 3 turning point nature ( 3, 5) 3 maximum turning point 10 1 solve equation 1 x = 1, x = correct substitution

31 3 4 evaluation discriminant solve for 1 root complete solution 3 x = x = 6 1 (rounding not required) 1 1 correct substitution evaluate discriminant #. interpret result 13 1 calculates and adds squares of two short sides 1 (6) #. equal roots since b 4ac = = squares longest side 8 = 64 #. interprets result 14 1 radius and tangent 3 subtract correct answer 15 1 use volume scale factor #. so and hence triangle is not rightangled using converse of Pythagoras. Angle DAB is not a right angle. either angle ZWX or angle ZYX = ( ) (9/1) 3 50 correct answer 4 cm 3 16 #.1 use a valid strategy #.1 eg centre angles 360/10 = 36 each 1 correct answer correct amplitude and period correctly annotated graph complete with roots and amplitude. 1 3 / 3 and 360 Correct graph 18 1 correct period solve for tan x solve for x 3 complete solution 1 tan x =

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