Ms Qe 2004 Paper 1

Maktab Sains Paduka Seri Begawan Sultan General Certificate of Education Advanced Subsidiary Level Qualifying Examination MATHEMATICS

9709/01

Paper 1 Pure Mathematics ( P1)

Time: 1 hour 45 minutes

Additional materials: Answer paper and graph paper List of Formulae

Instructions to candidates: Write in dark blue or black pen on both sides of the paper You may use a soft pencil for any diagrams or graphs. Give non –exact numerical answer correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. If possible try not to use correction fluid. Answer ALL questions

Information to candidates: At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 75. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers.

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1.  (4 x  1) 5 dx

[4]

0

2. (i) Find the first three terms in the expansion of the following binomial expressions (a) (b) (1  2 x) 5 [4] (2  3x) 5

(ii) Hence or otherwise find the first three terms in the expansion of (6 x 2  7 x  2) 5 , in ascending powers of x. [3]

3. An arithmetic progression has sum of the first 10 terms and 12 terms equal to 40 and 50 respectively. Find (i) the first term and the common difference

[4]

(ii) the 20th term.

[2]

2 and the line y  2 x  1 intersect at two points. Find x (i) the coordinates of these two points [4]

4. The curve y  4 

(ii) the equation of the tangents at these two points

[4]

5. (i) Show that the equation sin 2   sin   3 cos 2  can be written as a quadratic equation in sin  . [2] (ii) Hence or otherwise, solve the equation in part (i) for 0 o    360 o .[4]

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6. Functions f and g are defined as follows : f ( x)  x 2  2 x  3 x  R g ( x)  x  4 xR (i) Find the minimum value of the function f

[3]

(ii) Find the range of f and the value of a such that f will have an inverse for x  a . [2] (iii) Solve the equation gf(x) = 0.

[4]

(iv) Sketch , in a single diagram , the graph of y  g ( x) and y  g 1 ( x) , making clear the relationship between the graphs. [2]

7.

OAB is a sector of a circle centre O and radius 6 cm. The mid – point of OA and OB are C and D respectively. CD is a straight line . Given that AOB  0.9 radians , calculate (i) the length of the arc AB

[2]

(ii) the area of the shaded region

[3]

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(i) The figure above shows a trapezium ABCD in which the angles A and D are right angles , AB = 6 cm , AD = 10 cm and DC = 26 cm. A rectangle PQRD is inscribed in the trapezium. Denoting QR by x cm, show that RC = 2x and express [2] (a) DR , and

[1]

(b) area of PQRD , in terms of x

[2]

(ii) Find the maximum area of PQRD.

[3]

 2  3     9. The position vector of the points A , B , C and D are given by OA   3  ; OB   3  ; 1  2      5   2     OC   2  ; and OD   1  where n is a constant. Find 1  n      (i) the unit vector in the direction of BA

[3]

(ii) the angle AOC

[3]

(iii) the value of n for which OB is perpendicular to CD

[4]

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10.

12 . x2 The line x  2 cuts the curve at P and the normal at P cuts the x - axis at Q. Find

The diagram shows part of the curve y 

(i) the coordinates of the point P

[2]

(ii) the equation of the normal at P and the coordinates of the point at Q [4] (iii) the volume obtained , in terms of  , when the shaded region is rotated through 360 0 about the x – axis. [4]

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