Q. If f(x) = x^4 - 4x^3, find f'(2). (2023)
Solution
f'(x) = 4x^3 - 12x^2; thus, f'(2) = 4(2^3) - 12(2^2) = 32 - 48 = -16.
Correct Answer:
C
— 16
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Q. If f(x) = x^4 - 4x^3, what is f'(2)? (2019)
Solution
f'(x) = 4x^3 - 12x^2. f'(2) = 4(2^3) - 12(2^2) = 32 - 48 = -16.
Correct Answer:
B
— 8
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Q. If f(x) = x^4 - 8x^2 + 16, what is the minimum value of f(x)? (2023)
Solution
Finding the derivative f'(x) = 4x^3 - 16x. Setting it to zero gives x = 0, ±2. The minimum value occurs at x = 2, f(2) = 0.
Correct Answer:
A
— 0
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Q. If G = [[2, 3], [5, 7]], find the eigenvalues of G.
-
A.
1, 8
-
B.
2, 7
-
C.
3, 5
-
D.
4, 6
Solution
The eigenvalues are found by solving the characteristic equation det(G - λI) = 0. The eigenvalues are λ = 1, 8.
Correct Answer:
A
— 1, 8
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Q. If H = [[0, 1], [-1, 0]], what is the determinant of H? (2019)
Solution
The determinant of H is calculated as (0*-0) - (1*-1) = 0 + 1 = 1.
Correct Answer:
C
— -1
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Q. If H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant of H. (2022)
Solution
Using the determinant formula for 3x3 matrices, det(H) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = 0 - 40 - 15 = -55.
Correct Answer:
A
— -24
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Q. If H = [[1, 2], [2, 4]], what is the rank of H?
Solution
The rank of H is 1 because the second row is a multiple of the first row.
Correct Answer:
A
— 1
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Q. If H = [[2, 3], [5, 7]], find the eigenvalues of H. (2023)
-
A.
1, 8
-
B.
2, 7
-
C.
3, 5
-
D.
4, 5
Solution
The eigenvalues are found by solving the characteristic equation: det(H - λI) = 0, which gives λ^2 - 9λ + 1 = 0.
Correct Answer:
A
— 1, 8
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Q. If h(x) = e^(2x), what is h'(x)? (2019)
-
A.
2e^(2x)
-
B.
e^(2x)
-
C.
2xe^(2x)
-
D.
e^(x)
Solution
Using the chain rule, h'(x) = 2e^(2x).
Correct Answer:
A
— 2e^(2x)
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Q. If I = [[1, 1], [1, 1]], what is the rank of I? (2022)
Solution
The rank of I is 1 because all rows are linearly dependent.
Correct Answer:
B
— 1
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Q. If I = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant of I.
Solution
Using the determinant formula for 3x3 matrices, det(I) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = 0 - 40 - 15 = -55.
Correct Answer:
A
— -24
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Q. If I = [[2, 1], [1, 2]], what is the trace of I?
Solution
The trace of a matrix is the sum of its diagonal elements. Thus, trace(I) = 2 + 2 = 4.
Correct Answer:
C
— 3
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Q. If it is 10:00 AM at 90°W, what is the time at 60°E?
-
A.
1:00 PM
-
B.
2:00 PM
-
C.
3:00 PM
-
D.
4:00 PM
Solution
The total difference is 90° + 60° = 150°. Therefore, the time difference is 150° / 15° = 10 hours. So, it is 10:00 AM + 10 hours = 8:00 PM.
Correct Answer:
C
— 3:00 PM
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Q. If it is 10:00 AM at 90°W, what time is it at 30°E?
-
A.
2:00 PM
-
B.
3:00 PM
-
C.
4:00 PM
-
D.
5:00 PM
Solution
The total distance is 90° + 30° = 120°. The time difference is 120° * 4 minutes = 480 minutes = 8 hours. Therefore, 10:00 AM + 8 hours = 6:00 PM.
Correct Answer:
C
— 4:00 PM
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Q. If it is 12:00 noon at the Prime Meridian (0 degrees longitude), what time is it at 45 degrees East longitude? (2020)
-
A.
6:00 AM
-
B.
12:00 PM
-
C.
6:00 PM
-
D.
3:00 PM
Solution
At 45 degrees East, it is 45/15 = 3 hours ahead. Therefore, it is 12:00 PM + 3 hours = 3:00 PM.
Correct Answer:
C
— 6:00 PM
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Q. If it is 12:00 noon at the Prime Meridian (0° longitude), what time is it at 120°E longitude? (2020)
-
A.
4:00 PM
-
B.
6:00 PM
-
C.
8:00 PM
-
D.
10:00 PM
Solution
At 120°E, the time is 120° / 15° = 8 hours ahead of GMT. Therefore, it is 12:00 noon + 8 hours = 8:00 PM.
Correct Answer:
C
— 8:00 PM
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Q. If it is 12:00 noon at the Prime Meridian (0° longitude), what time is it at 45°E longitude?
-
A.
2:30 PM
-
B.
3:00 PM
-
C.
12:30 PM
-
D.
1:00 PM
Solution
Each degree of longitude represents 4 minutes of time. Therefore, at 45°E, it is 45 * 4 = 180 minutes or 3 hours ahead, making it 3:00 PM.
Correct Answer:
B
— 3:00 PM
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Q. If it is 12:00 noon at the Prime Meridian (0° longitude), what time is it at 75°W longitude?
