C100/SQP321. Course Assessment Specification 2. Specimen Question Paper 1 5. Specimen Question Paper Specimen Marking Instructions Paper 1 23

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1 C00/SQP Maths Higher NTIONL QULIFICTIONS Contents Page Course ssessment Specification Specimen Question Paper 5 Specimen Question Paper 7 Specimen Marking Instructions Paper Specimen Marking Instructions Paper 9 Summary nalysis [C00/SQP]

2 Course ssessment Specification Mathematics Higher The purpose of this document is to provide: details of the structure of the Question Papers in this Course guidance to centres on how to use information gathered from the Question Papers in this Course to estimate candidate performance. Part This part of the Course ssessment Specification details the structure of the Question Papers in this Course. The Course assessment consists of two Question Papers. Paper Paper Time llocation Mark llocation hour 0 mins 70 hour 0 mins 60 In this paper no access to a calculator is allowed. ll questions within this paper are classified as non-calculator. This paper has two sections, and B. Section total marks 40 This section of the paper consists of Objective Test Questions. There are 0 questions, each worth marks. t least 75% of these questions will provide opportunities at grade C. Section B total marks 0 Questions in this section of the paper will be short and extended response. There will be approximately three to five questions. [C00/SQP] Page two

3 Paper total marks 60 In this paper, access to a calculator is allowed. Questions within this paper are classified as calculator required (CR) or calculator neutral (CN). In calculator neutral questions, there is no advantage to using a calculator. Questions in this the paper will be short and extended response. There will be approximately six to eight questions. The Question Papers will assess the candidate s ability to retain and integrate mathematical knowledge across the Units of the Course. The Question Papers are designed to fit the following criteria: there should be as wide a syllabus coverage as possible approximately 65% of the total marks will provide opportunities at Grade C at least one quarter of the marks should be allocated to each of the three Units at least one quarter of the marks should be defined under each of the headings selecting a strategy, processing data, interpreting and communicating information. [C00/SQP] Page three

4 Part This part of the Course ssessment Specification provides guidance on how to use assessment information gathered from the Question Papers to estimate candidate performance. The Course award is based on two Question Papers with a total mark of 0. In National Qualifications, cut-off scores should be set at approximately 70% for Grade and 50% for Grade C with Grade B falling midway. For a total mark range of 0-0, the following gives an indication of the cut-off scores: Grade Band Mark Range B B C C D N N These cut-off scores may be lowered if the Question Papers turn out to be more demanding, or raised if the Question Papers are less demanding. Worked example In a centre s own prelim, a candidate scores 79/0. The prelim covers only two Units of the Course and the centre s view is that it is slightly less demanding than the SQ s examination. Using the mark range, a realistic estimate may be band 5 rather than band 4. [C00/SQP] 4 Page four

5 [C00/SQP] Mathematics Higher Paper Specimen Question Paper (for examinations from Diet 008 onwards) NTIONL QULIFICTIONS Read carefully Section Questions 0 (40 marks) Instructions for completion of Section are given on page two. For this section of the examination you must use an HB pencil. Section B (0 marks) Calculators may NOT be used in this paper. Full credit will be given only where the solution contains appropriate working. nswers obtained by readings from scale drawings will not receive any credit. [C00/SQP] 5

6 Read carefully Check that the answer sheet provided is for Mathematics Higher (Section ). For this section of the examination you must use an HB pencil and, where necessary, an eraser. Check that the answer sheet you have been given has your name, date of birth, SCN (Scottish Candidate Number) and Centre Name printed on it. Do not change any of these details. 4 If any of this information is wrong, tell the Invigilator immediately. 5 If this information is correct, print your name and seat number in the boxes provided. 6 The answer to each question is either, B, C or D. Decide what your answer is, then, using your pencil, put a horizontal line in the space provided (see sample question below). 7 There is only one correct answer to each question. 8 ny rough working should be done on the question paper or the rough working sheet, not on your answer sheet. 9 t the end of the exam, put the answer sheet for Section inside the front cover of this answer book. Sample Question curve has equation y = x 4x. What is the gradient at the point where x =? 8 B 4 C 0 D 4 The correct answer is 8. The answer has been clearly marked in pencil with a horizontal line (see below). B C D Changing an answer If you decide to change your answer, carefully erase your first answer and using your pencil, fill in the answer you want. The answer below has been changed to D. B C D [C00/SQP] 6 Page two

