New York is 5 hours behind London, so 3:00 PM + 5 hours = 8:00 PM in London.

The time difference is 30 + 120 = 150 degrees. 150/15 = 10 hours. So, 6:00 AM - 10 hours = 8:00 PM previous day.

135°E is 135/15 = 9 hours ahead. Therefore, 6:00 AM + 9 hours = 3:00 PM.

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.

If the office is open, then it cannot be a holiday, as the office would be closed on holidays.

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.

If it rains, the ground gets wet. Since it is raining, we can conclude that the ground is wet.

The conclusion is valid because if the ground is not wet, it logically follows that it did not rain.

Det(J) = (1*1) - (1*1) = 1 - 1 = 0.

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

Determinant of J = 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.

Using the determinant formula, det(J) = 1*(1*0 - 3*1) - 2*(0*0 - 3*2) + 1*(0*1 - 1*2) = 1*(-3) - 2*(-6) + 1*(-2) = -3 + 12 - 2 = 7.

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

Det(J) = (1*4) - (2*2) = 4 - 4 = 0.

Det(J) = (3*4) - (1*2) = 12 - 2 = 10.

L is last as it is scheduled after K, which is after J.

Given J is between K and L, and M is opposite K, we can conclude that P is next to L.

Since J is between K and L, L is sitting next to K.

If J is three places away from K, there are two people sitting between J and K.

K is the sister of L, making J the niece of L.

J and L are both children of K, making them siblings.

J is the wife of L as they are the parents of K.

Since John finishes before 3 PM, he will go to the gym.

Tom is John's brother.

John is the son of Mary, who is the daughter of Sarah, making John the grandson of Sarah.