National Quali cations

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1 H 2017 X747/76/11 FRIDAY, 5 MAY 9:00 AM 10:10 AM National Quali cations Mathematics Paper 1 (Non-Calculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. *X * A/PB

2 FORMULAE LIST Circle: The equation x 2 + y 2 + 2gx + 2fy + c = 0 represents a circle centre ( g, f ) and radius 2 2 g + f c The equation ( x a) + ( y b) = r represents a circle centre (a, b) and radius r. Scalar Product: a.b = a b cos θ, where θ is the angle between a and b or a1 b1 a.b = ab 1 1+ ab 2 2+ ab 3 3where a = a2 and b = b2. a b 3 3 Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A 2 2 cos 2 A = cos A sin A = 2cos 2 A 1 = 1 2 sin 2 A Table of standard derivatives: f( x ) f ( x) sin ax cos ax a cos ax asin ax Table of standard integrals: f( x ) f ( x) dx sin ax cos ax 1 cos ax + c a 1 sin a ax + c Page 02

3 ATTEMPT ALL QUESTIONS Total marks 60 MARKS 1. Functions f and g are defined on suitable domains by f ( x) = 5x and g( x) 2cos ( ) (a) Evaluate f g ( 0). (b) Find an expression for g( f ( x )). = x The point P ( 2, 1) lies on the circle 2 2 x y x y Find the equation of the tangent to the circle at P = Given y ( 4x 1) 12 =, find dy dx Find the value of k for which the equation x 2 x ( k ) = 0 has equal roots Vectors u and v are and respectively. (a) Evaluate u.v. 1 (b) u π 3 w π Vector w makes an angle of with u and w = 3. 3 Calculate u.w. 3 Page 03 [Turn over

4 MARKS h x 6. A function, h, is defined by ( ) 3 1 Determine an expression for h ( x) = x + 7, where x A ( 3, 5), B (7, 9) and C (2, 11) are the vertices of a triangle. Find the equation of the median through C Calculate the rate of change of d() t =, t 0, when t = 5. 2t 3 9. A sequence is generated by the recurrence relation u 1 = mu + 6 where m is a constant. n+ n (a) Given u 1 = 28 and u 2 = 13, find the value of m. 2 (b) (i) Explain why this sequence approaches a limit as n. (ii) Calculate this limit. 1 2 Page 04

5 10. Two curves with equations y = x 3 4x 2 + 3x + 1 and y = x 2 3x + 1 intersect as shown in the diagram. MARKS y y = x 3 4x 2 + 3x + 1 y = x 2 3x + 1 O 2 x (a) Calculate the shaded area. 5 The line passing through the points of intersection of the curves has equation y = 1 x. y y = x 3 4x 2 + 3x + 1 y = x 2 3x + 1 O 2 x y = 1 x (b) Determine the fraction of the shaded area which lies below the line y = 1 x. 4 [Turn over Page 05

6 11. A and B are the points ( 7, 2) and (5, a). AB is parallel to the line with equation 3y 2x = 4. Determine the value of a. MARKS Given that log a 36 log a 4 =, find the value of a Find 1 ( ) 1 5 4x 2 dx, 5 x < (a) Express 3 sin x cos x where k > 0 and 0 < a < 360. in the form k ( x a) sin, (b) Hence, or otherwise, sketch the graph with equation y = 3 sin x cos x, 0 x Use the diagram provided in the answer booklet. Page 06

7 15. A quadratic function, f, is defined on, the set of real numbers. MARKS Diagram 1 shows part of the graph with equation y = f ( x). The turning point is (2, 3). Diagram 2 shows part of the graph with equation y h( x) The turning point is (7, 6). =. y y (7, 6) (2, 3) ( ) y = h x y = f ( x) O 1 3 x O 6 8 x Diagram 1 Diagram 2 (a) Given that h( x) = f ( x+ a) + b. Write down the values of a and b. 2 3 (b) It is known that ( ) f x dx = Determine the value of h ( x ) dx. 6 (c) Given f () 1 = 6, state the value of h ( 8). 1 1 [END OF QUESTION PAPER] Page 07

8 [BLANK PAGE] WRITE ON PAGE Page 08

9 FOR OFFICIAL USE H National Quali cations 2017 X747/76/01 Mark Mathematics Paper 1 (Non-Calculator) Answer Booklet FRIDAY, 5 MAY 9:00 AM 10:10 AM *X * Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Date of birth Day Month Year Scottish candidate number Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper. *X * A/PB

10 1.(a) 1.(b) 2. *X * Page 02

11 3. 4. *X * Page 03

12 5.(a) 5.(b) 6. *X * Page 04

13 7. 8. *X * Page 05

14 9.(a) 9.(b) (i) 9.(b) (ii) *X * Page 06

15 10.(a) *X * Page 07

16 10.(b) *X * Page 08

17 *X * Page 09

18 (a) *X * Page 10

19 14.(b) An additional diagram, if required, can be found on Page 13. y x *X * Page 11

