*2500/405* 2500/405. MATHEMATICS STANDARD GRADE Credit Level Paper 1 (Non-calculator) 1 You may NOT use a calculator.

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1 C 500/05 NATIONAL QUALIFICATIONS 007 THURSDAY, MAY.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper (Non-calculator) You may NOT use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. *500/05* LI 500/05 6/570

2 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/05] Page two

3 . Evaluate Evaluate. 6. There are 00 people in a studio audience. The probability that a person chosen at random from this audience is male 5 is. 8 How many males are in this audience?. ( m ) P = Change the subject of the formula to m. 5. Remove brackets and simplify (x + ) (x 6). [Turn over [500/05] Page three

4 6. A taxi fare consists of a call-out charge plus a fixed amount per kilometre. The graph shows the fare, f pounds for a journey of d kilometres. f pounds (5, 6) 0 kilometres d The taxi fare for a 5 kilometre journey is 6. Find the equation of the straight line in terms of d and f. 7. Remove brackets and simplify a a ( ). [500/05] Page four

5 8. Mick needs an ironing board. He sees one in a catalogue with measurements as shown in the diagram below. length 75 cm 0 cm 8 cm When the ironing board is set up, two similar triangles are formed. Mick wants an ironing board which is at least 80 centimetres in length. Does this ironing board meet Mick s requirements? Show all your working. 9. A square of side x centimetres has a diagonal 6 centimetres long. 6cm x cm Calculate the value of x, giving your answer as a surd in its simplest form. [Turn over [500/05] Page five

6 0. k A relationship between T and L is given by the formula, T = a constant. L where k is When L is doubled, what is the effect on T?. (a) A cinema has 00 seats which are either standard or deluxe. Let x be the number of standard seats and y be the number of deluxe seats. Write down an algebraic expression to illustrate this information. (b) A standard seat costs and a deluxe seat costs 6. When all the seats are sold the ticket sales are 80. Write down an algebraic expression to illustrate this information. (c) How many standard seats and how many deluxe seats are in the cinema? [500/05] Page six

7 . The diagram shows water lying in a length of roof guttering. The cross-section of the guttering is a semi-circle with diameter 0 centimetres. The water surface is 8 centimetres wide. 0 cm 8cm d Calculate the depth, d, of water in the guttering. [Turn over for Questions and on Page eight [500/05] Page seven

8 . Part of the graph of y = cos bx + c is shown below. y O x Write down the values of b and c.. The sum S n of the first n terms of a sequence, is given by the formula S n = n. (a) Find the sum of the first terms. (b) When S n = 80, calculate the value of n. [END OF QUESTION PAPER] [500/05] Page eight

9 C 500/06 NATIONAL QUALIFICATIONS 007 THURSDAY, MAY.5 PM.05 PM MATHEMATICS STANDARD GRADE Credit Level Paper You may use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. *500/06* LI 500/06 6/570

10 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/06] Page two

11 . Alistair buys an antique chair for 600. It is expected to increase in value at the rate of. 5% each year. How much is it expected to be worth in years?. Solve the equation x x 0 = 0. Give your answer correct to significant figures.. (a) During his lunch hour, Luke records the number of birds that visit his bird-table. The numbers recorded last week were: Find the mean and standard deviation for this data. (b) Over the same period, Luke s friend, Erin also recorded the number of birds visiting her bird-table. Erin s recordings have a mean of 5 and a standard deviation of 5. Make two valid comparisons between the friends recordings.. Solve the inequality x < 5. [Turn over [500/06] Page three

12 5. Mark takes some friends out for a meal. The restaurant adds a 0% service charge to the price of the meal. The total bill is What was the price of the meal? 6. Brunton is 0 kilometres due North of Appleton. From Appleton, the bearing of Carlton is 065. From Brunton, the bearing of Carlton is 5. N B 0 km A 65 C Calculate the distance between Brunton and Carlton. [500/06] Page four

13 7. A fan has four identical plastic blades. 5cm 6 Each blade is a sector of a circle of radius 5 centimetres. The angle at the centre of each sector is 6. Calculate the total area of plastic required to make the blades. 8. In triangle PQR: QR = 6 centimetres angle PQR = 0 area of triangle PQR = 5 square centimetres. P Q 0 6cm R Calculate the length of PQ. [Turn over [500/06] Page five

