Q. The entropy of a perfect crystal at absolute zero is given by:
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Solution
According to the third law of thermodynamics, the entropy of a perfect crystal at absolute zero is zero.
Correct Answer:
B
— 0
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Q. The entropy of a perfect crystal at absolute zero is:
A.
Maximum
B.
Minimum
C.
Undefined
D.
Infinite
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Solution
According to the third law of thermodynamics, the entropy of a perfect crystal at absolute zero is zero, which is the minimum value.
Correct Answer:
B
— Minimum
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Q. The entropy of a perfect crystal at absolute zero temperature is given by which law?
A.
Third law of thermodynamics
B.
First law of thermodynamics
C.
Second law of thermodynamics
D.
Zeroth law of thermodynamics
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Solution
The third law of thermodynamics states that the entropy of a perfect crystal approaches zero as the temperature approaches absolute zero.
Correct Answer:
A
— Third law of thermodynamics
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Q. The entropy of a perfect crystal at absolute zero temperature is given by:
A.
0
B.
1
C.
Infinity
D.
Depends on the substance
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Solution
According to the third law of thermodynamics, the entropy of a perfect crystal at absolute zero is exactly zero.
Correct Answer:
A
— 0
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Q. The entropy of a perfect crystalline substance at absolute zero is given by which law?
A.
Third law of thermodynamics
B.
First law of thermodynamics
C.
Second law of thermodynamics
D.
Zeroth law of thermodynamics
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Solution
The third law of thermodynamics states that the entropy of a perfect crystalline substance approaches zero as the temperature approaches absolute zero.
Correct Answer:
A
— Third law of thermodynamics
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Q. The entropy of a perfect crystalline substance at absolute zero is given by: (2021)
A.
0 J/K
B.
1 J/K
C.
2 J/K
D.
Depends on the substance
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Solution
According to the third law of thermodynamics, the entropy of a perfect crystalline substance at absolute zero is exactly zero.
Correct Answer:
A
— 0 J/K
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Q. The entropy of a perfect crystalline substance at absolute zero is:
A.
Zero
B.
Maximum
C.
Undefined
D.
Infinite
Show solution
Solution
According to the third law of thermodynamics, the entropy of a perfect crystalline substance at absolute zero is zero.
Correct Answer:
A
— Zero
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Q. The equation of a line parallel to y = 2x + 3 and passing through (1, 1) is?
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 2x + 3
D.
y = 2x - 3
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Solution
Parallel lines have the same slope. Using point-slope form: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer:
A
— y = 2x - 1
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Q. The equation of a line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of a line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of a line with slope 2 passing through the point (1, 3) is?
A.
y = 2x + 1
B.
y = 2x + 2
C.
y = 2x + 3
D.
y = 2x - 1
Show solution
Solution
Using point-slope form: y - 3 = 2(x - 1) => y = 2x + 1.
Correct Answer:
C
— y = 2x + 3
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Q. The equation of a parabola is given by x^2 = 16y. What is the length of the latus rectum?
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Solution
The length of the latus rectum for the parabola x^2 = 4py is given by 4p. Here, 4p = 16, so p = 4. Thus, the length of the latus rectum is 4p = 16.
Correct Answer:
B
— 8
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Q. The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?
A.
x^2 = -12y
B.
y^2 = -12x
C.
x^2 = 12y
D.
y^2 = 12x
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Solution
The distance from the vertex to the directrix is 3, so the equation is x^2 = -12y.
Correct Answer:
A
— x^2 = -12y
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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?
A.
x^2 = 12y
B.
y^2 = 12x
C.
x^2 = 6y
D.
y^2 = 6x
Show solution
Solution
The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.
Correct Answer:
A
— x^2 = 12y
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Q. The equation of an ellipse is given by 4x^2 + 9y^2 = 36. What is the eccentricity of the ellipse?
A.
0.5
B.
0.6
C.
0.7
D.
0.8
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Solution
Rewriting gives x^2/9 + y^2/4 = 1. Here, a^2 = 9, b^2 = 4, c = √(a^2 - b^2) = √(9 - 4) = √5. Eccentricity e = c/a = √5/3 ≈ 0.6.
Correct Answer:
B
— 0.6
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Q. The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is given by?
