Q. What is the second derivative of f(x) = e^x?
Solution
The second derivative f''(x) = e^x.
Correct Answer: A — e^x
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Q. What is the second derivative of f(x) = x^4 - 4x^3 + 6x^2?
-
A.
12x - 12
-
B.
12x - 8
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C.
12x - 6
-
D.
12x - 4
Solution
f''(x) = 12x - 12.
Correct Answer: A — 12x - 12
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Q. What is the slope of the line passing through the points (1, 2) and (3, 6)?
Solution
Slope = (6-2)/(3-1) = 4/2 = 2.
Correct Answer: A — 2
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Q. What is the slope of the line passing through the points (2, 3) and (4, 7)?
Solution
Slope = (7-3)/(4-2) = 4/2 = 2.
Correct Answer: A — 2
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Q. What is the slope of the line perpendicular to the line y = -2x + 3?
-
A.
-1/2
-
B.
1/2
-
C.
2
-
D.
-2
Solution
The slope of the given line is -2. The slope of the perpendicular line is the negative reciprocal, which is 1/2.
Correct Answer: B — 1/2
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Q. What is the slope of the line perpendicular to the line y = -2x + 4?
-
A.
0.5
-
B.
2
-
C.
-0.5
-
D.
-2
Solution
The slope of the given line is -2. The slope of the perpendicular line is the negative reciprocal, which is 1/2.
Correct Answer: B — 2
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Q. What is the slope of the line perpendicular to the line y = -3x + 4?
-
A.
1/3
-
B.
-1/3
-
C.
3
-
D.
-3
Solution
The slope of the given line is -3. The slope of the perpendicular line is the negative reciprocal: 1/3.
Correct Answer: C — 3
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Q. What is the slope of the line represented by the equation 2x - 3y + 6 = 0?
-
A.
2/3
-
B.
3/2
-
C.
-2/3
-
D.
-3/2
Solution
Rearranging gives y = (2/3)x + 2, slope = 2/3.
Correct Answer: C — -2/3
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Q. What is the slope of the line represented by the equation 3x - 4y + 12 = 0?
-
A.
3/4
-
B.
4/3
-
C.
-3/4
-
D.
-4/3
Solution
Rearranging gives y = (3/4)x + 3. Slope = 3/4.
Correct Answer: C — -3/4
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Q. What is the slope of the line represented by the equation 5y - 10x = 20?
Solution
Rearranging to slope-intercept form: 5y = 10x + 20 => y = 2x + 4. The slope is 2.
Correct Answer: C — 0.5
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Q. What is the slope of the line that passes through the points (0, 0) and (4, 8)?
Solution
The slope m = (8 - 0) / (4 - 0) = 2.
Correct Answer: C — 2
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Q. What is the slope of the lines represented by the equation 5x^2 - 10xy + 5y^2 = 0?
Solution
The equation can be factored to find the slopes, which are both 1.
Correct Answer: A — 1
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Q. What is the slope of the lines represented by the equation x^2 - 6xy + 9y^2 = 0?
Solution
Factoring gives (x - 3y)^2 = 0, indicating a double root, hence the slope is 3.
Correct Answer: A — 3
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Q. What is the slope of the tangent line to the curve y = x^2 + 2x at the point (1, 3)?
Solution
The derivative y' = 2x + 2. At x = 1, y' = 2(1) + 2 = 4, which is the slope of the tangent line.
Correct Answer: B — 2
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Q. What is the solution for the inequality 4 - x < 2?
-
A.
x > 2
-
B.
x < 2
-
C.
x > 4
-
D.
x < 4
Solution
4 - x < 2 => -x < -2 => x > 2.
Correct Answer: D — x < 4
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Q. What is the solution for the inequality 4x + 1 < 3x + 5?
-
A.
x < 4
-
B.
x > 4
-
C.
x < 3
-
D.
x > 3
Solution
4x + 1 < 3x + 5 => x < 4.
Correct Answer: A — x < 4
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Q. What is the solution for the inequality 4x - 1 < 3x + 2?
-
A.
x < 3
-
B.
x > 3
-
C.
x < 1
-
D.
x > 1
Solution
4x - 1 < 3x + 2 => x < 3.