14 9. To make carat gold, copper and pure gold are mixed in the ratio 5:7. A jeweller has 60 grams of copper and 5 grams of pure gold. What is the maximum weight of carat gold that the jeweller can make? 0. Solve algebraically the equation 5cosx + = 0, 0 x < 60.. (a) A decorator s logo is rectangular and measures 0 centimetres by 6 centimetres. It consists of three rectangles: one red, one yellow and one blue. 0 cm x cm yellow 6cm red blue x cm The yellow rectangle measures 0 centimetres by x centimetres. The width of the red rectangle is x centimetres. Show that the area, A, of the blue rectangle is given by the expression A = x 6x (b) The area of the blue rectangle is equal to Calculate the value of x. 5 of the total area of the logo. [500/06] Page six

15 . (a) A cylindrical paperweight of radius centimetres and height centimetres is filled with sand. cm cm Calculate the volume of sand in the paperweight. (b) Another paperweight, in the shape of a hemisphere, is filled with sand. r It contains the same volume of sand as the first paperweight. Calculate the radius of the hemisphere. [The volume of a hemisphere with radius r is given by the formula, V = πr ]. [Turn over for Question on Page eight [500/06] Page seven

16 . The profit made by a publishing company of a magazine is calculated by the formula y = x (0 x), where y is the profit (in pounds) and x is the selling price (in pence) of the magazine. The graph below represents the profit y against the selling price x. y O x Find the maximum profit the company can make from the sale of the magazine. [END OF QUESTION PAPER] [500/06] Page eight

17 C 500/05 NATIONAL QUALIFICATIONS 008 THURSDAY, 8 MAY.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper (Non-calculator) You may NOT use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. LI 500/05 6/070 *500/05*

18 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/05] Page two

19 . Evaluate Factorise fully 5x 5.. W = BH. Change the subject of the formula to H.. A straight line cuts the x-axis at the point (9, 0) and the y-axis at the point (0, 8) as shown. y 8 O 9 x Find the equation of this line. [Turn over [500/05] Page three

20 5. Express as a single fraction in its simplest form +. p ( p+ 5) 6. Jane enters a two-part race. (a) She cycles for hours at a speed of (x + 8) kilometres per hour. Write down an expression in x for the distance cycled. (b) She then runs for 0 minutes at a speed of x kilometres per hour. Write down an expression in x for the distance run. (c) The total distance of the race is 6 kilometres. Calculate Jane s running speed. 7. The th term of each number pattern below is the mean of the previous three terms. (a) When the first three terms are, 6, and 8, calculate the th term. (b) When the first three terms are x, (x + 7) and (x + ), calculate the th term. (c) When the first, second and fourth terms are x, (x + 5),, (x + ), calculate the rd term. [500/05] Page four

21 8. The curved part of the letter A in the Artwork logo is in the shape of a parabola. The equation of this parabola is y = (x 8)( x). y y = (x 8)( x) h h O Q R x 6 (a) Write down the coordinates of Q and R. (b) Calculate the height, h, of the letter A. 9. Simplify m m. [Turn over [500/05] Page five

22 0. Part of the graph of y = a x, where a > 0, is shown below. y y = a x C B(, 6) O x The graph cuts the y-axis at C. (a) Write down the coordinates of C. B is the point (, 6). (b) Calculate the value of a.. A right angled triangle has dimensions as shown. A 50 C B Calculate the length of AC, leaving your answer as a surd in its simplest form. [500/05] Page six

23 . Given that x 0x + 8 = (x a) + b, find the values of a and b.. A new fraction is obtained by adding x to the numerator and denominator of the fraction 7. This new fraction is equivalent to Calculate the value of x.. [END OF QUESTION PAPER] [500/05] Page seven

24 C 500/06 NATIONAL QUALIFICATIONS 008 THURSDAY, 8 MAY.5 PM.05 PM MATHEMATICS STANDARD GRADE Credit Level Paper You may use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. LI 500/06 6/070 *500/06*

25 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/06] Page two

26 . A local council recycles 000 tonnes of waste a year. The council aims to increase the amount of waste recycled by 8% each year. How much waste does it expect to recycle in years time? Give your answer to three significant figures.. In a class, 0 pupils sat a test. The marks are illustrated by the stem and leaf diagram below. Test Marks n = 0 6 = 6 (a) Write down the median and the modal mark. (b) Find the probability that a pupil selected at random scored at least 0 marks.. In a sale, all cameras are reduced by 0%. A camera now costs 5. Calculate the original cost of the camera. NOW 5 [Turn over [500/06] Page three

