Q. Find the family of curves represented by the equation y = mx + c, where m and c are constants.
A.
Straight lines with varying slopes and intercepts
B.
Parabolas with varying vertices
C.
Circles with varying radii
D.
Ellipses with varying axes
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Solution
The equation y = mx + c represents straight lines where m is the slope and c is the y-intercept.
Correct Answer: A — Straight lines with varying slopes and intercepts
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Q. Find the focus of the parabola defined by the equation x^2 = 12y.
A.
(0, 3)
B.
(0, -3)
C.
(3, 0)
D.
(-3, 0)
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Solution
The equation x^2 = 12y can be rewritten as (y - 0) = (1/3)(x - 0)^2, indicating the focus is at (0, 3).
Correct Answer: A — (0, 3)
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Q. Find the focus of the parabola given by the equation y^2 = 12x.
A.
(3, 0)
B.
(0, 3)
C.
(0, 6)
D.
(6, 0)
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Solution
The standard form of a parabola is y^2 = 4px. Here, 4p = 12, so p = 3. The focus is at (p, 0) = (3, 0).
Correct Answer: C — (0, 6)
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Q. Find the length of the latus rectum of the parabola y^2 = 16x.
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Solution
The length of the latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 16, so p = 4. Therefore, the length of the latus rectum is 4 * 4 = 16.
Correct Answer: B — 8
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Q. Find the length of the line segment joining the points (-1, -1) and (2, 3).
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Solution
Length = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.
Correct Answer: A — 5
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Q. Find the length of the line segment joining the points (1, 1) and (4, 5).
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Solution
Length = √[(4-1)² + (5-1)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Correct Answer: C — 5
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Q. Find the length of the line segment joining the points (1, 2) and (1, 5).
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Solution
Length = |5 - 2| = 3.
Correct Answer: A — 3
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Q. Find the midpoint of the line segment joining the points (1, 2) and (3, 4).
A.
(2, 3)
B.
(1, 2)
C.
(3, 4)
D.
(4, 5)
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Solution
Midpoint = ((1+3)/2, (2+4)/2) = (2, 3).
Correct Answer: A — (2, 3)
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Q. Find the midpoint of the line segment joining the points (2, 3) and (4, 7).
A.
(3, 5)
B.
(2, 5)
C.
(4, 3)
D.
(5, 6)
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Solution
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 4)/2, (3 + 7)/2) = (3, 5).
Correct Answer: A — (3, 5)
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Q. Find the point of intersection of the lines y = 2x + 1 and y = -x + 4.
A.
(1, 3)
B.
(2, 5)
C.
(3, 7)
D.
(4, 9)
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Solution
Setting 2x + 1 = -x + 4 gives 3x = 3, thus x = 1. Substituting x back gives y = 3, so the point is (1, 3).
Correct Answer: A — (1, 3)
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Q. Find the point of intersection of the lines y = x + 1 and y = -x + 5.
A.
(2, 3)
B.
(3, 2)
C.
(1, 2)
D.
(0, 1)
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Solution
Set x + 1 = -x + 5. Solving gives x = 2, y = 3. Thus, the point is (2, 3).
Correct Answer: A — (2, 3)
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Q. Find the slope of the line passing through the points (2, 3) and (4, 7).
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Solution
The slope m is given by (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2.
Correct Answer: A — 2
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Q. Find the slope of the line represented by the equation 2x - 3y + 6 = 0.
A.
2/3
B.
-2/3
C.
3/2
D.
-3/2
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Solution
Rearranging gives y = (2/3)x + 2, so slope = 2/3.
Correct Answer: B — -2/3
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Q. Find the slope of the line that passes through the points (0, 0) and (5, 5).
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Solution
The slope m = (5 - 0) / (5 - 0) = 1.
Correct Answer: B — 1
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Q. Find the slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0.
A.
-1, -2
B.
-3, -1
C.
1, 2
D.
2, 3
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Solution
The slopes can be found by solving the quadratic equation for m in terms of x and y.
