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Q. For the hyperbola x^2/25 - y^2/16 = 1, what is the distance between the foci?

A. 10
B. 12
C. 8
D. 6

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?

A. -1
B. 0
C. 1
D. 2

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?

A. -3
B. 0
C. 3
D. 1

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?

A. -6/5
B. 0
C. 6/5
D. 1

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?

A. -5/6
B. 5/6
C. 0
D. 1

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 \( |B| \)?

A. 0
B. 1
C. 2
D. 4

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:

A. 0
B. 1
C. 2
D. 3

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?

A. 5
B. 6
C. 7
D. 8

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?

A. 2
B. -2
C. 3
D. -3

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?

A. 5
B. 10
C. 0
D. 25

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?

A. 6
B. 9
C. 12
D. 0

Solution

Setting the discriminant to zero: (-6)^2 - 4*1*k = 0 gives k = 9.

Correct Answer: B — 9

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