Q. If the points A(1, 2), B(3, 4), and C(5, 6) are collinear, what is the area of triangle ABC?
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Solution
Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 0.
Correct Answer: A — 0
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Q. If the vertex of the parabola y = ax^2 + bx + c is at (1, -2), what is the value of a if b = 4 and c = -6?
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Solution
The vertex form of a parabola is given by x = -b/(2a). Here, 1 = -4/(2a) => 2a = -4 => a = -2.
Correct Answer: A — 1
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Q. The angle between the lines represented by the equation 2x^2 + 3xy + y^2 = 0 is:
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
60 degrees
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Solution
Using the angle formula, we find the angle between the lines is 60 degrees.
Correct Answer: D — 60 degrees
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Q. The angle between the lines represented by the equation 3x^2 - 4xy + 2y^2 = 0 is:
A.
30 degrees
B.
45 degrees
C.
60 degrees
D.
90 degrees
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Solution
Using the formula tan(θ) = |(m1 - m2) / (1 + m1*m2)|, we find that the angle is 60 degrees.
Correct Answer: C — 60 degrees
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Q. The area of a rectangle with vertices at (1, 1), (1, 4), (5, 1), and (5, 4) is:
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Solution
Length = 5 - 1 = 4, Width = 4 - 1 = 3. Area = Length * Width = 4 * 3 = 12.
Correct Answer: B — 16
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Q. The condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to be parallel is:
A.
h^2 = ab
B.
h^2 > ab
C.
h^2 < ab
D.
a + b = 0
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Solution
The lines are parallel if h^2 = ab.
Correct Answer: A — h^2 = ab
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Q. The condition for the lines represented by the equation x^2 + 2xy + y^2 = 0 to be coincident is:
A.
Discriminant > 0
B.
Discriminant = 0
C.
Discriminant < 0
D.
None of the above
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Solution
For the lines to be coincident, the discriminant must be equal to zero.
Correct Answer: B — Discriminant = 0
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Q. The condition for the lines represented by the equation x^2 + y^2 + 2xy = 0 to be coincident is:
A.
Discriminant = 0
B.
Discriminant > 0
C.
Discriminant < 0
D.
None of the above
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Solution
For the lines to be coincident, the discriminant of the quadratic must be zero.
Correct Answer: A — Discriminant = 0
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Q. The condition for the lines represented by the equation x^2 + y^2 - 4x - 6y + 9 = 0 to be coincident is:
A.
Discriminant = 0
B.
Discriminant > 0
C.
Discriminant < 0
D.
None of the above
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Solution
For the lines to be coincident, the discriminant of the quadratic must equal zero.
Correct Answer: A — Discriminant = 0
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Q. The coordinates of the centroid of a triangle with vertices at (0, 0), (6, 0), and (3, 6) are:
A.
(3, 2)
B.
(3, 3)
C.
(2, 3)
D.
(0, 0)
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Solution
Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3) = (9/3, 6/3) = (3, 2).
Correct Answer: B — (3, 3)
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Q. The coordinates of the centroid of a triangle with vertices at (2, 3), (4, 5), and (6, 1) are:
A.
(4, 3)
B.
(4, 4)
C.
(3, 3)
D.
(5, 3)
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Solution
Centroid = ((2+4+6)/3, (3+5+1)/3) = (4, 3).
Correct Answer: A — (4, 3)
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Q. The coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (3, 6) are:
A.
(3, 2)
B.
(2, 3)
C.
(3, 3)
D.
(0, 0)
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Solution
Centroid = ((0+6+3)/3, (0+0+6)/3) = (3, 2).
Correct Answer: A — (3, 2)
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Q. The coordinates of the centroid of the triangle with vertices (2, 3), (4, 5), and (6, 7) are:
A.
(4, 5)
B.
(3, 4)
C.
(5, 6)
D.
(6, 5)
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Solution
Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).
Correct Answer: B — (3, 4)
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Q. The distance from the point (1, 2) to the line 2x + 3y - 6 = 0 is:
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Solution
Distance = |2(1) + 3(2) - 6| / √(2² + 3²) = |2 + 6 - 6| / √13 = 2/√13.
