This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the questions: 1. (a) Given that R = \x : 0 x < 10, x is an even number\, list the elements of set R. Step 1: Identify the conditions for the elements of set R. The elements must be even numbers, and they must be greater than or equal to 0 and strictly less than 10. Step 2: List the even numbers that satisfy these conditions. The even numbers in the range 0 x < 10 are 0, 2, 4, 6, 8. R = \0, 2, 4, 6, 8\ The elements of set R are 0, 2, 4, 6, 8 . 1. (b) A point F(-4, 1) is mapped onto a point G(5, -4) by a translation T. Express T as a column vector. Step 1: Understand the relationship between the original point, translation vector, and image point. If a point F is translated by a vector T to an image point G, then F + T = G. Let F = -4 \\ 1 and G = 5 \\ -4 . Let the translation vector be T = a \\ b . Step 2: Set up the vector equation. -4 \\ 1 + a \\ b = 5 \\ -4 Step 3: Solve for the components of the translation vector. For the x-component: -4 + a = 5 a = 5 + 4 = 9 For the y-component: 1 + b = -4 b = -4 - 1 = -5 The translation vector T is 9 \\ -5 . 2. In the following diagram, NS is the Prime Meridian. The positions of towns A and B are (30^ S, 30^ W) and (30^ N, 30^ W) respectively. (a) Find the distance between A and B in nautical miles. Step 1: Identify the coordinates of A and B and the type of distance. Town A is at (30^ S, 30^ W). Town B is at (30^ N, 30^ W). Since both towns are on the same longitude (30^ W), the distance between them is along a meridian. Step 2: Calculate the difference in latitude. The difference in latitude is the sum of the absolute values of their latitudes because they are in different hemispheres (North and South). Difference in latitude = 30^ N + 30^ S = 60^ Step 3: Calculate the distance in nautical miles. The distance along a meridian is given by 60 × difference in latitude in degrees nautical miles. Distance = 60 × 60^ = 3600 nautical miles The distance between A and B is 3600 nautical miles . (b) If the local time at B is 09 35 hours, what is the local time at the Prime Meridian? Step 1: Determine the longitude difference between B and the Prime Meridian. Town B is at 30^ W. The Prime Meridian is at 0^ longitude. Longitude difference = 30^ - 0^ = 30^ Step 2: Convert the longitude difference to time difference. Every 1^ of longitude corresponds to 4 minutes of time. Time difference = 30^ × 4 minutes/degree = 120 minutes Time difference = (120)/(60) hours = 2 hours Step 3: Calculate the local time at the Prime Meridian. Since Town B is West of the Prime Meridian, its time is earlier than the Prime Meridian. To find the time at the Prime Meridian, we add the time difference to B's local time. Local time at Prime Meridian = Local time at B + Time difference Local time at Prime Meridian = 09 35 hours + 2 hours Local time at Prime Meridian = 11 35 hours The local time at the Prime Meridian is 11 35 hours . 3 done, 2 left today. You're making progress.