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1. ALGEBRA AND FUNCTION
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Part B: Q1: Let f(x)=1/x^2, and g(x)=sqrt(x). Determine:
Q1a)f+g and fg
Answers: f+g =1/x^2+ sqrt(x)=(1+sqrt(x^5))/x^2. fg= [1/x^2]
sqrt(x)=1/sqrt(x^3).
Q5: Obtain the sum of the first 10 terms: {2,5,8,11,14,……}.
Answer: Using formulae Sn = n/2[ 2a+(n-1)d], when a=2, d=3 and n=10, S10 =
10/2[ 2(2)+(10-1)3]=155.

2. ALGEBRA AND FUNCTION
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Q2, Solve 1/(x+1) > 1/(x-1).
Answer: 1/(x+1)- 1/(x-1) > 0, (x-1)-(x+1)/(x+1)(x-1) >0, -2/(x+1)(x-1)>0,For x< -
1, we gain negative value, for -1<x<1, we gain positive value and for x>1, we gain
negative value. So, for 1/(x+1)- 1/(x-1) > 0, we choose positive value, then the
solution is in the range -1< x < 1.
Q3, determine the inverse matrix of [0 0 1, 1 0 0, 0 1 0].
Answer: By using ERO method, [ A / I ] =[0 0 1, 1 0 0, 0 1 0 / 1 0 0, 0 1 0, 0 0 1], A12….
[1 0 0, 0 0 1, 0 1 0 / 0 1 0, 1 0 0, 0 0 1],
A23…[1 0 0, 0 1 0, 0 0 1 / 0 1 0, 0 0 1, 1 0 0]= [ I / A-1],
So A-1 = [0 1 0, 0 0 1, 1 0 0].

3. ALGEBRA AND FUNCTION
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Q1c) the domain and the range of g-1.
Answers: For g-1, domain is x is all real numbers and range is g-1(x)>or =0.
Q1b) f o g
Answers: f o g = 1/(sqrt(x))^2=1/x

4. ALGEBRA AND FUNCTION
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No 5:
Given the sequence n^2.
(a) write the 5 th term.
Answer: T 5 = 5^2= 25.
No 4:
Write the system in the form of matrix equation.
x+ z=0,
y+ z=0,
x+y=0.
Answer:
[ 1 0 1, 0 1 1, 1 1 0] [x , y, z] = [ 0, 0, 0 ].

5. ALGEBRA AND FUNCTION
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(b)Determine its inverse.
its inverse is the matrix itself , ie [ 1 0, 0 1].
Note: I x I = I.
(a)Write the identity matrix.
Answer:
the identity matrix 2 x 2 is [ 1 0, 0 1],

6. ALGEBRA AND FUNCTION
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Solve (x+1)/(x+2)(x+1) >0.
Answer:
The roots are -1 or -2.
Consider the interval, -2<x, -2<x<-1, x>-1.
Construct the interval table:
-2<x -2<x<-1 x>-1.
(x+1) - - +
(x+2) - + +
(x+1) - - +
Result - + +
Since LHS >0, choose the positive value.
Conclusion : the solution is x> -2.

7. ALGEBRA AND FUNCTION
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(d) All integers are natural numbers.
Answer: FALSE. ( Not all integers are natural numbers, example: integer -2
is not a natural number )
( c) All natural numbers are integers.
Answer: TRUE. ( Set of natural numbers is a subset of set of intergers).

8. ALGEBRA AND FUNCTION
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(b) All rational numbers are integers.
Answer: FALSE. ( Not all rational numbers are intergers , example: rational number
1.5=3/2 is a fraction )
Part A
No1:
(a) All intergers are rational numbers.
Answer: TRUE. ( Set of intergers is a subset of set of rational numbers)

9. ALGEBRA AND FUNCTION
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