Q. What is the slope of the line that passes through the points (4, 5) and (6, 9)?
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Solution
The slope m = (9 - 5) / (6 - 4) = 4/2 = 2.
Correct Answer:
A
— 2
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Q. What is the slope of the tangent line to f(x) = x^2 + 2x at x = 1? (2023)
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Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.
Correct Answer:
B
— 3
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Q. What is the slope of the tangent line to the curve y = x^2 - 4x + 5 at x = 3? (2023)
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Solution
The slope is given by f'(x) = 2x - 4. At x = 3, f'(3) = 2(3) - 4 = 2.
Correct Answer:
C
— 2
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Q. What is the slope of the tangent to the curve y = x^2 + 2x at x = 1? (2023)
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Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.
Correct Answer:
B
— 3
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Q. What is the solution of the differential equation dy/dx = y^2?
A.
y = 1/(C - x)
B.
y = C/(x + 1)
C.
y = Cx
D.
y = e^(x + C)
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Solution
Separating variables and integrating gives 1/y = x + C, leading to y = 1/(C - x).
Correct Answer:
A
— y = 1/(C - x)
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Q. What is the solution of the differential equation y' = 2y + 3?
A.
y = Ce^(2x) - 3/2
B.
y = Ce^(2x) + 3/2
C.
y = 3e^(2x)
D.
y = 2e^(x) + C
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Solution
The integrating factor is e^(-2x). Solving gives y = Ce^(2x) + 3/2.
Correct Answer:
B
— y = Ce^(2x) + 3/2
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Q. What is the solution of the differential equation y' = 5y + 3?
A.
y = (3/5) + Ce^(5x)
B.
y = Ce^(5x) - (3/5)
C.
y = (3/5)e^(5x)
D.
y = Ce^(3x) + 5
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Solution
Using the integrating factor method, we find the general solution to be y = Ce^(5x) - (3/5).
Correct Answer:
B
— y = Ce^(5x) - (3/5)
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Q. What is the solution of the equation dy/dx = 3x^2?
A.
y = x^3 + C
B.
y = 3x^3 + C
C.
y = x^2 + C
D.
y = 3x^2 + C
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Solution
Integrating both sides gives y = ∫3x^2 dx = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. What is the solution of the equation dy/dx = 4y + 2? (2021)
A.
y = Ce^(4x) - 1/2
B.
y = Ce^(-4x) + 1/2
C.
y = 2e^(4x) + C
D.
y = 4e^(4x) + C
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Solution
Using an integrating factor, the solution is y = Ce^(4x) - 1/2.
Correct Answer:
A
— y = Ce^(4x) - 1/2
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Q. What is the solution of the equation dy/dx = 6 - 2y? (2021)
A.
y = 3 - Ce^(-2x)
B.
y = 3 + Ce^(-2x)
C.
y = 2 - Ce^(2x)
D.
y = 6 - Ce^(2x)
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Solution
Rearranging gives dy/(6 - 2y) = dx. Integrating both sides leads to y = 3 - Ce^(-2x).
Correct Answer:
A
— y = 3 - Ce^(-2x)
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Q. What is the solution of the equation y' + 4y = 0?
A.
y = Ce^(-4x)
B.
y = Ce^(4x)
C.
y = 4Ce^x
D.
y = Ce^(x/4)
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Solution
This is a separable equation. The solution is y = Ce^(-4x).
Correct Answer:
A
— y = Ce^(-4x)
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Q. What is the solution of the equation y' = -ky, where k is a constant?
A.
y = Ce^(kt)
B.
y = Ce^(-kt)
C.
y = -Ce^(kt)
D.
y = -Ce^(-kt)
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Solution
This is a separable equation. Integrating gives y = Ce^(-kt).
Correct Answer:
B
— y = Ce^(-kt)
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Q. What is the solution to the differential equation dy/dx = -y/x?
A.
y = Cx
B.
y = C/x
C.
y = Cx^2
D.
y = Cx^(-1)
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Solution
This is a separable equation. Separating variables and integrating gives y = C/x.
Correct Answer:
B
— y = C/x
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Q. What is the solution to the differential equation y' = 5y + 3?
A.
y = (3/5) + Ce^(5x)
B.
y = (5/3) + Ce^(5x)
C.
y = Ce^(5x) - 3
D.
y = Ce^(3x) + 5
Show solution
Solution
Using the integrating factor method, we find the solution to be y = (3/5) + Ce^(5x).
Correct Answer:
A
— y = (3/5) + Ce^(5x)
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Q. What is the solution to the equation dy/dx = -5y?
A.
y = Ce^(-5x)
B.
y = -5Ce^x
C.
y = Ce^(5x)
D.
y = 5Ce^(-x)
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Solution
This is a separable differential equation. The solution is y = Ce^(-5x), where C is a constant.
Correct Answer:
A
— y = Ce^(-5x)
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Q. What is the solution to the equation dy/dx = y^2? (2022)
A.
y = 1/(C - x)
B.
y = C/(x - 1)
C.
y = Cx^2
D.
y = ln(Cx)
Show solution
Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer:
A
— y = 1/(C - x)
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Q. What is the solution to the equation y' + 2y = 0?
A.
y = Ce^(-2x)
B.
y = Ce^(2x)
C.
y = 2Ce^x
D.
y = Ce^x
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Solution
This is a separable equation. The solution is y = Ce^(-2x).
