# 2014-1-JOH-SULTAN ISMAIL Flipbook PDF

## 2014-1-JOH-SULTAN ISMAIL

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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