Q. The general form of the family of exponential curves is given by:
A.
y = a^x
B.
y = ax^2 + bx + c
C.
y = mx + c
D.
y = log(x)
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Solution
The equation y = a^x represents an exponential function where a is a constant.
Correct Answer: A — y = a^x
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Q. The lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:
A.
Coincident
B.
Parallel
C.
Intersecting
D.
Perpendicular
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Solution
To determine the nature of the lines, we can analyze the discriminant of the quadratic equation.
Correct Answer: C — Intersecting
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Q. The lines represented by the equation 4x^2 - 12xy + 9y^2 = 0 are:
A.
Parallel
B.
Coincident
C.
Intersecting
D.
Perpendicular
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Solution
The lines are perpendicular if the product of their slopes is -1. We can find the slopes from the equation and check this condition.
Correct Answer: D — Perpendicular
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Q. The lines represented by the equation 5x^2 - 6xy + 5y^2 = 0 are:
A.
Parallel
B.
Perpendicular
C.
Coincident
D.
Intersecting
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Solution
The discriminant is negative, indicating that the lines are perpendicular.
Correct Answer: B — Perpendicular
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Q. The lines represented by the equation 5x^2 - 6xy + 5y^2 = 0 intersect at:
A.
(0,0)
B.
(1,1)
C.
(2,2)
D.
(3,3)
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Solution
The lines intersect at the origin (0,0) as derived from the equation.
Correct Answer: A — (0,0)
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Q. The lines represented by the equation 5x^2 - 6xy + y^2 = 0 intersect at which point?
A.
(0,0)
B.
(1,1)
C.
(2,2)
D.
(3,3)
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Solution
The lines intersect at the origin, which can be verified by substituting x = 0 and y = 0 into the equation.
Correct Answer: A — (0,0)
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Q. The lines represented by the equation 6x^2 - 5xy + y^2 = 0 are:
A.
Parallel
B.
Coincident
C.
Intersecting
D.
Perpendicular
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Solution
The lines are perpendicular if the product of their slopes is -1, which can be verified from the equation.
Correct Answer: D — Perpendicular
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Q. The lines represented by the equation x^2 + 2xy + y^2 = 0 are:
A.
Parallel
B.
Intersecting
C.
Coincident
D.
Perpendicular
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Solution
The lines intersect at the origin and are not parallel, hence they are intersecting.
Correct Answer: B — Intersecting
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Q. The lines represented by the equation x^2 - 6x + y^2 - 8y + 9 = 0 are:
A.
Parallel
B.
Coincident
C.
Intersecting
D.
Perpendicular
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Solution
Completing the square shows that the lines intersect at two distinct points.
Correct Answer: C — Intersecting
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Q. The lines represented by the equation x^2 - 6xy + 9y^2 = 0 are:
A.
Coincident
B.
Parallel
C.
Intersecting
D.
Perpendicular
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Solution
The equation can be factored as (x - 3y)^2 = 0, indicating that the lines are coincident.
Correct Answer: A — Coincident
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Q. The pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0 has slopes:
A.
-1, -2
B.
1, 2
C.
0, ∞
D.
1, -1
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Solution
The slopes can be found by solving the quadratic equation in terms of m, yielding slopes -1 and -2.
Correct Answer: A — -1, -2
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Q. The pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0 has:
A.
Two distinct real roots
B.
One real root
C.
No real roots
D.
Two complex roots
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Solution
The discriminant of the quadratic equation is positive, indicating two distinct real roots.
Correct Answer: A — Two distinct real roots
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Q. The pair of lines represented by the equation 2x^2 - 3xy + y^2 = 0 has slopes m1 and m2. What is the product m1*m2?
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Solution
The product of the slopes of the lines is given by m1*m2 = c/a = 1/2 = -2.
Correct Answer: A — -2
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Q. The pair of lines represented by the equation 4x^2 - 12xy + 9y^2 = 0 are:
A.
Parallel
B.
Intersecting
C.
Coincident
D.
Perpendicular
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Solution
Factoring gives (2x - 3y)(2x - 3y) = 0, indicating the lines are coincident.
Correct Answer: D — Perpendicular
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Q. The pair of lines represented by the equation 5x^2 + 6xy + 2y^2 = 0 has:
A.
Two distinct real roots
B.
One real root
C.
No real roots
D.
Infinite roots
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Solution
The discriminant of the quadratic equation is positive, indicating two distinct real roots.
