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Q. If the roots of the equation x^2 + 3x + k = 0 are real and distinct, what is the range of k?

A. k < 9
B. k > 9
C. k < 0
D. k > 0

Solution

For real and distinct roots, the discriminant must be positive: 3^2 - 4*1*k > 0 => 9 - 4k > 0 => k < 9.

Correct Answer: A — k < 9

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Q. If the roots of the equation x^2 + 4x + k = 0 are real and distinct, what is the condition on k?

A. k < 16
B. k > 16
C. k = 16
D. k <= 16

Solution

The discriminant must be greater than zero: 4^2 - 4*1*k > 0 => 16 - 4k > 0 => k < 4.

Correct Answer: A — k < 16

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Q. If the roots of the equation x^2 + 4x + k = 0 are real and equal, what is the minimum value of k?

A. 0
B. -4
C. -8
D. -16

Solution

For real and equal roots, the discriminant must be zero: 16 - 4k = 0, thus k = 4.

Correct Answer: B — -4

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Q. If the roots of the equation x^2 + 5x + 6 = 0 are a and b, what is the value of a + b?

A. 5
B. 6
C. 4
D. 3

Solution

Using Vieta's formulas, the sum of the roots is -b/a = -5/1 = -5.

Correct Answer: A — 5

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Q. If the roots of the equation x^2 + 5x + k = 0 are -2 and -3, find k.

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

Solution

Using Vieta's formulas, k = (-2)(-3) = 6.

Correct Answer: B — 6

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Q. If the roots of the equation x^2 + 6x + k = 0 are -2 and -4, what is the value of k?

A. 4
B. 8
C. 10
D. 12

Solution

Using the sum and product of roots: -2 + -4 = -6 and -2*-4 = k => k = 8.

Correct Answer: C — 10

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Q. If the roots of the equation x^2 + mx + n = 0 are -2 and -3, what is the value of m + n?

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

Solution

The sum of the roots is -(-2 - 3) = 5, so m = 5. The product of the roots is (-2)(-3) = 6, so n = 6. Thus, m + n = 5 + 6 = 11.

Correct Answer: C — -7

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Q. If the roots of the equation x^2 + px + q = 0 are -2 and -3, what is the value of p + q?

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

Solution

Using Vieta's formulas, p = -(-2 - 3) = 5 and q = (-2)(-3) = 6. Therefore, p + q = 5 + 6 = 11.

Correct Answer: C — -7

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Q. If the roots of the equation x^2 + px + q = 0 are -2 and -3, what is the value of p?

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

Solution

Using Vieta's formulas, p = -(-2 - 3) = 5.

Correct Answer: A — 5

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Q. If the roots of the equation x^2 + px + q = 0 are 1 and -1, what is the value of p?

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

Solution

The sum of the roots is 0, hence p = -sum = 0.

Correct Answer: A — 0

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Q. If the roots of the equation x^2 + px + q = 0 are equal, what is the relationship between p and q?

A. p^2 = 4q
B. p^2 > 4q
C. p^2 < 4q
D. p + q = 0

Solution

For equal roots, the discriminant must be zero: p^2 - 4q = 0, hence p^2 = 4q.

Correct Answer: A — p^2 = 4q

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Q. If the roots of the equation x^2 - 5x + k = 0 are equal, what is the value of k?

A. 0
B. 5
C. 6
D. 25

Solution

For the roots to be equal, the discriminant must be zero. Thus, b^2 - 4ac = 0 => 25 - 4k = 0 => k = 25.

Correct Answer: C — 6

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Q. If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the minimum value of k?

A. 0
B. 5
C. 6
D. 10

Solution

The minimum value of k is 6, as the discriminant must be zero.

Correct Answer: C — 6

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Q. If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the value of k?

A. 0
B. 5
C. 6
D. 25

Solution

For real and equal roots, the discriminant must be zero: 25 - 4k = 0, thus k = 6.

Correct Answer: C — 6

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Q. If the roots of the equation x^2 - 6x + k = 0 are 2 and 4, find the value of k.

A. 8
B. 10
C. 12
D. 14

Solution

Using Vieta's formulas, k = 2 * 4 = 8.

Correct Answer: B — 10

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Q. If the roots of the equation x^2 - 7x + p = 0 are 3 and 4, what is the value of p?