27 . Aaron saves 50 pence and 0 pence coins in his piggy bank. Let x be the number of 50 pence coins in his bank. Let y be the number of 0 pence coins in his bank. (a) There are 60 coins in his bank. Write down an equation in x and y to illustrate this information. (b) The total value of the coins is Write down another equation in x and y to illustrate this information. (c) Hence find algebraically the number of 50 pence coins Aaron has in his piggy bank. 5. A circle, centre the origin, is shown. P is the point (8, ). y T O P(8, ) x (a) Calculate the length of OP. The diagram also shows a tangent from P which touches the circle at T. The radius of the circle is 5 units. (b) Calculate the length of PT. [500/06] Page four

28 6. The distance, d kilometres, to the horizon, when viewed from a cliff top, varies directly as the square root of the height, h metres, of the cliff top above sea level. From a cliff top 6 metres above sea level, the distance to the horizon is kilometres. A boat is 0 kilometres from a cliff whose top is 0 metres above sea level. Is the boat beyond the horizon? Justify your answer A telegraph pole is 6. metres high. 6. m The wind blows the pole over into the position as shown below.. 9m. 9m B 0 A C AB is. 9 metres and angle ABC is 0. Calculate the length of AC. [Turn over [500/06] Page five

29 8. A farmer builds a sheep-pen using two lengths of fencing and a wall. 5 m 70 8 m The two lengths of fencing are 5 metres and 8 metres long. (a) Calculate the area of the sheep-pen, when the angle between the fencing is 70. (b) What angle between the fencing would give the farmer the largest possible area? 9. Contestants in a quiz have 5 seconds to answer a question. This time is indicated on the clock. The tip of the clock hand moves through the arc AB as shown. A O B (a) Calculate the size of angle AOB. (b) The length of arc AB is 0 centimetres. Calculate the length of the clock hand. [500/06] Page six

30 0. To hire a car costs 5 per day plus a mileage charge. The first 00 miles are free with each additional mile charged at pence. CAR HI 5 per day first 00 miles free each additional mile only p (a) Calculate the cost of hiring a car for days when the mileage is 60 miles. (b) A car is hired for d days and the mileage is m miles where m > 00. Write down a formula for the cost C of hiring the car.. The minimum number of roads joining towns to each other is 6 as shown. The minimum number of roads, r, joining n towns to each other is given by the formula r = n( n ). (a) State the minimum number of roads needed to join 7 towns to each other. (b) When r = 55, show that n n 0 = 0. (c) Hence find algebraically the value of n. [500/06] [Turn over for Question on Page eight Page seven

31 . The diagram shows part of the graph of y = tan x. The line y = 5 is drawn and intersects the graph of y = tan x at P and Q. y y = tan x 5 P Q R O x (a) Find the x-coordinates of P and Q. (b) Write down the x-coordinate of the point R, where the line y = 5 next intersects the graph of y = tan x. [END OF QUESTION PAPER] [500/06] Page eight

32 C 500/05 NATIONAL QUALIFICATIONS 009 WEDNESDAY, 6 MAY.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper (Non-calculator) You may NOT use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. LI 500/05 6/90 *500/05*

33 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/05] Page two

34 . Evaluate (86 0) Evaluate.. Given that (a) evaluate f( ) f(x) = x +, (b) find t when f(t) = 5.. (a) Factorise x y. (b) Expand and simplify (c) Expand (x )(x + ). ( + ). x x x [Turn over [500/05] Page three

35 5. In triangle ABC: angle ACB = 90 AB = 8 centimetres AC = centimetres. cm C A 8cm B Calculate the length of BC. Give your answer as a surd in its simplest form. 6. There are girls and boys in a class. A child is chosen at random and is asked to roll a die, numbered to 6. Which of these is more likely? A: the child is female. OR B: the child rolls a 5. Justify your answer. 7. This year, Ben paid 60 for his car insurance. This is an increase of 0% on last year s payment. How much did Ben pay last year? [500/05] Page four

