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```Part A
45 marks
1.
(a)
(b)
Solve the equation 1  20 x  2 x  1  3 .
The function f and g are defined by
f : x  ln  x  1 , x  1
[4]
g : x  x  1, x  1
2.
(i)
Give a reason why the f 1 exists. Hence, find f 1 .
[4]
(ii)
Find g  f
[3]
If f  r  
r
 r  1 r  2 
and its range.
, simplify f  r   f  r  1 .
r 1
n
Hence, find
1
 3  r  1 r  2  r  3 .
[3]
[4]
r 1
3.
(a)
(b)
4
3 

k  3 1  is singular. Find the possible values of k.
5
1 
[3]
Solve the following system of linear equations using Gaussian elimination.
 x  3 y  2z  1
[5]
x  2y  z  2
 2k

Given that matrix A   2
 2

2x  3y  4z  5
4.
The cubic polynomial p  x   6 x3  7 x 2  ax  b, where a and b are constants, has a factor
 x  1 ,
5.
6.
and has a remainder 72 when divided by  x  2  .
(a)
Find the values of a and b and hence, factorise p  x  completely.
[5]
(b)
Find the set of values of x which satisfies p  x   0.
[3]
1
 r 1 
Given that ur  ln 
, find, in term of n, an expression S n , where

 r  r  r  1
S n  u1  u2  u3  ...  un .
[5]
 p 3 4


Given that matrix M   2 1 q  , where p and q are constants.
 3 5 1 


If M  27,
(a)
2
1
q
find p
3
4 using determinant properties.
[3]
6 10 2
(b)
Express p in terms of q.
[3]
```
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