Mathematics Advanced Extension Award. Candidates may NOT use a calculator in answering this paper.

Transcription

1 Paper Reference(s) 9801/01 Edecel Mathematics Advanced Etension Award Monday 7 June 011 Afternoon Time: 3 hours Materials required for eamination Answer book (AB16) Graph paper (ASG) Mathematical Formulae (Pink) Items included with question papers Nil Candidates may NOT use a calculator in answering this paper. Instructions to Candidates Intheboesontheanswerbookprovided,writethenameoftheeaminingbody(Edecel),your centrenumber,candidatenumber,thepapertitle(mathematics),thepaperreference(9801),your surname, initials and signature. Check that you have the correct question paper. Answers should be given in as simple a form as possible. e.g. π, 6, 3. 3 Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. The marks for individual questions and the parts of questions are shown in round brackets: e.g. (). There are 7 questions in this question paper. The total mark for this paper is 100, of which 7 marks are for style, clarity and presentation. There are 8 pages in this question paper. Any blank pages are indicated. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Eaminer. Answers without working may not gain full credit. Printer s Log. No. P38167A W850/R9801/ /5/4/3 *P38167A* Turn over This publication may be reproduced only in accordance with Edecel Limited copyright policy. 011 Edecel Limited.

2 1. Solve for 0 θ 180 o o tan ( θ+ 35 )= cot θ 53 o ( ) (Total 4 marks). Given that π 1 ( 1+ ) d = + 0 tan a ln b find the value of a and the value of b. (Total 7 marks) 3. A sequence { u n }isgivenby u1 = k u = u p n n n 1 1 un+ 1 = un q n 1 where k, p and q are positive constants with pq 1 (a) Write down the first 6 terms of this sequence. (b) Show that n k 1+ p 1 ( pq) ur = 1 pq r= 1 ( ) n ( ) (3) (6) In part (c) [ ] means the integer part of, so for eample [ 73. ]=, [ 4]= 4 and [ 0]= 0 4 (c) Find 6 3 r= 1 r 3 5 r 1 (4) (Total 13 marks) P38167A

3 4. The curve C has parametric equations = cos t y = costsin t where 0 t < (a) Show that C is a circle and find its centre and its radius. (5) y P R C O Figure 1 ( ) 0 π Figure 1 shows a sketch of C. The point P, with coordinates cos α, cosα sin α, < α <, lies on C. The rectangle R has one side on the -ais, one side on the y-ais and OP as a diagonal, where O is the origin. (b) Show that the area of R is sinα cos 3 α (1) (c) Find the maimum area of R, as α varies. (7) (Total 13 marks) P38167A 3 Turn over

4 5. y U O y = 1 = = Figure Figure shows a sketch of the curve C with equation y = The curve cuts the y-ais at U. 4 and ±. (a) Write down the coordinates of the point U. (1) The point P with -coordinate a ( a 0) lies on C. (b) Show that the normal to C at P cuts the y-ais at the point 0, a a ( ) a (6) The circle E, with centre on the y-ais, touches all three branches of C. (c) (i) Show that a ( a 4) ( a 4) a = + ( a 4) (ii) Hence, show that (iii) Find the centre and radius of E. a 4 1 ( ) = (10) (Total 17 marks) P38167A 4

5 6. The line L has equation 13 5 r = 3 + t The point P has position vector 7. 7 The point P is the reflection of P in L. (a) Find the position vector of P. (6) (b) Show that the point A with position vector lies on L. (1) (c) Show that angle PA P = 10. (3) P A 10 B L P Figure 3 The point B lies on L and APBP forms a kite as shown in Figure 3. The area of the kite is 50 3 (d) Find the position vector of the point B. (e) Show that angle BPA = 90. (5) () The circle C passes through the points A, P, P and B. (f) Find the position vector of the centre of C. () (Total 19 marks) P38167A 5 Turn over

6 7. y O A B = 3 Figure 4 (a) Figure 4 shows a sketch of the curve with equation y = f( ), where 5 f( ) =, 3, 3 The curve has a minimum at the point A, with -coordinate α, and a maimum at the point B, with -coordinate β. Find the value of α, the value of β and the y-coordinates of the points A and B. (5) P38167A 6

7 (b) The functions g and h are defined as follows where p is a constant. g: + p h: y C O D Figure 5 Figure 5 shows a sketch of the curve with equation y = hfg ( ( ) + q),, 0, where q is a constant. The curve is symmetric about the y-ais and has minimum points at C and D. (i) Find the value of p and the value of q. (ii) Write down the coordinates of D. (5) (c) The function m is given by m( ) = 5,, α 3 where α is the -coordinate of A as found in part (a). (i) Find m 1 (ii) Write down the domain of m 1 (iii) Find the value of t such that m() t m () t (10) (Total 0 marks) FOR STYLE, CLARITY AND PRESENTATION: 7 MARKS TOTAL FOR PAPER: 100 MARKS END P38167A 7

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