This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
2 hours
(20 MAY 2005 (p.m.»)
Each section consists of 5 questions.The maximum mark for each section is 40.The maximum mark for this examination is 120.This examination paper consists of 6 pages.
3. Unless otherwise stated in the question, any numerical answer that isnot exact MUST be written correct to three significant figures.
Mathematical formulae and tablesElectronic calculatorGraph paper
1. The diagram below, not drawn to scale, shows the graph ofj(x) = x3 + JU2 - 8x + k where h, kare constants.
x2 - 21 x 1-3 < O.
(b) Show that if x and y are real numbers such that x < y, then for any real number k < 0,kx > ky. [ 4 marks)
4
";11- ";"7
Given that x + _1 = 1, by considering (x + _1 )2x x
show that x2 + \- = -1.x
Hence, or otherwise, find the value of .x3 + ~.x
x - 2y = -3x2 + 3y = 7
6. In the diagram below (not drawn to scale),M is the mid-point of AB. MN is perpendicular tothe straight line through A, M and B.
(a) Find
(i)
[ 2 marks]
[ 2 marks]
(b) The point P on AB divides AB internally such that the ratio AP : PB is 3 : 1. Find thecoordinates of P. [ 2 marks]
Express j (fJ) = .../2 cos e - sin e in the form R cos (e + a).
(c) Determine the value of e, 0::; e::; 2n, at which the minimum value ofj(fJ) occurs.[ 2 marks]
Find the range of values of k for which the quadratic equation x2 + 2kx + 9 = 0 hascomplex roots. [ 4 marks]
Express the complex number 2 + 3~ in the form x + yi, where x and y are real numbers.3 + 41 [ 4 marks]
9. Three points, A, Band C, have coordinates (1,2), (2,5) and (0, - 4) respectively relative to theorigin O.
(a) Express the position vector of EACH of A, B and C in terms of i andj. 3 marks]~ ~
(b) If AB = CD, find the position vector of D in terms of i and j. [ 6 marks]
10. Find the values of e, 0 ::; e::; 2n, for which the vectors cos e i + ...J} j and ii + sin ej areparallel.
Total 7 marks
Jimh ~ 0
v;+h - Ii =h
1
2~
(b) Deduce, from first principles, the derivative with respect to x of y = -IX.[lmark]
X
x2 - 2x - 8
the value of the constant k
d2
the value of ~ at P
Find the coordinates of the stationary points of the functionf x ~ x3 - 3x2 - 9x + 6.[ 6 marks]
15. Three points, P, Q and R, on the curve y = x2 - 2x are shown in the diagram (not drawn toscale) below.
(b) Find the TOTAL area bounded by the curve shown above, the x-axis and the linesx = -1 and x = 2. [ 4 marks]
PURE MA THEMA TICS
UNIT 1 - PAPER 02
( 25 MAY 2005 (p.m.»)
Each section consists of 2 questions.The maximum mark for each section is 40.The maximum mark for this examination is 120.This examination consists of 6 pages.
3. Unless otherwise stated in the question, any numerical answer that is notexact MUST be written correct to three significant figures.
Mathematical formulae and tablesElectronic calculatorGraph paper