36 8. In triangle PQR: PQ = x centimetres PR = 5x centimetres QR = y centimetres. P 5x R x y Q (a) The perimeter of the triangle is centimetres. Write down an equation in x and y to illustrate this information. (b) PR is centimetres longer than QR. Write down another equation in x and y to illustrate this information. (c) Hence calculate the values of x and y. 9. A formula used to calculate the flow of water in a pipe is f = kd. 0 Change the subject of the formula to d. [Turn over [500/05] Page five

37 0. The diagram below shows the path of a rocket which is fired into the air. The height, h metres, of the rocket after t seconds is given by h h(t) = t(t ). 0 t (a) For how many seconds is the rocket in flight? (b) What is the maximum height reached by the rocket? [500/05] Page six

38 . In triangle ABC: BC = 6 metres AC = 0 metres angle ABC = 0. A 0 m B 0 6m C Given that sin 0 = 0. 5, show that sin A = 0.. [END OF QUESTION PAPER] [500/05] Page seven

39 C 500/06 NATIONAL QUALIFICATIONS 009 WEDNESDAY, 6 MAY.5 PM.05 PM MATHEMATICS STANDARD GRADE Credit Level Paper You may use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. LI 500/06 6/90 *500/06*

40 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/06] Page two

41 . One atom of gold weighs. 7 0 grams. How many atoms will there be in one kilogram of gold? Give your answer in scientific notation correct to significant figures.. Lemonade is to be poured from a litre bottle into glasses. Each glass is in the shape of a cylinder of radius centimetres and height 8 centimetres. litres How many full glasses can be poured from the bottle?. Solve the quadratic equation x x 6 = 0. Give your answers correct to decimal place. [Turn over [500/06] Page three

42 . Two fridge magnets are mathematically similar. One magnet is centimetres long and the other is 0 centimetres long. cm 0 cm The area of the smaller magnet is 8 square centimetres. Calculate the area of the larger magnet. 5. Tom looked at the cost of 0 different flights to New York. He calculated that the mean cost was 60 and the standard deviation was 7. A tax of is then added to each flight Write down the new mean and standard deviation. [500/06] Page four

43 6. Teams in a quiz answer questions on film and sport. This scatter graph shows the scores of some of the teams. y sport score film score x A line of best fit is drawn as shown above. (a) Find the equation of this straight line. (b) Use this equation to estimate the sport score for a team with a film score of (a) The air temperature, t º Celsius, varies inversely as the square of the distance, d metres, from a furnace. Write down a formula connecting t and d. (b) At a distance of metres from the furnace, the air temperature is 50 ºC. Calculate the air temperature at a distance of 5 metres from the furnace. [Turn over [500/06] Page five

44 8. A company makes large bags of crisps which contain 90 grams of fat. The company aims to reduce the fat content of the crisps by 50%. They decide to reduce the fat content by 0% each year. Will they have achieved their aim by the end of the rd year? Justify your answer. 9. Jane is taking part in an orienteering competition. B N 60 m A 0 º 60 m C She should have run 60 metres from A to B on a bearing of 0 º. However, she actually ran 60 metres from A to C on a bearing of 05 º. (a) Write down the size of angle BAC. (b) Calculate the length of BC. (c) What is the bearing from C to B? [500/06] Page six

45 0. The weight, W kilograms, of a giraffe is related to its age, M months, by the formula ( +7 ). W = M M At what age will a giraffe weigh 8 kilograms?. A cone is formed from a paper circle with a sector removed as shown. The radius of the paper circle is 0 cm. Angle AOB is 00 º. 0 cm O 00 º B A (a) Calculate the area of paper used to make the cone. (b) Calculate the circumference of the base of the cone. [Turn over for Question on Page eight [500/06] Page seven

46 . The n th term, T n of the sequence,, 6, 0,... is given by the formula: st T n( n ) T ( ) n = + term = + = nd T ( ) term = + = rd T ( ) term = + = 6 (a) Calculate the 0 th term, T 0. n+ ( ) (b) Show that T = n + n+. (c) Show that T n + T n+ is a square number. [END OF QUESTION PAPER] [500/06] Page eight

47 C 500/05 NATIONAL QUALIFICATIONS 00 WEDNESDAY, 5 MAY MATHEMATICS.0 PM.5 PM STANDARD GRADE Credit Level Paper (Non-calculator) You may NOT use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. LI 500/05 6/60 *500/05*