7 FORMULE LIST Circle: The equation x + y + gx + fy + c = 0 represents a circle centre ( g, f) and radius The equation (x a) + (y b) = r represents a circle centre (a, b) and radius r. g + f c. Scalar Product: a.b = a b cos θ, where θ is the angle between a and b or a.b = a b + a b + a b where a = a a a b and b =. b b Trigonometric formulae: sin ( ± B) = sin cos B ± cos sin B ± cos ( ± B) = cos cos B sin sin B sin = sin cos cos = cos sin = cos = sin Table of standard derivatives: fx ( ) f ( x) sin ax cos ax acosax asinax Table of standard integrals: sin ax a cosax + C fx ( ) fxdx ( ) cosax a sin ax + C [C00/SQP] 7 Page three

8 SECTION LL questions should be attempted.. = dy If y x x, what is? x dx x x B + x C x + x D x x. Functions f and g are given by f(x) = x and g(x) = x. Find an expression for g(f(x)). g(f(x)) = 4x x + 9 B g(f(x)) = x C g(f(x)) = 4x 9 D g(f(x)) = x (x ). Find dx. x 4 x + c B x c C x + c 4 D x + c 4 [C00/SQP] 8 Page four

9 4. and B have coordinates (,, ) and (, 4, 0) respectively. What is the distance between and B? 6 B 4 C 54 D 6 5. sequence is defined by the recurrence relation u n+ = u n 4, u 0 =. What is the value of u? 5 B 9 C 7 D [C00/SQP] 9 Page five

10 6. The diagram shows a sketch of y = f(x). Which of the diagrams below shows a sketch of y = f(x)? O y x y O x B y O x C y O x D y O x 7. Which of the following describes the stationary point on the curve with equation y = (x 4) 5? minimum at (4, 5) B maximum at (4, 5) C minimum at (4, 5) D maximum at (4, 5) [C00/SQP] 0 Page six

11 8. The diagram shows a right-angled triangle with sides of, and. What is the value of sin x? B C x D 9. a and b are angles as shown in the diagram. What is the value of sin(a b)? 5 a 4 7 b B C D 5 7 [C00/SQP] Page seven

12 0. circle has equation x + y + 8x 6y = 0. What is the radius of this circle? B 9 C 7 D. The points P(,, 7), Q(5,, ) and R(s,, 5) are collinear as shown in the diagram. What is the value of s? 8 B 9 C D P Q R. If x x + is expressed in the form (x b) + c, what is the value of c? 6 B 7 C D dy. The curve y = f(x) is such that = x + 9x + and the curve passes through the origin. dx What is the equation of the curve? y = x x x B y = 9x + 8x + C y = x + 9x + x D y = 6x + 9 [C00/SQP] Page eight

13 4. For what value of k does the equation x x + k = 0 have equal roots? B C D The point P(, ) lies on the circle with equation x + y 6x 8y + 5 = 0. What is the gradient of the tangent at P? B C 7 6 D π 6. What is the value of 6 4cos xdx? 0 4 B C D [C00/SQP] Page nine

14 7. The graph shown in the diagram has equation of the form y = sin(px) + q. What are the values of p and q? y p q O π π x B C D 8. The vectors a, b and c are represented by the sides of a right-angled triangle as shown in the diagram. a = and c =5. Here are two statements about these vectors: I a.c = 9 II a.b = Which of the following is true? a c b B C D neither statement is correct only statement I is correct only statement II is correct both statements are correct 9. If log t = + log 5, what is the value of t? 0 B 4 C 5 D 45 [C00/SQP] 4 Page ten

15 0. If k = e 4, find an expression for k. k = 4 e B k = 4 log e C k = 4 / log e D k = / log e [END OF SECTION ] [C00/SQP] 5 Page eleven

16 SECTION B LL questions should be attempted. Marks. firm cleans the factory floor on a daily basis with disinfectant. It has a choice of two products, either or B. Product removes 70% of all germs but during the next 4 hours, 00 new germs per sq unit are estimated to appear. Product B removes 80% of all germs but during the next 4 hours, 50 new germs per sq unit are estimated to appear. For product, let u n represent the number of germs per sq unit on the floor immediately before disinfecting. For product B, let v n represent the number of germs per sq unit on the floor immediately before disinfecting. (a) Write down a recurrence relation for each product to show the number of germs per sq unit present prior to disinfecting. (b) Determine which product is more effective in the long term. 4. (a) Find the stationary points on the curve with equation y = x 9x + 4x 0 and justify their nature. (b) (i) Show that (x ) (x 5) = x 9x + 4x 0. (ii) Hence sketch the graph of y = x 9x + 4x The diagram shows a sketch of functions f and g where f(x) = x + 5x 6x + and g(x) = x + x +. y = f(x) The two graphs intersect at the points, B and C. Determine the x-coordinate of each of these three points. O B 8 C y = g(x) 4. Find the solution(s) of the equation sin p sin p + = cos p for π < p < π. 5 [END OF SECTION B] [C00/SQP] 6 [END OF QUESTION PPER] Page twelve

