M thematics. National 5 Practice Paper A. Paper 1. Duration 1 hour. Total marks 40

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1 N5 M thematics Natinal 5 Practice Paper A Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper A Last updated 18/03/14

2 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah x x 2 Standard deviatin: s = =, where is the sample size. n 1 x 2 n 1 x 2 /n Natinal 5 Practice Paper A Last updated 18/03/14

3 1. Evaluate Factrise Find the equatin f this straight line in the frm = Express = 2 + in the frm = and hence state the crdinates f the turning pint = 3 Change the subject f the frmula t. 3 Natinal 5 Practice Paper A Last updated 18/03/14

4 6. Tw vectrs are defined as = ( 2 ) and = ( 4 3 ). (a) Find the resultant vectr (b) Find Part f the graph f = cs is shwn in the diagram. State the value f Find the pint f intersectin f the straight lines with equatins 2 + = and 3 = A parabla has equatin = Using the discriminant, determine the nature f its rts. 3 Natinal 5 Practice Paper A Last updated 18/03/14

5 10. A straight line has the equatin 3 =. A secnd line is parallel t this and passes thrught the pint 3. Write dwn the equatin f the secnd line The equatin f the parabla in the diagram abve is = 2 2. (a) State the crdinates f the minimum turning pint f the parabla. 2 (b) Find the crdinates f. 2 (c) is the pint 1 0. State the crdinates f. 1 Natinal 5 Practice Paper A Last updated 18/03/14

6 12. The square and rectangle shwn belw have the same perimeter. Shw that the length f the rectangle is centimetres (a) Express as a single fractin in its simplest frm. 3 x x 0 x 2 x + 2 (b) Express as a surd in its simplest frm. 3 [End f questin paper] Natinal 5 Practice Paper A Last updated 18/03/14

7 N5 M thematics Natinal 5 Practice Paper A Paper 2 Duratin 1 hur and 30 minutes Ttal marks 50 Yu may use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper A Last updated 18/03/14

8 1. The ppulatin f a city is increasing at a steady rate f 2.4% per annum. The current ppulatin is What is the expected ppulatin in 4 years? Give yur answer t the nearest thusand Tw grups f 6 students are given the same test. (a) The marks f Grup A are: Use an apprpriate frmula t calculate the mean and the standard deviatin. Shw clearly all yur wrking. 4 (b) In Grup B, the mean is 60 and the standard deviatin is Cmpare the results f the tw grups Multiply ut the brackets and cllect like terms Natinal 5 Practice Paper A Last updated 18/03/14

9 4. Grdn and Brian leave a hstel at the same time. Grdn walks n a bearing f 045 at a speed f 4.4 kilmetres per hur. Brian walks n a bearing f 100 at a speed f 4.8 kilmetres per hur. If they bth walk at stead speeds, hw far apart will they be after 2 hurs? 5 5. The diagram shws a mirrr which has been designed fr a new htel. The shape cnsists f a sectr f a circle and a kite AOCB. The circle, centre O, has a radius f 50 centimetres. Angle AOC = 140 AB and CB are tangents t the circle at A and C respectively. Find the perimeter f the mirrr. 5 Natinal 5 Practice Paper A Last updated 18/03/14

10 6. A drinks cntainer is in the shape f a cylinder with radius 20 centimetres and height 50 centimetres. (a) Calculate the vlume f the drinks cntainer. Give yur answer in cubic centimetres, crrect t tw significant figures. 3 (b) Liquid frm the full cntainer can fill 800 cups, in the shape f cnes, each f radius 3 centimetres. What will be the height f liquid in each cup? 4 7. A regular pentagn is drawn in a circle, centre, with radius 10 centimetres. Calculate the area f the regular pentagn. 5 Natinal 5 Practice Paper A Last updated 18/03/14

11 8. (a) Express 2 (2 2 + ) in its simplest frm. 2 (b) Use an apprpriate frmula t slve the quadratic equatin = 0. Give yur answers crrect t 1 decimal place (a) Slve the equatin 4 n + = 0, (b) Shw that n cs = sin. 2 Natinal 5 Practice Paper A Last updated 18/03/14

12 10. A rectangular wall vent is 30 centimetres lng and 10 centimetres wide. It is t be enlarged by increasing bth the length and the width by centimetres. (a) Shw that the area, square centimetres, f the new vent is given by = The area f the new vent must be at least 75% mre than the riginal area. (b) Find the minimum dimensins f the new vent. 5 [End f questin paper] Natinal 5 Practice Paper A Last updated 18/03/14

13 N5 Mαthematics Natinal 5 Practice Paper B Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper B Last updated 27/01/15

14 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah Standard deviatin: s = (x x )2 n 1 = x2 ( x) 2 /n n 1, where n is the sample size. Natinal 5 Practice Paper B Last updated 27/01/15

