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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level MATHEMATICS Paper 1 Pure Mathematics 1 (P1) October/November 2004 1 hour 45 minutes
Additional materials: Answer Booklet/Paper Graph paper List of Formulae (MF9)

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READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer
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4 (i) Sketch and label, on the same diagram, the graphs of y = 2 sin x and y = cos 2x, for the interval [4] 0 ≤ x ≤ π. (ii) Hence state the number of solutions of the equation 2 sin x = cos 2x in the interval 0 ≤ x ≤ π . [1]


The equation of a curve is y = x2 − 4x + 7 and the equation of a line is y + 3x = 9. The curve and the line intersect at the points A and B.
(i) The mid-point of AB is M . Show that the coordinates of M are
1 , 2

71 . 2


(ii) Find the coordinates of the point Q on the curve at which the tangent is parallel to the line y + 3x = 9. [3] (iii) Find the distance MQ.



The function f : x → 5 sin2 x + 3 cos2 x is defined for the domain 0 ≤ x ≤ π .
(i) Express f(x) in the form a + b sin2 x, stating the values of a and b. (ii) Hence find the values of x for which f(x) = 7 sin x. (iii) State the range of f.

[2] [3] [2]

3 7

A curve is such that

dy 6 =√ and P (3, 3) is a point on the curve. dx (4x − 3)

(i) Find the equation of the normal to the curve at P, giving your answer in the form ax + by = c. [3] (ii) Find the equation of the curve.



The points A and B have position vectors i + 7j + 2k and −5i + 5j + 6k respectively, relative to an origin O.
(i) Use a scalar product to calculate angle AOB, giving your answer in radians correct to 3 significant figures. [4]

−→ − −→ − − − → (ii) The point C is such that AB = 2BC. Find

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