St. Maria Goretti Secondary School Katende

ST. MARIA GORETTI SECONDARY SCHOOL KATENDE INTERNAL MOCK EXAM JUNE 2012 S. 6SC MATHEMATICS PAPER I P425/1 TIME 3 HOURS INSTRUCTIONS

Answer all questions in section A and any five questions from section B

SECTION A (40 MARKS)

25 1a) Given that z  3  4i , find the value of the expression z  z z  1 b) Given that  2 find the locus of the complex number z  1

2. By row reducing to echelon form, solve the system of equations x  2y  2z  1 2x  y  4z  1 4x  3y  z  11 1  x 2  cos 1   3. Find in the simplest form, the derivative of  2  1  x  4. Prove by induction that 23n  1 is divisible by 7. For all positive integral values of n .

5. Given that the roots of the equation x 2  2x  10  0 are  and  , determine 1 1 the equation whose roots are and 2    2    x 2  7x  10 6. Solve the inequality  2x  7 x  1  cos x 7. Evaluate 2 dx 0 1  sin x

8. Find the Cartesian equation of the line through the points x  1 y  2 z  2 1, 2, 1 4,  2,  2   and is given by 3  4  3

SECTION B (60 marks) 9a) Find (i)  xsec2 x dx 3 (ii)  x 5e x dx

10a) Given that a  c  k , show that k  a  c . Hence solve the equations x  4z  y  z  3x  y , and 4x  2y  5z  20 b) Solve the equation e 2x  4e x  3  0

z  z  8 11a) Solve the simultaneous equations: 1 2 4z1  3iz2  26  8i

3 b) Find the value of: ( 3  i) 2

12a) A and B are points with position vectors a and b respectively. D is a point on the line joining A to B such that AD : DB  3 : 4 . Find the position vector of D in terms of a and b . b) Find the symmetric equation of the straight line passing through the point (1,2,1) and is normal to the plane 2x  3y  z  2 . c) Find the point of intersection of the line in (b) above with the plane x  y  2z  9 .

2 2 5 0 0 13a) Solve the equation. 3sin x  2cos x  2 tan x , for 0  x  360 b) Express 10sin x cos x  12cos 2x in the form Rsin2x   . Hence or otherwise solve 10sin x cos x  12cos 2x  7  0 in the range 0o  x  360o .

14a) Given that the circles x 2  y 2  ax  by  c  0 and x 2  y 2  bx  ay  c  0 , are orthogonal, prove that ab  2c . b) Show that the normal to the curve x 2  y 2  2y  2  0 at the point (1,1) is a tangent to the curve y  x 2  6x  7, and find the coordinates of the point of contact of the tangent. 11  1  15a) Find the 5th term in the expansion of 2x 2    3x  1 5x b) Expand , as far as and including term in x3 . Taking the first three 1 5x 1 terms and x  , evaluate 14 correct to 4 significant figures. 9

16. In a certain chemical reaction, in which compound X is formed from a compound Y , the masses of X , Y present at time t are x, y respectively. The sum of the masses of X andY is a , where a is constant, and at any time the rate at which x is increasing is propotional to the product of the two masses at dx a that time. Show that  kx(a  x) where k is constant. If x  at t  0 , and dt 5 a 2 x  at t  In2 , show that k  . Hence, find t , correct to three significant 2 a 99a figures when x  . 100