Correct Answer: A — x < 3
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Q. What is the solution for the inequality 6 - 2x ≥ 0?
-
A.
x ≤ 3
-
B.
x > 3
-
C.
x < 3
-
D.
x ≥ 3
Solution
6 - 2x ≥ 0 => -2x ≥ -6 => x ≤ 3.
Correct Answer: A — x ≤ 3
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Q. What is the solution for the inequality 7 - 4x < 3?
-
A.
x > 1
-
B.
x < 1
-
C.
x > -1
-
D.
x < -1
Solution
7 - 4x < 3 => -4x < -4 => x > 1.
Correct Answer: B — x < 1
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Q. What is the solution for the inequality 7x + 2 ≤ 5x + 10?
-
A.
x ≤ 4
-
B.
x ≥ 4
-
C.
x ≤ 3
-
D.
x ≥ 3
Solution
7x + 2 ≤ 5x + 10 => 2x ≤ 8 => x ≤ 4.
Correct Answer: A — x ≤ 4
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Q. What is the solution for the inequality 7x + 3 < 24?
-
A.
x < 3
-
B.
x > 3
-
C.
x < 4
-
D.
x > 4
Solution
7x + 3 < 24 => 7x < 21 => x < 3.
Correct Answer: C — x < 4
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Q. What is the solution for the inequality 8 - 3x > 2?
-
A.
x < 2
-
B.
x > 2
-
C.
x < 3
-
D.
x > 3
Solution
8 - 3x > 2 => -3x > -6 => x < 2.
Correct Answer: C — x < 3
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Q. What is the solution for the inequality: 2x + 5 > 3x - 1?
-
A.
x < 6
-
B.
x > 6
-
C.
x < 4
-
D.
x > 4
Solution
2x + 5 > 3x - 1 => 6 > x => x < 6.
Correct Answer: D — x > 4
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Q. What is the solution of the differential equation dy/dx = (x^2 + 1)y?
-
A.
y = Ce^(x^3/3 + x)
-
B.
y = Ce^(x^2 + 1)
-
C.
y = Ce^(x^2/2)
-
D.
y = Ce^(x^3)
Solution
This is a separable equation. Integrating gives y = Ce^(x^3/3 + x).
Correct Answer: A — y = Ce^(x^3/3 + x)
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Q. What is the solution of the equation y'' - 4y' + 4y = 0?
-
A.
y = (C1 + C2x)e^(2x)
-
B.
y = C1 e^(2x) + C2 e^(-2x)
-
C.
y = C1 e^(4x) + C2 e^(-4x)
-
D.
y = C1 cos(2x) + C2 sin(2x)
Solution
The characteristic equation has a repeated root, leading to the solution form (C1 + C2x)e^(2x).
Correct Answer: A — y = (C1 + C2x)e^(2x)
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Q. What is the solution set for the inequality -x + 6 > 0?
-
A.
x < 6
-
B.
x > 6
-
C.
x < 0
-
D.
x > 0
Solution
-x + 6 > 0 => -x > -6 => x < 6.
Correct Answer: A — x < 6
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Q. What is the solution set for the inequality 4x + 1 < 9?
-
A.
x < 2
-
B.
x > 2
-
C.
x < 1
-
D.
x > 1
Solution
4x + 1 < 9 => 4x < 8 => x < 2.
Correct Answer: A — x < 2
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Q. What is the solution set for the inequality 4x - 1 < 3x + 2?
-
A.
x < 3
-
B.
x > 3
-
C.
x < 1
-
D.
x > 1
Solution
4x - 1 < 3x + 2 => x < 3.
Correct Answer: A — x < 3
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Q. What is the solution set for the inequality 5 - 2x < 1?
-
A.
x > 2
-
B.
x < 2
-
C.
x > 3
-
D.
x < 3
Solution
5 - 2x < 1 => -2x < -4 => x > 2.
Correct Answer: A — x > 2
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Q. What is the solution set for the inequality 6x - 4 < 2x + 8?
-
A.
x < 3
-
B.
x > 3
-
C.
x < 2
-
D.
x > 2
Solution
6x - 4 < 2x + 8 => 4x < 12 => x < 3.
Correct Answer: B — x > 3
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