CFE National 5 Resource

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1 Pegasys Educational Publishing CFE National 5 Resource Unit Applications Homework Exercises Homework exercises covering all the Unit topics + Answers + Marking Schemes Pegasys 0

2 TRIGONOMETRY () AREA of TRIANGLE, SINE RULE and COSINE RULE. Calculate the area of the triangle in the diagram. () 7m 60 9m. Calculate the length of the shortest side in the triangle shown.. (4) 5 6 7cm. A metal rod 8cm long is bent to form an angle of 5 at a point 7cm from one end. 7cm 5 o How far apart are the two ends of the rod now? (4). 4. The three sides of a triangle are cm, 4 cm and 0 4cm. Calculate the size of the largest angle in the triangle. (4) 4 marks Pegasys 0

3 TRIGONOMETRY () PROBLEMS. Three oil platforms, Alpha, Gamma and Delta are situated in the North Sea as shown in the diagram. N The distances between the oil platforms are shown in the diagram. N 90 km Gamma If the bearing of Delta from Alpha is 5 o, Alpha what is the bearing of Gamma from Alpha? 60 km (6) 75 km N Delta. On an orienteering course, Ian follows the direct route through a forest from A to C while Kate follows the road which goes from A to B and then from B to C. A 7 o 000 m 40 o B C Calculate the total distance which Kate has to travel from A to C. (8). A small boat race travels round a set of three buoys to cover a total distance 5 km. Q P 9 km km R Pegasys 0 (a) Calculate the size of angle PQR. (b) Calculate the area of triangle PQR. (6) 0 marks

4 WORKING with VECTORS. The diagram shows vectors a, b and c. a b c (a) Write down the components of vectors a, b and c () (b) Draw diagrams on squared paper to represent: (i) a + b (ii) a c (iii) b + c (iv) (a + b) + c (v) a + ( b c) (9) (c) For the resultant vectors in (i) and (iii) from part (b), state the components and calculate its magnitude correct to one decimal place. (4). PQRS is a parallelogram. PQ is represented by vector a and PS is represented by vector b as shown in the diagram. M is the mid point of SQ. S R b M P a Q Express, in terms of a and b: (a) PR (b) SQ (c) SM (6). State the coordinates of each vertex of the rectangular based pyramid shown in the diagram. (4) z y P S 5 R 0 O 8 Q x 4 Two forces are acting on an object. Pegasys 0 They are represented by the vectors F = i + j k and F = i + j. Find the components and magnitude of the resultant force F F. (4) 0 marks

5 WORKING with PERCENTAGES. John has just put 700 into a savings account where the rate of interest is 4% per annum. How much will his savings be worth after years? (). Mary puts 00 into an account where the annual rate of interest is 5 5%. How long will it be before she has at least 400 in her account? (5). My new car has just cost me 8,000. Its value will depreciate by 0% every year. How much will it be worth when I trade it in years from now? () 4. The pressure in my car tyre should be 0psi, but a nail in it is causing it to lose pressure at the rate of 5% every mile that I drive. How far can I drive before the pressure falls below 0psi? (5) 5. Hassan has been told his hourly pay is to increase by 6% to Calculate his hourly rate before the increase. () 6. Due to fire damage, the value of a painting has fallen by 4% and is now valued at 458. What was its value before the damage? () marks Pegasys 0

6 WORKING with FRACTIONS. Work out the answers to the following: (a) 7 (b) (c) (d) (e) (f) (g) (h) (8). Siobhan likes to go to the gym. Last week she spent a total of 8 hours there. If she went on 6 days, calculate the mean number of hours she spent in the gym each day. (). Calculate: (a) 7 4 (b) (c) () Billy is a long distance lorry driver. One day he had to drive to Birmingham. He drove for hours at an average speed of 76 km/h and then for hours at an average speed of 8km/h before arriving at his destination. (a) (b) (c) (d) How far did he drive during the first part of his journey? How far did he drive during the second part? How far did he travel altogether? How many hours did it take him in total? (e) What was his average speed over the whole journey (6) 5. Laura has applied to join the RAF and has to sit an Entrance Test. Part of it includes some problems with fractions. Work out the answers. (a) 5 5 (b) (c) (d) (e) (f) A plank of wood metres long is cut up into 5 equal pieces. How long is each piece? 4 Each cow in a herd of 5 produces 4 litres of milk. How much milk is this in total? (6) 5 marks Pegasys 0

7 COMPARING DATA SETS using STATISTICS. Find the median, the upper and lower quartiles and the interquartile range for: (a) (4) (b) (5). A set of test marks is shown below Use an appropriate formula to calculate the mean and standard deviation. (5). (a) A quality control examiner on a production line measures the weight in grams of cakes coming off the line. In a sample of eight cakes the weights were Calculate the mean and standard deviation. (5) (b) On a second production line, a sample of 8 cakes gives a mean of 49 and a standard deviation of 6. Compare the distribution of the cakes produced on the two production lines. () 0 marks Pegasys 0

8 Fitness level on scale of -5(f) FORMING a LINEAR MODEL from a given SET of DATA. Copy these graphs and use your ruler to draw what you think is the line of best fit. (). A health visitor measured the fitness level of a group of teenagers and recorded B the number of hours they watched television in a week. She then drew this graph and the line of best fit. () f Hours spent watching TV(h) h Find the equation of the line of best fit drawn.. The data below shows the marks gained by seven pupils in two class tests. Maths Physics (a) Show the data on a scattergraph and draw the line of best fit. () (b) Find the equation of your line of best fit. () (c) Use your equation to estimate the Physics mark of a pupil whose Maths mark was 50. () Pegasys 0 marks