15 1. Evaluate Evaluate Slve the inequality 5 x > 2(x + 1) 3 4. Given that f(x) = x 2 + 5x, evaluate f( 3) Vectr u has cmpnents ( 2 ) and vectr v has cmpnents ( 4 ). 1 1 Calculate 4u 2v (a) Factrise p 2 4q 2. 1 p (b) Hence simplify 2 4q 2. 3p + 6q 2 Natinal 5 Practice Paper B Last updated 27/01/15

16 7. Find the equatin f the straight line shwn in the diagram. Give yur answer in the frm y = mx + c Part f the graph f y = cs x is shwn abve. If cs 60 = 0.5, state tw values fr x fr which cs x = 0.5, 0 x Multiply ut the brackets and cllect like terms. (x 3)(x 2 + 4x 1) 3 Natinal 5 Practice Paper B Last updated 27/01/15

17 10. A sample f students was asked hw many times each had visited the cinema in the last three mnths. The results are shwn belw (a) Frm the abve data, find the median, the lwer quartile and the upper quartile. 3 (b) Calculate the semi-interquartile range. 1 (c) The same sample f students was asked hw many times each had attended a ftball match in the same three mnths. The data had a median f 5 and a semi-interquartile range f 3. Make tw apprpriate cmments cmparing students visiting the cinema and students attending a ftball match Tw functins are given belw. f(x) = x 2 + 2x 1 g(x) = 5x + 3 Find the values f x fr which f(x) = g(x) Express in its simplest frm y 8 (y 3 ) 2 2 Natinal 5 Practice Paper B Last updated 27/01/15

18 13. y x The equatin f the parabla in the abve diagram is y = (x 1) (a) State the crdinates f the minimum turning pint f the parabla. 2 (b) State the equatin f the axis f symmetry f the parabla (a) Express as a surd in its simplest frm. 2 (b) Express as a fractin in its simplest frm 1 x x, x 0 2 [End f questin paper] Natinal 5 Practice Paper B Last updated 27/01/15

19 N5 Mαthematics Natinal 5 Practice Paper B Paper 2 Duratin 1 hur and 30 minutes Ttal marks 50 Yu may use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper B Last updated 27/01/15

20 1. A spider weighs apprximately kilgrams. A humming bird is 18 times heavier. Calculate the weight f the humming bird. Give yur answer in scientific ntatin A micrwave ven is sld fr 150. This price includes VAT at 20%. Calculate the price f the micrwave ven withut VAT (a) The price, in pence, f a cartn f milk in six different supermarkets is shwn belw Use an apprpriate frmula t calculate the mean and standard deviatin f these prices. Shw clearly all yur wrking. 4 (b) In six lcal shps, the mean price f a cartn f milk is 73 pence with a standard deviatin f 17.7 pence. Cmpare the supermarket prices with thse f the lcal shps. 2 Natinal 5 Practice Paper B Last updated 27/01/15

21 4. A pendulum travels alng an arc f a circle, centre C. The length f the pendulum is 20 centimetres. The pendulum swings frm A t B. The length f the arc AB is 28.6 centimetres. Find the angle thrugh which the pendulum swings frm A t B A cntainer t hld chclates is in the shape f part f a cne with dimensins as shwn belw. Calculate the vlume f the cntainer. Give yur answer crrect t ne significant figure. 5 Natinal 5 Practice Paper B Last updated 27/01/15

22 6. Slve the equatin 2x 2 + 3x 1 = 0. Give yur answers crrect t ne decimal place The diagram belw shws a circular crss-sectin f a cylindrical il tank. In the figure belw, O represents the centre f the circle. PQ represents the surface f the il in the tank. PQ is 3 metres. The radius OP is 2.5 metres. Find the depth, d metres, f il in the tank. 4 Natinal 5 Practice Paper B Last updated 27/01/15

23 8. The ppulatin f Newtwn is The ppulatin f Newtwn is increasing at a steady rate f 5% per annum. The ppulatin f Auldtwn is The ppulatin f Auldtwn is decreasing at a steady rate f 20% per annum. Hw many years will it take until the ppulatin f Newtwn is greater than the ppulatin f Auldtwn? 5 9. A TV signal is sent frm a transmitter (T) via a satellite (S) t a village (V), as shwn in the diagram. The village is 500 kilmetres frm the transmitter. The signal is sent ut at an angle f 35 and is received in the village at an angle f 40. Calculate the height f the satellite abve the grund Change the subject f the frmula t p. r = 3p 2t 2 Natinal 5 Practice Paper B Last updated 27/01/15