A. 12
B. 15
C. 16
D. 20

Solution

Using Vieta's formulas, the sum of the roots is 7 and the product is p. Thus, 3 * 4 = p, so p = 12.

Correct Answer: C — 16

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Q. If the roots of the equation x^2 - 7x + p = 0 are in the ratio 3:4, what is the value of p?

A. 12
B. 16
C. 20
D. 24

Solution

Let the roots be 3k and 4k. Then, 3k + 4k = 7 => 7k = 7 => k = 1. The product of the roots is 3k * 4k = 12k^2 = p => p = 12.

Correct Answer: C — 20

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Q. If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?

A. 4
B. 8
C. 16
D. 0

Solution

For equal roots, the discriminant must be zero: k^2 - 4*1*8 = 0, solving gives k = 4.

Correct Answer: A — 4

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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are 3 and -2, what is the value of c if a = 1 and b = -1?

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

Solution

Using the product of the roots, c = 3 * (-2) = -6.

Correct Answer: A — -6

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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are equal, what is the condition on a, b, and c?

A. b^2 - 4ac > 0
B. b^2 - 4ac = 0
C. b^2 - 4ac < 0
D. a + b + c = 0

Solution

The condition for equal roots is given by the discriminant b^2 - 4ac = 0.

Correct Answer: B — b^2 - 4ac = 0

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Q. If the roots of the quadratic equation x^2 + mx + n = 0 are 3 and 4, what is the value of m?

A. 7
B. 12
C. 10
D. 8

Solution

The sum of the roots is 3 + 4 = 7, hence m = -7.

Correct Answer: A — 7

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Q. If the roots of the quadratic equation x^2 + px + q = 0 are equal, what is the relationship between p and q?

A. p^2 = 4q
B. p^2 > 4q
C. p^2 < 4q
D. p + q = 0

Solution

For equal roots, the discriminant must be zero: p^2 - 4q = 0, hence p^2 = 4q.

Correct Answer: A — p^2 = 4q

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Q. If the roots of the quadratic equation x^2 - 3x + p = 0 are 1 and 2, what is the value of p?

A. 1
B. 2
C. 3
D. 6

Solution

Using Vieta's formulas, sum of roots = 1 + 2 = 3 and product of roots = 1*2 = 2. Thus, p = 2.

Correct Answer: D — 6

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Q. If the scalar product of two vectors A and B is 0, what can be said about the vectors?

A. They are parallel
B. They are orthogonal
C. They are equal
D. They are collinear

Solution

If A · B = 0, then the vectors are orthogonal.

Correct Answer: B — They are orthogonal

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Q. If the scalar product of vectors A = (x, y, z) and B = (2, -1, 3) is 10, what is the equation?

A. 2x - y + 3z = 10
B. 2x + y + 3z = 10
C. 2x - y - 3z = 10
D. 2x + y - 3z = 10

Solution

A · B = 2x - y + 3z = 10.

Correct Answer: A — 2x - y + 3z = 10

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Q. If the scores of 10 students are: 50, 60, 70, 80, 90, 100, 50, 60, 70, 80, what is the mode?

A. 50
B. 60
C. 70
D. 80

Solution

Mode is the value that appears most frequently. Here, 50, 60, 70, and 80 all appear twice, but 50 is the smallest.

Correct Answer: A — 50

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Q. If the scores of 5 students are 10, 20, 30, 40, and x, and the mean is 30, what is the value of x?

A. 30
B. 40
C. 50
D. 60

Solution

Mean = (10 + 20 + 30 + 40 + x) / 5 = 30. Solving gives x = 50.

Correct Answer: C — 50

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Q. If the scores of 7 students are 50, 60, 70, 80, 90, 100, and 110, what is the median score?

A. 70
B. 80
C. 90
D. 100

Solution

Arranging the scores: 50, 60, 70, 80, 90, 100, 110. Median = 80 (4th value).

Correct Answer: B — 80

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Q. If the scores of a student are 50, 60, 70, 80, and 90, what is the mode?

A. 50
B. 60
C. 70
D. No mode

Solution

All scores occur only once, so there is no mode.

Correct Answer: D — No mode

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Q. If the scores of a student in five subjects are 60, 70, 80, 90, and 100, what is the median score?

A. 70
B. 80
C. 90
D. 100

Solution

Arranging the scores: 60, 70, 80, 90, 100. Median = 80 (middle value).

Correct Answer: B — 80

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