Condensed. Mathematics. General Certificate of Education Advanced Level Examination June Unit Pure Core 3. Time allowed * 1 hour 30 minutes

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1 General Certificate of Education Advanced Level Eamination June 01 Mathematics MPC3 Unit Pure Core 3 Thursda 31 Ma am to am For this aer ou must have: the blue AQA booklet of formulae and statistical tables. You ma use a grahics calculator. Time allowed 1 hour 30 minutes Instructions Use black ink or black ball-oint en. Pencil should onl be used for drawing. Fill in the boes at the to of this age. Answer all questions. Write the question art reference (eg (a), (b)(i) etc) in the left-hand margin. You must answer each question in the sace rovided for that question. If ou require etra sace, use an AQA sulementar answer book; do not use the sace rovided for a different question. Do not write outside the bo around each age. Show all necessar working; otherwise marks for method ma be lost. Do all rough work in this book. Cross through an work that ou do not want to be marked. Condensed Information The marks for questions are shown in brackets. The maimum mark for this aer is 75. Advice Unless stated otherwise, ou ma quote formulae, without roof, from the booklet. You do not necessaril need to use all the sace rovided. 6/6/6/ MPC3

2 1 Use the mid-ordinate rule with four stris to find an estimate for giving our answer to three decimal laces. ð 1: 0: cotð Þ d, ( marks) For 0 <, the curves with equations ¼ ln and ¼ ffiffi single oint where ¼ a. intersect at a (a) Show that a lies between 0.5 and 1.5. (b) Show that the equation ln ¼ ffiffi ¼ e can be rearranged into the form ffiffi (1 mark) (c) Use the iterative formula nþ1 ¼ e ffiffiffi n (d) with 1 ¼ 0:5 to find the values of and 3, giving our answers to three decimal laces. Figure 1, on the age 3, shows a sketch of arts of the grahs of ¼ e ¼, and the osition of 1. ffiffi and n Figure 1, draw a cobweb or staircase diagram to show how convergence takes lace, indicating the ositions of and 3 on the -ais. (0)

3 3 Figure 1 ¼ ¼ e ffiffi 1 3 A curve has equation ¼ 3 ln. (a) Find d d. (b) (i) (ii) Find an equation of the tangent to the curve ¼ 3 ln at the oint on the curve where ¼ e. This tangent intersects the -ais at the oint A. Find the eact value of the -coordinate of the oint A. Turn over s (03)

4 (a) B using integration b arts, find ð e 6 d. ( marks) (b) The diagram shows art of the curve with equation ¼ ffiffi e 3. R 1 The shaded region R is bounded b the curve ¼ ffiffi e 3, the line ¼ 1 and the -ais from ¼ 0to¼1. Find the volume of the solid generated when the region R is rotated through 360 about the -ais, giving our answer in the form ðe 6 þ qþ, where and q are rational numbers. 5 The functions f and g are defined with their resective domains b fðþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 5, for 5 :5 gðþ ¼ 10, for real values of, 6¼ 0 (a) State the range of f. (b) (i) Find fgðþ. (1 mark) (ii) Solve the equation fgðþ ¼5. (c) The inverse of f is f 1. (i) Find f 1 ðþ. (ii) Solve the equation f 1 ðþ ¼7. (0)

5 5 ð Use the substitution u ¼ þ to find the value of 0 ð d, giving our þ Þ answer in the form ln q þ r, where, q and r are rational numbers. (6 marks) 7 The sketch shows art of the curve with equation ¼ fðþ (a) n Figure on age 6, sketch the curve with equation ¼jfðÞj. (b) n Figure 3 on age 6, sketch the curve with equation ¼ fðj jþ. (c) Describe a sequence of two geometrical transformations that mas the grah of ¼ fðþ onto the grah of ¼ 1 fð þ 1Þ. ( marks) (d) The maimum oint of the curve with equation ¼ fðþ has coordinates ð 1, 10Þ. Find the coordinates of the maimum oint of the curve with equation ¼ 1 fð þ 1Þ. Turn over s (05)

6 6 (a) Figure (b) Figure (06)

7 7 8 (a) Show that the equation can be written in the form 1 1 þ cos þ 1 1 cos ¼ 3 cosec ¼ 16 ( marks) (b) Hence, or otherwise, solve the equation 1 1 þ cosð 0:6Þ þ 1 1 cosð 0:6Þ ¼ 3 giving all values of in radians to two decimal laces in the interval 0 < <. (5 marks) 9 (a) Given that ¼ sin, use the quotient rule to show that cos d d ¼ sec (b) Given that tan ¼ 1, use a trigonometrical identit to show that sec ¼ þ (c) Show that, if ¼ tan 1 ð 1Þ, then d d ¼ 1 þ (1 mark) (d) A curve has equation ¼ tan 1 ð 1Þ ln. (i) Find the value of the -coordinate of each of the stationar oints of the curve. ( marks) (ii) Find d d. (iii) Hence show that the curve has a minimum oint which lies on the -ais. Coright Ó 01 AQA and its licensors. All rights reserved. (07)