24 11. Lk at the cubid shwn n the crdinate diagram. The crdinates f pint E are (5,3,1) z D y E (a) State the crdinates f F A O G C B F x (b) State the crdinates f G (c) What is the shrtest distance between pints D and C? At the carnival, the height, H metres, f a carriage n the big wheel abve the grund is given by the frmula H = sin t, t secnds after starting t turn. (a) Find the height f the carriage abve the grund after 10 secnds. 2 (b) Find the tw times during the first turn f the wheel when the carriage is 12.5 metres abve the grund. 4 [End f questin paper] Natinal 5 Practice Paper B Last updated 27/01/15

25 N5 Mαthematics Natinal 5 Practice Paper C Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper C Last updated 27/01/15

26 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah Standard deviatin: s = (x x )2 n 1 = x2 ( x) 2 /n n 1, where n is the sample size. Natinal 5 Practice Paper C Last updated 27/01/15

27 1. Evaluate Evaluate 2 7 ( ). 3. Simplify 3(2x 4) 4(3x + 1) 3 4. f(x) = 7 4x (a) Evaluate f( 2). 1 (b) Given that f(t) = 9, find t Slve, by factrising 7 + 6x x 2 = 0. 3 Natinal 5 Practice Paper C Last updated 27/01/15

28 6. A htel bks taxis frm a cmpany called Quick-Cars. The receptinist ntes the waiting time fr every taxi rdered ver a perid f tw weeks. These times, in minutes, are shwn belw (a) Fr the given data, calculate: (i) the median 1 (ii) the lwer quartile 1 (iii) the upper quartile 1 (b) Calculate the semi-interquartile range. 1 In anther tw week perid, the htel bks taxis frm a cmpany called Fast-Cabs. The median waiting time fr Fast-Cabs is fund t be 27.5 minutes and the semi-interquartile range fr Fast-Cabs is fund t be 2.5 minutes. (c) Use this infrmatin t cmpare the tw cmpanies Part f the graph f y = asin(x + b) is shwn in the diagram. y 4 O x -4 State the values f a and b. 2 Natinal 5 Practice Paper C Last updated 27/01/15

29 8. In the diagram belw, A is the pint ( 1, 7) and B is the pint (4,3). (a) Find the gradient f the line AB. 1 (b) AB cuts the y-axis at the pint (0, 5). Write dwn the equatin f the line AB. 2 (c) The pint (3k, k) lies n AB. Find the value f k f(x) = x 2 + 6x 7 (a) Write f(x) in the frm (x + a) 2 + b. 2 (b) State the crdinates f the turning pint f f(x). 1 Natinal 5 Practice Paper C Last updated 27/01/15

30 10. Andrew and Daisy each bk in at the Sleepwell Ldge. (a) (b) Andrew stays fr 3 nights and has breakfast n 2 mrnings. His bill is 145. Write dwn an algebraic equatin t illustrate this infrmatin. 1 Daisy stays fr 5 nights and has breakfast n 3 mrnings. Her bill is 240. Write dwn an algebraic equatin t illustrate this infrmatin. 1 (c) Find the cst f ne breakfast (a) Evaluate 2 24 (b) Simplify 2 2 2x + 2 (c) Simplify 2 (x + 1) 2 [End f questin paper] Natinal 5 Practice Paper C Last updated 27/01/15

31 N5 Mαthematics Natinal 5 Practice Paper C Paper 2 Duratin 1 hur and 30 minutes Ttal marks 50 Yu may use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper C Last updated 27/01/15

32 1. Bacteria in a test-tube increase at the rate f 4.6% per hur. At 12 nn, there are bacteria. At 5 pm, hw many bacteria will be present? Give yur answer crrect t 3 significant figures The tangent, MN, tuches the circle, centre O, at L. Angle JLN = 47 Angle KPL = 31 Find the size f angle JLK Change the subject f the frmula y = ax 3 + c t x. 3 Natinal 5 Practice Paper C Last updated 27/01/15

33 4. A mug is in the shape f a cylinder with diameter 10 centimetres and height 14 centimetres. (a) Calculate the capacity f the mug. 2 (b) 600 millilitres f cffee are pured in. Calculate the depth f the cffee in the mug The diagram belw shws a big wheel at the fairgrund. The wheel has 16 chairs equally spaced n its circumference. The radius f the wheel is 9 metres. As the wheel rtates in an anticlckwise directin, find the distance a chair travels in mving frm psitin T t psitin P in the diagram. 4 Natinal 5 Practice Paper C Last updated 27/01/15

34 6. Find the rts f the equatin 2x 2 + 4x 9 = 0. Give yur answers crrect t ne decimal place Tw perfume bttles are mathematically similar in shape. The smaller ne is 6 centimetres high and hlds 30 millilitres f perfume. The larger ne is 9 centimetres high. What vlume f perfume will the larger ne hld? 3 8. Determine the nature f the rts f the equatin (x 2) 2 5x = 0. 4 Natinal 5 Practice Paper C Last updated 27/01/15

