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.
ScanSolve AI Answer
Here are the solutions to the problems: 2 (i) The scores are: 70, 80, 78, 98, 84, 67, 98, 70, 80, 100, 87, 83, 70, 70, 88, 91, 70, 78, 88, 88. First, arrange the scores in ascending order: 67, 70, 70, 70, 70, 70, 78, 78, 80, 80, 83, 84, 87, 88, 88, 88, 91, 98, 98, 100. There are 20 scores in total. a) Represent the above scores on a frequency distribution table. Step 1: Create a table with scores and their frequencies. |c|c| Score & Frequency \\ 67 & 1 \\ 70 & 5 \\ 78 & 2 \\ 80 & 2 \\ 83 & 1 \\ 84 & 1 \\ 87 & 1 \\ 88 & 3 \\ 91 & 1 \\ 98 & 2 \\ 100 & 1 \\ Total & 20 \\ b) The mode of the distribution Step 1: Identify the score with the highest frequency from the table. The score 70 has the highest frequency of 5. The mode is 70. c) The median score Step 1: Since there are 20 scores (an even number), the median is the average of the 10th and 11th scores in the ordered list. The 10th score is 80. The 11th score is 83. Step 2: Calculate the average of these two scores. Median = (80 + 83)/(2) = (163)/(2) = 81.5 The median score is 81.5. d) The mean score Step 1: Calculate the sum of all scores. scores = 67(1) + 70(5) + 78(2) + 80(2) + 83(1) + 84(1) + 87(1) + 88(3) + 91(1) + 98(2) + 100(1) scores = 67 + 350 + 156 + 160 + 83 + 84 + 87 + 264 + 91 + 196 + 100 = 1638 Step 2: Divide the sum of scores by the total number of scores (20). Mean = scoresNumber of scores = (1638)/(20) = 81.9 The mean score is 81.9. 2 (ii) A straight line l_1 passes through the points P(3,4) and Q(1,-2). a) Determine the equation of the line l_1. Step 1: Calculate the gradient (slope) m of the line using the points P(x_1, y_1) = (3,4) and Q(x_2, y_2) = (1,-2). m = (y_2 - y_1)/(x_2 - x_1) = (-2 - 4)/(1 - 3) = (-6)/(-2) = 3 Step 2: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), with point P(3,4) and m=3. y - 4 = 3(x - 3) y - 4 = 3x - 9 Step 3: Rearrange the equation into the slope-intercept form y = mx + c. y = 3x - 9 + 4 y = 3x - 5 The equation of the line l_1 is y = 3x - 5. b) The coordinates of the point R. Step 1: The line meets the y-axis at point R. On the y-axis, the x-coordinate is 0. Substitute x=0 into the equation of the line y = 3x - 5. y = 3(0) - 5 y = -5 The coordinates of the point R are (0, -5). Send me the next one 📸