20 15.(a) 15.(b) 15.(c) *X * Page 12

21 ADDITIONAL SPACE FOR ANSWERS Additional diagram for Question 14(b). 4 y x *X * Page 13

22 ADDITIONAL SPACE FOR ANSWERS *X * Page 14

23 ADDITIONAL SPACE FOR ANSWERS *X * Page 15

24 For Marker s Use Question No Marks/Grades *X * Page 16

25 H 2017 X747/76/12 National Quali cations Mathematics Paper 2 FRIDAY, 5 MAY 10:30 AM 12:00 NOON Total marks 70 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. *X * A/PB

26 FORMULAE LIST Circle: The equation x 2 + y 2 + 2gx + 2fy + c = 0 represents a circle centre ( g, f ) and radius 2 2 g + f c The equation ( x a) + ( y b) = r represents a circle centre (a, b) and radius r. Scalar Product: a.b = a b cos θ, where θ is the angle between a and b or a1 b1 a.b = ab 1 1+ ab 2 2+ ab 3 3where a = a2 and b = b2. a b 3 3 Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A 2 2 cos 2 A = cos A sin A = 2cos 2 A 1 = 1 2 sin 2 A Table of standard derivatives: f( x ) f ( x) sin ax cos ax a cos ax asin ax Table of standard integrals: f( x ) f ( x) dx sin ax cos ax 1 cos ax + c a 1 sin a ax + c Page 02

27 Attempt ALL questions Total marks 70 MARKS 1. Triangle ABC is shown in the diagram below. The coordinates of B are (3,0) and the coordinates of C are (9, 2). The broken line is the perpendicular bisector of BC. y A O B x C (a) Find the equation of the perpendicular bisector of BC. (b) The line AB makes an angle of 45 with the positive direction of the x-axis. Find the equation of AB. (c) Find the coordinates of the point of intersection of AB and the perpendicular bisector of BC (a) Show that ( x 1) is a factor of ( ) (b) Hence, or otherwise, solve f ( x ) = 0. f x = 2x 5x + x [Turn over Page 03

28 The line y = 3x intersects the circle with equation ( x ) ( y ) = 25. MARKS y O x Find the coordinates of the points of intersection (a) Express 3x 24x 50 a x+ b + c. + + in the form ( ) (b) Given that f ( x) = x + 12x + 50x 11, find f ( x). (c) Hence, or otherwise, explain why the curve with equation y f ( x) increasing for all values of x. = is strictly Page 04

29 MARKS 5. In the diagram, PR = 9i+ 5j+ 2k and RQ = 12i 9j+ 3k. Q S P R (a) Express PQ in terms of i, j and k. 2 The point S divides QR in the ratio 1:2. (b) Show that PS = i j+ 4k. (c) Hence, find the size of angle QPS Solve 5sin x 4 = 2cos 2x for 0 x < 2π (a) Find the x-coordinate of the stationary point on the curve with equation y x x 3 = (b) Hence, determine the greatest and least values of y in the interval 1 x 9. 3 [Turn over Page 05

30 8. Sequences may be generated by recurrence relations of the form u = n ku n, u = 5 0 where k R. 2 (a) Show that u = 5k 20k (b) Determine the range of values of k for which u2 < u0. MARKS Two variables, x and y, are connected by the equation y n = kx. The graph of log 2 y against log 2 x is a straight line as shown. log 2 y 3 12 O log 2 x Find the values of k and n. 5 Page 06

31 10. (a) Show that the points A( 7, 2), B(2, 1) and C(17, 6) are collinear. MARKS 3 Three circles with centres A, B and C are drawn inside a circle with centre D as shown. A B D C The circles with centres A, B and C have radii r A, r B and r C respectively. r A = 10 r = 2r B A rc = ra + rb (b) Determine the equation of the circle with centre D (a) Show that sin 2x 2 3 sin xcos x sin x 2cos x =, where 0 π < x <. 2 (b) Hence, differentiate sin 2x 2 sin xcos x 2cos x, where 0 π < x < [END OF QUESTION PAPER] Page 07

32 [BLANK PAGE] WRITE ON PAGE Page 08

33 FOR OFFICIAL USE H National Quali cations Mark 2017 X747/76/02 Mathematics Paper 2 Answer Booklet FRIDAY, 5 MAY 10:30 AM 12:00 NOON *X * Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Date of birth Day Month Year Scottish candidate number Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper. *X * A/PB

34 1.(a) 1.(b) 1.(c) *X * Page 02

35 2.(a) 2.(b) *X * Page 03

36 3. *X * Page 04

37 4.(a) 4.(b) 4.(c) *X * Page 05

38 5.(a) 5.(b) *X * Page 06

39 5.(c) *X * Page 07

40 6. *X * Page 08

41 7.(a) 7.(b) *X * Page 09

42 8.(a) 8.(b) *X * Page 10

43 9. *X * Page 11

44 10.(a) 10.(b) *X * Page 12

45 11.(a) 11.(b) *X * Page 13

46 ENTER OF QUESTION ADDITIONAL SPACE FOR ANSWERS *X * Page 14

47 ENTER OF QUESTION ADDITIONAL SPACE FOR ANSWERS *X * Page 15

48 For Marker s Use Question No Marks/Grades *X * Page 16

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