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
Learn More →
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
Learn More →
Q. If the roots of the equation x^2 + 4x + k = 0 are real and equal, what is the minimum value of k?
Solution
For real and equal roots, the discriminant must be zero: 16 - 4k = 0, thus k = 4.
Correct Answer: B — -4
Learn More →
Q. If the roots of the equation x^2 + 5x + 6 = 0 are a and b, what is the value of a + b?
Solution
Using Vieta's formulas, the sum of the roots is -b/a = -5/1 = -5.
Correct Answer: A — 5
Learn More →
Q. If the roots of the equation x^2 + 5x + k = 0 are -2 and -3, find k.
Solution
Using Vieta's formulas, k = (-2)(-3) = 6.
Correct Answer: B — 6
Learn More →
Q. If the roots of the equation x^2 + 6x + k = 0 are -2 and -4, what is the value of k?
Solution
Using the sum and product of roots: -2 + -4 = -6 and -2*-4 = k => k = 8.
Correct Answer: C — 10
Learn More →
Q. If the roots of the equation x^2 + mx + n = 0 are -2 and -3, what is the value of m + n?
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
Learn More →
Q. If the roots of the equation x^2 + px + q = 0 are -2 and -3, what is the value of p + q?
Solution
Using Vieta's formulas, p = -(-2 - 3) = 5 and q = (-2)(-3) = 6. Therefore, p + q = 5 + 6 = 11.
Correct Answer: C — -7
Learn More →
Q. If the roots of the equation x^2 + px + q = 0 are -2 and -3, what is the value of p?
Solution
Using Vieta's formulas, p = -(-2 - 3) = 5.
Correct Answer: A — 5
Learn More →
Q. If the roots of the equation x^2 + px + q = 0 are 1 and -1, what is the value of p?
Solution
The sum of the roots is 0, hence p = -sum = 0.
Correct Answer: A — 0
Learn More →
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
Learn More →
Q. If the roots of the equation x^2 - 5x + k = 0 are equal, what is the value of k?
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
Learn More →
Q. If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the minimum value of k?
Solution
The minimum value of k is 6, as the discriminant must be zero.
Correct Answer: C — 6
Learn More →
Q. If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the value of k?
Solution
For real and equal roots, the discriminant must be zero: 25 - 4k = 0, thus k = 6.
Correct Answer: C — 6
Learn More →
Q. If the roots of the equation x^2 - 6x + k = 0 are 2 and 4, find the value of k.
Solution
Using Vieta's formulas, k = 2 * 4 = 8.
Correct Answer: B — 10
Learn More →
Q. If the roots of the equation x^2 - 7x + p = 0 are 3 and 4, what is the value of p?
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
Learn More →
Q. If the roots of the equation x^2 - 7x + p = 0 are in the ratio 3:4, what is the value of p?
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
Learn More →
Q. If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*8 = 0, solving gives k = 4.
Correct Answer: A — 4
Learn More →
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?
Solution
Using the product of the roots, c = 3 * (-2) = -6.
Correct Answer: A — -6
Learn More →
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
Learn More →
Q. If the roots of the quadratic equation x^2 + mx + n = 0 are 3 and 4, what is the value of m?
Solution
The sum of the roots is 3 + 4 = 7, hence m = -7.
Correct Answer: A — 7
Learn More →
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
Learn More →
Q. If the roots of the quadratic equation x^2 - 3x + p = 0 are 1 and 2, what is the value of p?
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
Learn More →
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
Learn More →
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
Learn More →
Q. If the scores of 10 students are: 50, 60, 70, 80, 90, 100, 50, 60, 70, 80, what is the mode?
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
Learn More →
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?
Solution
Mean = (10 + 20 + 30 + 40 + x) / 5 = 30. Solving gives x = 50.
Correct Answer: C — 50
Learn More →
Q. If the scores of 7 students are 50, 60, 70, 80, 90, 100, and 110, what is the median score?
Solution
Arranging the scores: 50, 60, 70, 80, 90, 100, 110. Median = 80 (4th value).
Correct Answer: B — 80
Learn More →
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
Learn More →
Q. If the scores of a student in five subjects are 60, 70, 80, 90, and 100, what is the median score?
Solution
Arranging the scores: 60, 70, 80, 90, 100. Median = 80 (middle value).
Correct Answer: B — 80
Learn More →
Showing 4051 to 4080 of 10700 (357 Pages)