48 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/05] Page two

49 . Evaluate 0% of Evaluate Change the subject of the formula to s. t = 7s +.. Two functions are given below. f(x) = x x g(x) = x + 7 (a) If f(x) = g(x), show that x 6x 7 = 0. (b) Hence find algebraically the values of x for which f(x) = g(x). [Turn over [500/05] Page three

50 5. A bag contains 7 marbles. Some are black and some are white. The probability that a marble chosen at random is black is. 9 (a) What is the probability that a marble chosen at random is white? (b) How many white marbles are in the bag? 6. Cleano washing powder is on special offer. Each box on special offer contains 0% more powder than the standard box. A box on special offer contains 900 grams of powder. How many grams of powder does the standard box contain? [500/05] Page four

51 7. A straight line has equation y = mx + c, where m and c are constants. (a) The point (, 7) lies on this line. Write down an equation in m and c to illustrate this information. (b) A second point (, 7) also lies on this line. Write down another equation in m and c to illustrate this information. (c) Hence calculate the values of m and c. (d) Write down the gradient of this line. 8. (a) Simplify 8. (b) Simplify (c) Hence show that =. + 8 [Turn over [500/05] Page five

52 9. Part of the graph of the straight line with equation y = x +, is shown below. y y = x + B O x (a) Find the coordinates of the point B. (b) For what values of x is y < 0? [500/05] Page six

53 0. A number pattern is shown below. = + = + + = (a) Write down a similar expression for (b) Write down a similar expression for n. (c) Hence evaluate Two triangles have dimensions as shown. x (x ) The triangles are equal in area. Calculate the value of x. [END OF QUESTION PAPER] [500/05] Page seven

54 C 500/06 NATIONAL QUALIFICATIONS 00 WEDNESDAY, 5 MAY.5 PM.05 PM MATHEMATICS STANDARD GRADE Credit Level Paper You may use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. LI 500/06 6/60 *500/06*

55 FORMULAE LIST The roots of ax + bx + c = 0 are x = ( ) b± b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/06] Page two

56 . It is estimated that an iceberg weighs tonnes. As the iceberg moves into warmer water, its weight decreases by 5% each day. What will the iceberg weigh after days in the warmer water? Give your answer correct to three significant figures.. Expand fully and simplify x(x ).. A machine is used to put drawing pins into boxes. A sample of 8 boxes is taken and the number of drawing pins in each is counted. The results are shown below: (a) Calculate the mean and standard deviation of this sample. (b) A sample of 8 boxes is taken from another machine. This sample has a mean of 0 and a standard deviation of.. Write down two valid comparisons between the samples.. Use the quadratic formula to solve the equation, x + 5x 7 = 0. Give your answers correct to decimal place. [Turn over [500/06] Page three

57 5. A concrete ramp is to be built. The ramp is in the shape of a cuboid and a triangular prism with dimensions as shown. 0. 5m m m m x (a) Calculate the value of x. (b) Calculate the volume of concrete required to build the ramp. 6. A circle, centre O, has radius 6 centimetres. Part of this circle is shown. Angle AOB = 0 º. 6 cm A O 0 º Calculate the length of arc AB. B [500/06] Page four

58 7. Shampoo is available in travel size and salon size bottles. The bottles are mathematically similar. cm h cm travel salon The travel size contains 00 millilitres and is centimetres in height. The salon size contains 600 millilitres. Calculate the height of the salon size bottle. [Turn over [500/06] Page five

59 8. As part of their training, footballers run around a triangular circuit DEF. E 6. m 6. m D 8 F EDF = º DFE = 8 º DE = 6. metres EF = 6. metres How many complete 000 metres? circuits must they run to cover at least 9. The ratio of sugar to fruit in a particular jam is 5 :. It is decided to: decrease the sugar content by 0% increase the fruit content by 0%. Calculate the new ratio of sugar to fruit. Give your answer in its simplest form. [500/06] Page six

60 0. In triangle PQR: Q PQ = 5 centimetres PR = 6 centimetres 5cm area of triangle PQR = square centimetres angle QPR is obtuse. P 6cm R Calculate the size of angle QPR.. The height, h, of a square-based pyramid varies directly as its volume, V, and inversely as the square of the length of the base, b. h (a) Write down an equation connecting h, V and b. b A square-based pyramid of height centimetres has a volume of 56 cubic centimetres and length of base 8 centimetres. (b) Calculate the height of a square-based pyramid of volume 600 cubic centimetres and length of base 0 centimetres. [Turn over for Questions and on Page eight [500/06] Page seven

