Engineering & Architecture Admissions
Q. What is the solution to the differential equation dy/dx = (x^2 + 1)y?
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A.
y = Ce^(x^3/3 + x)
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B.
y = Ce^(x^2 + 1)
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C.
y = Ce^(x^2/2)
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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)
Q. What is the solution to the differential equation dy/dx = 2y + 3?
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A.
y = Ce^(2x) - 3/2
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B.
y = Ce^(2x) + 3/2
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C.
y = 3e^(2x)
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D.
y = 3/2e^(2x)
Solution
Using the integrating factor method, we find y = Ce^(2x) + 3/2.
Correct Answer: B — y = Ce^(2x) + 3/2
Q. What is the solution to the differential equation dy/dx = y^2?
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A.
y = 1/(C - x)
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B.
y = C/(x - 1)
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C.
y = Cx
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D.
y = e^(x + C)
Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
Q. What is the solution to the differential equation y' = y^2 - 1?
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A.
y = tan(x + C)
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B.
y = 1/(C - x)
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C.
y = C/(1 - Cx)
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D.
y = C/(x - 1)
Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: B — y = 1/(C - x)
Q. What is the solution to the first-order linear differential equation dy/dx + y = e^x?
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A.
y = e^x + Ce^(-x)
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B.
y = e^x - Ce^(-x)
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C.
y = Ce^x - e^x
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D.
y = Ce^(-x) + e^x
Solution
Using the integrating factor e^x, we solve to get y = e^x + Ce^(-x).
Correct Answer: A — y = e^x + Ce^(-x)
Q. What is the solution to the inequality -2x + 4 > 0?
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A.
x < 2
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B.
x > 2
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C.
x ≤ 2
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D.
x ≥ 2
Solution
-2x + 4 > 0 => -2x > -4 => x < 2.
Correct Answer: B — x > 2
Q. What is the solution to the inequality -2x + 5 > 1?
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A.
x < 2
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B.
x > 2
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C.
x < 3
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D.
x > 3
Solution
-2x + 5 > 1 => -2x > -4 => x < 2.
Correct Answer: B — x > 2
Q. What is the solution to the inequality 3(x - 1) < 2(x + 2)?
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A.
x < 5
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B.
x > 5
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C.
x < 4
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D.
x > 4
Solution
3(x - 1) < 2(x + 2) => 3x - 3 < 2x + 4 => x < 7.
Correct Answer: C — x < 4
Q. What is the solution to the inequality 4 - x < 2?
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A.
x > 2
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B.
x < 2
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C.
x > 4
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D.
x < 4
Solution
4 - x < 2 => -x < -2 => x > 2.
Correct Answer: D — x < 4
Q. What is the solution to the inequality 4x + 1 > 3x + 5?
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A.
x < 4
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B.
x > 4
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C.
x < 3
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D.
x > 3
Solution
Subtract 3x from both sides: x > 4.
Correct Answer: B — x > 4
Q. What is the solution to the inequality 4x + 1 ≤ 3?
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A.
x ≤ 0.5
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B.
x < 0.5
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C.
x ≥ 0.5
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D.
x > 0.5
Solution
Subtract 1 from both sides: 4x ≤ 2. Then divide by 4: x ≤ 0.5.
Correct Answer: A — x ≤ 0.5
Q. What is the solution to the inequality 4x + 5 < 3?
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A.
x < -0.5
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B.
x > -0.5
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C.
x < -2
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D.
x > -2
Solution
4x + 5 < 3 => 4x < -2 => x < -0.5.
Correct Answer: A — x < -0.5
Q. What is the solution to the inequality 4x - 1 ≤ 3x + 2?
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A.
x ≤ 3
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B.
x ≥ 3
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C.
x < 3
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D.
x > 3
Solution
4x - 1 ≤ 3x + 2 => x ≤ 3.
Correct Answer: A — x ≤ 3
Q. What is the solution to the inequality 4x - 5 < 3x + 2?
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A.
x < 7
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B.
x > 7
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C.
x < -7
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D.
x > -7
Solution
4x - 5 < 3x + 2 => x < 7.
Correct Answer: A — x < 7
Q. What is the solution to the inequality 4x - 5 ≥ 3?
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A.
x < 2
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B.
x > 2
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C.
x ≤ 2
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D.
x ≥ 2
Solution
4x - 5 ≥ 3 => 4x ≥ 8 => x ≥ 2.
