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

Transcription

1 N5 Practice Paper C MATHEMATICS ational Qualifications - ational 5 Paper 1 (non-calculator) Covering all Units Time allowed - 1 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 014 National 5 Practice Paper C

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 1 Volume of a cone: Volume = π r h 1 Volume of a Pyramid: Volume = Ah Standard deviation: s = ( x x) n 1 = x ( x) n 1 / n, where n is the sample size. Pegasys 014 National 5 Practice Paper C

3 All questions should be attempted Marks 1. Lauren and Vicky record the amount of time they spend in the gym every day. The Standard Deviation of the time Lauren spends is 4 4 and for Vicki is 5 7. Compare the two sets of data making particular reference to the spread of the times the girls spend in the gym. 1. James is looking to buy a new rug. The two rugs below are mathematically similar in shape. 10 ft length 8 ft 9 6 ft He is hoping that the length of the large rug will be enough to make the area of the large rug at least 10 square feet. Does the large rug have the required area? You must show appropriate working and give a reason for your answer. 4. Evaluate of 5 7 Pegasys 014 National 5 Practice Paper C

4 Marks 4. The diagram below shows the graph of y o = a cosbx for 0 60 x. y x Write down the values of a and b. a a 5. (a) Simplify a 5 (b) Evaluate 15 Pegasys 014 National 5 Practice Paper C

5 Marks 6. The local riding stables buy in 48 tonnes of hay to feed the horses during the winter season, which lasts for 9 days. After 16 days they have 40 tonnes of hay left. The graph below illustrates the situation. W 48 Weight of hay (tonnes.) (16, 40) Number of days d (a) Find the equation of the line shown above. (b) If the horses continue to consume the hay at rate, will it last to the end of the winter season? Pegasys 014 National 5 Practice Paper C

6 7. A graph has equation of the form y = ax + bx + c. Marks Given that a > 0 and b 4ac < 0, draw a possible graph for y. y 0 x 8. The diagram below shows the end view of a scale model of a garden shed. x cm 7 cm 9 cm 4 cm Calculate the exact value of x, giving your answer as a surd in its simplest form. 4 Pegasys 014 National 5 Practice Paper C

7 Marks 9. Calculate the volume of sphere which has radius m. [Take π = 14 ] m 10. A quadratic graph has equation y = 5 ( x+ ). (a) What are the coordinates and nature of the turning point of the graph? (b) Which of the following is the equation of its axis of symmetry? 1 A x = B x = ` C x = 5 D x = 5 Pegasys 014 National 5 Practice Paper C

8 Marks 11. P and Q are points on the circumference of circle with centre O. PR is a tangent to the circle and angle AOB = 16 o. Calculate the size of angle PQR, the shaded area in the diagram. P 16 o O Q R 1. Solve the equation x ( x ) = 10 4 End of Question Paper Pegasys 014 National 5 Practice Paper C

9 ational 5 Practice Paper C Paper 1 Marking Scheme Qu Give one mark for each Illustrations for awarding mark 1 ans : comment 1 mark compares Standard Deviations Vicki s times are less consistent than Lauren s [or equivalent]. ans : will not be big enough 4 marks finds linear scale factor finds length of missing side finds area of large rug valid conclusion Area scale factor can also be used finds linear scale factor finds area scale factor finds area of large rug 4 valid conclusion 55 1 ans: = marks know order of calculations multiply fractions correctly add fractions correctly 4 ans: a = ; b = 4 marks L.S.F. = 9 6/8 = 1 [or equivalent] 1 10 = realises area < 10 square feet L.S.F. = 9 6/8 = 1 [or equivalent] A.S.F. = (1 )² = realises area < 10 square feet (a) states value of a states value of b ans : a 10 marks a = b = 4 simplifies numerator simplifies expression a 7 a 10 5(b) ans: 5 marks 6(a) interprets index evaluates ans: W = 0 5d + 48 marks 15 = 5 15 identifies y - intercept calculates gradient states equation c = m = = W = 0 5d (b) ans: Yes, days spare marks correct strategy solves equation correct conclusion 0 5 d + 48 = 0 d = 96 yes, days to spare Pegasys 014 National 5 Practice Paper C

10 Qu Give one mark for each Illustrations for awarding mark 7 ans : suitable graph drawn marks correct shape correct nature of turning point no roots 8 ans : 5 4 marks parabolic shape [accept any] minimum turning point graph above x axis assembles facts in R A T knows to use Pythagoras finds length as surd simplifies 9 ans : 11 04cm marks 4 subs values in correct formula starts to evaluate answer 10 ans: (, 5); maximum marks x = (4² + ²) x = 0 4 x = 5 4 V = 14 evidence of carrying out part calculation 11 04cm 1 states x coordinate of T.P. states y coordinate of T.P. identifies nature (,., 5) maximum 1 ans: A 1 mark correct axis of symmetry 11 ans : 117 o marks 1 recognises isosceles triangle recognises right angle 1 ans : x = 5 or x = 4 marks 1 A ABO = 7 o ABC = = 117 o 4 multiplies brackets/collects terms to LHS factorises equate each bracket to zero solves for x x² x = 10; x² x 10 = 0 (x 5)(x + ) = 0 (x 5) = 0; (x + ) = 0 4 x = 5 or x = Total 40 marks Pegasys 014 National 5 Practice Paper C

11 N5 Practice Paper C MATHEMATICS ational Qualifications - ational 5 Paper (Calculator) Covering all Units Time allowed - 1 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 014 National 5 Practice Paper C

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 1 Volume of a cone: Volume = π r h 1 Volume of a Pyramid: Volume = Ah Standard deviation: s = ( x x) n 1 = x ( x) n 1 / n, where n is the sample size. Pegasys 014 National 5 Practice Paper C

