Intermediate 2 NATIONAL QUALIFICATIONS. Mathematics Specimen Question Paper 1 (Units 1, 2, 3) Non-calculator Paper [C056/SQP105] Time: 45 minutes

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1 [C056/SQP105] Intermediate Time: 45 minutes Mathematics Specimen Question Paper 1 (Units 1,, 3) Non-calculator Paper NATIONAL QUALIFICATIONS 1 Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. 3 You may NOT use a calculator. 4 Square-ruled paper is provided.

2 FORMULAE LIST ( 4 ) b b ac The roots of ax + bx+ c= 0 are x = ± a Sine rule: a b c = = sin A sin B sin C Cosine rule: a = b + c bccos A or cos A = b + c a bc Area of a triangle: Area = Wab sin C Volume of a sphere: Volume = 4 πr 3 3 Volume of a cone: Volume = 1 πr h 3 Volume of a cylinder: Volume = πr h Standard deviation: s = ( x x) n 1 Page two

3 1. A spinner has eight edges numbered as shown in the diagram. When it is spun it comes to rest on one edge. What is the probability that it comes to rest on an even number? Marks () Factorise x 7x 8. () 3. (a) The boxplot, drawn below, shows the daily rainfall (in millimetres) recorded over a number of days State the upper and lower quartiles. (1) (b) The daily rainfall (in millimetres) for a town was recorded over seven days. The boxplot, shown below, was drawn for this data set Write down a possible data set which fits this boxplot. (3) Page three

4 4. The diagram opposite shows the graph of y = f(x). y y = f(x) Marks Write down the equation of the graph in the form y = (x + a) + b x () 5. y 8 B 6 4 A x Find the equation of the straight line AB. (3) Page four

5 6. A district council decided to plant maple and rowan trees on a piece of ground with area 0. 5 hectares. The trees were planted using a recommended spacing of 70 per hectare. (a) Let m be the number of maple trees and r the number of rowan trees planted. Write down an equation in m and r which satisfies the above condition. Marks () (b) The district council spent 1500 on the trees. Each maple tree cost 9 and each rowan tree cost Write down an equation in m and r which satisfies this condition. () (c) How many trees of each kind did the council plant? (3) 7. (a) Express 3 as a fraction with a rational denominator. 5 () (b) Express b 1 3 b b in its simplest form. () (c) Express as a single fraction in its simplest form 5 3, x 0 or x. x ( x ) (3) [END OF QUESTION PAPER] Page five

6 [C056/SQP105] Intermediate Mathematics Paper 1 (Non-calculator) Units 1, and Applications of Mathematics Specimen Question Paper Time: 45 minutes NATIONAL QUALIFICATIONS 1 Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. 3 You may NOT use a calculator. 4 Square-ruled paper is provided. [C056/SQP105] 1

7 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b 4ac a Sine rule: a b c = = sin A sin B sin C Cosine rule: a = b + c bc cos A or cos A = b + c a bc Area of a triangle: Area = 1 ab sinc Volume of a sphere: Volume = 4 πr 3 3 Volume of a cone: Volume = 1 πr h 3 Volume of a cylinder: Volume = πr h Standard deviation: x x x x n s = ( ) ( ) / =, n 1 n 1 where n is the sample size. [C056/SQP105] Page two

8 1. A spinner has eight edges numbered as shown in the diagram. When it is spun it comes to rest on one edge. What is the probability that it comes to rest on an even number? Marks () Factorise x 7x 8. () 3. (a) The boxplot, drawn below, shows the daily rainfall (in millimetres) recorded over a number of days State the upper and lower quartiles. (1) (b) The daily rainfall (in millimetres) for a town was recorded over seven days. The boxplot, shown below, was drawn for this data set Write down a possible data set which fits this boxplot. (3) [C056/SQP105] 3 Page three

9 4. The flowchart shown below gives instructions on how to calculate travelling expenses. The total expenses claimed depends on the engine capacity of the car being used and the number of miles travelled. Marks START Yes Does the engine capacity exceed 1199cc? No Rate is 30p per mile Rate is 5p per mile Yes Does the number of miles travelled exceed 100? No Allowance = 100 rate + (number of miles 100) 7p Allowance = number of miles rate STOP Use the flowchart to calculate the expenses which can be claimed for travelling 19 miles in a car with an engine capacity of 998 cc. (4) [C056/SQP105] 4 Page four