A.
x^2/a^2 + y^2/b^2 = 1
B.
y^2/a^2 + x^2/b^2 = 1
C.
x^2/b^2 + y^2/a^2 = 1
D.
y^2/b^2 + x^2/a^2 = 1
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Solution
The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is y^2/a^2 + x^2/b^2 = 1.
Correct Answer:
B
— y^2/a^2 + x^2/b^2 = 1
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Q. The equation of motion for a simple harmonic oscillator is given by x(t) = A cos(ωt + φ). What does A represent?
A.
Angular frequency
B.
Phase constant
C.
Amplitude
D.
Displacement
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Solution
A represents the amplitude of the oscillation, which is the maximum displacement from the mean position.
Correct Answer:
C
— Amplitude
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Q. The equation of motion for a simple harmonic oscillator is given by x(t) = A cos(ωt + φ). What does φ represent?
A.
Amplitude
B.
Phase constant
C.
Angular frequency
D.
Time period
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Solution
In the equation of motion for simple harmonic motion, φ is the phase constant, which determines the initial position of the oscillator.
Correct Answer:
B
— Phase constant
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Q. The equation of state for an ideal gas is given by: (2022)
A.
PV = nRT
B.
PV = NkT
C.
PV = mRT
D.
PV = kT
Show solution
Solution
The equation of state for an ideal gas is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.
Correct Answer:
A
— PV = nRT
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Q. The equation of the directrix of the parabola y^2 = 8x is?
A.
x = -2
B.
x = 2
C.
y = -4
D.
y = 4
Show solution
Solution
The directrix of the parabola y^2 = 8x is given by x = -2.
Correct Answer:
A
— x = -2
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Q. The equation of the line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of the line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = 4x - 2
D.
y = x + 1
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer:
A
— y = 2x
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Q. The equation of the pair of lines through the origin is given by y = mx. If m1 and m2 are the slopes, what is the condition for them to be perpendicular?
A.
m1 + m2 = 0
B.
m1 * m2 = 1
C.
m1 - m2 = 0
D.
m1 * m2 = -1
Show solution
Solution
For two lines to be perpendicular, the product of their slopes must equal -1.
Correct Answer:
D
— m1 * m2 = -1
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is given by:
A.
y = mx
B.
y^2 = mx
C.
x^2 + y^2 = 0
D.
x^2 - 2mxy + y^2 = 0
Show solution
Solution
The correct form of the equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.
Correct Answer:
D
— x^2 - 2mxy + y^2 = 0
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is:
A.
y = m1x + m2x
B.
y = (m1 + m2)x
C.
y = m1x - m2x
D.
y = m1x * m2x
Show solution
Solution
The equation of the lines can be expressed as y = (m1 + m2)x, representing the sum of the slopes.
Correct Answer:
B
— y = (m1 + m2)x
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Q. The equation of the tangent line to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x
B.
y = 4x - 4
C.
y = 4x - 8
D.
y = x + 2
Show solution
Solution
The slope of the tangent at x = 2 is f'(x) = 2x, so f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 4x - 8.
Correct Answer:
C
— y = 4x - 8
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Q. The equation of the tangent to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x - 4
B.
y = 2x
C.
y = x + 2
D.
y = x^2 - 2
Show solution
Solution
The derivative f'(x) = 2x. At x = 2, f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 2x - 4.
Correct Answer:
A
— y = 2x - 4
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Q. The equation x^2 + 2x + 1 = 0 can be factored as:
A.
(x + 1)(x + 1)
B.
(x - 1)(x - 1)
C.
(x + 2)(x + 1)
D.
(x - 2)(x - 1)
Show solution
Solution
This is a perfect square: (x + 1)^2 = 0.
Correct Answer:
A
— (x + 1)(x + 1)
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Q. The equation x^2 + 4x + 4 = 0 has:
A.
Two distinct roots
B.
One repeated root
C.
No real roots
D.
None of these
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Solution
The discriminant is 0, indicating one repeated root.
Correct Answer:
B
— One repeated root
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Q. The equation x^2 - 2x + 1 = 0 has how many distinct roots?
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Solution
The discriminant is 0, indicating that there is exactly one distinct root.
Correct Answer:
B
— 1
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