13 All questions should be attempted Marks 1. A patient in a hospital is injected with 00 mg of a drug. (a) It is known that for each hour after the injection the number of milligrams of the drug left in the body is 15% less than at the beginning of that hour. How many milligrams of the drug are left in the patient s body at the end of hours? (b) The patient is given a second drug. It is known that, for second drug, at the end of each hour the number of milligrams of the drug left in the body is 1% less than at the beginning of that hour. At the end of one hour the patient had 1 mg of the second drug left in his body. Calculate the size of the initial dose, of second drug, given to the patient.. Vector a has components u = 4 k. If u = 6, calculate the values of k. 4 Pegasys 014 National 5 Practice Paper C

14 Marks. The graph below shows the relationship between the number of hours (h) a swimmer trains per week and the number of races (R) they have won. R Number of races won (R) Number of hours training per week (h) h A best fitting straight line has been drawn. (a) Use information from the graph to find the equation of line of best fit. (b) Use the equation to predict how many races a swimmer who trains hours per week should win. 1 Pegasys 014 National 5 Practice Paper C

15 Marks 4. A building company has to fence off a triangular piece of waste ground. The plan of the ground is shown below. All lengths are in metres. 16m 56 o 0m If the fence costs per metre to erect, how much will the company have to pay in total to fence off piece of ground? Give your answer to the nearest ten pounds Determine the nature of the roots of the quadratic equation x² x + 7 = 0 Pegasys 014 National 5 Practice Paper C

16 6. Some friends stopped at a roadside café. Marks (a) Peter bought bacon rolls and cups of tea which cost him a total of Taking the cost of a bacon roll as x pence and y as the cost of a cup of tea, write an equation to illustrate. 1 At the same café, Colin bought bacon rolls and 1 cup of tea and was charged.15. (b) Construct a second equation to illustrate. 1 (c) How much did Stewart pay for 4 bacon rolls and cups of tea? 4 7. Simplify fraction 10x 17x+ 4x 9 Pegasys 014 National 5 Practice Paper C

17 Marks 8. Shown is a children s play tunnel which has been fitted with a rectangular insulating mat. The end of the tunnel consists of part of a circle, centre C, with diameter 1 metres. The height of the tunnel is 0 9 metres. 1 m Calculate the area of the mat if the tunnel is C 0 9m 7 metres long. 4 Pegasys 014 National 5 Practice Paper C

18 Marks 9. A hand fan is made of wooden slats with material on the outer edge. 9cm 15cm 105 Calculate the area of material needed for the hand fan Simplify 1 cos x cos x Pegasys 014 National 5 Practice Paper C

19 Marks 11. The diagram below shows the graph of y = sin x o + 1 for 0 x 70. The line y = has also been drawn on the diagram. y 4 A A y = 0 70 x Find the coordinates of point A. 4 Pegasys 014 National 5 Practice Paper C

20 1. The diagram show the graph of y = x + 7x y Marks x y = x + 7x Find the x - coordinates of the points where the graph crosses the x - axis giving your answers correct to 1 decimal place. 5 End of Question Paper Pegasys 014 National 5 Practice Paper C

21 ational 5 Practice Paper C Paper Marking Scheme Qu Give one mark for each Illustrations for awarding mark 1a ans : 1 85 mg marks correct multiplier knows how to decrease for hours answer mg b ans: 140 mg marks method answer ans : 4 or 4 4 marks 88% = mg a knows how to find magnitude equates to 6² 4 removes roots signs and simplifies solves ans : R = ½h + 4 marks ( 4) + () + k ( 4) + () + k = 6 0+ k = 6; k = 16 4 k =± 4 finds gradient of line finds y intercept states equation of line m = ½ (0, 4) R = ½h + 4 b ans: 15 races won 1 mark subs into equation and evaluates 4 ans: marks knows to use Cosine rule subs values into formula evaluates finds perimeter 5 calculates cost of fencing to nearest 5 ans : no real roots marks 4 ½() + 4 = 15 evidence x = cos m = 70 9 m = 1 10 to nearest 10 6a knows to find discriminant finds discriminant valid conclusion ans: x + y = mark evidence of finding b 4ac ()² 4 7 = 47 no real roots b 1 states equation ans: x + y = 15 1 mark 1 states equation 1 x + y = x + y = 15 Pegasys 014 National 5 Practice Paper C

22 Qu Give one mark for each Illustrations for awarding mark c ans: marks 1 knows to use sim. equations prepares equations finds value for x and y 4 calculates cost 1 evidence equates x or y coefficients x = 1 0; y = = ans : (5x 1)/(x + ) marks factorises numerator factorises denominator simplifies 8 ans : 7m 4 marks interpret information in rt. Triangle (5x 1)(x ) (x )(x + ) (5x 1)/(x + ) 0 m 0 6m calculate missing side state breadth of mat calculate area 9 ans : cm 5 marks 4 knows how to calculate area of sector calculates area of large sector calculates radius of smaller sector calculates area of small sector subtracts areas 10 ans : tan x o marks 4 5 replaces 1 cos x simplifies 11 ans: A(150 o, ) 4 marks equates equation to solves for sin x o 4 finds solution(s) states coordinates of A 1 ans : 0 4; 9 5 marks ( ) = 0 5 breadth = 1m 4 7m² 105 / 60 π / 60 π 15 = cm 15 9 = 6cm / 60 π 6 = 99 cm = cm sin x tan x o = sin x sin x o = x = 0 o or 150 o 4 A(150 o, ) m 4 5 equates equation to zero knows to use quadratic formula evaluates discriminant finds roots [no rounding] rounds correctly x + 7x = 0 evidence of substituting values 7 4 ( 7 + 7) 4; ( 7 7) ; 9 Total 50 marks Pegasys 014 National 5 Practice Paper C