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

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1 N5 Practice Paper B 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 B

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 B

3 All questions should be attempted Marks. Factorise fully 4x 0x 6.. Two lines have equations y + x = 0 y + 5x = 8 Find, algebraically, the point where these two lines meet. 4. Given that kq P= express r in terms of P, Q and k. r Pegasys 04 National 5 Practice Paper B

4 Marks 4. The graph shown has equation y = (x + )(x 5). y 0 5 x C (a) Find the coordinates of the turning point. (b) State the equation of the axis of symmetry of the parabola. 5. Express as a single fraction in its simplest form. x x 5 Pegasys 04 National 5 Practice Paper B

5 Marks 6. Sparks Electrical are having their annual clearance sale where everything is reduced by 40%. A Flat screen TV cost 480 in the sale. How much did the TV originally cost? 7. (a) A function is given as 6 f ( x) =, where x > 0. x Find the exact value of f (8), giving your answer as a surd in its simplest form and with a rational denominator. 4 5 p 8 p (b) Express p in its simplest form. Pegasys 04 National 5 Practice Paper B

6 8. The diagram shows the graph of a function of the form y = ax + bx + c. Marks y 0 x Write down a possible value for a and a possible value for b 4ac. 9. A function is given as f ( x) = x x. (a) Find f ( ). (b) Given that f ( p) = 5, find the two values of p. Pegasys 04 National 5 Practice Paper B

7 Marks 0. (a) Calculate the value of cos ABC in triangle. B 7cm 5cm A 0cm C (b) Without actually calculating the size of the angle a pupil was able to say that angle ABC was obtuse. By referring to your answer in (a), explain why the pupil was able to do. Pegasys 04 National 5 Practice Paper B

8 Marks. A goldfish bowl is filled with water to a depth of 8 cm. A cross section through the centre of the bowl is circular. w cm 0 cm 8 cm If the radius of the cross section is 0 cm, find the width of the water, w cm, in the bowl. 4 End of Question Paper Pegasys 04 National 5 Practice Paper B

9 ational 5 Practice Paper B Paper Marking Scheme Qu Give one mark for each Illustrations for awarding mark x + x marks ans: ( )( ) extract common factor attempt to factorise trinomial complete factorisation ans: (, ) 4 marks ( x 5x ) ( x +) ( x ) 4a 4 knows to use system of equations finds value for y finds value for x states coordinates of intersection kq ans: r= marks P remove fraction manipulate formula solve for r ans: (, 9) marks scales equations y = x = 4 (, ) Pr = kq kq r = P answer correct x - coordinate correct y - coordinate (,.. 9) b ans: x = mark 5 ans: states equation x 5 x x 5 ( ) common denominator correct numerator simplify fraction marks 6 ans: 800 marks correct strategy uses correct ratio calculations correct x = x ( x 5) ( x 5) x( x 5) answer x 60% = answer Pegasys 04 National 5 Practice Paper B

10 Qu Give one mark for each Illustrations for awarding mark 7a ans : 4 marks substitutes for x 6 f ( x) = 8 simplifies 8 and expression 6 f ( x) = = multiplies by 4 simplifies 4 b ans: 4p 9 marks simplifies numerator simplifies fraction 8 ans: a < 0; b 4ac > 0 marks 8p 6 4p 9 9a correct value for a correct value for b 4ac ans : 4 marks any value of a < 0 any value of b 4ac > 0 substitutes evaluates f(x) = ( ) ( ) 4 b ans : 5/ or marks 0 equates to 5 and rearranges factorises solves 6 ans : [or equivalent] 70 marks p p = 5; p p 5 = 0 (p 5)(p + ) = 0 p = 5/ or knows to use the cosine rule substitutes values answer evidence [or equivalent] 70 ans: cosine is negative mark gives valid reason ans: cm 4 marks 4 assembles facts in RAT knows to use Pythagoras calculates unknown side states width of water cosine is negative so angle is obtuse (0² 8²) 6cm 4 cm Total marks Pegasys 04 National 5 Practice Paper B

11 N5 Practice Paper B 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 B

12 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 B

13 All questions should be attempted Marks. Remove the brackets and simplify (x ) + x(4x ).. Halley's comet travels in a wide loop around our solar system. At its closest point to the earth it is travelling at an 5 average speed of 4 0 miles per hour. At speed how far, in miles, will it travel in a week? Give your answer in scientific notation correct to significant figures. 4 Pegasys 04 National 5 Practice Paper B

14 Marks. (a) Point U has coordinates (6,, 7) in the cuboid OPQR STUV shown below. z S T (6,, 7) V y U O R P Q x Write down the coordinates of point S (b) Calculate the length of vector a defined as a= i+ 5 j k. 4. The area of the rectangle in the diagram is m x Calculate the value of x giving your answer correct to decimal place. 5 (x ) Pegasys 04 National 5 Practice Paper B