10 5. y 8 B Marks 6 4 A x Find the equation of the straight line AB. (3) 6. A district council decided to plant maple and rowan trees on a piece of ground with area 0. 5 hectares. The trees were planted using a recommended spacing of 70 per hectare. (a) Let m be the number of maple trees and r the number of rowan trees planted. Write down an equation in m and r which satisfies the above condition. () (b) The district council spent 1500 on the trees. Each maple tree cost 9 and each rowan tree cost Write down an equation in m and r which satisfies this condition. () (c) How many trees of each kind did the council plant? (3) [C056/SQP105] 5 Page five

11 7. (a) A network is traversable if it can be drawn by going over every line once and only once without lifting your pencil. Marks The network shown opposite can be traversed by the route: P Q P R S P Q R T S R T Write down a route by which the network below can be traversed. P Q T S R (1) (b) A delivery van leaves its depot in Allbridge to make deliveries to Beckworth, Castlehill and Downfield. The diagram shown below gives the distances between the depots. Beckworth 19 km 14 km Allbridge 0 km Downfield 17 km 15 km Castlehill (i) Draw a tree diagram to show the possible delivery routes. () (ii) [C056/SQP105] 6 Which is the shortest route? [END OF QUESTIONS] Page six ()

12 [C056/SQP105] Intermediate Time: 1 hour 30 minutes Mathematics Specimen Question Paper (Units 1,, 3) NATIONAL QUALIFICATIONS 1 Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. 3 You may use a calculator. 4 Square-ruled paper is provided.

13 FORMULAE LIST ( 4 ) b b ac The roots of ax + bx+ c= 0 are x = ± a Sine rule: a b c = = sin A sin B sin C Cosine rule: a = b + c bccos A or cos A = b + c a bc Area of a triangle: Area = Wab sin C Volume of a sphere: Volume = 4 πr 3 3 Volume of a cone: Volume = 1 πr h 3 Volume of a cylinder: Volume = πr h Standard deviation: s = ( x x) n 1 Page two

14 1. A golfer recorded the following scores for six games of golf. Marks 69, 75, 74, 78, 70, 7 Use the formula to calculate the standard deviation. Show clearly all your working. (4). A storage barn is prism shaped, as shown below. 5m 7m 1 m The cross-section of the storage barn consists of a rectangle measuring 7 metres by 5 metres and a semi-circle of radius 3. 5 metres. (a) Find the volume of the storage barn. Give your answer in cubic metres, correct to significant figures. (4) (b) An extension to the barn is planned to increase the volume by 00 cubic metres. 3m 5m 1 m width The uniform cross-section of the extension consists of a rectangle and a right-angled triangle. Find the width of the extension. (3) Page three

15 3. Two parachutists, X and Y, jump from two separate aircrafts at different times. The graph shows how their height above the ground changes over a period of time. Marks Parachutist X Height of parachutist above the ground Parachutist Y Time (a) Which parachutist jumped first? (b) Which parachutist did not open his parachute immediately after jumping? Explain your answer clearly. (3) Page four

16 4. Figure 1 shows the circular cross-section of a tunnel with a horizontal floor. Marks figure 1 In figure, AB represents the floor. AB is. 4 metres. The radius, OA, of the cross-section is. 5 metres. Find the height of the tunnel.. 5m O height of the tunnel A. 4m B figure (4) Page five

17 5. The table shows the emission levels of harmful gases at different places in a city. Marks Emission Levels City Sq Albert Sq Wellgate Centre Bus Station High St 111 units 41 units 161 units 146 units 114 units Health regulations state that the emission levels of harmful gases should be less than 135 units. The city council plan to reduce the levels in such a way that for each of the next 3 years the emission levels will be 5% less than the level in the previous year. Will all the places listed in the table meet the health regulations in 3 years time? Show clearly all your working. (4) 6. Solve the equation x + x 6 = 0 giving your answers correct to 1 decimal place. (4) Page six

18 7. The table shows the distribution of absentees per class on a particular day in a secondary school. Marks Number of absentees Frequency (a) Make a cumulative frequency table from the above data. (b) Find the median and the lower and upper quartiles for this distribution. (1) (3) 8. The boat on a carnival ride travels along an arc of a circle, centre C. C C 6m A B A B The boat is attached to C by a rod 6 metres long. The rod swings from position CA to position CB. The length of the arc AB is 7 metres. Find the angle through which the rod swings from position A to position B. (4) Page seven

19 9. Change the subject of the formula to k d = k m. t Marks () 10. The diagram shows two positions of a student as she views the top of a tower. T A B From position B, the angle of elevation to T at the top of the tower is 64. From position A, the angle of elevation to T at the top of the tower is 69. The distance AB is 4. 8 metres and the height of the student to eye level is 1. 5 metres. Find the height of the tower. (6) Page eight

