Q. What is the angle between the lines y = 2x + 3 and y = -1/2x + 1?
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A.
90 degrees
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B.
45 degrees
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C.
60 degrees
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D.
30 degrees
Solution
The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan^(-1) |(m1 - m2) / (1 + m1*m2)| = tan^(-1)(5/0) = 90 degrees.
Correct Answer: A — 90 degrees
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Q. What is the condition for the lines 2x + 3y = 6 and 4x + 6y = 12 to be parallel?
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A.
They have the same slope
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B.
They intersect
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C.
They are identical
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D.
None of the above
Solution
Both lines can be rewritten in slope-intercept form. The first line has slope -2/3 and the second line has the same slope, hence they are parallel.
Correct Answer: A — They have the same slope
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Q. What is the condition for two lines ax + by + c1 = 0 and ax + by + c2 = 0 to be parallel?
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A.
c1 = c2
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B.
a/b = c1/c2
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C.
a/b = c2/c1
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D.
a = 0
Solution
Two lines are parallel if their coefficients of x and y are proportional, which means c1 must equal c2.
Correct Answer: A — c1 = c2
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Q. What is the condition for two lines to be parallel?
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A.
m1 = m2
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B.
m1 + m2 = 0
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C.
m1 * m2 = -1
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D.
m1 - m2 = 0
Solution
Two lines are parallel if their slopes are equal, i.e., m1 = m2.
Correct Answer: A — m1 = m2
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Q. What is the condition for two lines to be perpendicular?
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A.
m1 * m2 = -1
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B.
m1 + m2 = 0
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C.
m1 - m2 = 1
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D.
m1 * m2 = 1
Solution
Two lines are perpendicular if the product of their slopes m1 and m2 is -1.
Correct Answer: A — m1 * m2 = -1
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Q. What is the equation of the line parallel to y = 3x + 4 that passes through the point (0, -2)?
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A.
y = 3x - 2
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B.
y = -3x - 2
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C.
y = 3x + 2
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D.
y = -3x + 4
Solution
Parallel lines have the same slope. The slope is 3, so using point-slope form: y + 2 = 3(x - 0) => y = 3x - 2.
Correct Answer: A — y = 3x - 2
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Q. What is the equation of the line parallel to y = 3x - 2 and passing through the point (2, 5)?
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A.
y = 3x + 1
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B.
y = 3x - 1
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C.
y = 3x + 2
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D.
y = 3x - 2
Solution
The slope of the given line is 3. Using point-slope form: y - 5 = 3(x - 2) gives y = 3x + 1.
Correct Answer: A — y = 3x + 1
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Q. What is the equation of the line parallel to y = 3x - 2 that passes through the point (2, 5)?
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A.
y = 3x + 1
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B.
y = 3x - 1
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C.
y = 3x + 2
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D.
y = 3x - 2
Solution
Since parallel lines have the same slope, the equation is y - 5 = 3(x - 2) which simplifies to y = 3x + 1.
Correct Answer: A — y = 3x + 1
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Q. What is the equation of the line parallel to y = 4x - 5 and passing through (2, 3)?
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A.
y = 4x - 5
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B.
y = 4x - 1
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C.
y = 4x + 5
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D.
y = 4x + 3
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 4(x - 2) => y = 4x - 5.
Correct Answer: B — y = 4x - 1
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Q. What is the equation of the line parallel to y = 4x - 5 that passes through the point (2, 3)?
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A.
y = 4x - 5
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B.
y = 4x - 1
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C.
y = 4x + 5
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D.
y = 4x + 3
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 4(x - 2) => y = 4x - 8 + 3 => y = 4x - 5.
Correct Answer: B — y = 4x - 1
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Q. What is the equation of the line parallel to y = 5x - 2 and passing through the point (2, 3)?
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A.
y = 5x - 7
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B.
y = 5x + 7
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C.
y = 5x - 2
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D.
y = 5x + 2
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 5(x - 2) gives y = 5x - 7.
Correct Answer: A — y = 5x - 7
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Q. What is the equation of the line that is perpendicular to y = 3x + 1 and passes through the point (2, 3)?
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A.
y = -1/3x + 4
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B.
y = 3x - 3
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C.
y = -3x + 9
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D.
y = 1/3x + 2
Solution
The slope of the given line is 3, so the perpendicular slope is -1/3. Using point-slope form: y - 3 = -1/3(x - 2) gives y = -1/3x + 4.