Correct Answer: A — Two distinct real roots
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Q. The pair of lines represented by the equation 5x^2 + 6xy + 5y^2 = 0 are:
A.
Real and distinct
B.
Imaginary
C.
Coincident
D.
Real and coincident
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Solution
The discriminant of the quadratic equation is negative, indicating imaginary lines.
Correct Answer: B — Imaginary
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Q. The pair of lines represented by the equation x^2 - 4x + y^2 - 4y = 0 are:
A.
Parallel
B.
Perpendicular
C.
Coincident
D.
Intersecting
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Solution
Rearranging gives (x-2)^2 + (y-2)^2 = 0, which represents two intersecting lines.
Correct Answer: D — Intersecting
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Q. The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 8 = 0 are:
A.
Parallel
B.
Intersecting
C.
Coincident
D.
Perpendicular
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Solution
To determine the nature of the lines, we can rewrite the equation in the form of (x - a)^2 + (y - b)^2 = r^2 and analyze the discriminant.
Correct Answer: B — Intersecting
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Q. The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 9 = 0 are:
A.
Parallel
B.
Intersecting
C.
Coincident
D.
Perpendicular
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Solution
Rearranging gives (x-2)^2 + (y-3)^2 = 0, which represents a single point, hence the lines are coincident.
Correct Answer: B — Intersecting
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Q. The pair of lines represented by the equation x^2 - 4xy + 3y^2 = 0 are:
A.
Parallel
B.
Perpendicular
C.
Intersecting
D.
Coincident
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Solution
To determine the nature of the lines, we can find the slopes from the equation. The product of the slopes will help us conclude if they are perpendicular.
Correct Answer: B — Perpendicular
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Q. The pair of straight lines represented by the equation x^2 - 4xy + y^2 = 0 are:
A.
Parallel
B.
Perpendicular
C.
Coincident
D.
Intersecting at a point
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Solution
The given equation can be factored as (x - 2y)(x - 2y) = 0, indicating that the lines are perpendicular.
Correct Answer: B — Perpendicular
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Q. The parabola y = -3(x - 2)^2 + 5 opens in which direction?
A.
Upwards
B.
Downwards
C.
Left
D.
Right
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Solution
Since the coefficient of (x - 2)^2 is negative, the parabola opens downwards.
Correct Answer: B — Downwards
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Q. The slope of the line represented by the equation 2x - 3y + 6 = 0 is:
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. The slope of the line represented by the equation 3x - 4y + 12 = 0 is:
A.
3/4
B.
4/3
C.
-3/4
D.
-4/3
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Solution
Rearranging gives y = (3/4)x + 3. Slope = 3/4.
Correct Answer: C — -3/4
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Q. The slopes of the lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:
A.
-1, -2
B.
1, 2
C.
-1, 1
D.
2, -2
Show solution
Solution
The slopes can be found by solving the quadratic equation derived from the given equation.
Correct Answer: A — -1, -2
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Q. The slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0 are given by:
A.
-3/5 and -2/5
B.
2/5 and -5/2
C.
1/2 and -2
D.
None of the above
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Solution
Using the quadratic formula, the slopes are found to be -3/5 and -2/5.
Correct Answer: A — -3/5 and -2/5
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Q. The slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0 are:
A.
-3/5, -2/5
B.
2/5, 3/5
C.
1, -1
D.
0, ∞
Show solution
Solution
The slopes can be calculated using the quadratic formula, yielding -3/5 and -2/5.
Correct Answer: A — -3/5, -2/5
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Q. The vertices of the ellipse 9x^2 + 16y^2 = 144 are located at?
A.
(±4, 0)
B.
(0, ±3)
C.
(±3, 0)
D.
(0, ±4)
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Solution
The vertices of the ellipse 9x^2 + 16y^2 = 144 are located at (±3, 0).
Correct Answer: C — (±3, 0)
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Q. What is the angle between the lines 2x + 3y - 6 = 0 and 4x - y + 1 = 0?
A.
45 degrees
B.
60 degrees
C.
90 degrees
D.
30 degrees
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Solution
The slopes of the lines are -2/3 and 4. The angle θ can be found using tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
Correct Answer: C — 90 degrees
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Q. What is the angle between the lines represented by the equation 2x^2 + 3xy - 2y^2 = 0?
A.
45 degrees
B.
60 degrees
C.
90 degrees
D.
30 degrees
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Solution
Using the formula for the angle between two lines, we find that the angle is 90 degrees.
Correct Answer: C — 90 degrees
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