Q. What is the shape of the methane (CH4) molecule? (2023)
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A.
Linear
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B.
Trigonal planar
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C.
Tetrahedral
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D.
Octahedral
Solution
Methane (CH4) has a tetrahedral shape due to the four pairs of bonding electrons around the central carbon atom.
Correct Answer: C — Tetrahedral
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Q. What is the shape of the s orbital? (2021) 2021
-
A.
Spherical
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B.
Dumbbell
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C.
Double dumbbell
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D.
Linear
Solution
The s orbital has a spherical shape.
Correct Answer: A — Spherical
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Q. What is the SI unit of electric charge? (2020)
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A.
Coulomb
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B.
Ampere
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C.
Volt
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D.
Ohm
Solution
The SI unit of electric charge is Coulomb (C).
Correct Answer: A — Coulomb
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Q. What is the significance of the wave function in quantum mechanics? (2023)
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A.
It describes the position of a particle
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B.
It gives the probability of finding a particle
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C.
It determines the mass of a particle
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D.
It is irrelevant
Solution
The wave function provides the probability distribution of a particle's position.
Correct Answer: B — It gives the probability of finding a particle
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Q. What is the simplest ketone?
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A.
Acetaldehyde
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B.
Acetone
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C.
Butanone
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D.
Propionaldehyde
Solution
Acetone (CH3COCH3) is the simplest ketone.
Correct Answer: B — Acetone
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Q. What is the slope of the line passing through the points (1, 2) and (3, 6)? (2020) 2020
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 perpendicular to the line 3x + 4y = 12?
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A.
-3/4
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B.
4/3
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C.
3/4
-
D.
-4/3
Solution
The slope of the line 3x + 4y = 12 is -3/4. The slope of the perpendicular line is the negative reciprocal, which is 4/3.
Correct Answer: B — 4/3
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Q. What is the slope of the line perpendicular to the line 4x + 3y = 12?
-
A.
-3/4
-
B.
4/3
-
C.
3/4
-
D.
-4/3
Solution
The slope of the line 4x + 3y = 12 is -4/3. The slope of the perpendicular line is the negative reciprocal, which is 3/4.
Correct Answer: D — -4/3
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Q. What is the slope of the line perpendicular to the line 5x + 2y = 10?
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A.
-2/5
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B.
5/2
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C.
2/5
-
D.
-5/2
Solution
The slope of the line is -5/2. The slope of the perpendicular line is the negative reciprocal, which is 2/5.
Correct Answer: A — -2/5
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Q. What is the slope of the line perpendicular to the line 7x - 2y + 3 = 0?
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A.
1/2
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B.
-1/2
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C.
2
-
D.
-2
Solution
The slope of the line is 7/2. The slope of the perpendicular line is the negative reciprocal, which is -2/7.
Correct Answer: B — -1/2
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Q. What is the slope of the line perpendicular to the line 7x - 2y = 14?
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A.
1/7
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B.
-1/7
-
C.
2/7
-
D.
-2/7
Solution
The slope of the line is 7/2. The slope of the perpendicular line is the negative reciprocal, which is -2/7.
Correct Answer: B — -1/7
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Q. What is the slope of the line perpendicular to the line with slope 3?
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A.
-1/3
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B.
1/3
-
C.
3
-
D.
-3
Solution
Slope of perpendicular line = -1/m = -1/3.
Correct Answer: A — -1/3
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Q. What is the slope of the line perpendicular to the line y = 3x + 1? (2023)
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A.
-1/3
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B.
1/3
-
C.
3
-
D.
-3
Solution
Slope of perpendicular line = -1/(slope of original line) = -1/3.
Correct Answer: A — -1/3
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Q. What is the slope of the line that passes through the points (4, 5) and (6, 9)?
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)
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)
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)
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?
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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 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
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B.
y = Ce^(2x) + 3/2
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C.
y = 3e^(2x)
-
D.
y = 2e^(x) + C
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)
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B.
y = Ce^(5x) - (3/5)
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C.
y = (3/5)e^(5x)
-
D.
y = Ce^(3x) + 5
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?
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A.
y = x^3 + C
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B.
y = 3x^3 + C
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C.
y = x^2 + C
-
D.
y = 3x^2 + C
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
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B.
y = Ce^(-4x) + 1/2
-
C.
y = 2e^(4x) + C
-
D.
y = 4e^(4x) + C
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)
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A.
y = 3 - Ce^(-2x)
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B.
y = 3 + Ce^(-2x)
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C.
y = 2 - Ce^(2x)
-
D.
y = 6 - Ce^(2x)
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)
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)
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B.
y = Ce^(-kt)
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C.
y = -Ce^(kt)
-
D.
y = -Ce^(-kt)
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)
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
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)
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)
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
Solution
This is a separable equation. The solution is y = Ce^(-2x).
Correct Answer: A — y = Ce^(-2x)
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