Tuesda 8 June 0 Morning A GCE MATHEMATICS (MEI) 475/0 Methods for Advanced Mathematics (C) QUESTION PAPER * 4 7 5 6 6 0 6 * Candidates answer on the Printed Answer Book. OCR supplied materials: Printed Answer Book 475/0 MEI Eamination Formulae and Tables (MF) Other materials required: Scientific or graphical calculator Duration: hour 0 minutes INSTRUCTIONS TO CANDIDATES These instructions are the same on the Printed Answer Book and the Question Paper. The Question Paper will be found in the centre of the Printed Answer Book. Write our name, centre number and candidate number in the spaces provided on the Printed Answer Book. Please write clearl and in capital letters. Write our answer to each question in the space provided in the Printed Answer Book. Additional paper ma be used if necessar but ou must clearl show our candidate number, centre number and question number(s). Use black ink. HB pencil ma be used for graphs and diagrams onl. Read each question carefull. Make sure ou know what ou have to do before starting our answer. Answer all the questions. Do not write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Final answers should be given to a degree of accurac appropriate to the contet. INFORMATION FOR CANDIDATES This information is the same on the Printed Answer Book and the Question Paper. The number of marks is given in brackets [ ] at the end of each question or part question on the Question Paper. You are advised that an answer ma receive no marks unless ou show sufficient detail of the working to indicate that a correct method is being used. The total number of marks for this paper is 7. The Printed Answer Book consists of 6 pages. The Question Paper consists of 8 pages. An blank pages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR Do not send this Question Paper for marking; it should be retained in the centre or reccled. Please contact OCR Copright should ou wish to re-use this document. OCR 0 [M/0/65] DC (RW/SW) 6557/ OCR is an eempt Charit Turn over
Section A (6 marks) Fig. shows the graphs of and a + b, where a and b are constants. The intercepts of a + b with the - and -aes are _-, 0i and a0, k respectivel. a + b O Fig. (i) Find a and b. [] (ii) Find the coordinates of the two points of intersection of the graphs. [4] (i) Factorise full n - n. [] (ii) Hence prove that, if n is an integer, n - n is divisible b 6. [] OCR 0
The function f ( ) is defined b f ( ) - sin for - r G G r. Fig. shows the curve f( ). O - r r Fig. (i) Write down the range of the function f ( ). [] - (ii) Find the inverse function f ( ). [] (iii) Find f l ( 0). Hence write down the gradient of f - ( ) at the point _, 0i. [] 4 Water flows into a bowl at a constant rate of 0 cm s (see Fig. 4). h cm Fig. 4 When the depth of water in the bowl is h cm, the volume of water is V cm, where V rh. Find the rate at which the depth is increasing at the instant in time when the depth is 5 cm. [5] J 5 Given that lnk L - N O, show that + P d d - - +. [4] r sin 6 Using a suitable substitution or otherwise, show that d ln. [5] + cos 0 OCR 0 Turn over
4 7 (i) Show algebraicall that the function f( ) - is odd. [] Fig. 7 shows the curve f( ) for 0 G G 4, together with the asmptote. O 4 Fig. 7 (ii) Use the cop of Fig. 7 to complete the curve for - 4 G G 4. [] OCR 0
5 Section B (6 marks) 8 Fig. 8 shows the curve f( ), where f( ) _ - i, with its turning point P. e P f() O Fig. 8 (i) Write down the coordinates of the intercepts of f( ) with the - and -aes. [] (ii) Find the eact coordinates of the turning point P. [6] 4 - (iii) Show that the eact area of the region enclosed b the curve and the - and -aes is _ e i. [5] The function g ( ) is defined b g( ) fa k. (iv) Epress g ( ) in terms of. Sketch the curve g( ) on the cop of Fig. 8, indicating the coordinates of its intercepts with the - and -aes and of its turning point. [4] (v) Write down the eact area of the region enclosed b the curve g( ) and the - and -aes. [] OCR 0 Turn over
6 9 Fig. 9 shows the curve with equation. It has an asmptote a and turning point P. - P O a Fig. 9 (i) Write down the value of a. [] (ii) Show that d d 4 - _ - i. Hence find the coordinates of the turning point P, giving the -coordinate to significant figures. [9] (iii) Show that the substitution u - transforms - d to u u du 4 _ + - i. Hence find the eact area of the region enclosed b the curve and 4. 5., the -ais and the lines - [8] OCR 0
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