75°W to 15°E is 90° apart, which is 6 hours. So, 6:00 AM + 6 hours = 12:00 PM.

New York is 5 hours behind London. Therefore, if it is 6:00 AM in New York, it is 6:00 AM + 5 hours = 11:00 AM in London.

New Delhi is 5 hours and 30 minutes ahead of GMT, so 6:00 PM - 5:30 = 12:30 PM.

The time difference is 77° * 4 = 308 minutes or approximately 5 hours and 8 minutes. Therefore, 6:00 PM - 5 hours = 1:00 PM.

The time difference is 5 hours. Therefore, it is 8:00 AM + 5 hours = 1:00 PM in London.

90°W to 0° is 6 hours ahead, and 0° to 30°E is 2 hours ahead, total = 8 hours ahead, so 9:00 AM + 8 hours = 5:00 PM.

90° East is 90/15 = 6 hours ahead of the Prime Meridian. Therefore, if it is noon at the Prime Meridian, it is 6 PM at 90° East.

The rank of J is 1 because both rows are linearly dependent (they are identical).

The determinant of J is 0, as the rows are linearly dependent.

log_10(0.01) = log_10(10^-2) = -2, so x = -2.

log_10(10^x) = x. Therefore, x = 2.

log_10(20) = log_10(2) + log_10(10) = 0.301 + 1 = 1.301.

log_2(8) = 3 and log_2(4) = 2, thus x = 3 + 2 = 5.

log_2(x) + 2 = 6 implies log_2(x) = 4, so x = 2^4 = 16.

log_3(81) = log_3(3^4) = 4.

log_3(x) = 4 implies x = 3^4 = 81.

log_4(16) = log_4(4^2) = 2.

log_4(256) = log_4(4^4) = 4.

log_a(1) = 0 is true for any base a > 0, except when a = 1.

log_a(16) = 4 implies a^4 = 16. Since 16 = 2^4, we have a^4 = 2^4, thus a = 2.

log_a(3) = 0.5 implies a^0.5 = 3, thus a = 3^2 = 9.

log_a(64) = 6 implies a^6 = 64. Since 64 = 2^6, we have a = 2.

log_b(25) = 2 implies b^2 = 25. Therefore, b = 5.

log_x(64) = 6 implies x^6 = 64. Since 64 = 2^6, we have x = 2.

The sum of the roots is 3 + (-1) = 2, hence p = -2.

Substituting x = 2 into the equation gives 2^2 - 4*2 + k = 0, which simplifies to k = 4.

Using the root 3 in the equation: 3^2 - 7*3 + k = 0, we get k = 6.

The double angle formula for cosine is cos 2A = cos²A - sin²A.

Using the double angle formula, sin 2A = 2(1/2)(√(1 - (1/2)²)) = 2(1/2)(√(3/4)) = 1.

sin 2(30°) = sin 60° = √3/2.