35 9. A pny shelter is part f a cylinder as shwn in figure 1. It is 6 metres wide and 2 metres high. The crss-sectin f the shelter is a segment f a circle with centre O, as shwn in figure 2. OB is the radius f the circle. Calculate the length f OB The diagram shws a parallelgram, PQRS. (a) Calculate the size f angle PQR. D nt use a scale drawing. 3 (b) Calculate the area f the parallelgram. 3 Natinal 5 Practice Paper C Last updated 27/01/15

36 11. (a) Slve the equatin 2 tan x + 7 = 0, 0 x (b) Prve that sin 3 x + sin x cs 2 x = sin x (a) A driver travels frm A t B, a distance f x kilmetres, at a cnstant speed f 75 kilmetres per hur. Find the time taken fr this jurney in terms f x. 1 (b) The time taken fr the jurney frm B t A is x 50 hurs. Calculate the average speed fr the whle jurney. 4 [End f questin paper] Natinal 5 Practice Paper C Last updated 27/01/15

37 N5 M thematics Natinal 5 Practice Paper D Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper D Last updated 27/01/15

38 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah x x 2 Standard deviatin: s = =, where is the sample size. n 1 x 2 n 1 x 2 /n Natinal 5 Practice Paper D Last updated 27/01/15

39 1. Find the gradient f the straight line jining pints A(3, -7) and B(-5, 3) Evaluate 3 5 f Find the equatin f the straight line AB. 3 Natinal 5 Practice Paper D Last updated 27/01/15

40 4. The marks scred by a grup f students in their Octber test are listed belw (a) Calculate: (i) The median mark; (ii) The semi-interquartile range fr the data. 2 The teacher arranges extra revisin classes fr the students befre their next test in December. In this test the median mark is 67 and the semi-interquartile range is 7. (b) Make tw apprpriate cmments cmparing the marks in the Octber and Nvember tests Marmalade is n special ffer. Each jar n special ffer cntains 12.5% mre than the standard jar. A jar n special ffer cntains 450 g f marmalade. Hw much des the standard jar cntain? 3 Natinal 5 Practice Paper D Last updated 27/01/15

41 6. The diagram shws a square based pyramid. Express in terms f and William Watsn Fast Fds use a lg based n parts f three identical parablas. The lg is represented n the diagram belw. The first parabla has turning pint P and equatin = (a) State the crdinates f P. 2 (b) If R is the pint (2,0), find the crdinates f Q, the minimum turning pint f the secnd parabla. 1 (c) Find the equatin f the parabla with turning pint S. 2 Natinal 5 Practice Paper D Last updated 27/01/15

42 3 8. Write as a single fractin in its simplest frm. 3 m + 4 m Part f the graph f = cs is shwn in the diagram. State the values f and Evaluate Express as a surd in its simplest frm. 3 Natinal 5 Practice Paper D Last updated 27/01/15

43 12. A piece f gld wire 10 centimetres lng is made int a circle. The circumference f the circle is equal t the length f the wire. Shw that the area f the circle is exactly 2 square centimetres Tw variables and are cnnected by the relatinship = +. Sketch a pssible graph f against t illustrate this relatinship when and are bth less than zer. 3 [End f questin paper] Natinal 5 Practice Paper D Last updated 27/01/15

44 N5 M thematics Natinal 5 Practice Paper D Paper 2 Duratin 1 hur and 30 minutes Ttal marks 50 Yu may use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper D Last updated 27/01/15

45 Q1. The average Scttish huse price is The average price is expected t rise by 2.5% per mnth. What will the average Scttish huse price be in 3 mnths? Give yur answer crrect t three significant figures. 3 Q2. The heights, in millimetres, f six seedlings are given belw (a) Calculate: (i) the mean; 1 (ii) the standard deviatin; 3 f these heights. Shw clearly all yur wrking. (b) Later the same six seedlings are measured again. Each has grwn by 4 millimetres. State: (i) the mean; 1 (ii) the standard deviatin; 1 f these new heights. Q3. Slve algebraically the equatin 4 sin + 1 = Natinal 5 Practice Paper D Last updated 27/01/15

