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P425/1 PURE MATHEMATICS PAPER 1 22ND JULY, 2012 3 HOURS. UGANDA ADVANCED CERTIFICATE OF EDUCATION INTERNAL MOCK EXAMINATIONS PURE MATHEMATICS PAPER 1 3 HOURS INSTRUCTIONS. Attempt ALL the EIGHT questions in section A and only FIVE from section B. Begin each answer on a fresh sheet of paper. No paper should be given for rough work. Mathematical tables and squared paper are provided. Silent, non-programmable calculators may be used. State the degree of accuracy at the end of each answer attempted a calculator or table; and indicate CAL for calculator or TAB for mathematical tables. © INTERNAL MOCK EXAMINATIONS 2012 1 TURN OVER SECTION A (40 marks) 1. Given that A is an acute angle such that Sin 2 A Cos 2 A 2Sin 2 A 0 , find the exact value of tan A . (5 marks) 2. The tangent to curve at a point x, y passes through the point x,0 . If the curve passes through the point 1,4 , show that the equation of the curve is y 2 16 x . (5 marks) 3. Prove by induction; 1 2 2 2 3 2 ... n(2 0 1 2 n 1 ) 1 (n 1)2 n . (5 marks) 4. Find e x ln 1 e x dx . (5 marks) 5. Find the distance of the point P(4,1,2) from the line; 3 5 r 4 4 1 3 (5 marks) 6. If the line 3x – 4y – 12 = 0 is the tangent to the circle with a centre at (1, 1). Find the equation of that circle. (5 marks) 7. Expand 1 4 x up to the term in x 3 . State the range of values of x for which the expansion is valid. Taking x 400 , estimate 11 to 3dp . (5 marks) 8. Given that y e tan 1 x , show that d2y dy (1 x ) 2 (2 x 1) 0. dx dx (5 marks) 2 SECTION B (60 marks) 9. a) Given that log a b c m, log b a c n ; prove that log c a 1 n . 1 mn (6 marks) b) i) Find the number of arrangements of all the letters of the word CALCULUS. ii) How many of the arrangements in (i) above are the A and S separate? (6 marks) 2 3 10. a) Given that z 4 z 5 is a factor of z az b ; find the values of the real numbers a, b. (6 marks) Z b) Given that Z x iy , and the quantity is a real number, find and sketch Z 4 the locus of the point representing Z. (6 marks) © INTERNAL MOCK EXAMINATIONS 2012 2 TURN OVER 11. a) Solve the equation Sin 30 0 Cos 60 0 Cos ; for 0 0 360 0 . b) A and B are points a metres apart on level ground. A tower OT stands, vertically, at point O in the line AB such that the angles of elevation of T from A and B are and respectively. C is a point in AB such that CB b , prove that a the elevation of T from C is tan 1 a b Cot bCot (6 marks) 12. a) Differentiate with respect to x ; i) e 3 x Sin 3 x ii) x ( x 1 2 x 2 Cot 2 x ) (5 marks) b) A cylinder of height h is inscribed in a right pyramid of square base 12cm 12cm , and of slant length of 9cm, given that the upper rim of the cylinder touches the slant faces of the pyramid, show that its volume is (7 marks) 4h3 2h . Hence find the maximum volume. x 2 3x 4 does not x 8 lie, hence determine the coordinates of the stationary points; hence sketch the curve. (12 marks) 13. Find the range of values of y within which the curve y 14. a) Evaluate; 2 Sin 3 2 xdx . (5 marks) 4 b) A(1,2) is a point on the curve y x 2 1 , the area bounded by the curve, the y - axis and the tangent to the curve at A is rotated about the y - axis. Find the volume generated. (7 marks) 15. a) Show that the equation of the tangent at P ct , c on the hyperbola xy c is 2 t x t 2 y 2ct . (4 marks) b) The normal, to the tangent at P , through the origin O (0,0) meets the tangent at N, and Q is the point such that ONPQ is a rectangle; find; i) the coordinates of N and Q in terms of t. ii) the loci of N and Q. (8 marks) © INTERNAL MOCK EXAMINATIONS 2012 3 TURN OVER dy y 1 dx b) In a certain type of chemical reaction a substance A is continuously transformed into a substance B. Throughout the reaction the sum of masses of A and B remains constant and equal to m. The mass of B present at time t after the commencement of the reaction is denoted by x . At any instant, the rate of increase of the mass of B is proportional to the mass of A. i) Write down the differential equation relating x and t . 1 ii) Given that x 0 when t 0 and also x m when t In2 , show that at 2 16. a) Find the general solution of the differential equation x( x 1) time t , x m 1 e t . iii) Hence find the value of t when x 3 m 4 (12 marks) END 4