61 . A right-angled triangle has dimensions, in centimetres, as shown. x x + 8 x + 7 Calculate the value of x. 5. The depth of water, D metres, in a harbour is given by the formula D = sin 0 h º where h is the number of hours after midnight. (a) Calculate the depth of water at 5 am. (b) Calculate the maximum difference in depth of the water in the harbour. Do not use a trial and improvement method. [END OF QUESTION PAPER] [500/06] Page eight

62 C 500/05 NATIONAL QUALIFICATIONS 0 WEDNESDAY, MAY MATHEMATICS.0 PM.5 PM STANDARD GRADE Credit Level Paper (Non-calculator) You may NOT use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. If you make use of this, you should write your name on it clearly and put it inside your answer booklet. LI 500/05 6/0 *500/05*

63 FORMULAE LIST ± The roots of ax + bx + c = 0 are x = ( b b ac 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 = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/05] Page two

64 . Evaluate Factorise fully m 8.. Given that f(x) = 5 x, evaluate f( ).. Solve the equation x + = x 5. [Turn over [500/05] Page three

65 5. Jamie is going to bake cakes for a party. He needs 5 of a block of butter for cake. He has 7 blocks of butter. How many cakes can Jamie bake? 6. A driving examiner looks at her diary for the next 0 days. She writes down the number of driving tests booked for each day as shown below. Number of tests booked Frequency 9 0 (a) Find the median for this data. (b) Find the probability that more than tests are booked for one day. [500/05] Page four

66 7. (a) Brian, Molly and their four children visit Waterworld. The total cost of their tickets is 56. Let a pounds be the cost of an adult s ticket and c pounds the cost of a child s ticket. Write down an equation in terms of a and c to illustrate this information. (b) Sarah and her three children visit Waterworld. The total cost of their tickets is 6. Write down another equation in terms of a and c to illustrate this information. (c) (i) Calculate the cost of a child s ticket. (ii) Calculate the cost of an adult s ticket. [Turn over [500/05] Page five

67 8. A square, OSQR, is shown below. Q is the point (8, 8). y T S P Q (8, 8) O R x The straight line TR cuts the y-axis at T (0, ) and the x-axis at R. (a) Find the equation of the line TR. The line TR also cuts SQ at P. (b) Find the coordinates of P. 9. (a) Simplify a a. (b) Solve for x, x + 8 =. [500/05] Page six

68 0. In triangle ABC AC = centimetres BC = 0 centimetres angle BAC = 50 A 50 cm C B 0 cm Given that sin 0 =, show that sin B =. 5. F varies directly as s and inversely as the square of d. (a) Write down a relationship connecting F, s and d. (b) What is the effect on F when s is halved and d is doubled?. The sums, S, S and S of the first, and natural numbers are given by: S = + = ( ) = S = + + = ( ) = 6 S = = ( 5) = 0 (a) Find S 0, the sum of the first 0 natural numbers. (b) Write down the formula for the sum, S n, of the first n natural numbers. [END OF QUESTION PAPER] [500/05] Page seven

69 C 500/06 NATIONAL QUALIFICATIONS 0 WEDNESDAY, MAY.5 PM.05 PM MATHEMATICS STANDARD GRADE Credit Level Paper You may use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. If you make use of this, you should write your name on it clearly and put it inside your answer booklet. LI 500/06 6/0 *500/06*

70 FORMULAE LIST Sine rule: ± The roots of ax + bx + c = 0 are x = a = b = c sin A sin B sin C ( b b ac a ( Cosine rule: a = b + c bc cos A or cos A = b + c a bc Area of a triangle: Area = ab sin C Standard deviation: ( x x) x ( x) / n s = =, where n is the sample size. n n [500/06] Page two

71 . Olga normally runs a total distance of 8 miles per week. She decides to increase her distance by 0% a week for the next four weeks. How many miles will she run in the fourth week?. Expand and simplify (x + )(x 5x + ).. Solve the equation x + x 7 = 0. Give your answers correct to significant figures.. A car is valued at 780. This is 6% less than last year s value. What was the value of the car last year? [Turn over [500/06] Page three

