MATHEMATICS ational Qualifications - ational 5 Paper 1 (non-calculator) Covering all Units

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1 N5 Practice Paper A MATHEMATICS ational Qualifications - ational 5 Paper (non-calculator) Covering all Units Time allowed - hour Fill in these boxes and read carefully what is printed below Full name of centre Town Forename(s) Surname Date of birth Day Month Year Candidate number Seat number Total marks You may NOT use a calculator.. Use blue or black ink. Pencil may be used for graphs and diagrams only.. Write your working and answers in the spaces provided. Additional space for answers is provided at the end of the booklet. If you use space, write clearly the number of the question you are attempting. 4. Square ruled paper is provided. 5. Full credit will be given only where the solution contains appropriate working. 6. State the units for your answer where appropriate. 7. Before leaving the examination room you must give up booklet to the invigilator. If you do not, you may lose all the marks for paper. Pegasys 04 National 5 Practice paper A

2 FORMULAE LIST The roots of ax + bx + c = 0 are x = b± ( b 4ac) a Sine rule: a sin A = b sin B = c sin C Cosine rule: a = b + c bc cos A or cos A = b + c a bc Area of a triangle: Area = ½ ab sin C Volume of a sphere: Volume = 4 π r Volume of a cone: Volume = π r h Volume of a Pyramid: Volume = Ah Standard deviation: s = ( x x) n = x ( x) n / n, where n is the sample size. Pegasys 04 National 5 Practice paper A

3 All questions should be attempted Marks. Evaluate 6 of ( ) Find the equation of the line in the diagram. y 7 O 4 x. A formula is given as V + = u t. Change the subject of the formula to t. Pegasys 04 National 5 Practice paper A

4 Marks 4. (a) Factorise 5x 45 (b) Factorise 6x 7x 0 5. Vectors p and q have components as follows: p = 4 5 and q = 4 5 (a) Find the components of the vector represented by p + q. (b) Calculate the magnitude of the vector represented by p + q leaving your answer as a surd in its simplest form. Pegasys 04 National 5 Practice paper A

5 Marks 6. Solve the system of equations 8x+ y= 0 y= x 7. The diagram shows part of the graph of y 4sin x o = for 0 x 60. y B 0 A C What numbers should be in positions A, B and C. 8. Remove the brackets and simplify (x ) x(4x ) Pegasys 04 National 5 Practice paper A

6 Marks 9. Express as a single fraction in its simplest form: 5 x + x 0. Simplify the following fraction, giving your answer in positive index form. x x 7 x 4 y Pegasys 04 National 5 Practice paper A

7 Marks. The diagram shows a semi-circle with centre C. R RT is a tangent to the semi-circle at T. S C 64 o T (a) Explain why angle RTC is a right angle. (b) Calculate the size of the shaded angle. 4. Simplify cos x sin x. A function is given as f ( x) = x 0. Find f (). End of Question Paper Pegasys 04 National 5 Practice paper A

8 ational 5 Practice Paper A Paper Marking Scheme Qu Give one mark for each Illustrations for awarding mark ans : / marks evaluates bracket knows how to complete calculation completes calculation ans : y = 7/4x + 7 marks /0 0/ /0 / finds gradient states y intercept states equation of line v u ans: t = marks m = 7/4 c = 7 y = 7/4x + 7 4a subtracts u from both sides divides both sides by takes square root of both sides ans: 5(x )(x + ) marks t = v u v u t = v u t = takes common factor factorises difference of two squares 5(x 9) 5(x )(x + ) b ans: (x + 4)(x 5) marks 5a b first factor correct second factor correct ans: 4-5 states components ans: 5 mark marks (x + 4)..(x 5) 4-5 knows how to find magnitude evaluates correct simplification ( 5 ) 50 5 Pegasys 04 National 5 Practice paper A