17 [C00/SQP] Mathematics Higher Paper Specimen Question Paper (for examinations from Diet 008 onwards) NTIONL QULIFICTIONS Read Carefully Calculators may be used in this paper. Full credit will be given only where the solution contains appropriate working. nswers obtained by readings from scale drawings will not receive any credit. [C00/SQP] 7

18 FORMULE LIST Circle: The equation x + y + gx + fy + c = 0 represents a circle centre ( g, f) and radius The equation (x a) + (y b) = r represents a circle centre (a, b) and radius r. g + f c. Scalar Product: a.b = a b cos θ, where θ is the angle between a and b or a.b = a b + a b + a b where a = a a a b and b =. b b Trigonometric formulae: sin ( ± B) = sin cos B ± cos sin B ± cos ( ± B) = cos cos B sin sin B sin = sin cos cos = cos sin = cos = sin Table of standard derivatives: fx ( ) f ( x) sin ax cos ax acosax asinax Table of standard integrals: sin ax a cosax + C fx ( ) fxdx ( ) cosax a sin ax + C [C00/SQP] 8 Page two

19 LL questions should be attempted. Marks. Triangle BC has coordinates (, ), B(0, ) and C(4, 7). C(4, 7) D (a) Find the equation of the median CN. (b) Find the equation of the altitude D. (c) The median from (a) and the altitude from (b) intersect at P. Find the coordinates of P. (, ) N B(0, ) (d) The point Q lies on B and has coordinates (8, ). Show that PQ is parallel to BC.. The diagram shows a wire framework in the shape of a cuboid with the edges parallel to the axes. Relative to these axes,, B, C and H have coordinates (,, 4), (,, 4), (, 7, 4) and (, 7, 9) respectively. H(, 7, 9) E D F G C(, 7, 4) (a) State the lengths of B, D and E. (,, 4) B(,, 4) (b) Write down the components of HB and HC and hence or otherwise calculate the size of angle BHC. 7. (a) Express 5sin x cos x in the form k sin(x a) where k > 0 and 0 < a < 60. (b) Hence solve the equation 5sin x cos x = 6. 5 in the interval 0 < x < [C00/SQP] 9 Page three

20 Marks 4. The diagram shows a parabola with equation y = x x +. tangent to the parabola has been drawn at P(, ). y P(, ) (a) Find the equation of this tangent. 4 circle has equation x + y + 8y + = 0. (b) Show that the line from (a) is also a tangent to this circle and state the coordinates of the point of contact Q. O x 6 (c) Determine the ratio in which the y-axis cuts the line QP. 5. The diagram shows a curve with equation y = x and a straight line with equation y = 6x + 6 intersecting the curve at P and Q. (a) Calculate the exact value of the area enclosed by the curve and the straight line. y y = x y = 6x + 6 P O Q x 7 The second diagram shows a third point, R, lying on the curve between P and Q. (b) The area,, of triangle PQR, is given by (x) = 5x + 0x Determine the maximum area of this triangle, and express your answer as a fraction of the area enclosed by the curve and the straight line. y y = x Q y = 6x + 6 P R(x, y) O x 4 6. Radium decays exponentially and its half-life is 600 years. If 0 represents the amount of radium in a sample to start with and (t) represents the amount remaining after t years, then (t) = 0 e kt. (a) Determine the value of k, correct to 4 significant figures. (b) Hence find what percentage, to the nearest whole number, of the original amount of radium will be remaining after 00 years. [C00/SQP] 0 Page four

21 Marks 7. Triangle BC is right-angled at and BD is the bisector of angle BC. C B = 6 units and CB = 0 units. Determine the exact value of BD, expressing your answer in its simplest form. D 0 x x 6 B 5 [END OF QUESTION PPER] [C00/SQP] Page five

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23 [C00/SQP] Mathematics Higher Paper Specimen Marking Instructions (for examinations from Diet 008 onwards) NTIONL QULIFICTIONS Note: In the Specimen Marking Instructions the Marking Scheme indicates which marks awarded are strategy marks (ss), which marks awarded are processing marks (pd) and which marks awarded are interpretation and communication marks (ic). [C00/SQP]