9 ANSWERS TRIGONOMETRY (). 7 m². 5 6cm. 7 8cm o TRIGONOMETRY () PROBLEMS. 084 o. metres. (a) 8 o (b) 5 5 km² WORKING with VECTORS. (a) a = 4 (b) b = (c) c = (b) (i) (ii) b a a + b a c a c (iii) b + c b c (iv) a + b (a + b) + c c (v) b c b c b c a a + (b c) Pegasys 0

10 7 (c) (i) ; 5 = 7 (ii) ; 0 =. (a) b + a (b) a b (c) ½ (a b). Q(8, 0, 0); R(8, 0, 0); S(0, 0, 0); P(4, 5, 5) ; WORKING with PERCENTAGES years to reach at least miles WORKING with FRACTIONS. (a) 9 5 (b) 5 (c) 44 (d) (e) (f) (g) (h) (a) 69 (b) (c) (a) 90 km (b) 97 km (c) 487 km (d) 6 (e) (a) (b) 4 (c) 6 0 (d) (e) metre (f) 6 litres 4 Pegasys 0

11 Physics Mark(P) COMPARING DATA SETS using STATISTICS. (a) Q = ; Q = 7; Q = 0; IQR = 8 (b) Q = 4; Q = ; Q = 5 5; IQR = 5. Mean = 7; SD = 8. (a) Mean = 48; SD = (b) On average the second line produces cakes where the weights are less consistent. FORMING a LINEAR MODEL from a given SET of DATA. (a) and (b) Any reasonable lines drawn. f h 7 5. (a) Maths Mark(M) (b) 4 P h 0 (c) approx 48 Pegasys 0

12 National 5 Trigonometry () Homework Marking Scheme APPS. A 7 9 sin 60 = 7 m² [ marks]. Shortest side opposite 5 o. Third angle is 8 o. A 7 9 sin 60 7 cm Use Sine rule x 7 sin 5 sin 8 7sin 5 x sin 8 = 5 6cm [4 marks]. 7cm 5 o 45cm Cosine rule for finding a side x x cm 7 45cos5 [4 marks] 4. Largest angle is opposite largest side. Use Cosine rule for finding an angle Cos [4 marks] Total: 4 marks Pegasys 0

13 National 5 Trigonometry () Homework Marking Scheme APPS. Find angle A Use cosine rule for angle cos A Evaluates cos A A = 4 4 o Bearing = 5 4 o = 084 o [6 marks]. Strategy: find AB Use sine rule AB 000 sin7 sin40 AB = 569 m Strategy: find BC Use cosine rule or Sine rule for side cos Or BC 000 sin sin40 54 m = m [8 marks]. (a) length of third side is 4km Use cosine rule for angle 9 4 cos A 9 A = 8 o (b) A 9 sin km³ [6 marks] Total: 0 marks Pegasys 0

14 National 5 Working with Vectors Homework Marking Scheme APPS. (a) a = 4 (b) b = (c) c = each [ marks] (b) (i) (ii) b a a + b a c a c (iii) b + c b c (iv) a + b (a + b) + c c (v) b c b c b c a a + (b c) (i) (iv) marks each for individual vectors for resultant vectors (v) marks finding b c for second diagram for resultant vector [9 marks] (c) (i) 7 ; 5 = 7 (ii) ; 0 = marks each for components for magnitude [4 marks]. (a) PR = PS + SR = b + a (b) SQ = SP + PQ = a b (c) SM = ½ SQ = ½ (a b) marks each for path for vectors [6 marks]. Q(8, 0, 0); R(8, 0, 0); S(0, 0, 0); P(4, 5, 5) each point [4 marks] knowing to add + answer knowing how to find magnitude answer [4 marks] Pegasys 0 Total: 0 marks

15 National 5 Working with Percentages Homework Marking Scheme APPS correct multiplier correct power [ marks]. strategy = = = years to reach at least 400 [5 marks] correct multiplier correct power 96 [ marks] 4. strategy = = = miles [5 marks] = [ marks] = [ marks] Total: marks National 5 Working with Fractions Homework Marking Scheme APPS Pegasys 0

16 . (a) 9 5 (b) 5 (c) 44 (d) (e) (f) (g) (h) 5 [8 marks]. 8 6 dividing. (a) 5 [ marks] 69 (b) (a) 90 km (c) (b) 97 km (c) 487 km 4 [ marks] 94 (d) (e) km/h [6 marks] 7 5. (a) (b) 4 (c) 6 0 (d) (e) metre (f) 6 litres [6 marks] 4 Total: 5 marks Pegasys 0

17 National 5 Comparing Data Sets Homework Marking Scheme APPS. (a) Q = ; Q = 7; Q = 0; IQR = 8 each [4 marks] (b) ordering Q = 4; Q = ; Q = 5 5; IQR = 5 each [5 marks] SD = 770 SD = 8 [5 marks]. (a) SD = 66 7 SD = [5 marks] (b) Weights on the second production line are less consistent [ mark] Total: 0 marks National 5 Forming a Linear Model Homework Marking Scheme APPS Pegasys 0

18 Physics Mark(P). (a) and (b) Any reasonable lines drawn [ marks]. m 7 5 f h 7 5 [ marks]. (a) points plotted line drawn [ marks] (b) Maths Mark(M) 4 m 0 4 P h 0 [ marks] (c) approx 77 [ mark] Total: marks Pegasys 0

TRIGONOMETRY (1) AREA of TRIANGLE, SINE RULE and COSINE RULE cm

TRIGONOMETRY (1) AREA of TRIANGLE, SINE RULE and COSINE RULE 1. Calculate the area of the triangle in the diagram. () 7m 60 9m. Calculate the length of the shortest side in the triangle shown.. (4) 35