Download answers -One-dimensional mathematics

Self-assessment A
One-dimensional Mathematics
1. Consider the following numbers:
-8
0
0.124373971…

0.23232323….

5
128
15
4
8
8
15
, 128, 8___
4
b) Name the irrational numbers: __ 5 , 0.124373971…, 
a) Name the rational numbers: ___-8, 0, 0.23232323…, 
c) Name the integers: __-8, 0, 128, 8________________________________
15
d) Name the real numbers: __ ___-8, 0, 0.23232323…,  , 128, 8, 5 , 0.124373971…, 
4
e) Name the imaginary numbers: ____  8 ____________________
f) Name the natural or counting numbers: ___128, 8 ____________________
g) Name the complex numbers: ___All of them are complex______________________
h) Name the whole numbers: ___ 0, 8, 128 _____________________________
2. Convert the following decimals to an equivalent fraction in reduced form. You can use a
calculator.
33
a) 0.033
a)
100
113
____________
900
b) 0.12555555….
b) _
c) 0.12424242424242424…
c) __
41
_______________
330
3. Find the solution of the equation |3 – 5x| = x -3
3. _________________
Solution: |3 – 5x| = x -3 is equivalent to 3-5x = (x-3) or 3-5x = -(x-3) where x – 3 can not be
negative.
 3 – 5x = x – 3 or 3 – 5x = -x +3  -6x = -6 or -4x = 0
 x = 1 or x = 0.
But x = 0 does not check because x – 3 = 0 – 3 = -3
Answer: The solution is x = 1
-1-
4. Write each of the following in interval notation;
Problem
All the real numbers x, such that -3 < x < 7
All the real numbers that are at least -4 but less than 15
All the real numbers between -8 and 30
All the real numbers that are at least 4
All the non-negative real numbers
All the real numbers from 5 to 12
Answer
[-3, 7)
[-4, 15)
(-8, 30 )
[ 4,  )
[ 0,  )
[5, 12]
5. Perform the operations required to simplify and write the answers in the forma a + bi
Problem
1. (-3 + 5i) - 3(2 -4i)
Solution: -3 + 5i -6 +12i = -9 + 17i
2. (2 + 4i)(-3-5i)
Solution:
Answer:
-9 + 17i
14 – 22i
(2  4i)(-3 - 5i)  -6 - 10i - 12i - 20i 2  6  22i  20( 1)
 6  22i  20  14  22i
3. ( 2  3i ) 3
3
2
( 2  3i )  ( 2  3i )( 2  3i )( 2  3i )  ( 2  3i )( 4  6i  6i  9i )
-46-9i
Solution:  ( 2  3i )( 4  12i  9)  ( 2  3i )( 5  12i )  10  24i  15i  36i 2
 10  9i  36   46  9i
4. 2i13  5i 34  6i19  6
Solution:
11-4i
2i13  5i 34  6i19  6  2i  5i 2  6i 3  6  2i  5  6i  6  11  4i
5. 20i 7
Solution: 20i
6.
7
20
20
20 20 i 20i 20i




 

 20i
7
3
 i  i i  i2
1
i
i
6  8i
 2i
0+20i
-4 + 3i
Solution:
6  8i 2(3  4i ) 3  4i 3  4i i 3i  4i 2 3i  4( 1)  4  3i



 


 2i
 2i
i
i i
 ( 1)
1
 i2
7.
 15  10i
2i
-8 + i
Solution:
 15  10i  15  10i 2  i  30  15i  20i  10i 2  40  5i




 8  i
2i
2i
2i
5
4  2i  2i  i 2
-2-
8. Find the solution of the equation 6x 2  x  2  0
Solution:
6x 2  x  2  0  (3x  2)( 2x  1)  0  3x  2  0 or 2x  1  0
 3x  2 or 2x  1  x  
 2 1
Answer  , 
 3 2
2
1
or x 
3
2
9. Solve the equation:
Answer: ___x = -1___________-
4
3
7


2
x  2 x  5 x  3x  10
Solution:
4
3
7
4
3
7





, x  2, x  5
x  2 x  5 x 2  3x  10
x  2 x  5 ( x  5)( x  2)
4
3
7

( x  2)( x  5) 
( x  2)( x  5) 
( x  2)( x  5)
x2
x5
( x  5)( x  2)


4( x  5)  3( x  2)  7 
x  1, it checks
4x  20  3x  6  7 
7x   7
Answer : {1}
10. Consider the quadratic equation 4x 2  24x  30  7
a) Give the values of a=4 ___ b=-24 _____ c= 37______
b) Find the value of the discriminant b 2  4ac
Answer: -16
b 2  4ac  (24) 2  4(4)(37)  576  592  16
c) According to the value of the discriminant, which of the following is true?
1. The equation has two real rational solutions
2. The equation has two equal rational solutions
3. The equation has two irrational solutions
4. The equation has no real solution
Answer: 4. The equation has no real solution
d) Find the solutions of the equation.
 b  b 2  4ac  ( 24)   16 24  4i 24 4
1
x



 i  3 i
2a
2( 4)
8
8 8
2
Answer : x  3 
1
1
i or 3  i
2
2
1
3
11. Find the solution of the inequality  2  1  x  . Give answer in interval notation.
3
2
1
3
 2  1 x 
  12  6  2x  9   12  6  2x  9  6
3
2
Solution:
3
3
3 
 18  2x  3  9  x 
  x  9 Answer :  , 9 
2
2
2 
12. Use the formula x 2  x1  x 2 x  x1x 2  0 to find the quadratic equation in each of the
following cases. Give answer in the form ax 2  bx  c  0 where a, b and c are integers.
-3-
The roots or solution of the equation are given below
1) -6 and 5
Answer: Equation
Solution: x 2  x1  x 2 x  x1x 2  0
x 2  x  30  0
Solution: x 2  x1  x 2 x  x1x 2  0
x 2  10x  29  0
x 2   6  5x  ( 6)(5)  0  x 2  x  30  0
2) 5 – 2i and 5 + 2i
x 2  5  2i  5  2i x  (5  2i )(5  2i )  0  x 2  10x  29  0
3) The only solution is -6
Only one solution means that the two solutions are the same: -6, -6
Solution: x 2  x1  x 2 x  x1x 2  0
x 2   6  6x  (6)( 6)  0  x 2  12x  36  0
2
4
4) The solutions are
and 
5
3
x 2  12x  36  0
15x 2  14x  8  0
Solution: x 2  x1  x 2 x  x1x 2  0
4   2  4 
14
8
2
x 2      x       0  x 2  x 
0
3   5  3 
15
15
5
 15x 2  14x  8  0
13. Find the solution of the inequality
| 1- 4x | - 7 <
-2 . Give answer in interval notation.
-2  | 1 -4x | < 5
Solve the equation: 1 – 4x = 5 or 1 – 4x = -5  -4x = 4 or -4x = -6  x = -1 or
6 3
x    1.5
4 2
3
Investigate which interval determined by x = -1 and x   1.5 work
2
1)For x = -2, | 1-4(-2) | -7 = 9 – 7 =2 which is not < -2, therefore the interval  ,  1 is not a
| 1-4x | - 7 <
solution.
2)For x = 0, | 1-4(0) | -7 = 1 – 7 =-6 which is < -2, therefore the interval (-1, 1.5) is a solution.
3) 1)For x = 2, | 1-4(2) | -7 = 7 – 7 =0 which is not < -2, therefore the interval 1.5,   is not a
solution.
The solution is the interval [-1, 1.5]
-1
1.5
-4-