20 11. (a) Solve the equation 7cos x = 0, for 0 x< 360. Marks (3) (b) Show that sin x = tan x. 1 sin x () 1. A rectangular sheet of plastic 18 cm by 100 cm is used to make a gutter for draining rain water. The gutter is made by bending the sheet of plastic as shown below in diagram cm 18 cm diagram 1 (a) The depth of the gutter is x centimetres as shown in diagram below. Write down an expression in x for the width of the gutter. (1) x cm width x cm diagram 100 cm (b) Show that the volume, V cubic centimetres, of this gutter is given by V = 1800x 00x. () (c) Find the dimensions of the gutter which has the largest volume. Show clearly all your working. (4) [END OF QUESTION PAPER] Page nine

21 [C056/SQP105] Intermediate Mathematics Paper Units 1, and Applications of Mathematics Specimen Question Paper Time: 1 hour 30 minutes NATIONAL QUALIFICATIONS 1 Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. 3 You may use a calculator. 4 Square-ruled paper is provided. [C056/SQP105] 1

22 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b 4ac a Sine rule: a b c = = sin A sin B sin C Cosine rule: a = b + c bc cos A or cos A = b + c a bc Area of a triangle: Area = 1 ab sinc Volume of a sphere: Volume = 4 πr 3 3 Volume of a cone: Volume = 1 πr h 3 Volume of a cylinder: Volume = πr h Standard deviation: x x x x n s = ( ) ( ) / =, n 1 n 1 where n is the sample size. [C056/SQP105] Page two

23 1. A golfer recorded the following scores for six games of golf. Marks 69, 75, 74, 78, 70, 7 Use the formula to calculate the standard deviation. Show clearly all your working. (4). A storage barn is prism shaped, as shown below. 5m 7m 1 m The cross-section of the storage barn consists of a rectangle measuring 7 metres by 5 metres and a semi-circle of radius 3. 5 metres. (a) Find the volume of the storage barn. Give your answer in cubic metres, correct to significant figures. (4) (b) An extension to the barn is planned to increase the volume by 00 cubic metres. 3m 5m 1 m width The uniform cross-section of the extension consists of a rectangle and a right-angled triangle. Find the width of the extension. (3) [C056/SQP105] 3 Page three

24 3. Two parachutists, X and Y, jump from two separate aircrafts at different times. The graph shows how their height above the ground changes over a period of time. Marks Parachutist X Height of parachutist above the ground Parachutist Y Time (a) Which parachutist jumped first? (b) Which parachutist did not open his parachute immediately after jumping? Explain your answer clearly. (3) [C056/SQP105] 4 Page four

25 4. Figure 1 shows the circular cross-section of a tunnel with a horizontal floor. Marks figure 1 In figure, AB represents the floor. AB is. 4 metres. The radius, OA, of the cross-section is. 5 metres. Find the height of the tunnel.. 5m O height of the tunnel A. 4m B figure (4) [C056/SQP105] 5 Page five

26 5. The table shows the emission levels of harmful gases at different places in a city. Marks Emission Levels City Sq Albert Sq Wellgate Centre Bus Station High St 111 units 41 units 161 units 146 units 114 units Health regulations state that the emission levels of harmful gases should be less than 135 units. The city council plan to reduce the levels in such a way that for each of the next 3 years the emission levels will be 5% less than the level in the previous year. Will all the places listed in the table meet the health regulations in 3 years time? Show clearly all your working. (4) 6. The income tax rates are shown in the table below. 10% on the first 1500 of taxable income 3% on the next of taxable income 40% on taxable income over Jenny White earns per year. Her annual tax allowances total 50. Calculate Jenny s annual salary after tax has been deducted. (5) [C056/SQP105] 6 Page six

27 7. The table shows the distribution of absentees per class on a particular day in a secondary school. Marks Number of absentees Frequency (a) Make a cumulative frequency table from the above data. (b) Find the median and the lower and upper quartiles for this distribution. (1) (3) 8. The boat on a carnival ride travels along an arc of a circle, centre C. C C 6m A B A B The boat is attached to C by a rod 6 metres long. The rod swings from position CA to position CB. The length of the arc AB is 7 metres. Find the angle through which the rod swings from position A to position B. (4) [C056/SQP105] 7 Page seven

28 9. A metal bar expands when it is heated. The new length, L centimetres, of the metal bar is given by the formula Marks L = B(1 + kt) where B centimetres is the length of the bar before heating t C is the rise in temperature k depends on the type of metal. (a) Calculate L when B = 0, t = 15 and k = (3) (b) Find B when L = 53, t = 0 and k = () 10. The diagram shows two positions of a student as she views the top of a tower. T A B From position B, the angle of elevation to T at the top of the tower is 64. From position A, the angle of elevation to T at the top of the tower is 69. The distance AB is 4. 8 metres and the height of the student to eye level is 1. 5 metres. Find the height of the tower. (6) [C056/SQP105] 8 Page eight

