Download MAC 1105 Self- Assessment Exam 1 Sanchez

College Algebra
Self-assessment C - answer
Mathematical Relations
Name: ___________________________________
Multiple choice
1. Every real number that can be written as a fraction is called
a) digit
b) rational
c) irrational
d) imaginary
Answer
b) rational
e) complex
2. i 3 =
a) 0
b) i
c) -1
d) i
e) -i
3. If the solutions of a quadratic equation are not real numbers, then the
discriminant is
a) positive
b) negative
c) zero
d) undefined
4. The slope of a line perpendicular to the line 5x – 7y = 21 is
a)
7
5
b) 
7
5
c)
5
7
d) 
5
7
c) -i
b) negative
b) 
e) none of these
7
5
5. The interval that represents all the non-negative real numbers is
a) [0,  )
6. Which of the following points is symmetric to the point (-6, 7) with respect to
the x – axis.
a) (6, 7)
b) (6, -7)
c) (-6, -7) d) (7, -6) e) (-7, 6)
7) If one solution of a quadratic equation with rational coefficients is -5 + 3i,
then the other solution is
a) 5 + 3i
b) 5 – 3i
c) -5 – 3i
d) 3 – 5i
e) -3 +5i
8) The decimal 0.2222222….. is equivalent to which of the following fractions
c) (-6, -7)
a) [0,  )
b) (0,  )
2
a)
10
2
b)
11
c) ( , 0)
22
c)
100
d ) ( , 0]
2
d)
9
c) -5 – 3i
d)
e) none of these
2
9
9) The two solutions of a quadratic equation are -3 and 7. Which of the following
Hint: x 2  ( x1  x 2 )x  x1x 2  0
is the quadratic equation?
a)
a) x 2  4x  21  0 b) x 2  4x  21  0
c) x 2  4x  21  0 d) x 2  4x  21  0
10) Which of the following intervals indicates all the real numbers which are greater
a) (5, 10.6]
than 5,
but at most 10.6?
a) (5, 10. 6]
b ) [5, 10.6)
c) (5, 10.6)
d ) [5, 10.6]
11) Which of the following intervals indicates all the real numbers which are
less
than 5 or at least 7 ?
a) (5, 7]
b) [5, 7)
c) ( ,  )
d ) ( . 5)  [7,  )
d) ( . 5)  [7,  )
12) Which of the following is the solution of the inequality 5 – 3x > 2?
a) x> 1
b) x < 1
c) x > -1 d) x < -1
e) none of these
13) Which of the following is equivalent to the expression |-7 +2|
a) 9
b) -9
c) -5
d) 5
e) none of these
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b) x < 1
d) 5
14) The center of the circle (x  3) 2  ( y  2) 2  16 is
a) (3, -2)
b) (-3, 2) c) (2, -3) d) (-2, 3)
b) (-3, 2)
15) The domain of the circle (x  3) 2  ( y  2) 2  16 is
a) (-7, 1)
b) (-2, 6)
c) [-7, 1]
d) [-2, 6] e) [-3, 2]
c) [-7, 1]
Problem 1 . Find the solutions of the equation
Solution: 15 – 2x = 7 or 15 – 2x = -7
-2x = -8 or -2x = -22
x = 4 or x =11
|15 – 2x|
= 7
answer: __x = 4 or x =11
|
Find the solution of the inequality |15 – 2x < 7 . Give answer in interval notation.
(Solve by investigating the different interval on the number line determined by the solutions of
the equation.
______________________________________________
4
11
For x =0, the inequality is false, therefore the interval ( , 4) is not a solution.
For x = 5, the inequality is true, therefore the interval (4, 11) is a solution.
For x =12, the inequality is false, therefore the interval ( 4,  ) is not a solution.
The solution of the inequality is [4, 11]
4
11
2. Problem 2. Consider the linear relation 3x + 4y =8.
a) Find the slope m of the relation.
_A
3
m

B
4
a) m 
_A
3
  __________
B
4
b) Find the y-intercept of the relation
When x = 0 , y = 2
b) __ 2 ____________
c) Use the slope and the fact that (0, 2) is a point of the
relation to find another two points.
(0 + 4, 2 – 3) = (4, -1)
(4 + 4, -1 -3) = (8, -4)
c) (4, -1) and (8, -4) or many
other possible answers.
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d) Use the points found to draw the line in the following Cartesian system.
X = -2
y-axis
(0, 2)
Y=1
1
-1
1
x-axis
(4, -1)
e) After you draw the line, shade the region that represents the inequality 3x + 4y < 8.
The point (0, 0) belongs to the inequality
f) Draw the horizontal line y = 1
g) Draw the vertical line x = -2
Problem 3. Perform the operations required to simplify and write the answers in the forma a + bi
a) (3 – 4i)(-5+3i)
a) -3 + 29i
Solution:
 15  9i  20i  12i 2  15  29i  12( 1)  15  29i  12
 3  29i
b) (-2+3i) – (-5+6i)
=-2 + 3i +5 -6i
3 – 3i
b) 3 – 3i
Problem 4. Find the solution of the equation 3x 2  x  2  4
Use the quadratic formula. a = _3____
b = __-1____ c = -2__ Careful!
2
 b  b 2  4ac  1  ( 1)  4(3)( 2) 1  25 1  5 6
4
2
x



 or
 1 or 
2a
2(3)
6
6
6
6
3
Problem 5. Find the equation of the circle with center (-1, 3) and radius r = 9.
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Give the domain and range of the circle
Equation: __ ( x  1) 2  ( y  3) 2  81 _____________________
Domain: ___[-1-9, -1+9] = [-10, 8]_________
Range: _____[3 – 9, 3 + 9] = [-6, 12]__________
Problem 6. Find an equation of the line that contains the points (-4, 1) and (3, -1)
H int : y  y1  m( x  x1 )
y  y1  1  1  2
m 2


x 2  x1 3  4
7
y  y1  m( x  x1 )  y  1 
2
( x  4)
7
 7 y  7  2x  8
 2x  7 y  1
Problem 7. Consider the linear relation: y = 5x – 7
a) Give the slope of the relation
a) m = 5
b) Give the y-intercept of the relation
b) b = -7
c) Give the equation of the transpose relation
x = 5y -7
d) Give the equation of the relation that is
symmetrical to y = 5x – 7 with respect to the
y-axis.
y = 5(-x) - 7
e) Give the equation of the relation that is
symmetrical to y = 5x – 7 with respect to the
x-axis.
- y = 5x – 7 or y = -5x + 7
f) Give the equation of the relation that is
symmetrical to y = 5x – 7 with respect to the
origin
-y = 5(-x) – 7 or y = 5x + 7
c) x = 6y -7
d) y = -5x - 7
e) y = 5x - 7
f) y = 5x + 7
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