page 07 MARKS 15. (a) Solve the equation sin cos + = x x 2 ° 6 ° 0 for ≤ < x 0 360. (b) Hence solve sin cos + = x x 4 ° 6 2 ° 0 for ≤ < x 0 360. 16. The point P has coordinates (4,k). C is the centre of the circle with equation ( ) ( ) − + + = x y 2 2 1 2 25. (a) Show that the distance between the points P and C is given by + + k k 2 4 13. (b) Hence, or otherwise, find the range of values of k such that P lies outside the circle. 17. (a) Express ( ) sin cos − x x 2 in the form sin + p q rx where p, q and r are integers. (b) Hence, find ( ) sin cos . − ∫ x x dx 2 [END OF QUESTION PAPER] 4 1 2 4 3 2 page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer is not an indication of how much to write. You do not need 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. FOR OFFICIAL USE X847/76/01 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 National Qualications 2019 THURSDAY, 2 MAY 9:00 AM – 10:30 AM A/PB H page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1. 2. page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3. 4.(a) 4.(b) 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. 8.(b) 8.(a) page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 9.(a) (i) 9.(a) (ii) 9.(b) page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10.(a) 10.(b) 11. page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(a) 12.(b) page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 13.(a) (i) 13.(a) (ii) 13.(b) page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 14.(a) 14.(b) page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15.(a) 15.(b) page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 16.(a) 16.(b) page 13 17.(a) 17.(b) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS 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 2019 H Total marks — 80 Attempt ALL questions. You may use a calculator. To earn full marks you must show your working in your answers. State the units for your answer where appropriate. You will not earn marks for answers obtained by readings from scale drawings. Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer is not an indication of how much to write. You do not need 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. X847/76/12 Mathematics Paper 2 THURSDAY, 2 MAY 11:00 AM – 12:45 PM 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 +