Correct Answer:
A
— y = Ce^(-2x)
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Q. What is the solution to the equation y' + 3y = 0?
A.
y = Ce^(-3x)
B.
y = Ce^(3x)
C.
y = 3Ce^(-x)
D.
y = Ce^(-x/3)
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Solution
This is a first-order linear differential equation. The solution is y = Ce^(-3x).
Correct Answer:
A
— y = Ce^(-3x)
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Q. What is the solution to the equation y' = 3y + 6?
A.
y = Ce^(3x) - 2
B.
y = Ce^(3x) + 2
C.
y = 2e^(3x)
D.
y = 3Ce^(x)
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Solution
This is a first-order linear equation. The integrating factor is e^(3x), leading to the solution y = Ce^(3x) + 2.
Correct Answer:
B
— y = Ce^(3x) + 2
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Q. What is the solution to the equation y'' + 4y = 0?
A.
y = C1 cos(2x) + C2 sin(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 e^(4x) + C2 e^(-4x)
D.
y = C1 sin(4x) + C2 cos(4x)
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Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The general solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer:
A
— y = C1 cos(2x) + C2 sin(2x)
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Q. What is the solution to the equation y'' - 3y' + 2y = 0?
A.
y = C1 e^(2x) + C2 e^(x)
B.
y = C1 e^(x) + C2 e^(2x)
C.
y = C1 e^(-x) + C2 e^(-2x)
D.
y = C1 + C2x
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Solution
The characteristic equation r^2 - 3r + 2 = 0 has roots 1 and 2, leading to y = C1 e^(x) + C2 e^(2x).
Correct Answer:
B
— y = C1 e^(x) + C2 e^(2x)
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Q. What is the specific heat at constant volume (Cv) for a monatomic ideal gas? (2019)
A.
3R/2
B.
5R/2
C.
R
D.
2R
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Solution
For a monatomic ideal gas, the specific heat at constant volume (Cv) is 3R/2.
Correct Answer:
A
— 3R/2
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Q. What is the specific heat capacity of a substance if it requires 500 J to raise the temperature of 2 kg of the substance by 5°C? (2021)
A.
20 J/kg°C
B.
50 J/kg°C
C.
100 J/kg°C
D.
200 J/kg°C
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Solution
Specific heat capacity (c) = Q / (m * ΔT) = 500 J / (2 kg * 5°C) = 50 J/kg°C.
Correct Answer:
B
— 50 J/kg°C
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Q. What is the specific heat capacity of a substance if it requires 500 J to raise the temperature of 2 kg of the substance by 10°C? (2021)
A.
25 J/kg°C
B.
50 J/kg°C
C.
100 J/kg°C
D.
200 J/kg°C
Show solution
Solution
Specific heat capacity (c) = Q / (m * ΔT) = 500 J / (2 kg * 10°C) = 25 J/kg°C.
Correct Answer:
B
— 50 J/kg°C
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Q. What is the specific heat capacity of a substance if it takes 500 J to raise the temperature of 2 kg of the substance by 5°C? (2021)
A.
20 J/kg°C
B.
50 J/kg°C
C.
100 J/kg°C
D.
200 J/kg°C
Show solution
Solution
Specific heat capacity (c) = Q / (m * ΔT) = 500 J / (2 kg * 5°C) = 50 J/kg°C
Correct Answer:
B
— 50 J/kg°C
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Q. What is the specific heat capacity of air at constant pressure? (2022)
A.
1.005 J/kg·K
B.
0.718 J/kg·K
C.
4.186 J/kg·K
D.
2.093 J/kg·K
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Solution
The specific heat capacity of air at constant pressure is approximately 1.005 J/kg·K.
Correct Answer:
A
— 1.005 J/kg·K
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Q. What is the specific heat capacity of air? (2022)
A.
1.005 J/kg·K
B.
4.18 J/kg·K
C.
2.09 J/kg·K
D.
0.9 J/kg·K
Show solution
Solution
The specific heat capacity of air is approximately 1.005 J/kg·K.
Correct Answer:
A
— 1.005 J/kg·K
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Q. What is the specific heat capacity of iron? (2023)
A.
0.45 J/g·K
B.
1.00 J/g·K
C.
2.00 J/g·K
D.
4.18 J/g·K
Show solution
Solution
The specific heat capacity of iron is approximately 0.45 J/g·K.
Correct Answer:
A
— 0.45 J/g·K
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Q. What is the specific heat capacity of water at constant pressure (C_p)? (2020)
A.
4.18 J/g°C
B.
2.09 J/g°C
C.
1.00 J/g°C
D.
3.00 J/g°C
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Solution
The specific heat capacity of water at constant pressure is approximately 4.18 J/g°C.
Correct Answer:
A
— 4.18 J/g°C
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Q. What is the specific heat capacity of water if it takes 4200 J to raise the temperature of 1 kg of water by 1°C? (2021)
A.
2100 J/kg°C
B.
4200 J/kg°C
C.
8400 J/kg°C
D.
1000 J/kg°C
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Solution
The specific heat capacity of water is defined as the amount of heat required to raise the temperature of 1 kg of water by 1°C, which is given as 4200 J/kg°C.
Correct Answer:
B
— 4200 J/kg°C
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