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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You're on a roll — Here are the solutions to the problems: 15. The table below shows marks obtained by a form three class in a certain school. Marks (x): 2, 6, 8, 3, 1 No. of students (y): 2, 6, 8, 3, 1 Use the table to represent the information on a histogram. (3mks) To represent this data on a histogram: Step 1: Draw two axes. The horizontal axis (x-axis) represents the 'Marks (x)', and the vertical axis (y-axis) represents the 'Number of students (y)' (frequency). Step 2: Choose an appropriate scale for both axes. For the marks, you can use intervals of 1 unit. For example, mark 1 can be represented by a bar from 0.5 to 1.5, mark 2 from 1.5 to 2.5, and so on. Step 3: For each mark, draw a rectangular bar. The width of each bar should be uniform (e.g., 1 unit). The height of each bar should correspond to the frequency (number of students) for that mark. • For Mark 1, draw a bar of height 1. • For Mark 2, draw a bar of height 2. • For Mark 3, draw a bar of height 3. • For Mark 6, draw a bar of height 6. • For Mark 8, draw a bar of height 8. Step 4: Ensure that the bars touch each other where the class intervals are continuous (e.g., the bar for mark 1 ends at 1.5, and the bar for mark 2 starts at 1.5). There will be gaps where there are no marks (e.g., between mark 3 and mark 6, and between mark 6 and mark 8). --- 16. Find the inverse of the matrix and hence solve the simultaneous equations below. (4mks) x + y = 21 2x + y = 21 Step 1: Write the simultaneous equations in matrix form. 1 & 1 \\ 2 & 1 x \\ y = 21 \\ 21 Let A = 1 & 1 \\ 2 & 1 , X = x \\ y , and B = 21 \\ 21 . Step 2: Find the determinant of matrix A. The determinant of a 2 × 2 matrix a & b \\ c & d is ad - bc. (A) = (1)(1) - (1)(2) = 1 - 2 = -1 Step 3: Find the inverse of matrix A. The inverse A^-1 is given by (1)/((A)) d & -b \\ -c & a . A^-1 = (1)/(-1) 1 & -1 \\ -2 & 1 A^-1 = -1 & 1 \\ 2 & -1 The inverse of the matrix is -1 & 1 \\ 2 & -1 . Step 4: Solve the simultaneous equations using the inverse matrix. To solve for X, we use the formula X = A^-1B. x \\ y = -1 & 1 \\ 2 & -1 21 \\ 21 Perform the matrix multiplication: x = (-1)(21) + (1)(21) = -21 + 21 = 0 y = (2)(21) + (-1)(21) = 42 - 21 = 21 The solution to the simultaneous equations is x = 0 and y = 21. Send me the next one 📸