29 11. A record was kept of the number of packets of crisps sold each day in a school shop. The results are shown below. Marks Number of packets Number of days Calculate the mean number of packets sold. (4) 1. Jim Grant borrows 3000 over 4 months with personal loan protection. Amount MONTHLY REPAYMENT TABLE 36 months 4 months W W/o W W/o W = With personal loan protection W/o = Without personal loan protection Use the loan repayment table shown above to calculate how much his loan will cost. (3) [END OF QUESTIONS] [C056/SQP105] 9 Page nine

30 [C056/SQP105] Intermediate Mathematics Specimen Marking Instructions 1 (Units 1,, 3) Non-calculator Paper NATIONAL QUALIFICATIONS

31 Mathematics Intermediate (Paper 1) Qu Marking Scheme Illustrations of evidence Give 1 mark for each for awarding a mark at each 1. ans : 3/8 strategy: know to write probability as х/total number of outcomes 8 marks process: find probability 3_ 8. ans : (x 8) (x + 1) process: factorise binomial expression one correct factor process: complete factorisation nd correct factor marks 3a. ans : 9 and 3 communicate: state upper and lower quartiles 9 and 3 1 mark 3b. ans : (possible) 13, 15, 15, 16, 17, 19, strategy: identify upper, lower quartiles and median 15, 16, 19 strategy: identify end points of the range 13, 3 communicate: complete process correctly 3 any two valid measurements 3 marks Page

32 Qu Marking Scheme Illustrations of evidence Give 1 mark for each for awarding a mark at each 4. ans : (x 4) - 3 interpret: identify a a = - 4 interpret: identify b b = - 3 marks 5. ans : у = x + 1 process: identify у intercept or evaluate c in у = m x + c 1 process: find gradient 3 communicate: state equation of straight line 3 у = x marks 6a. ans : m + r = 180 interpret: start to interpret text m + r interpret: complete interpretation m + r = 180 marks Page 3

33 Qu Marking Scheme Illustrations of evidence Give 1 mark for each for awarding a mark at each 6b. ans : 9m + 7.5r = 1500 interpret: start to interpret text 9 m r interpret: complete interpretation 9 m r = 1500 marks 6c. ans : 100 maple trees; 80 rowan trees strategy: know to solve system of equations m + r = 180 9m + 7.5r = process: solve system of equations communicate: state result m = 100 r = 80 3 marks maple 80 rowan Page 4

34 Qu Marking Scheme Illustrations of evidence Give 1 mark for each for awarding a mark at each 7a. ans strategy: know how to rationalise denominator process: simplify fraction marks 7b. ans : b process: add indices in numerator b process: subtract indices b marks 7c. x 10 ans : x( x ) process: state valid denominator x (x ) process: find numerators of equivalent fractions 5(x ), 3x 3 process: state answer in simplest form 3 x 10 x (x ) 3 marks [END OF MARKING INSTRUCTIONS] Page 5

35 [C056/SQP105] Intermediate Mathematics Specimen Marking Instructions Paper 1 (Applications of Mathematics questions) Non-calculator Paper NATIONAL QUALIFICATIONS [C056/SQP105] 5

36 Mathematics Intermediate (Paper 1) Applications of Mathematics questions Qu 4 Marking Scheme Give 1 mark for each interpret: identify rate per mile Illustrations of evidence for awarding a mark at each 5p interpret: identify relevant formula use of relevant formula 3 process: start to evaluate formula p 4 process: calculate expenses (a) interpret/communicate: state route Q S R Q P S T (b) (i) strategy: start to draw tree diagram 4 branches of tree strategy: complete diagram correctly completed tree diagram A B C C D B D D C D B (ii) strategy: calculate distance for each route evidence of calculations communicate: identify shortest route A C D B [END OF MARKING INSTRUCTIONS] [C056/SQP105] 6 Page two

37 [C056/SQP105] Intermediate Mathematics Specimen Marking Instructions (Units 1,, 3) NATIONAL QUALIFICATIONS

38 Mathematics Intermediate (Paper ) 1. ans : s = 3.35 process: calculate x 73 process: calculate ( x ) x 16, 4, 1, 5, 9, 1 3 process: substitute in formula process: calculate standard deviation 4 s = 3.35 (disregard rounding) 4 marks a. ans : 650 m 3 strategy: know how to calculate volume of barn volume of cuboid + volume of 1 cylinder process: substitute into formulae for cuboid and half cylinder 7 X 1 X X π X 3.5 X 1 3 process: calculate total volume m 3 4 process: round answer to significant figures m 3 4 marks Page

