page 08 MARKS 13. The right-angled triangle in the diagram is such that sin x = 2 11 and x π < < 0 4 . 2 x 11 (a) Find the exact value of: (i) sin 2x (ii) cos 2x. (b) By expressing sin 3x as ( ) sin x x + 2 , find the exact value of sin 3x. 14. Evaluate ( ) dx x − + ⌠ ⎮ ⌡ 9 2 3 4 1 2 9 . 15. A cubic function, f, is defined on the set of real numbers. • ( ) x + 4 is a factor of ( ) f x • x = 2 is a repeated root of ( ) f x • ( ) f − ′ = 2 0 • ( ) f x ′ > 0 where the graph with equation ( ) y f x = crosses the y-axis Sketch a possible graph of ( ) y f x = on the diagram in your answer booklet. [END OF QUESTION PAPER] 3 1 3 5 4 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. FOR OFFICIAL USE X747/76/01 Mathematics Paper 1 (Non-Calculator) Answer Booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth H National Qualications 2018 THURSDAY, 3 MAY 9:00 AM – 10:10 AM A/PB page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1. 2. page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3. 4. page 04 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(a) 6. 5.(b) page 05 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7.(a) 7.(c) 7.(b) page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 8. 9.(a) 9.(b) page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10. *X74776010* page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(a) 11.(b) An additional diagram, if required, can be found on page 13. y x 1 O (3,1) log y x = 3 page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(a) 12.(b) page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 13.(a) (i) 13.(a) (ii) 13.(b) page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 14. page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15. –6 –5 –4 –3 –2 –1 O 1 2 3 4 5 6 An additional diagram, if required, can be found on page 14. x y page 13 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS Additional diagram for Question 11(a). y x 1 O (3,1) log y x = 3 page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS Additional diagram for Question 15. –6 –5 –4 –3 –2 –1 O 1 2 3 4 5 6 x y page 15 ADDITIONAL SPACE FOR ANSWERS DO NOT WRITE IN THIS MARGIN QUESTION NUMBER page 16 For Marker’s Use Question No Marks/Grades © National Qualications 2018 H 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. X747/76/12 Mathematics Paper 2 THURSDAY, 3 MAY 10:30 AM – 12:00 NOON A/PB page 02 FORMULAE LIST Circle: The equation 2 2 2 2 0 x y gx fy c + + + + = represents a circle centre (−g, −f ) and radius 2 2 g f c + − . The equation ( ) ( ) 2 2 2 x a y b r − + − = represents a circle centre (a, b) and radius r. Scalar Product: cos , where is the angle between and θ θ = a.b a b a b or 1 1 1 1 2 2 3 3 2 2 3 3 where and . a b a b a b a b a b a b ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + + = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ a.b a b 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 A sin AcosA 2 2 = cos A cos A sin A 2 2 2 = − cos A 2 2 1 = − sin A 2 1 2 = − Table of standard derivatives: ( ) f x ( ) f x ′ sin ax cosax cos a ax sin a ax − Table of standard integrals: ( ) f x ( ) f x dx ∫ sin ax cosax cos 1 ax c a − + sin 1 ax c a +