46 Q4. A sprts centre charges different entrance fees fr adults and children. (a) One evening 14 adults and 4 children visited the sprts centre. The ttal cllected in entrance fees was Let punds be the adult entrance fee and punds be the child s entrance fee. Write dwn an equatin in and t describe the abve infrmatin. 1 (b) The fllwing evening 13 adults and 6 children visited the sprts centre. The ttal cllected in entrance fees was Write dwn a secnd equatin in and. 1 (c) Calculate the entrance fee fr an adult and the entrance fee fr a child. 4 Q5. The diagram belw shws part f a parabla with equatin f the frm = (a) Write dwn the equatin f the axis f symmetry f the graph. 1 (b) Write dwn the equatin f the parabla. 2 (c) Find the crdinates f C. 2 Natinal 5 Practice Paper D Last updated 27/01/15

47 Q6. A garden, in the shape f a quadrilateral, is represented in the diagram. Calculate: (a) (b) the length f the diagnal BD; D nt use a scale drawing. 3 the area f the garden. Q7. The curved part f a drway is an arc f a circle with radius 500 millimetres and centre C. The height f the drway t the tp f the arc is 2000 millimetres. The vertical edge f the drway is 1800 millimetres. Calculate the width f the drway. 5 Natinal 5 Practice Paper D Last updated 27/01/15

48 Q8. Find the values f such that the equatin = 0 has equal rts. 4 Q9. A gift bx, 8 centimetres high, is prism shaped. The unifrm crss-sectin is a regular pentagn. Each vertex f the pentagn is 10 centimetres frm the centre O. Calculate the vlume f the bx. 5 Q10. Express 4 3, x 3, x 0, x 3 x as a fractin in its simplest frm. 3 Natinal 5 Practice Paper D Last updated 27/01/15

49 Q11. A rectangular lawn has a path, 1 metre wide, n 3 sides as shwn. The breadth f the lawn is metres. The length f the lawn is three times its breadth. The area f the lawn equals the area f the path. (a) Shw that = 0. 3 (b) Hence find the length f the lawn. 4 [End f questin paper] Natinal 5 Practice Paper D Last updated 27/01/15

50 N5 M thematics Natinal 5 Practice Paper E Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper E Last updated 08/04/14

51 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah x x 2 Standard deviatin: s = =, where is the sample size. n 1 x 2 n 1 x 2 /n Natinal 5 Practice Paper E Last updated 08/04/14

52 1. Evaluate f Multiply ut the brackets and cllect like terms In an experiment invlving tw variables, the fllwing values fr and were recrded The results were pltted and a straight line was drawn thrugh the pints. Find the gradient f the line and write dwn its equatin Slve the equatin 2 x + 9 = Given = 0 shw that x = 1 ± Natinal 5 Practice Paper E Last updated 08/04/14

53 6. The diagram belw shws part f the graph f = (a) State the crdinates f the maximum turning pint. 2 (b) State the equatin f the axis f symmetry. 1 The line = 20 is drawn. It cuts the graph f = at R and S as shwn belw. (c) S is the pint (6, 20). Find the crdinates f R. 2 Natinal 5 Practice Paper E Last updated 08/04/14

54 7. A badge is made frm a circle f radius 5 centimetres. Segments are taken ff the tp and bttm f the circle as shwn. The straight edges are parallel. The badge measures 7 centimetres frm the tp t the bttm. The tp is 8 centimetres wide. Calculate the width f the base Sketch the graph f = sin = (a) Find the value f 2 as a surd in its simplest frm. 3 (b) Find the value f given that = Natinal 5 Practice Paper E Last updated 08/04/14

55 10. The height f a triangle is 2 centimetres and the base is 2 centimetres. The area f the triangle is 7 square centimetres. Calculate the value f 5 [End f questin paper] Natinal 5 Practice Paper E Last updated 08/04/14

56 N5 M thematics Natinal 5 Practice Paper E Paper 2 Duratin 1 hur and 30 minutes Ttal marks 50 Yu may use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper E Last updated 08/04/14

57 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah x x 2 Standard deviatin: s = =, where is the sample size. n 1 x 2 n 1 x 2 /n Natinal 5 Practice Paper E Last updated 08/04/14

58 1. = 2. Find the value f when = and = 3 10 Give yur answer in scientific ntatin Calculate the area f triangle PQR In the evening, the temperature in a greenhuse drps by 10.4% per hur. At 8 p.m. the temperature was 28 Celsius. Find the temperature at 11 p.m. 3 Natinal 5 Practice Paper E Last updated 08/04/14

59 4. Relative t crdinate axes, the pint A has crdinates (2, 4, 6). (a) Find the crdinates f C and D. 2 (b) Write dwn the crdinates f B Shamp is available in travel size and saln size bttles. The bttles are mathematically similar. The travel size cntains 200 millilitres and is 12 centimetres in height. The saln size cntains 1600 millilitres. Calculate the height f the saln size bttle. 3 Natinal 5 Practice Paper E Last updated 08/04/14