72 5. A spiral staircase is being designed. B C. m A Each step is made from a sector of a circle as shown. The radius is metres. Angle BAC is. For the staircase to pass safety regulations, the arc BC must be at least 0 9 metres. Will the staircase pass safety regulations? 6. Two rectangular solar panels, A and B, are mathematically similar. Panel A has a diagonal of 90 centimetres and an area of 00 square centimetres. A 90 cm B 5 cm A salesman claims that panel B, with a diagonal of 5 centimetres, will be double the area of panel A. Is this claim justified? Show all your working. [500/06] Page four

73 7. ABCDE is a regular pentagon with each side centimetre. Angle CDF is 7. EDF is a straight line. B cm cm A C cm cm 7 E cm D F (a) Write down the size of angle ABC. (b) Calculate the length of AC. 8. A pipe has water in it as shown. A B A O r B 5cm 8 cm The depth of the water is 5 centimetres. The width of the water surface, AB, is 8 centimetres. Calculate r, the radius of the pipe. [Turn over [500/06] Page five

74 9. A flower planter is in the shape of a prism. The cross-section is a trapezium with dimensions as shown. 68 cm 0 cm cm 0 cm cm (a) Calculate the area of the cross-section of the planter. (b) The volume of the planter is 56 litres. l centimetres Calculate the length, l centimetres, of the planter. 0. Tom and Samia are paid the same hourly rate. Harry is paid more per hour than Tom. Tom worked 5 hours, Samia worked 8 hours and Harry worked hours. They were paid a total of 9. How much was Tom paid? [500/06] Page six

75 . Paper is wrapped round a cardboard cylinder exactly times. The cylinder is 70 centimetres long. 70 cm The area of the paper is 000 square centimetres. Calculate the diameter of the cylinder.. Part of the graph of y = sin x is shown below. y P O Q R 60 x y = sin x The graph cuts the x-axis at Q and R. P is the maximum turning point. (a) Write down the coordinates of P. (b) Calculate the x-coordinates of Q and R. [Turn over for Question on Page eight [500/06] Page seven

76 . The diagram shows the path of a flare after it is fired. The height, h metres above sea level, of the flare is given by h = 8 + 8t t where t is the number of seconds after firing. h h = 8 + 8t t O t Calculate, algebraically, the time taken for the flare to enter the sea. [END OF QUESTION PAPER] [500/06] Page eight

77 C 500//0 NATIONAL QUALIFICATIONS 0 WEDNESDAY, MAY.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper (Non-calculator) You may NOT use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. If you make use of this, you should write your name on it clearly and put it inside your answer booklet. LI 500//0 6/60 *500//0*

78 FORMULAE LIST The roots of ax + bx + c = 0 are ( ) b b ac x = ± 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 = ab sin C Standard deviation: s = ( x x) x x n = ( ) /, n n where n is the sample size. [500//0] Page two

79 . Evaluate Expand and simplify (x )(x + x + 5).. Change the subject of the formula to m. L = m k R. In the diagram, 0 cm PQ is the diameter of the circle PQ = centimetres PR = 0 centimetres. P cm Q Calculate the length of QR. Give your answer as a surd in its simplest form. [Turn over [500//0] Page three

80 5. Mike is practising his penalty kicks. Last week, Mike scored 8 out of 0. This week, he scored 6 out of 5. Has his scoring rate improved? Give a reason for your answer. 6. The diagram shows part of the graph of y = 5 + x x. y A O B x y = 5 + x x A is the point (, 0). B is the point (5, 0). (a) State the equation of the axis of symmetry of the graph. (b) Hence, find the maximum value of y = 5+ x x. [500//0] Page four

81 7. Given x x = 0, show that x = ± 8. The graph below shows two straight lines. y = x x + y = y y = x P O x x + y = The lines intersect at the point P. Find, algebraically, the coordinates of P. [Turn over for Questions 9, 0 and on Page six [500//0] Page five

82 9. Each day, Marissa drives 0 kilometres to work. (a) On Monday, she drives at a speed of x kilometres per hour. Find the time taken, in terms of x, for her journey. (b) On Tuesday, she drives 5 kilometres per hour faster. Find the time taken, in terms of x, for this journey. (c) Hence find an expression, in terms of x, for the difference in times of the two journeys. Give this expression in its simplest form. 0. (a) Evaluate ( ). (b) Hence find n, when ( ) n =. 6. The sum of consecutive even numbers can be calculated using the following number pattern: (a) Calculate = = = 5 = = 5 6 = 0 (b) Write down an expression for n. (c) Hence or otherwise calculate [END OF QUESTION PAPER] [500//0] Page six