24 B x x, x, + x SECTION marks g(f(x)) = g(x ) = (x ) = 4x x + 9 marks C x x x + c, +, marks 4 D d = ( ) + ( 4) + ( 0) = 6; d = 6 marks 5 u = ( ) 4 = 7, u = ( 7) 4 = 5 marks 6 marks 7 C minimum at (4, 5) marks 8 sin x =, sin x = sin x cos x = = 4 9 marks 9 B sin acos b cos asin b =. 4. = marks 0 C r = 4 + ( ) ( ) = 7, r = 7 marks C 4 s 5 PQ = 0, QR = 0 ; 6 s 5 = 4, s = marks B (x ) 8 +, c = 7 marks y = x x + c, c = 0 x marks 4 D ( ) 4k = 0; 4k = 9; k = 9 4 marks 5 C = (, 4),m CO =, m tgt = marks 6 C sinx, sin π sin0, 0 = marks [C00/SQP] 4 Page two

25 7 period = π /, p = ; q =+ marks 8 9 B D a.c = 5 5 = 9 so () T, a.b = 0 so () F () log t log 5 =, log t =, t =, t = marks marks 0 C log e ( k ) = log e (e 4 ), klog e () = 4log e (e), k = 4 / log e () marks [END OF SECTION ] [C00/SQP] 5 Page three

26 Qu Marking Scheme Give mark for each SECTION B Illustrations of evidence for awarding a mark at each (a) marks (b) 4 marks ic: interpret a and b in au n + b ic: interpret a and b in av n + b ic: state limit conditions 4 ss: know how to find limit 5 pd: process 6 pd: process and decide u n + = 0. u n + 00 v n + = 0. v n + 50 limits are valid since < 0. < and < 0. < 4 : L = 0. L : L = 48 or 49 6 : L = 47 or 48 and product more effective (a) 7 marks (b) 4 marks ss: know to differentiate pd: differentiate ss: set derivative to zero 4 pd: solve for x 5 pd: evaluate corresponding y s 6 ss: know to justify eg use nature table 7 ic: interpret the stationary points 8 pd: expand and complete 9 ic: state x-axis intersections 0 ic: state y-axis intersections ic: complete sketch dy dx =..( term correct) dy = x 8x + 4 dx dy = 0 dx 4 x = or x = 4 5 y = 0 or y = 4 x dy dx 7 max. at (, 0), min at (4, 4) 8 y = (x 4x + 4)(x 5) and complete 9 x-axis: at (,0, (5, 0) 0 y-axis: at (0, 0) sketch [C00/SQP] 6 Page four

27 Qu Marking Scheme Give mark for each Illustrations of evidence for awarding a mark at each 8 marks ss: equate functions pd: rearrange in standard form ss: try a particular value of x 4 ic: interpret a zero value of equation 5 ss: start to factorise 6 pd: obtain quadratic factor 7 pd: factorise completely 8 ic: interpret roots x + 5x 6x + = x + x + x + 6x 7x + 0 = 0 say f(x) = x + 6x 7x + 0 try evaluating eg f() =... 4 f() = = 0 so (x ) is a factor 5 (x )(x...) 6 (x )(x + 7x 0) 7 (x )(x + 0)(x ) = 0 8 x = 0, x B = x C = OR rem. = 0 so x is factor 6 x + 7x marks ss: know to substitute for cos x pd: express in standard form pd: factorise 4 pd: solve for sin p 5 ic: solve for p sin p sin p + = sin p sin p sin p = 0 sinp(sin p ) = 0 4 sinp =, or sin p = 0 5 p = 5 π, no solution in given interval 6 [END OF SECTION B] [END OF SPECIMEN MRKING INSTRUCTIONS] [C00/SQP] 7 Page five

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29 [C00/SQP] Mathematics Higher Paper Specimen Marking Instructions (for examinations from Diet 008 onwards) NTIONL QULIFICTIONS Note: In the Specimen Marking Instructions the Marking Scheme indicates which marks awarded are strategy marks (ss), which marks awarded are processing marks (pd) and which marks awarded are interpretation and communication marks (ic). [C00/SQP] 9