60 6. A jeweller uses tw different arrangements f bead and pearls. The first arrangement cnsists f 2 beads and 5 pearls and has an verall length f 5.2 centimetres. The secnd arrangement cnsists f 3 beads and 2 pearls and has an verall length f 5.6 centimetres. Find the length f ne bead and the length f ne pearl A pharmaceutical cmpany makes vitamin pills in the shape f spheres f radius 0.5 centimetres. (a) Calculate the vlume f ne pill. 3 Give yur answer t 3 significant figures. The cmpany decides t change the shape f each pill t a cylinder. The new pill has the same vlume as the riginal and its diameter is 1.4 centimetres. (b) Calculate the height f the new pill. 3 Natinal 5 Practice Paper E Last updated 08/04/14

61 8. David walks n a bearing f 050 frm hstel A t viewpint V, 5 kilmetres away. Hstel B is due east f hstel A. Susie walks n a bearing f 294 frm hstel B t the same viewpint. N V 50 5 km A B Calculate the length f AB, the distance between the tw hstels The chain f a demlitin ball is 12.5 metres lng. When vertical, the end f the chain is 1.5 metres frm the grund. It swings t a maximum height f 2.5 metres abve the grund n bth sides. (a) At this maximum height, shw that the angle which the chain makes with the vertical, is apprximately (b) Calculate the maximum length f the arc thrugh which the end f the chain swings. Give yur answer t 3 significant figures. 4 Natinal 5 Practice Paper E Last updated 08/04/14

62 10. Find the range f values f such that the equatin = 0 0 has real rts (a) Slve algebraically the equatin 3 sin 1 = (b) Simplify n cs 2 [End f questin paper] Natinal 5 Practice Paper E Last updated 08/04/14

63 N5 M thematics Natinal 5 Practice Paper F Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper F Last updated 11/04/14

64 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah x x 2 Standard deviatin: s = =, where is the sample size. n 1 x 2 n 1 x 2 /n Natinal 5 Practice Paper F Last updated 11/04/14

65 1. Evaluate (a) Factrise (b) Hence simplify 4x 2 y 2 6x + 3y A grup f peple attended a curse t help them stp smking. The fllwing table shws the statistics befre and after the curse. Mean number f cigarettes smked per Standard deviatin persn per day Befre After Make tw valid cmments abut these results. 2 Natinal 5 Practice Paper F Last updated 11/04/14

66 4. Teams in a quiz answer questins n film and sprt. This scatter graph shws the scres f sme f the teams. A line f best fit is drawn as shwn abve. (a) Find the equatin f this straight line. 3 (b) Use this equatin t estimate the sprt scre fr a team with a film scre f Given that = ( 0 ) calculate. 3 Give yur answer as a surd in its simplest frm. 3 Natinal 5 Practice Paper F Last updated 11/04/14

67 6. Slve the inequatin The graph f = 2 has been mved t the psitin shwn in the diagram. Write dwn the equatin f the graph shwn A straight line is represented by the equatin 2 + = 6. (a) Find the gradient f this line. 2 (b) This line crsses the -axis at 0. Find the value f. 1 Natinal 5 Practice Paper F Last updated 11/04/14

68 9. The tangent SV tuches the circle, centre O, at T. Angle PTQ is 37. Angle VTR is 68. Calculate the size f angle PQR The graph shwn belw has an equatin f the frm = cs. Write dwn the value f. 1 Natinal 5 Practice Paper F Last updated 11/04/14

69 11. Clean washing pwder is n special ffer. Each bx n special ffer cntains 20% mre pwder than the standard bx. A bx n special ffer cntains 900 grams f pwder. Hw many grams f pwder des the standard bx cntain? A parabla has equatin = (a) Shw that the parabla has n real rts. 2 (b) Write the equatin in the frm = (c) Sketch the graph f = , shwing the crdinates f the turning pint and the pint f intersectin with the -axis. 3 Natinal 5 Practice Paper F Last updated 11/04/14

70 13. Triangle ABC is right-angled at B. The dimensins are shwn. (a) Calculate the area f triangle ABC. 1 BD, the height f triangle ACB is drawn as shwn. (b) Use yur answer t part (a) t calculate the height BD. 3 [End f questin paper] Natinal 5 Practice Paper F Last updated 11/04/14

71 N5 M thematics Natinal 5 Practice Paper F Paper 2 Duratin 1 hur and 30 minutes Ttal marks 50 Yu may use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain apprpriate wrking. State the units fr yur answer where apprpriate. Natinal 5 Practice Paper F Last updated 11/04/14

72 FORMULAE LIST The rts f are ax 2 + bx + c = 0 x = b ± b2 4ac 2a Sine rule: Csine rule: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cs A r cs A = b2 + c 2 a 2 2bc Area f a triangle: A = 1 ab sin C 2 Vlume f a Sphere: V = 4 3 πr3 Vlume f a cne: V = 1 3 πr2 h Vlume f a pyramid: V = 1 3 Ah x x 2 Standard deviatin: s = =, where is the sample size. n 1 x 2 n 1 x 2 /n Natinal 5 Practice Paper F Last updated 11/04/14

