Draw a cumulative frequency curve for The median mark (ii) the inter quartile range and semi interquartile range (iii) If only 5 % of the candidates attained the distinction level, estimate the lowest mark for this level.

Question

Asked on July 7, 2026Mathematics

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.

Accepted Answer · 2026-07-07T13:41:57.822Z

Here's the solution to the matrix problems. The data for the cumulative frequency curve is not provided, so that part cannot be solved. Part 7a: Given the matrix A = 4 & -3 \\ 1 & 2 , find the inverse matrix A^-1. Step 1: Calculate the determinant of matrix A. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (A) = (4)(2) - (-3)(1) = 8 - (-3) = 8 + 3 = 11 Step 2: Apply the formula for the inverse of a 2 × 2 matrix. The inverse of A = a & b \\ c & d is A^-1 = (1)/((A)) d & -b \\ -c & a . A^-1 = (1)/(11) 2 & -(-3) \\ -1 & 4 = (1)/(11) 2 & 3 \\ -1 & 4 Step 3: Multiply the scalar into the matrix. A^-1 = (2)/(11) & (3)/(11) \\ -(1)/(11) & (4)/(11) The inverse matrix is (2)/(11) & (3)/(11) \\ -(1)/(11) & (4)/(11) . Part 7b: Under the linear transformation P of a plane, the image of (2,1) is (4,2) and that of (-7,1) is (-1,7). Find the matrix P of the transformation. Step 1: Set up the matrix equation for the transformation. Let the transformation matrix be P = a & b \\ c & d . The transformation can be written as P x = x'. Given P 2 \\ 1 = 4 \\ 2 and P -7 \\ 1 = -1 \\ 7 . These two equations can be combined into a single matrix equation: P 2 & -7 \\ 1 & 1 = 4 & -1 \\ 2 & 7 Let X = 2 & -7 \\ 1 & 1 and Y = 4 & -1 \\ 2 & 7 . So, PX = Y. Step 2: Find the inverse of matrix X. First, calculate the determinant of X: (X) = (2)(1) - (-7)(1) = 2 - (-7) = 2 + 7 = 9 Now, find X^-1: X^-1 = (1)/(9) 1 & -(-7) \\ -1 & 2 = (1)/(9) 1 & 7 \\ -1 & 2 Step 3: Calculate P by multiplying Y by X^-1. Since PX = Y, then P = YX^-1. P = 4 & -1 \\ 2 & 7 (1)/(9) 1 & 7 \\ -1 & 2 P = (1)/(9) 4 & -1 \\ 2 & 7 1 & 7 \\ -1 & 2 Step 4: Perform the matrix multiplication. P = (1)/(9) (4)(1) + (-1)(-1) & (4)(7) + (-1)(2) \\ (2)(1) + (7)(-1) & (2)(7) + (7)(2) P = (1)/(9) 4 + 1 & 28 - 2 \\ 2 - 7 & 14 + 14 P = (1)/(9) 5 & 26 \\ -5 & 28 Step 5: Multiply the scalar into the matrix. P = (5)/(9) & (26)/(9) \\ -(5)/(9) & (28)/(9) The matrix P of the transformation is (5)/(9) & (26)/(9) \\ -(5)/(9) & (28)/(9) . Send me the next one 📸

Accepted Answer · 2026-07-07T13:41:57.822Z

Here's the solution to the matrix problems. The data for the cumulative frequency curve is not provided, so that part cannot be solved. Part 7a: Given the matrix A = 4 & -3 \\ 1 & 2 , find the inverse matrix A^-1. Step 1: Calculate the determinant of matrix A. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (A) = (4)(2) - (-3)(1) = 8 - (-3) = 8 + 3 = 11 Step 2: Apply the formula for the inverse of a 2 × 2 matrix. The inverse of A = a & b \\ c & d is A^-1 = (1)/((A)) d & -b \\ -c & a . A^-1 = (1)/(11) 2 & -(-3) \\ -1 & 4 = (1)/(11) 2 & 3 \\ -1 & 4 Step 3: Multiply the scalar into the matrix. A^-1 = (2)/(11) & (3)/(11) \\ -(1)/(11) & (4)/(11) The inverse matrix is (2)/(11) & (3)/(11) \\ -(1)/(11) & (4)/(11) . Part 7b: Under the linear transformation P of a plane, the image of (2,1) is (4,2) and that of (-7,1) is (-1,7). Find the matrix P of the transformation. Step 1: Set up the matrix equation for the transformation. Let the transformation matrix be P = a & b \\ c & d . The transformation can be written as P x = x'. Given P 2 \\ 1 = 4 \\ 2 and P -7 \\ 1 = -1 \\ 7 . These two equations can be combined into a single matrix equation: P 2 & -7 \\ 1 & 1 = 4 & -1 \\ 2 & 7 Let X = 2 & -7 \\ 1 & 1 and Y = 4 & -1 \\ 2 & 7 . So, PX = Y. Step 2: Find the inverse of matrix X. First, calculate the determinant of X: (X) = (2)(1) - (-7)(1) = 2 - (-7) = 2 + 7 = 9 Now, find X^-1: X^-1 = (1)/(9) 1 & -(-7) \\ -1 & 2 = (1)/(9) 1 & 7 \\ -1 & 2 Step 3: Calculate P by multiplying Y by X^-1. Since PX = Y, then P = YX^-1. P = 4 & -1 \\ 2 & 7 (1)/(9) 1 & 7 \\ -1 & 2 P = (1)/(9) 4 & -1 \\ 2 & 7 1 & 7 \\ -1 & 2 Step 4: Perform the matrix multiplication. P = (1)/(9) (4)(1) + (-1)(-1) & (4)(7) + (-1)(2) \\ (2)(1) + (7)(-1) & (2)(7) + (7)(2) P = (1)/(9) 4 + 1 & 28 - 2 \\ 2 - 7 & 14 + 14 P = (1)/(9) 5 & 26 \\ -5 & 28 Step 5: Multiply the scalar into the matrix. P = (5)/(9) & (26)/(9) \\ -(5)/(9) & (28)/(9) The matrix P of the transformation is (5)/(9) & (26)/(9) \\ -(5)/(9) & (28)/(9) . Send me the next one 📸

Draw a cumulative frequency curve for The median mark (ii) the inter quartile range and semi interquartile range (iii) If only 5 % of the candidates attained the distinction level, estimate the lowest mark for this level.

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Draw a cumulative frequency curve for The median mark (ii) the inter quartile range and semi interquartile range (iii) If only 5 % of the candidates attained the distinction level, estimate the lowest mark for this level.

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