Correct Answer: A — y = -1/3x + 4
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Q. What is the equation of the line that is perpendicular to y = 3x + 2 and passes through the point (2, 3)?
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A.
y = -1/3x + 4
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B.
y = 3x - 3
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C.
y = -3x + 9
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D.
y = 1/3x + 2
Solution
The slope of the perpendicular line is -1/3. Using point-slope form: y - 3 = -1/3(x - 2) gives y = -1/3x + 4.
Correct Answer: A — y = -1/3x + 4
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Q. What is the equation of the line that is perpendicular to y = 3x + 4 and passes through the origin?
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A.
y = -1/3x
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B.
y = 3x
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C.
y = -3x
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D.
y = 1/3x
Solution
The slope of the given line is 3. The slope of the perpendicular line is -1/3. Thus, the equation is y = -1/3x.
Correct Answer: A — y = -1/3x
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Q. What is the equation of the line with slope 3 that passes through the point (1, 2)?
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A.
y = 3x + 2
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B.
y = 3x - 1
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C.
y - 2 = 3(x - 1)
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D.
y = 2x + 1
Solution
Using point-slope form: y - y1 = m(x - x1) => y - 2 = 3(x - 1).
Correct Answer: C — y - 2 = 3(x - 1)
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Q. What is the equation of the line with slope 5 that passes through the point (1, 2)?
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A.
y = 5x - 3
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B.
y = 5x + 2
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C.
y = 5x + 1
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D.
y = 5x - 2
Solution
Using point-slope form: y - 2 = 5(x - 1) gives y = 5x - 3.
Correct Answer: C — y = 5x + 1
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Q. What is the length of the segment of the line 3x + 4y = 12 between the x-axis and y-axis?
Solution
The x-intercept is (4, 0) and the y-intercept is (0, 3). The length of the segment is sqrt((4-0)^2 + (0-3)^2) = sqrt(16 + 9) = 5.
Correct Answer: B — 6
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Q. What is the slope of the line perpendicular to the line y = -2x + 3?
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A.
-1/2
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B.
1/2
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C.
2
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D.
-2
Solution
The slope of the given line is -2. The slope of the perpendicular line is the negative reciprocal, which is 1/2.
Correct Answer: B — 1/2
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Q. What is the slope of the line perpendicular to the line y = -2x + 4?
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A.
0.5
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B.
2
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C.
-0.5
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D.
-2
Solution
The slope of the given line is -2. The slope of the perpendicular line is the negative reciprocal, which is 1/2.
Correct Answer: B — 2
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Q. What is the slope of the line perpendicular to the line y = -3x + 4?
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A.
1/3
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B.
-1/3
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C.
3
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D.
-3
Solution
The slope of the given line is -3. The slope of the perpendicular line is the negative reciprocal: 1/3.
Correct Answer: C — 3
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Q. What is the slope of the line represented by the equation 5y - 10x = 20?
Solution
Rearranging to slope-intercept form: 5y = 10x + 20 => y = 2x + 4. The slope is 2.
Correct Answer: C — 0.5
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Q. What is the slope of the line that passes through the points (0, 0) and (4, 8)?
Solution
The slope m = (8 - 0) / (4 - 0) = 2.
Correct Answer: C — 2
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Q. What is the x-intercept of the line 3x + 4y - 12 = 0?
Solution
To find the x-intercept, set y = 0. Thus, 3x - 12 = 0 gives x = 4.
Correct Answer: B — 3
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Q. What is the y-intercept of the line 5x + 2y - 10 = 0?
Solution
Setting x = 0 in the equation gives 2y - 10 = 0, thus y = 5.
Correct Answer: C — 2
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Q. What is the y-intercept of the line represented by the equation 5x + 2y = 10?
Solution
Set x = 0: 2y = 10 => y = 5. The y-intercept is (0, 5).
Correct Answer: B — 2
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Q. Which of the following lines is parallel to the line 4x - 5y + 10 = 0?
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A.
y = (4/5)x + 2
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B.
y = (5/4)x - 1
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C.
y = (4/5)x - 3
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D.
y = (-5/4)x + 1
Solution
The slope of the given line is 4/5. A line parallel to it must have the same slope, hence y = (4/5)x - 3.
Correct Answer: C — y = (4/5)x - 3
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