This document contains 6 problems in Section A and 2 problems in Section B of a math exam. Section A includes problems on inverse functions, complex numbers, matrices, trigonometric identities, power series expansions, and geometric series. Section B includes problems on factorizing a polynomial and finding properties of a parametric curve such as its Cartesian equation, the equation of a chord between two points on the curve, and where the curve intersects the line y=x.

1. Section A ( 45 marks )
Answer all question in this section
1. The function of f and g defined by , 𝑥 ∈ 𝑅 𝑎𝑛𝑑 𝑥 ≠ −1
(a ) Find the inverse function 𝑓−1
( 3 marks )
(b ) Find the solution set for which 𝑓( 𝑥 ) = 𝑓−1
(𝑥) ( 4 marks )
2. The complex number 𝑧 is given by 𝑧 = √3 + 𝑖
(a ) Find | 𝑧| and arg z. ( 3 marks )
(b ) Using de Moivre’s theorem, show that 𝑧5
= −16√3 +16i ( 3 marks)
(c ) Express
𝑧4
𝑧∗ in the form 𝑥 + 𝑦𝑖,where 𝑧∗
is the conjugate of z and x , 𝑦 ∈ 𝑅
( 3 marks )
3.(a) By using the elementary row operation, find the inverse of the matrix (
1 1 1
1 0 2
−1 1 1
)
( 6 marks )
(b) Solve the following system of linear equations
𝑥 + 𝑦 + 𝑧 = 4
𝑥 + 2𝑧 = 3
−𝑥 + 𝑦 + 𝑧 = 6 (4 marks )
4.Express 5sin 𝜃 − 8cos 𝜃 in the form r sin (𝜃 − 𝛼),Where 𝑟 > 0 and 0 <∝<
𝜋
2
.
State the maximum and minimum values of 5sin 𝜃 − 8cos 𝜃. ( 6 marks )
5. Find the first three terms of the expansion of √
1+2𝑥
1−2𝑥
in ascending power of x ( 4 marks )
(b) Hence , or otherwise , find and approximation for √51 correct to five significant
figure by substituting 𝑥 = 0.01 ( 4 marks )
6. For the geometric series 6 + 3 +
3
2
+…Obtain the smallest value of n if the difference
between the sum of the first n + 4 terms and the sum of the first n term is less than
45
64
( 6 marks )
𝑓 ∶ 𝑥 →
2
𝑥 + 1

2. Section B ( 15 marks )
Answer any one question in this section
7. Given that 𝑥2
− 𝑥 − 2 is quadratic factor of the polynomial
𝑃( 𝑥 ) = 𝑥4
+ 𝑎𝑥3
+ 2𝑥2
+ 𝑏𝑥 − 4 ; a , b ∈ 𝑍
(a ) Determine the values of a and b ( 5 marks )
(b ) Find the other quadratic factor of P(x ) and show that this factor is positive all the value of x.
( 6 marks )
(c ) Find the set value of x for which P(x )≥ ( 𝑥 − 1 )(𝑥 + 2 ) ( 4 marks )
8. The Parametric equations of the curve are given by
𝑥 =
6
(1+𝑡)2 and 𝑦 =
3𝑡
1+𝑡
, t≠ −1
M and N are two any points on the curve with parameter t=1 and t=2 respectively.
Find
(a ) The Cartesian equation of the curve ( 4 marks )
(b ) The equation of the chord ( 6 marks )
(c ) The coordinates of the point where the curve meets the line y=x ( 5 marks )