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  FREE-STANDING MATHEMATICS QUALIFICATIONADVANCED LEVEL Additional Mathematics  6993 QUESTION PAPER Candidates answer on the printed answer book. OCR supplied materials: ã Printed answer book 6993 Other materials required: ã Scientific or graphical calculator Monday 13 June 2011Morning Duration:  2 hours INSTRUCTIONS TO CANDIDATES These instructions are the same on the printed answer book and the question paper.ã The question paper will be found in the centre of the printed answer book.ã Write your name, centre number and candidate number in the spaces provided on the printedanswer book. Please write clearly and in capital letters. ã Write your answer to each question in the space provided in the printed answer book. Additional paper may be used if necessary but you must clearly show your candidate number,centre number and question number(s).ã Use black ink. Pencil may be used for graphs and diagrams only.ã Read each question carefully. Make sure you know what you have to do before starting youranswer.ã Answer  all  the questions.ã Do  not  write in the bar codes.ã You are permitted to use a scientific or graphical calculator in this paper.ã Final answers should be given correct to three significant figures where appropriate. INFORMATION FOR CANDIDATES This information is the same on the printed answer book and the question paper.ã The number of marks is given in brackets  [ ]  at the end of each question or part question on thequestion paper.ã You are advised that an answer may receive  no marks  unless you show sufficient detail of theworking to indicate that a correct method is being used.ã The total number of marks for this paper is  100 .ã The printed answer book consists of  20  pages. The question paper consists of  8  pages. Any blankpages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR ã Do not send this question paper for marking; it should be retained in the centre or destroyed.  © OCR 2011 [100/2548/0] OCR is an exempt Charity5R–0G15  Turn over  2Formulae Sheet: 6993 Additional MathematicsIn any triangle  ABC  Cosine rule  a 2 =  b 2 + c 2 − 2 bc cos  A  A BC b ac Binomial expansion When  n  is a positive integer ( a + b ) n =  a n +  n 1  a n − 1 b +  n 2  a n − 2 b 2 +  ...  +  nr    a n − r  b r  +  ...  + b n where  nr     =  n C  r   =  n ! r  ! ( n − r  ) !  © OCR 2011 6993 Jun11  3Answer all questions on the Printed Answer Book provided.Section A1  Determine whether the point  ( 5, 2 )  lies inside or outside the circle whose equation is  x  2 +  y 2 =  30.You must show your working.  [3]2  The equation of a curve is  y  =  x  3 −  x  2 − 2  x  − 3.Find the equation of the tangent to this curve at the point  ( 3, 9 ) .  [5]3  In the triangle PQR, PQ  =  8cm, RQ  =  9cm and RP  =  7cm. (i)  Find the size of the largest angle.  [4](ii)  Calculate the area of the triangle.  [3]4  Solve the equation 5sin2  x   =  2cos2  x   in the interval 0 ◦ ≤  x   ≤  360 ◦ .Give your answers correct to 1 decimal place.  [5]5  The coordinates of the points A, B and C are  (− 2, 1 ) ,  ( 5, 2 )  and  ( 4, 9 )  respectively. (a)  Find the coordinates of the midpoint, M, of the line AC.  [1](b)  Show that BM is perpendicular to AC.  [3](c) (i)  Use the result of part  (b)  to state the mathematical name of the triangle ABC.  [1](ii)  Prove this by another method.  [2]6  Solve the inequality  x  2 − 12  x  + 35  ≤  0.  [4]7 (a)  Determine whether or not each of the following is a factor of the expression  x  3 − 7  x  + 6.You must show your working. (i)  (  x  − 2 )  [2](ii)  (  x  + 1 )  [1](b) (i)  Factorise the function f  (  x  ) =  x  3 − 7  x  + 6.  [3](ii)  Solve the equation f  (  x  ) =  0.  [1]  © OCR 2011 6993 Jun11  Turn over  48 (i)  On the axes given, indicate the region for which the following inequalities hold. You shouldshade the region which is  not  required.5  x  + 3  y  ≥  303  x  +  y  ≥  12  y  ≥  0  x   ≥  0  [5](ii)  Find the minimum value of 6  x  +  y  subject to these conditions.  [2]9  The gradient function of a curve is given by d  y d  x   =  3  x  2 − 2  x  + 4.Find the equation of the curve, given that it passes through the point  ( 2, 2 ) .  [4]10  You are given that sin θ   =  25  with 0 ◦ ≤  θ   ≤  90 ◦ .Using the identity sin 2 θ   + cos 2 θ   =  1, find an exact value for cos θ  .  [3]  © OCR 2011 6993 Jun11

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Jul 23, 2017
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