73 1. The rbit f a planet arund a star is circular. The radius f the rbit is kilmetres. Calculate the circumference f the rbit. Given yur answer in scientific ntatin A bat was bught fr Its value decreases by 8% each year. Hw much will the bat be wrth after 4 years? 3 3. Change the subject f the frmula belw t. + = 2 4. Slve algebraically the system f equatins = = Natinal 5 Practice Paper F Last updated 11/04/14

74 5. A child s ty is in the shape f a hemisphere with a cne n tp, as shwn in the diagram. The ty is 10 centimetres wide and 16 centimetres high. 16 cm Calculate the vlume f the ty. Give yur answer crrect t tw significant figures The diagram shws the base f a ludspeaker stand which has the shape f part f a circle. The centre f the circle is O. EF is a chrd f the circle. EF is 18 centimetres. The radius, OF, f the circle is 15 centimetres. Find the width f the stand. 4 Natinal 5 Practice Paper F Last updated 11/04/14

75 7. There are three mring pints n Lake Srling. Frm A, the bearing f B is 074. Frm C, the bearing f B is 310. (a) Calculate the size f angle ABC. 2 B is 230 metres frm A and 110 metres frm C. (b) Calculate the direct distance frm A t C. Give yur answer t 3 significant figures Express as a single fractin in its simplest frm. 3 Natinal 5 Practice Paper F Last updated 11/04/14

76 9. A set f scales has a circular dial. The pinter is 9 centimetres lng. The tip f the pinter mves thrugh an arc f 2 centimetres fr each 100 grams f weight n the scales. A parcel, placed n the scales, mves the pinter thrugh an angle f 284. Calculate the weight f the parcel The number f diagnals,, in a plygn f sides is given by the frmula = (a) Hw many diagnals des a plygn f 7 sides have? 2 (b) A plygn has 65 diagnals. Shw that fr this plygn, = 0. 2 (c) Hence find the number f sides in this plygn. 3 Natinal 5 Practice Paper F Last updated 11/04/14

77 11. Emma ges n the Big Eye. Her height, h metres, abve the grund is given by the frmula h = 31 cs + 33 where is the number f secnds since the start. (a) Calculate Emma s height abve the grund 20 secnds after the start. 2 (b) When will Emma first reach a height f 60 metres abve the grund? 3 (c) When will she next be at a height f 60 metres abve the grund? 1 Natinal 5 Practice Paper F Last updated 11/04/14

78 12. In triangle ABC, BC = 8 centimetres AC = 6 centimetres PQ is parallel t BC M is the midpint f AC. Q lies n AC, centimetres frm M, as shwn in the diagram. (a) Write dwn an expressin fr the length f AQ. 1 (b) Shw that PQ = (4 + 3 ) centimetres. 3 [End f questin paper] Natinal 5 Practice Paper F Last updated 11/04/14

79 Natinal 5 Practice Paper A Answers Paper ( )( ) ( ), T.P.( ) 5. 6a. ( ) b ( ) 9. Therefre there are n real rts 10. ( ) 11a. ( ) b. ( ) c. ( ) 12. Prf 13a. ( ) b. Natinal 5 Practice Paper A - Answers Last updated 19/03/14

80 Natinal 5 Practice Paper A Answers Paper a. = 60, s = (2dp) 2b. On average the marks f bth grups are the same. Hwever, the marks frm Grup A are much mre cnsistent km centimetres 6a cm 3 b. 8.4 cm (using the answer t part a) cm 2 8a. b. = 1.1 r a. = , b. prf 10a. prf b. length = 35 cm, breadth = 15 cm Natinal 5 Practice Paper A - Answers Last updated 19/03/14

81 Natinal 5 Practice Paper B Answers Paper ( ) a. ( )( ) b a. median, Q1 Q3 = 4 b c. On average the students went t mre ftball matches than t the cinema. The number f times the students visited the cinema was mre cnsistent a. Min T.P. ( ) b. 14a. 14b. Natinal 5 Practice Paper B - Answers Last updated 01/03/15

82 Natinal 5 Practice Paper B Answers Paper kilgrams a. = 73, s = b. On average, milk is the same price in bth types f shp. The price f milk is mre cnsistent in supermarkets cm 3 6. = 0.3, = 0.5 metres 8. 3 years kilmetres a. F (5, 3, 0) b. G (0, 3, 0) c. 12a metres b. 30 secnds and 150 secnds Natinal 5 Practice Paper B - Answers Last updated 01/03/15