Correct Answer: D — x ≥ 2
Q. What is the solution to the inequality 4x - 7 < 9?
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A.
x < 4
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B.
x < 3
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C.
x > 4
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D.
x > 3
Solution
4x - 7 < 9 => 4x < 16 => x < 4.
Correct Answer: B — x < 3
Q. What is the solution to the inequality 5 - 2x < 1?
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A.
x > 2
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B.
x < 2
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C.
x > 3
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D.
x < 3
Solution
Subtract 5 from both sides: -2x < -4. Then divide by -2 (reverse the inequality): x > 2.
Correct Answer: A — x > 2
Q. What is the solution to the inequality 5 - 2x ≤ 1?
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A.
x ≥ 2
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B.
x ≤ 2
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C.
x < 2
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D.
x > 2
Solution
5 - 2x ≤ 1 => -2x ≤ -4 => x ≥ 2.
Correct Answer: A — x ≥ 2
Q. What is the solution to the inequality 6 - 3x ≥ 0?
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A.
x < 2
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B.
x > 2
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C.
x ≤ 2
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D.
x ≥ 2
Solution
6 - 3x ≥ 0 => -3x ≥ -6 => x ≤ 2.
Correct Answer: C — x ≤ 2
Q. What is the solution to the inequality 6x + 2 < 14?
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A.
x < 2
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B.
x > 2
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C.
x < 3
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D.
x > 3
Solution
6x + 2 < 14 => 6x < 12 => x < 2.
Correct Answer: A — x < 2
Q. What is the solution to the inequality 6x + 4 < 2x + 20?
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A.
x < 4
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B.
x > 4
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C.
x ≤ 4
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D.
x ≥ 4
Solution
6x + 4 < 2x + 20 => 4x < 16 => x < 4.
Correct Answer: A — x < 4
Q. What is the solution to the inequality 7 - 3x > 1?
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A.
x < 2
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B.
x > 2
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C.
x < 3
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D.
x > 3
Solution
7 - 3x > 1 => -3x > -6 => x < 2.
Correct Answer: A — x < 2
Q. What is the solution to the inequality 7 - 3x < 1?
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A.
x > 2
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B.
x < 2
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C.
x ≤ 2
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D.
x ≥ 2
Solution
7 - 3x < 1 => -3x < -6 => x > 2.
Correct Answer: A — x > 2
Q. What is the solution to the inequality 7x + 2 > 5?
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A.
x > 0.5
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B.
x < 0.5
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C.
x ≥ 0.5
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D.
x ≤ 0.5
Solution
7x + 2 > 5 => 7x > 3 => x > 0.5.
Correct Answer: A — x > 0.5
Q. What is the solution to the inequality 7x + 2 > 5x + 6?
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A.
x < 2
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B.
x > 2
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C.
x ≤ 2
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D.
x ≥ 2
Solution
7x + 2 > 5x + 6 => 2x > 4 => x > 2.
Correct Answer: B — x > 2
Q. What is the solution to the inequality 7x + 2 < 5?
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A.
x < -1
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B.
x > -1
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C.
x < 1
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D.
x > 1
Solution
7x + 2 < 5 => 7x < 3 => x < -1.
Correct Answer: A — x < -1
Q. What is the solution to the inequality 7x + 3 > 24?
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A.
x < 3
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B.
x > 3
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C.
x ≤ 3
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D.
x ≥ 3
Solution
7x + 3 > 24 => 7x > 21 => x > 3.
Correct Answer: B — x > 3
Q. What is the solution to the inequality 7x + 5 < 2x + 20?
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A.
x < 3
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B.
x > 3
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C.
x < 4
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D.
x > 4
Solution
7x + 5 < 2x + 20 => 5x < 15 => x < 3.
Correct Answer: A — x < 3
Q. What is the solution to the inequality 7x - 4 < 3x + 8?
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A.
x < 3
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B.
x > 3
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C.
x < 2
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D.
x > 2
Solution
7x - 4 < 3x + 8 => 4x < 12 => x < 3.
Correct Answer: B — x > 3
Q. What is the solution to the inequality 8 - 2x > 0?
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A.
x < 4
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B.
x > 4
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C.
x < 2
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D.
x > 2
Solution
8 - 2x > 0 => -2x > -8 => x < 4.
Correct Answer: B — x > 4
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