9 Qu Give one mark for each Illustrations for awarding mark 6 ans: x = ; y = 8 marks subs for y solves for x subs and solves for y 7 ans: A:80; B/C: 4/ 4 marks states value at A states values at B and C 8 ans: x + marks squares first bracket multiplies second bracket simplifies 9 ans : 4x 4x + 6 / (x ) marks correct denominator correct numerator solves 0 ans: y /x marks 8x + ( x) = 0 x = y = 8 A:80 B/C; 4/ 4 9x² 6x + 8x² + 6x x² + (x ) [or equivalent] 5 + (x ) 4x 4x + 6 / (x ) a simplifies numerator correct numerator correct denominator ans: reason mark 6x 6 y y x b gives reason ans: 8 o 4 marks tangent makes right angle with radius at point of contact finds angle CST finds angle RST finds angle STR finds required angle ans: tan x marks 4 factorises numerator replaces numerator and denominator correct denominator ans: 7 mark 4 angle CST = 64 o angle RST = 6 o angle STR = 6 o 80 (6 + 6) = 8 o ( cos²x) / sin²x sin²x / cos²x tan²x substitutes and evaluates 7 Total 40 marks Pegasys 04 National 5 Practice paper A

10 N5 Practice Paper A MATHEMATICS ational Qualifications - ational 5 Paper (Calculator) Covering all Units Time allowed - hour and 0 minutes Fill in these boxes and read carefully what is printed below Full name of centre Town Forename(s) Surname Date of birth Day Month Year Candidate number Seat number Total marks You may use a calculator.. Use blue or black ink. Pencil may be used for graphs and diagrams only.. Write your working and answers in the spaces provided. Additional space for answers If you use space, write clearly the number of the question you are attempting. is provided at the end of the booklet. 4. Square ruled paper is provided. 5. Full credit will be given only where the solution contains appropriate working. 6. State the units for your answer where appropriate. 7. Before leaving the examination room you must give up booklet to the invigilator. If you do not, you may lose all the marks for paper. Pegasys 04 National 5 Practice paper A

11 FORMULAE LIST The roots of ax + bx + c = 0 are x = b± ( b 4ac) a Sine rule: a sin A = b sin B = c sinc Cosine rule: a = b + c bc cos A or cos A = b + c a bc Area of a triangle: Area = ½ ab sin C Volume of a sphere: Volume = 4 π r Volume of a cone: Volume = π r h Volume of a Pyramid: Volume = Ah Standard deviation: s = ( x x) n = x ( x) n / n, where n is the sample size. Pegasys 04 National 5 Practice paper A

12 All questions should be attempted Marks. A bank pays interest of % per annum on a special investment account. Carly s parents invested 000 in account for her when she was years old and hoped that by the time she was she would have enough to pay a deposit of to buy a flat. Would Carly have enough for her deposit? You must show all your working and give a reason for your answer. 9. Uranium is a radioactive isotope which has a half-life of years. This means 9 that only half of the original mass will be radioactive after years. How long will it take for the radioactivity of a piece of Uranium to reduce to one eighth of its original level? Give your answer in scientific notation. Pegasys 04 National 5 Practice paper A

13 . In the diagram M is the mid point of QR. Q Marks a P M b R (a) Express QR in terms of a and b. (b) Show clearly that PM can be represented as ( b+ a) 4. Solve the quadratic equation x 6x+ = 0 using an appropriate formula. Give your answer(s) correct to decimal places. 4 Pegasys 04 National 5 Practice paper A

14 Marks 5. Calculate the area of triangle ABC. cm C B 5 o 6cm A 6. John bought an antique watch last year. Over the next year it increased in value by % and is now worth 680. By how much had the watch increased in value over that year? You must show all working. Pegasys 04 National 5 Practice paper A

15 Marks 7. Sam, Roisin and Fieza are studying Law at University. At the beginning of term Sam buys hardback notebooks and 4 loose leaf pads for 0.5. Roisin buys 6 hardback notebooks and loose leaf pads for.00. How much will Fieza pay for 5 hardback notebooks and loose leaf pad? 5 8. A child's spinning top is shown opposite. It is made from solid wood. The shape consists of a hemisphere base with a cone on top. 7cm Calculate the volume of the spinning top if the hemisphere has a diameter of 6 centimetres and the cone has a height of 7 centimetres. Give your answer correct to decimal place. 6cm 5 Pegasys 04 National 5 Practice paper A

