Q. For the hyperbola x^2/25 - y^2/16 = 1, what is the distance between the foci?
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
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?
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?
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
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
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
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?
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?
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
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
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 matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is the determinant \( |B| \)?
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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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)
Solution
The vertex of the parabola y^2 = 4px is at (0, 0).
Correct Answer: A — (0, 0)
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Q. For the parabola y = x^2 - 4x + 3, find the coordinates of the vertex.
-
A.
(2, -1)
-
B.
(1, 2)
-
C.
(2, 1)
-
D.
(1, -1)
Solution
To find the vertex, use x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x = 2 into the equation to find y = -1. Thus, the vertex is (2, -1).
Correct Answer: A — (2, -1)
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Q. For the parabola y^2 = 16x, what is the coordinates of the vertex?
-
A.
(0, 0)
-
B.
(4, 0)
-
C.
(0, 4)
-
D.
(0, -4)
Solution
The vertex of the parabola y^2 = 4px is at (0, 0).
Correct Answer: A — (0, 0)
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Q. For the parabola y^2 = 20x, what is the coordinates of the vertex?
-
A.
(0, 0)
-
B.
(5, 0)
-
C.
(0, 5)
-
D.
(10, 0)
Solution
The vertex of the parabola y^2 = 4px is at (0, 0). Here, p = 5, but the vertex remains at (0, 0).
Correct Answer: A — (0, 0)
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Q. For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the condition on k?
-
A.
k >= 0
-
B.
k <= 0
-
C.
k >= 2
-
D.
k <= 2
Solution
The discriminant must be non-negative: (-4)^2 - 4*2*k >= 0, which simplifies to k <= 2.
Correct Answer: C — k >= 2
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Q. For the quadratic equation ax^2 + bx + c = 0, if a = 1, b = -3, and c = 2, what are the roots?
-
A.
1 and 2
-
B.
2 and 1
-
C.
3 and 0
-
D.
0 and 3
Solution
The roots can be found using the quadratic formula: x = (3 ± √(9-8))/2 = 1 and 2.
Correct Answer: A — 1 and 2
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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the nature of the roots?
-
A.
Real and distinct
-
B.
Real and equal
-
C.
Complex
-
D.
None of the above
Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer: B — Real and equal
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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the vertex of the parabola?
-
A.
(-1, 0)
-
B.
(-1, 1)
-
C.
(0, 1)
-
D.
(1, 1)
Solution
The vertex can be found using the formula (-b/2a, f(-b/2a)). Here, vertex is (-1, 0).
Correct Answer: A — (-1, 0)
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Q. For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:
-
A.
< 0
-
B.
≥ 0
-
C.
≤ 0
-
D.
> 0
Solution
The discriminant must be negative: 2^2 - 4*1*k < 0 => 4 < 4k => k > 1.
Correct Answer: A — < 0
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Q. For the quadratic equation x^2 + 4x + 4 = 0, what is the nature of the roots?
-
A.
Real and distinct
-
B.
Real and equal
-
C.
Complex
-
D.
None of the above
Solution
The discriminant is 0 (b^2 - 4ac = 16 - 16 = 0), indicating real and equal roots.
Correct Answer: B — Real and equal
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Q. For the quadratic equation x^2 + 4x + k = 0 to have no real roots, k must be:
Solution
The discriminant must be negative: 4^2 - 4*1*k < 0 => 16 < 4k => k > 4.
Correct Answer: A — 0
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Q. For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the condition on k?
-
A.
k >= 4
-
B.
k <= 4
-
C.
k > 0
-
D.
k < 0
Solution
The discriminant must be non-negative: 4^2 - 4*1*k >= 0, which simplifies to k <= 4.
Correct Answer: A — k >= 4
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Q. For the quadratic equation x^2 + 6x + 8 = 0, what are the roots?
-
A.
-2 and -4
-
B.
-4 and -2
-
C.
2 and 4
-
D.
0 and 8
Solution
Factoring gives (x+2)(x+4) = 0, hence the roots are -2 and -4.
Correct Answer: B — -4 and -2
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Q. For the quadratic equation x^2 + 6x + 9 = 0, what is the nature of the roots?
-
A.
Two distinct real roots
-
B.
One real root
-
C.
No real roots
-
D.
Complex roots
Solution
The discriminant is 0, indicating one real root (a repeated root).
Correct Answer: B — One real root
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Q. For the quadratic equation x^2 + mx + n = 0, if the roots are 2 and 3, what is the value of n?
Solution
The product of the roots is n = 2 * 3 = 6.
Correct Answer: B — 6
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Q. For the quadratic equation x^2 + px + q = 0, if the roots are 1 and -3, what is the value of p?
Solution
The sum of the roots is 1 + (-3) = -2, hence p = -2.
Correct Answer: A — 2
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Q. For the quadratic equation x^2 - 10x + 25 = 0, what is the double root?
Solution
The equation can be factored as (x-5)^2 = 0, hence the double root is x = 5.
Correct Answer: A — 5
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Q. For the quadratic equation x^2 - 6x + k = 0 to have equal roots, what must be the value of k?
Solution
Setting the discriminant to zero: (-6)^2 - 4*1*k = 0 gives k = 9.
Correct Answer: B — 9
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