Data Loading...
2014-1-JOH-SULTAN ISMAIL Flipbook PDF
2014-1-JOH-SULTAN ISMAIL
247 Views
70 Downloads
FLIP PDF 35.35KB
1
2014-1-JOH-SULTAN ISMAIL Section A [45, Marks] Answer all questions in this section.
A functions f is defined as f : x a x 2 + 2, x ∈ [−5, −1] Find f −1 . Sketch f and f −1 on the same axes.
(a) (b)
[4 marks] [3 marks]
2 The second and the fifth terms of a geometric series are 24 and −3 respectively. Find the smallest value of n for which the difference between the sum of the first n terms and the sum to infinity is less than 0.005. [5 marks]
[6 marks]
1 2 2 −5 a 4 3 Given that P = 1 1 3 , Q = 8 −5 c and PQ = kI , where k is a constant and I 4 c 3 b 1 c [4 marks] is the 3 × 3 identity matrix. Find the values of a, b and c .
and hence, solve the following system of linear equations. −5 x + ay + 4 z = 11 8 x − 5 y + cz = −5 cx + 4 y + cz = 4
2π The complex number z is such that zz* = 64 and arg( z ) = . 3 [3 marks] Find z in polar form and deduce z 5 in polar form. Find the complex numbers w such that w3 = −4 + 4 3i in the form of a + bi . [5 marks]
Deduce Q −1
4
(a) (b)
[5 marks]
x2 y2 5 The equation of an ellipse is given by + =1. 9 25 (a) Find the centre, vertices and the foci of the ellipse and sketch the curve. [6 marks] (b) If the distances of a point on the ellipse from the foci are d1 and d 2 , show that (d1 + d 2 ) 2 is constant. [4 marks]
6
0 2 4 −1 Find the acute angle between the lines r = 1 + m 3 and r = 7 + t 2 . 4 −1 7 −2
7.
Section B [15 marks] Answer any one question in this section
(b)
Solve the equation log 3 x 2 + log 3 x = log 27 9 .
If
(a)
(c)
Solve the inequality cos( x − 17°13') < −0.5, − 180° ≤ x ≤ 180° .
x3 + 2 x 2 + 2 x + 3 Express in partial fractions. x2 + 2 x − 3 a = b + c , show that (a − b − c) 2 = 4bc .
(d)
Solve the following system of linear equations using Gaussian elimination.
[4 marks]
[3 marks]
[3 marks]
[5 marks]
[6 marks]
8. (a) The position vectors of the points A , B and C are i + 3 j + 5k , 4i − 9 j − k and pi − j + 3k respectively. find the unit vector parallel to AB . [3 marks] find the value of p if A, B and C are collinear. [3 marks] if p = 2 , find the position vectors of the points D such that DABC is a parallelogram. [3 marks] (i) (ii) (iii)
(b)
4 x + 2 y − 3z = 5 x + y − 2z = 2 2 x − y + 3z = 1