16 Marks 9. Solve algebraically the equation 0 cos x + = for 0 x A basket, B, containing medical supplies is descending vertically at a constant speed over a point X. An observer, O, notes that at a certain instant the angle of elevation to B is 45 o. B At the same instant a second observer, P, standing 80m from O notes the angle of elevation to B is 6 o. Calculate the height of the basket above the ground. i.e. the height BX in the diagram. X 45 o 6 o O 80m P 5 Pegasys 04 National 5 Practice paper A

17 . Fiona Baxter discovered that to make the best mango chutney the mango should weigh as close to 0 grams as possible. Less than 0g the mango becomes sour and more than 0g the mango becomes too sweet. Fruit-to-go have sent a sample of 8 mangoes, their weights are shown in the table below. Marks Mango Weight (g) (a) Calculate the mean and standard deviation of batch of mangoes, giving your answers correct to one decimal place where necessary. 4 (b) Burtlets Fruit also sent a sample of 8 mangoes. The mean weight of batch is 0g and the standard deviation is Which company should Fiona choose to supply her with mangoes? You must give a reason for your answer. Pegasys 04 National 5 Practice paper A

18 Marks. Find the value of p for which the quadratic equation px 6x+ = 0 has equal roots.. The flat wire framework below shows two similar triangles. 40 cm 4 cm 0 cm 0 cm 6 cm It is made from a single length of wire which has been bent to shape. Would a two metre length of wire be enough to make framework? You must show all your working and give a reason for your answer. 4 End of Question Paper Pegasys 04 National 5 Practice paper A

19 ational 5 Practice Paper A Paper Marking Scheme Qu Give one mark for each Illustrations for awarding mark ans: no; since 67 < 7000 marks correct multiplier knows how to find amount answer with conclusion 0 ans: 5 0 marks since 67 < 7000 a knowing to multiply by correctly multiplying leaving answer in scientific notation ans : b a mark answer answer b a b ans: proof marks states pathway realises QM is half QR simplifies to answer 4 ans : x = 7, marks 4 knows to use quadratic formula substitutes into quadratic formula correctly evaluates b 4ac evaluates values of x 5 ans : 5 cm² marks subs values into formula for area evaluates 6 ans : 80 marks knows that % = 680 knows to divide 680 by answer 7 ans : marks PM = PQ + QM or PM = PR + RM a + ½ (b a ) or b ½ (b a) ½ a + ½ b = ½ (a + b) evidence 6± ( 6) 4 x = 6± x = 4 4 x = 7 and 0 6 A = 6 sin 5 5 cm² % = % = 680 = create first equation create second equation begin to solve equations simultaneously correctly solve equations calculate cost H + 4L = 0.5 6H + L =.00 H =.75 4 L =.5 5 5H + L = 0.00 Pegasys 04 National 5 Practice paper A

20 Qu Give one mark for each Illustrations for awarding mark 8 ans : 5cm 5 marks 4 5 subs to find volume of cone subs to find volume of hemisphere finds both volumes adds to total correct rounding 9 ans: 09 5 o, 50 5 o marks rearranges to cos x o finds st solution finds nd solution 0 ans: 76 4 m 5 marks π 7 π and cm cos x = 09 5 o 50 5 o a finds third angle knows to use sine rule calculates side correctly 4 attempts to calculate height 5 calculates height correctly ans: mean = ; S.D. = 8 4 marks side 80m, angles 6 o, 5 o, 9 o evidence 80 BO = BO= 08 m sin9 sin 6 4 x sin 45 = 08 BX = 76 4m 5 finds mean finds ( x) and x substitutes into formula 4 answer = 0g = x 840, x = sd = 8 7 Or 4 5 [accept any correct rounding] b 4 finds mean finds deviations squared knows how to find SD answer ans: Burtlets with reasons marks = 0g = [accept any correct rounding] compares mean compares SD same mean interpret SD as spread of weights Pegasys 04 National 5 Practice paper A

21 Qu Give one mark for each Illustrations for awarding mark ans : p = 9 marks knows condition for equal roots substitutes values simplifies and solves for p ans : 0 cm short 4 marks b² 4ac = 0 [stated or implied] ( 6)² 4 p = 0 p = 9 4 finds scale factor of enlargement finds missing side finds total of sides and conclusion enlargement scale factor = / 40 / = 60 cm = 0 cm 4 not enough since 0 > 00 Total 50 marks Pegasys 04 National 5 Practice paper A