83 Natinal 5 Practice Paper C Answers Paper 1 Q Q Q3. 6x 16 Q4. a 15 b t = 1 2 Q5. x = 7, x = 1 Q6. a (i) 27.5 (ii) Q 1 = 13 (iii) Q 3 = 35 b 11 c On average, bth cmpanies have the same waiting time. Waiting times fr Fast-Cabs are mre cnsistent. Q7. a = 4, b = 30 Q8. a m AB = 2 b y = 2x 5 c k = 1 Q9. a f(x) = (x + 3) 2 16 b (-3, -16) Q10. a 3n + 2b = 145 b 5n + 3b = 240 c One breakfast csts 5 Q11. a 4 b 2 3 c 2 x+1 Natinal 5 Practice Paper C - Answers Last updated 27/01/15

84 Natinal 5 Practice Paper C Answers Paper 2 Q Q Q3. x = y c a Q4. a 1100 cm 3 b 7.64 cm (2dp) Q m Q6. x = 1.3, 3.3 Q ml Q8. b 2 4ac > 0, therefre tw real and distinct rts Q m Q10. a 78.6 b cm 2 Q11. x = 105.9, b prf Q12. a x 75 b 60 km/h Natinal 5 Practice Paper C - Answers Last updated 27/01/15

85 Natinal 5 Practice Paper D Answers Paper 1 Q1. m AB = Q2. r Q3. Q4. a(i) Median = 58.5 (ii) SIQR = 11 b On average, after the extra revisin classes, there was an imprvement in the students marks. After the extra classes the marks were mre cnsistent. Q grams Q6. r equivalent Q7. a P(-2, -16) b Q(6, -16) c ( ) Q8. ( ) Q9. a = 4, b = 3 Q10. Q11. Q12. Prf y Q13. x Natinal 5 Practice Paper D - Answers Last updated 27/01/15

86 Natinal 5 Practice Paper D Answers Paper 2 Q Q2. a (i) 16.5 mm (ii) 1.87 mm (2dp) Q2. b (i) 20.5 mm (ii) 1.87 mm (2dp) Q3. Q4. a b c Adult entrance fee Child entrance fee Q5. a b ( ) c C(0,5) Q6. a m b m 2 (1dp) Q mm Q8. Q cm 3 (1dp) Q10. ( ) Q11. a prf b The lawn is 6 metres in length. Natinal 5 Practice Paper D - Answers Last updated 27/01/15

87 Natinal 5 Practice Paper E Answers Paper 1 Q1. Q2. Q3. Q4. Q5. Prf (using quadratic frmula) Q6. (a) (2,36) (b) (c) R(-2,20) Q7. 6 centimetres Q8. 1 O Q9. (a) (b) Q10. Natinal 5 Practice Paper E - Answers Last updated 08/04/14

88 Natinal 5 Practice Paper E Answers Paper 2 Q Q cm 2 (1 decimal place) Q3. 20 C (nearest degree) Q4. (a) C(4,3,4), D(6,2,2) (b) B(6,4,2) Q5. 24 centimetres Q6. Bead 1.6 centimetres, pearl 0.4 centimetres Q7. (a) cm 3 (3 SF) (b) 0.34 cm (2 decimal places) Q8. AB = kilmetres (2 decimal places) Q9. (a) prf (b) 10.0 metres (3 SF) Q10. Q11. a (1 decimal place) b Prf using the fact that Natinal 5 Practice Paper E - Answers Last updated 08/04/14

89 Natinal 5 Practice Paper F Answers Paper 1 Q1. Q2. (a) ( )( ) (b) Q3. On average the number f cigarettes smked went dwn after the curse. Hwever, there was mre variability in the number f cigarettes smked after the curse. Q4. (a) ( ) r equivalent (b) 43 Q5. Q6. Q7. ( ) Q8. (a) (b) Q Q10. Q grams Q12. (a) therefre n real rts (b) ( ) (c) 5 Q13. (a) 150 m 2 ( ), (b) 12 metres Natinal 5 Practice Paper F - Answers Last updated 05/05/14

90 Natinal 5 Practice Paper F Answers Paper 2 Q kilmetres Q Q3. ( ) (r equivalent) Q4., (r, ) Q cubic centimetres (t 2 SF) Q6. 27 centimetres Q7. (a) 124 (b) 305 metres (t 3 SF) Q8. ( )( ) Q grams (t 1 decimal place) Q10. (a) 14 diagnals (b) prf (c) 13 sides Q11. (a) 3.87 metres (1 decimal place) (b) secnds (c) secnds Q12. (a) AQ = + 3 (b) PQ = ( ) as required. Natinal 5 Practice Paper F - Answers Last updated 05/05/14