Correct Answer: B — -3, -1
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Q. Find the slopes of the lines represented by the equation 6x^2 - 5xy + y^2 = 0.
A.
-1/6, 5
B.
1/6, -5
C.
5/6, -1
D.
1, -1
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Solution
The slopes can be calculated from the quadratic equation, yielding slopes of 5/6 and -1.
Correct Answer: C — 5/6, -1
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Q. Find the y-intercept of the line represented by the equation 5x - 2y = 10.
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Solution
Set x = 0: -2y = 10 => y = -5. The y-intercept is (0, -5).
Correct Answer: B — 2
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Q. For the ellipse defined by the equation 9x^2 + 16y^2 = 144, what are the lengths of the semi-major and semi-minor axes?
A.
3, 4
B.
4, 3
C.
6, 8
D.
8, 6
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Solution
The semi-major axis is 4 and the semi-minor axis is 3 after rewriting the equation in standard form.
Correct Answer: A — 3, 4
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Q. For the hyperbola x^2/25 - y^2/16 = 1, what is the distance between the foci?
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Solution
The distance between the foci of the hyperbola is 2c, where c = √(a^2 + b^2) = √(25 + 16) = √41, so the distance is 2√41.
Correct Answer: A — 10
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Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, find the slopes of the lines.
A.
-3/2, -1
B.
1, -1/3
C.
0, -1
D.
1, 1
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Solution
The slopes can be found by solving the quadratic equation derived from the given equation.
Correct Answer: A — -3/2, -1
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Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the product of the slopes?
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Solution
The product of the slopes of the lines can be found from the equation, which gives -1.
Correct Answer: A — -1
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Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the sum of the slopes?
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Solution
The sum of the slopes can be found using the relationship between the coefficients of the quadratic.
Correct Answer: A — -3
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Q. For the lines represented by the equation 3x^2 - 2xy + y^2 = 0 to be parallel, the condition is:
A.
3 + 1 = 0
B.
3 - 1 = 0
C.
2 = 0
D.
None of the above
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Solution
The condition for parallel lines is that the determinant of the coefficients must equal zero.
Correct Answer: A — 3 + 1 = 0
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Q. For the lines represented by the equation 4x^2 - 12xy + 9y^2 = 0, find the slopes of the lines.
A.
1, 3
B.
2, 4
C.
3, 1
D.
0, 0
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Solution
Factoring the equation gives the slopes as m1 = 1 and m2 = 3.
Correct Answer: A — 1, 3
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Q. For the lines represented by the equation 4x^2 - 4xy + y^2 = 0, the angle between them is:
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
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Solution
The lines are at an angle of 45 degrees as the determinant of the coefficients gives a non-zero value.
Correct Answer: B — 45 degrees
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Q. For the lines represented by the equation 5x^2 + 6xy + 5y^2 = 0, what is the sum of the slopes?
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Solution
The sum of the slopes is given by - (coefficient of xy)/(coefficient of x^2) = -6/5.
Correct Answer: A — -6/5
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Q. For the lines represented by the equation 6x^2 + 5xy + y^2 = 0, what is the sum of the slopes?
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Solution
The sum of the slopes of the lines is given by -b/a, which is -5/6.
Correct Answer: A — -5/6
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Q. For the lines represented by the equation x^2 - 2xy + y^2 = 0, find the slopes of the lines.
A.
1, -1
B.
2, -2
C.
0, 0
D.
1, 1
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Solution
The slopes can be found by solving the quadratic equation formed by the coefficients of x^2, xy, and y^2.
Correct Answer: A — 1, -1
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Q. For the lines represented by the equation x^2 - 2xy + y^2 = 0, the angle between them is:
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
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Solution
The angle can be calculated using the slopes derived from the equation.
Correct Answer: B — 45 degrees
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Q. For the parabola defined by the equation y^2 = 20x, what is the coordinates of the vertex?
A.
(0, 0)
B.
(5, 0)
C.
(0, 5)
D.
(10, 0)
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Solution
The vertex of the parabola y^2 = 4px is at (0, 0).
Correct Answer: A — (0, 0)
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