83 C 500//0 NATIONAL QUALIFICATIONS 0 WEDNESDAY, MAY.5 PM.05 PM MATHEMATICS STANDARD GRADE Credit Level Paper You may use a calculator. Answer as many questions as you can. Full credit will be given only where the solution contains appropriate working. Square-ruled paper is provided. If you make use of this, you should write your name on it clearly and put it inside your answer booklet. LI 500//0 6/60 *500//0*

84 FORMULAE LIST The roots of ax + bx + c = 0 are ( ) b b ac x = ± 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 = ab sin C Standard deviation: s = ( x x) x x n = ( ) /, n n where n is the sample size. [500//0] Page two

85 . There are 69 million vehicles in Scotland. It is estimated that this number will increase at a rate of % each year. If this estimate is correct, how many vehicles will there be in years time? Give your answer correct to significant figures.. Before training, athletes were tested on how many sit-ups they could do in one minute. The following information was obtained: lower quartile (Q ) median (Q ) 9 upper quartile (Q ) 5 (a) Calculate the semi-interquartile range. After training, the athletes were tested again. Both sets of data are displayed as boxplots. Performance before training Performance after training number of sit-ups (b) Make two valid statements to compare the performances before and after training. [Turn over [500//0] Page three

86 . A container for oil is in the shape of a prism. The width of the container is 9 centimetres. The uniform cross section of the container consists of a rectangle and a triangle with dimensions as shown. 8 cm 9 cm 0 cm Calculate the volume of the container, correct to the nearest litre.. A sector of a circle, centre O, is shown below. B m A 65 O The radius of the circle is metres. Angle AOB is 65. Find the length of the arc AB. [500//0] Page four

87 5. The depth, d, of water in a tank, varies directly as the volume, v, of water in the tank and inversely as the square of the radius, r, of the tank. When the volume of water is cm, the depth of water is 50 cm and the radius of the tank is 0 cm. Calculate the depth of the water, when the volume of water is cm and the radius of the tank is 5 cm. 6. The price for Paul s summer holiday is The price includes a % booking fee. What is the price of his holiday without the booking fee? 7. A heavy metal beam, AB, rests against a vertical wall as shown. The length of the beam is 8 metres and it makes an angle of 59 with the ground. B 8 m A 59 A cable, CB, is fixed to the ground at C and is attached to the top of the beam at B. The cable makes an angle of with the ground. B 8 m C A 59 Calculate the length of cable CB. [500//0] Page five [Turn over

88 8. A necklace is made of beads which are mathematically similar. 0 8 cm cm The height of the smaller bead is 0 8 centimetres and its area is 0 6 square centimetres. The height of the larger bead is centimetres. Find the area of the larger bead. 9. Paving stones are in the shape of a rhombus. 0 cm 0 The side of each rhombus is 0 centimetres long. The obtuse angle is 0. Find the area of one paving stone. [500//0] Page six

89 0. A taxi fare consists of a call-out charge of 80 plus a fixed cost per kilometre. A journey of kilometres costs The straight line graph shows the fare, f pounds, for a journey of d kilometres. f 6 60 pounds 80 0 kilometres d (a) Find the equation of the straight line. (b) Calculate the fare for a journey of 7 kilometres.. Quadrilateral ABCD with angle ABC = 90 is shown below. A m B 5 m 6 m AB = metres BC = 6 metres CD = 7 metres AD = 5 metres D 7 m C (a) Calculate the length of AC. (b) Calculate the size of angle ADC. [Turn over for Questions and on Page eight [500//0] Page seven

90 . f(x) = sin x, 0 x < 60 (a) Find f(70). (b) f(t) = 0 6. Find the two possible values of t.. Triangles PQR and STU are mathematically similar. The scale factor is and PR corresponds to SU. T Q 8 cm 6 cm P x cm R S (x + 5) cm U (a) Show that x 6x + 5 = 0. (b) Given QR is the shortest side of triangle PQR, find the value of x. [END OF QUESTION PAPER] [500//0] Page eight