30 Qu Marking Scheme Give mark for each Illustrations of evidence for awarding a mark at each (a) marks (b) marks (c) and (d) 5 marks ic: interpret median ss: find gradient of median ic: state equation of median 4 ss: know to find gradient of base 5 ss: find perpendicular gradient 6 ic: state equation of altitude 7 ss: juxtaposition of two equations 8 pd: solve for one variable 9 pd: solve for other variable 0 pd: find gradient of new line ic: know and use condition for parallel lines mid of B = (6, ) m 7 median = = 4 6 median : y 7 = (x 4) 4 m 7 BC = = m alt =+ 6 altitude : y = (x ) 7 y = x + 9 and y = x 8 x p =5 9 y p =4 0 m 4 PQ = = 5 8 m CB = and so PQ is parallel to CB (a) mark (b) 7 marks ic: interpret diagram ic: interpret diagram for components ic: interpret diagram for components 4 pd: evaluate scalar product 5 pd: evaluate length 6 pd: evaluate length 7 ss: know formula for angle 8 pd: process the angle B =, D = 4, E = 5 HB = 4 5 HC = HB.HC = = 6 5 magnitude of HB = 6 magnitude of HC = cos HBC = (6) / HBC ˆ = 8. or radians [C00/SQP] 0 Page two

31 Qu Marking Scheme Give mark for each Illustrations of evidence for awarding a mark at each (a) 4 marks (b) marks ss: know to expand and expand ic: compare coefficients pd: process k 4 pd: process a 5 ic: use result 6 pd: start to solve 7 pd: complete solving k sin x cos a k cos x sin a k cos a = 5, k sin a = k = 4 a = sin(x 67. 4) = x = 0 and x = x = 50 and x = (a) 4 marks (b) 6 marks (c) marks ss: know to differentiate pd: differentiate pd: evaluate gradient of tangent 4 ic: state equation of tangent 5 ic: express in standard form 6 ss: know to substitute 7 pd: process into standard form 8 ss: know how to prove tangency 9 ic: complete proof 0 pd: evaluate point of contact ss: know to find coord of intersection ss: choose approach (eg vectors) ic: interpret ratio (from eg vectors) dy dx =... dy = 4x dx m x= = 4 y = (x ) 5 y = x + 6 x + (x + ) + 8(x + ) + = 0 7 5x + 0x + 0 = 0 8 5(x + ) = 0 9 equal roots so line is tangent 0 Q = (, ) line cuts y axis at (0, )(= T, say) QT =, TP = 4 QT = TP and QT : TP = : [C00/SQP] Page three

32 Qu Marking Scheme Give mark for each Illustrations of evidence for awarding a mark at each 5 (a) 7 marks (b) 4 marks ss: know to equate pd: solve for x ss: know to integrate (upper lower) 4 ic: interpret limits 5 pd: integrate 6 ic: substitute limits 7 pd: process results 8 ss: set derivative to zero 9 pd: differentiate and solve 0 ic: know to justify (eg know of maximum parabola) ic: interpret maximum area and connection x = 6x + 6 (x 8)(x + ) = 0 and x = 8, x = = (6x+ 6 x ) dx 4 8 =... dx 5 x + 6x x d 0 dx = 9 x = 0 nature table or 5 maximum parabola triangle area max = 5 = of enclosed 4 area 6 5 marks ic: interpret half-life ss: take logarithms pd: complete evaluation of k 4 ic: substitute new data 5 pd: process 0 = 0 e 600k 600k = ln k = (00) = 0 e 00k 5 (00) = and 5% 7 5 marks ic: state value of trig function ss: substitute expression for cos x pd: process 4 ss: know to review info in diagram 5 pd: equate expressions and solve 6 cos(x) = 0 cos (x) = cos(x) = 4 cos(x) = 5 DB = DB = [END OF SPECIMEN MRKING INSTRUCTIONS] [C00/SQP] Page four

33 Specimen Question Paper Section Specimen Question Paper : Summary nalysis qu ans mk code calc source ss pd ic C B U U U B C CN CN 808 C C CN D G CN CN CN C 6 NC T8 CN B T9 CN C G9 CN 80 C G+,G CN 807 B 5 CN 800 8, CN D 8 CN G CN C C,T NC T NC B G6,G7 CN D 8 CN C 4 CN 804 totals Specimen Question Paper Section B qu part mk code calc source ss pd ic C B U U U a 0 CN 80 b 4 4,, 4 4 a 7 C8,9 NC b 4 C ,, NC T7,T NC totals Specimen Question Paper qu part mk code calc source ss pd ic C B U U U a G7,G6,G CN 806 b G7,G5,G CN c 5 G+, G CN 5 5 a CN 807 b 7 G7,G6,G8 CR a 4 T,T CR b T6 CR 4 a 4 C5,C4 CN b 6 G,G,7 CN 4 6 c G4 CN 5 a 7 C7,C5, CN b 4 C CN 6 5 0,4 CR T,T7 CN totals TOTL target ± : [C00/SQP]