15 5. A large triangular flag advertising a UFO conference is shown below. Marks 80 cm 50 cm 85 cm Calculate the area of the flag, giving your answer to the nearest square centimetre Find the value of k for which the quadratic equation kx + kx+ 6= 0 ; k 0 has equal roots. 4 Pegasys 04 National 5 Practice Paper B

16 Marks 7. The value of an industrial machine is expected to decrease each year by 4 % of of its value at the beginning of the year. If it was valued at 5500 at the beginning of 0, what will its expected value be at the end of 0? Give your answer correct to the nearest pound A survey of the number of hours that senior pupils had spent studying for a Maths exam gave a mean of 5 and a standard deviation of 4. The following year, after a programme of supported study, the mean number of hours remained the same but the standard deviation fell to. Make two comments about the effectiveness of the supported study programme. Pegasys 04 National 5 Practice Paper B

17 9. A clock has a pendulum swinging below it. When the clock is ticking the pendulum travels along an arc of a circle, centre O. The length of the connection cord OQ is 5 centimetres. The length of the arc followed by the pendulum as it swings from P to Q is 8cm. Marks O 5 cm P Q Find the size of angle POQ, the angle through which the pendulum swings from P to Q Solve algebraically the equation 7 cos x = cos x o o + 4, for 0 < 60 x. 4 Pegasys 04 National 5 Practice Paper B

18 Marks. A company sells boxed chocolates in two different sizes. The boxes are mathematically similar truncated cones, as shown in the diagram below. 0 cm 4 cm The cost of the chocolates should be in direct proportion to their weight. The chocolates in the larger box have been weighed and are priced at The company is considering pricing the smaller box at 5. Is a fair price? Your answer must be accompanied with appropriate working. Pegasys 04 National 5 Practice Paper B

19 Marks. A cone of ice with a base radius of 6cm and a height of 6cm is placed in a small rectangular glass tank as shown below. 6cm 6cm cm 5cm (a) Calculate the volume of the cone giving your answer correct to significant figures. (b) If the cone is left to melt away completely, calculate the depth of water in the tank once all the ice has melted. End of Question Paper Pegasys 04 National 5 Practice Paper B

20 ational 5 Practice Paper B Paper Marking Scheme Qu Give one mark for each Illustrations for awarding mark ans: 6x² x + 9 marks multiplies first bracket multiplies second bracket simplifies ans : miles 4 marks 4x² x x² 9x 6x² x + 9 a knows to multiply by 4 and 7 starts to evaluates rounds to sig. figs. 4 gives answer in Scientific notation ans : S(0,, 7) mark miles states coordinates of S S(0,, 7) b ans: 6 units finds components of vector a marks knows how to find magnitude answer 4 ans : 4m 5 marks equates areas knows to use quadratic formula evaluates discriminant 4 finds values of x 5 discards 5 ans : 956 cm 6 marks knows to find an angle and uses cosine rule subs values evaluates for cosine of angle finds angle knows how to find area answer properly rounded ( ) 6 units x(x ) = ; 6x x = 0 evidence could list values of a, b and c b 4ac = ( ) (4 6 ) = or 5 x = 4m evidence of cosine rule for angle or or or or o or 77 9 o or o 5 a = sin 5 or a = sin 77 9 or a = sin cm² Answers may vary depending on rounding. penalise premature rounding Pegasys 04 National 5 Practice Paper B

21 Qu Give one mark for each Illustrations for awarding mark 6 ans : k = 4 4 marks knows condition for equal roots substitutes values simplifies and factorises solves for k and chooses correct value 7 ans: marks 4 4 correct multiplier knows how to decrease over years answer correctly rounded 8 ans: comments marks b² 4ac = 0 [stated or implied] k² 4 6 k = 0 k² 4k = 0; k(k 4) = 0 4 k = comments on mean comments on distribution 9 ans : 68 8 o 5 marks on average hours studied same more consistent uses correct diameter calculates circumference sets up equal ratios starts to solve answer 0 ans : 4 o and 6 9 o 4 marks 4 5 evaluates cosx o takes inverse finds one value for x 4 finds second value for x ans: o, as.5 >.99 marks finding scale factor for reduction calculating cost comparing cost with.5 d = 0 cm [may be in formula] C = 4 0 = 94 8/94 = angle/60 4 angle = (8 60) / o cosx o = 4/5 cos (4/5) = 6 9 o 4 o o 0 linear S.F. = = cost = = answer a ans: 60 cm marks substitutes values in formula answer correct rounding π cm b ans: cm marks subs know values into formula knows how to find height answer 60 = 5 h h = cm Total 50 marks Pegasys 04 National 5 Practice Paper B