39 b. ans : width = 4. m strategy: know to find expression for volume of extension 1 3 w + w 1 process: equate with 00 and simplify 48 w = 00 3 communicate: state width 3 width = 4. m 3 marks 3a. ans : X interpret: interpret qualitative graphs X 1 mark 3b. ans : X - reason interpret: interpret qualitative graphs X communicate: state reason graph shows rates of fall marks Page 3

40 4. ans : 4.69 m strategy: marshal facts and know to use Pythagoras.5m 1.m x process: rearrange equation x = process: calculate x 3 x =.19 4 communicate: state the height m 4 marks 5. ans : NO + reason strategy: identify crucial aspect 161 strategy: know how to calculate level in 3 years x process: calculate within valid strategy communicate: give response based on previous evidence 4 NO, with reason 4 marks Page 4

41 6. ans : x = -3.6 and 1.6 strategy: know to use the quadratic formula quadratic formula process: substitute correctly into quadratic formula ± x = 3 process: calculate b - 4 ac communicate: state values of x correct to 1 decimal place and marks 7a. ans :, 7, 14, 18,, 4, 5 communicate: table with cumulative frequencies, 7, 14, 18,, 4, 5 7b. ans: median =, lower quartile = 1, upper quartile = 4 communicate: median communicate: lower quartile 1 3 communicate: upper quartile 3 4 Page 5

42 8. ans : 66.9 strategy: marshal facts evidence of link with circumference strategy: express arc as ratio of circumference 7 π 1 3 strategy: know how to find angle π 1 o 4 process: carry out all calculations marks 9. ans : k = dt + m process: "remove" denominator dt = k - m process: make k the subject k = dt + m marks Page 6

43 10. ans : 47.7 m interpret: find size of angle ATB ATB = 5 strategy: know to apply sine rule to find AT sine rule 3 process: substitute into sine rule 3 TA = 4.8 sin 64 sin 5 4 process: calculate TA 4 TA = 49.5 m 5 strategy: know to apply trigonometry to find TC (C on building horizontally aligned with A) 5 TC = sin 69 TA 6 process: calculate TC and height of building 6 TC = 46. m TA = 47.7 m Page 7

44 11a. ans : x = 73.4 and 86.6 process: solve equation for cos x cos x = 7 process: find one value for x x = process: final nd value for x 3 x = marks 11b. Proof strategy: substitute cos x for 1-sin x sin cos x x communicate: state valid explanation tan x marks Page 8

45 1a. ans : 18-x communicate: find formula 18 - x 1 mark 1b. ans: Proof communicate: state expression for volume in terms of x 100 π (18 - x) cm 3 process: demonstrate clearly the result 1800x - 00x marks 1c. ans: 100 cm by 9 cm by 4.5 cm strategy: equate 1800x - 00x to zero 1800x - 00x = 0 process: solve equation for x x = 0, x = 9 3 process: find x value for maximum volume 3 x = communicate: state the dimensions cm by 9 cm by 4.5 cm 4 marks [END OF MARKING INSTRUCTIONS] Page 9

46 [C056/SQP105] Intermediate Mathematics Specimen Marking Instructions Paper (Applications of Mathematics questions) NATIONAL QUALIFICATIONS [C056/SQP105] 11

47 Mathematics Intermediate (Paper ) Applications of Mathematics questions Qu 6 Marking Scheme Give 1 mark for each process: calculate taxable income Illustrations of evidence for awarding a mark at each strategy: know how to calculate lower rate of tax strategy: know how to calculate middle rate of tax ( ) 4 process: calculate tax at lower and middle rate and 77 5 process: calculate total amount of tax due (a) process: substitute into formula 0 ( ) process: start evaluation process: complete evaluation (b) process: substitute into formula 53 = B ( ) process: calculate B 50 [C056/SQP105] 1 Page two

48 Qu 11 Marking Scheme Give 1 mark for each process: calculate midpoints of intervals Illustrations of evidence for awarding a mark at each 3, 57, 8, 107, 13, 157, 18 process: multiply midpoints by frequency 64, 171, 38, 749, 130, 471, 18 3 process: find totals and 30 4 process: calculate mean interpret: identify monthly repayment process: calculate total repayment process: calculate cost of loan [END OF MARKING INSTRUCTIONS] [C056/SQP105] 13 Page three

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