-
A.
6:00 AM
-
B.
7:00 AM
-
C.
8:00 AM
-
D.
9:00 AM
Solution
Each degree of longitude represents 4 minutes of time. 75°W = 75 * 4 = 300 minutes = 5 hours. Therefore, it is 12:00 noon - 5 hours = 7:00 AM.
Correct Answer:
C
— 8:00 AM
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Q. If it is 12:00 noon at the Prime Meridian (0° longitude), what time is it at 90° East longitude? (2020)
-
A.
6:00 AM
-
B.
12:00 PM
-
C.
6:00 PM
-
D.
12:00 AM
Solution
90° East is 6 hours ahead of the Prime Meridian (90/15 = 6), so it is 6:00 PM.
Correct Answer:
C
— 6:00 PM
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Q. If it is 3 PM in New York (75° W), what time is it in London (0° longitude)? (2020)
-
A.
12 PM
-
B.
6 PM
-
C.
8 PM
-
D.
9 PM
Solution
New York is 75° W, which is 75/15 = 5 hours behind London. Therefore, 3 PM + 5 hours = 8 PM in London.
Correct Answer:
B
— 6 PM
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Q. If it is 3:00 PM at 90°W, what is the time at 60°E?
-
A.
1:00 AM
-
B.
1:00 PM
-
C.
9:00 PM
-
D.
9:00 AM
Solution
The difference is 90° + 60° = 150°, which is 10 hours ahead. So, 3:00 PM + 10 hours = 1:00 AM.
Correct Answer:
C
— 9:00 PM
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Q. If it is 3:00 PM in New Delhi (77 degrees East), what time is it in London (0 degrees)?
-
A.
8:30 AM
-
B.
9:30 AM
-
C.
10:30 AM
-
D.
11:30 AM
Solution
New Delhi is 77/15 = 5.13 hours ahead of GMT. Therefore, 3:00 PM - 5.13 hours = 9:30 AM.
Correct Answer:
B
— 9:30 AM
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Q. If it is 3:00 PM in New Delhi (77.1°E), what time is it in London (0° longitude)?
-
A.
9:30 AM
-
B.
8:30 AM
-
C.
10:30 AM
-
D.
7:30 AM
Solution
New Delhi is 77.1°E, which is 77.1 * 4 = 308.4 minutes ahead of UTC. Therefore, it is 3:00 PM - 5 hours and 8 minutes = 9:30 AM in London.
Correct Answer:
B
— 8:30 AM
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Q. If it is 3:00 PM in New Delhi (77.2°E), what time is it in London (0° longitude)?
-
A.
8:30 AM
-
B.
9:30 AM
-
C.
10:30 AM
-
D.
11:30 AM
Solution
New Delhi is 77.2°E, which is 77.2 * 4 minutes = 308.8 minutes ahead of GMT. Therefore, 3:00 PM - 5 hours and 8.8 minutes = 9:30 AM in London.
Correct Answer:
B
— 9:30 AM
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Q. If it is 3:00 PM in New Delhi (77°E), what time is it in London (0° longitude)?
-
A.
8:30 AM
-
B.
9:30 AM
-
C.
10:30 AM
-
D.
11:30 AM
Solution
New Delhi is 77°E, which is 77 * 4 = 308 minutes = 5 hours and 8 minutes ahead of GMT. Therefore, 3:00 PM - 5:08 = 9:30 AM.
Correct Answer:
B
— 9:30 AM
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Q. If it is 3:00 PM in New Delhi (77°E), what time is it in London (0°)?
-
A.
8:00 AM
-
B.
9:00 AM
-
C.
10:00 AM
-
D.
11:00 AM
Solution
New Delhi is 77°E, which is 77/15 = 5.13 hours ahead. Therefore, 3:00 PM - 5 hours = 10:00 AM in London.
Correct Answer:
A
— 8:00 AM
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Q. If it is 3:00 PM in New York (75°W), what time is it in London (0° longitude)?
-
A.
8:00 PM
-
B.
6:00 PM
-
C.
5:00 PM
-
D.
4:00 PM
Solution
New York is 5 hours behind London. Therefore, 3:00 PM + 5 hours = 8:00 PM in London.
Correct Answer:
B
— 6:00 PM
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Q. If it is 3:00 PM in New York (GMT-5), what time is it in London (GMT+0)? (2022)
-
A.
8:00 AM
-
B.
10:00 AM
-
C.
12:00 PM
-
D.
3:00 PM
Solution
New York is 5 hours behind London, so 3:00 PM + 5 hours = 8:00 PM in London.
Correct Answer:
B
— 10:00 AM
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Q. If it is 6:00 AM at 30° East longitude, what time is it at 120° West longitude? (2022)
-
A.
10:00 PM
-
B.
12:00 AM
-
C.
2:00 AM
-
D.
4:00 AM
Solution
The time difference is 30 + 120 = 150 degrees. 150/15 = 10 hours. So, 6:00 AM - 10 hours = 8:00 PM previous day.
Correct Answer:
A
— 10:00 PM
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Q. If it is 6:00 AM at 45°E longitude, what is the time at 135°E longitude?
-
A.
8:00 AM
-
B.
9:00 AM
-
C.
10:00 AM
-
D.
11:00 AM
Solution
135°E is 135/15 = 9 hours ahead. Therefore, 6:00 AM + 9 hours = 3:00 PM.
Correct Answer:
C
— 10:00 AM
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