Correct Answer: B — 2
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Q. The distance from the point (3, 4) to the line 2x + 3y - 6 = 0 is:
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Solution
Distance = |2(3) + 3(4) - 6| / √(2² + 3²) = |6 + 12 - 6| / √13 = 12/√13.
Correct Answer: B — 2
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Q. The eccentricity of an ellipse is defined as e = c/a. If a = 10 and c = 6, what is the eccentricity?
A.
0.6
B.
0.8
C.
0.4
D.
0.5
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Solution
Eccentricity e = c/a = 6/10 = 0.6.
Correct Answer: B — 0.8
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Q. The equation of a line parallel to y = 2x + 3 and passing through (1, 1) is?
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 2x + 3
D.
y = 2x - 3
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Solution
Parallel lines have the same slope. Using point-slope form: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer: A — y = 2x - 1
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Q. The equation of a line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
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Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of a line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
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Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of a parabola is given by x^2 = 16y. What is the length of the latus rectum?
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Solution
The length of the latus rectum for the parabola x^2 = 4py is given by 4p. Here, 4p = 16, so p = 4. Thus, the length of the latus rectum is 4p = 16.
Correct Answer: B — 8
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Q. The equation of an ellipse is given by 4x^2 + 9y^2 = 36. What is the eccentricity of the ellipse?
A.
0.5
B.
0.6
C.
0.7
D.
0.8
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Solution
Rewriting gives x^2/9 + y^2/4 = 1. Here, a^2 = 9, b^2 = 4, c = √(a^2 - b^2) = √(9 - 4) = √5. Eccentricity e = c/a = √5/3 ≈ 0.6.
Correct Answer: B — 0.6
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Q. The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is given by?
A.
x^2/a^2 + y^2/b^2 = 1
B.
y^2/a^2 + x^2/b^2 = 1
C.
x^2/b^2 + y^2/a^2 = 1
D.
y^2/b^2 + x^2/a^2 = 1
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Solution
The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is y^2/a^2 + x^2/b^2 = 1.
Correct Answer: B — y^2/a^2 + x^2/b^2 = 1
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Q. The equation of the directrix of the parabola y^2 = 8x is?
A.
x = -2
B.
x = 2
C.
y = -4
D.
y = 4
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Solution
The directrix of the parabola y^2 = 8x is given by x = -2.
Correct Answer: A — x = -2
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Q. The equation of the line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
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Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of the line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = 4x - 2
D.
y = x + 1
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of the pair of lines through the origin is given by y = mx. If m1 and m2 are the slopes, what is the condition for them to be perpendicular?
A.
m1 + m2 = 0
B.
m1 * m2 = 1
C.
m1 - m2 = 0
D.
m1 * m2 = -1
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Solution
For two lines to be perpendicular, the product of their slopes must equal -1.
Correct Answer: D — m1 * m2 = -1
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is given by:
A.
y = mx
B.
y^2 = mx
C.
x^2 + y^2 = 0
D.
x^2 - 2mxy + y^2 = 0
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Solution
The correct form of the equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.
Correct Answer: D — x^2 - 2mxy + y^2 = 0
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is:
A.
y = m1x + m2x
B.
y = (m1 + m2)x
C.
y = m1x - m2x
D.
y = m1x * m2x
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Solution
The equation of the lines can be expressed as y = (m1 + m2)x, representing the sum of the slopes.
Correct Answer: B — y = (m1 + m2)x
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Q. The family of curves defined by the equation x^2 + y^2 = r^2 represents:
A.
Ellipses
B.
Hyperbolas
C.
Circles
D.
Parabolas
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Solution
The equation x^2 + y^2 = r^2 represents a circle with radius r.
Correct Answer: C — Circles
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Q. The family of curves defined by the equation y = a(x - h)^2 + k represents which type of function?
A.
Linear
B.
Quadratic
C.
Cubic
D.
Rational
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Solution
The equation y = a(x - h)^2 + k represents a quadratic function in vertex form.
Correct Answer: B — Quadratic
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