Q. For the parabola y^2 = 20x, what is the coordinates of the vertex?
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A.
(0, 0)
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B.
(5, 0)
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C.
(0, 5)
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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?
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A.
k >= 0
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B.
k <= 0
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C.
k >= 2
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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?
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A.
1 and 2
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B.
2 and 1
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C.
3 and 0
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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?
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A.
Real and distinct
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B.
Real and equal
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C.
Complex
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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?
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A.
(-1, 0)
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B.
(-1, 1)
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C.
(0, 1)
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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:
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A.
< 0
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B.
≥ 0
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C.
≤ 0
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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?
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A.
Real and distinct
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B.
Real and equal
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C.
Complex
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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?
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A.
k >= 4
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B.
k <= 4
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C.
k > 0
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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?
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A.
-2 and -4
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B.
-4 and -2
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C.
2 and 4
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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?
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A.
Two distinct real roots
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B.
One real root
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C.
No real roots
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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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Q. For the set E = {1, 2, 3, 4}, how many subsets contain the element 1?
Solution
If 1 is included, we can choose from the remaining elements {2, 3, 4}, which has 2^3 = 8 subsets.
Correct Answer: C — 12
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Q. For the set E = {1, 2, 3, 4}, how many subsets have exactly 2 elements?
Solution
The number of ways to choose 2 elements from 4 is given by the combination formula C(4,2) = 6.
Correct Answer: B — 6
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Q. For the set F = {a, b, c}, how many subsets have exactly 2 elements?
Solution
The subsets with exactly 2 elements are {a, b}, {a, c}, and {b, c}. Total = 3.
Correct Answer: C — 3
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Q. For the vectors A = (1, 0, 0) and B = (0, 1, 0), what is the scalar product A · B?
Solution
A · B = 1*0 + 0*1 + 0*0 = 0.
Correct Answer: A — 0
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Q. For vectors A = (2, 3) and B = (4, 5), find the scalar product A · B.
Solution
A · B = 2*4 + 3*5 = 8 + 15 = 23.
Correct Answer: A — 23
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Q. For vectors A = (3, -2, 1) and B = (1, 4, -2), find A · B.
Solution
A · B = 3*1 + (-2)*4 + 1*(-2) = 3 - 8 - 2 = -7.
Correct Answer: A — -1
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Q. For what value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x >= 1 continuous at x = 1?
Solution
Setting 1^3 - 3(1) + b = 2(1) + 1 gives b = 2.
Correct Answer: B — 1
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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable at x = -1?
Solution
To ensure differentiability at x = -1, we find f'(-1) exists. Setting a = 0 ensures the derivative is defined.
Correct Answer: B — 0
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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?
Solution
The function is a polynomial and is differentiable for all real numbers, hence any value of a works.
Correct Answer: B — 0
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Q. For which value of a is the function f(x) = x^2 - ax + 2 differentiable at x = 1?
Solution
Setting the derivative f'(1) = 0 gives a = 1 for differentiability.
Correct Answer: B — 1
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Q. For which value of a is the function f(x) = x^2 - ax + 4 differentiable at x = 2?
Solution
f(x) is a polynomial and is differentiable for all a, hence any value of a works.
Correct Answer: A — 0
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Q. For which value of a is the function f(x) = x^3 - 3ax + 2 differentiable at x = 1?
Solution
Setting f'(1) = 0 gives a = 1, ensuring differentiability at that point.
Correct Answer: B — 1
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Q. For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiable at x = 1?
Solution
Setting f'(1) = 0 gives a = 1 for differentiability at x = 1.
Correct Answer: B — 1
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Q. For which value of a is the function f(x) = { 2x + a, x < 0; x^2 + 1, x >= 0 continuous at x = 0?
Solution
Setting a = 1 gives continuity at x = 0.
Correct Answer: B — 0
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Q. For which value of a is the function f(x) = { 3x + a, x < 2; 4x - 1, x >= 2 continuous at x = 2?
Solution
Setting 3(2) + a = 4(2) - 1 gives a = 1.
Correct Answer: A — -1
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