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 your questions: Problem 1: Statistics The table shows the number of children per house and the frequency (number of houses). | No. of children (x) | No. of houses (f) | fx | Cumulative Frequency (cf) | | :------------------ | :---------------- | :- | :------------------------ | | 0 | 2 | 0 | 2 | | 1 | 3 | 3 | 5 | | 2 | 4 | 8 | 9 | | 3 | 5 | 15 | 14 | | 4 | 1 | 4 | 15 | | 5 | 0 | 0 | 15 | | Total | f = 15 | fx = 30 | | Step 1: Find the mode. The mode is the value with the highest frequency. From the table, the highest frequency is 5, which corresponds to 3 children. The mode is 3 children. Step 2: Find the median. The total number of houses is f = 15. The position of the median is (N+1)/(2), where N is the total frequency. Median position = (15+1)/(2) = (16)/(2) = 8^th position. Looking at the cumulative frequency, the 8th value falls in the category of 2 children (since the cumulative frequency reaches 5 at 1 child, and 9 at 2 children). The median is 2 children. Step 3: Find the mean. The mean is calculated as ( fx)/( f). Mean = (30)/(15) = 2. The mean is 2 children. Problem 2: Geometry Given the diagram with triangle PST and line segment MN parallel to ST. Given: PM = 5 cm, MS = 10 cm. There is a typo in the problem statement "RN = 8cm". Based on the diagram and typical similar triangle problems, we will assume 'RN' is a typo for 'PN', so PN = 8 cm. Step 1: Name the triangle similar to triangle PST. Since MN is parallel to ST, by the property of similar triangles, PMN is similar to PST. The triangle similar to PST is PMN. Step 2: Find the length of NT. First, calculate the length of PS: PS = PM + MS = 5 cm + 10 cm = 15 cm Since PMN PST, the ratio of corresponding sides is equal: (PM)/(PS) = (PN)/(PT) Substitute the known values: (5)/(15) = (8)/(PT) Simplify the ratio: (1)/(3) = (8)/(PT) Solve for PT: PT = 3 × 8 = 24 cm Now, find NT: NT = PT - PN = 24 cm - 8 cm = 16 cm The length of NT is 16 cm. Problem 3: Vectors Given: AB = 3 \\ -4 and BC = 6 \\ 4 . Step 1: Express AC as a column vector. Using the triangle law of vector addition, AC = AB + BC. Step 2: Substitute the given column vectors and perform the addition. AC = 3 \\ -4 + 6 \\ 4 AC = 3+6 \\ -4+4 AC = 9 \\ 0 The column vector for AC is 9 \\ 0 . Send me the next one 📸