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's a breakdown of the problems and the completed table. Problem 1: Circle Equation The general equation of a circle is given by x^2 + y^2 + 2gx + 2fy + c = 0. The center of the circle is (-g, -f) and the radius is r = sqrt(g^2 + f^2 - c). The given equation is x^2 + y^2 - 4x - 6y + 9 = 0. Step 1: Identify the coefficients. Comparing the given equation with the general form: 2gx = -4x 2g = -4 g = -2 2fy = -6y 2f = -6 f = -3 c = 9 Step 2: Calculate the center of the circle. The center is (-g, -f). Center = (-(-2), -(-3)) = (2, 3) Step 3: Calculate the radius of the circle. The radius is r = sqrt(g^2 + f^2 - c). r = sqrt((-2)^2 + (-3)^2 - 9) r = sqrt(4 + 9 - 9) r = sqrt(4) r = 2 The student's work for the circle equation is correct. The center of the circle is (2, 3) and the radius is 2. --- Problem 2: Complete Table for Assumed Mean To complete the table and calculate the mean using the assumed mean method, we need to: 1. Choose an assumed mean (A). We will choose A=13 (the midpoint of the class with the highest frequency). 2. Determine the class width (c). The class intervals are not perfectly uniform (0-5 has width 6, 6-10 has width 5, 11-15 has width 5). We will use the most common class width, c=5, for the calculation of u. 3. Calculate the deviation d = x - A. 4. Calculate u = (x - A)/(c). 5. Calculate u^2. 6. Calculate Fu and Fu^2. 7. Sum the columns F, Fu, and Fu^2. 8. Calculate the mean using the formula x = A + ( Fu)/( F) × c. Here is the completed table: | Marks | F | x | d = x - A | u = (x - A)/c | u^2 | Fu | Fu^2 | C.B | | :---- | :-: | :-: | :--------: | :-------------: | :---: | :----: | :----: | :----------: | | 0-5 | 2 | 2.5 | 2.5 - 13 = -10.5 | (-10.5)/(5) = -2.1 | (-2.1)^2 = 4.41 | 2 × (-2.1) = -4.2 | 2 × 4.41 = 8.82 | -0.5 - 5.5 | | 6-10 | 8 | 8 | 8 - 13 = -5 | (-5)/(5) = -1 | (-1)^2 = 1 | 8 × (-1) = -8 | 8 × 1 = 8 | 5.5 - 10.5 | | 11-15 | 13 | 13 | 13 - 13 = 0 | (0)/(5) = 0 | 0^2 = 0 | 13 × 0 = 0 | 13 × 0 = 0 | 10.5 - 15.5 | | Total | 23 | | | | | -12.2 | 16.82 | | Step 1: Calculate the sum of frequencies. F = 2 + 8 + 13 = 23 Step 2: Calculate the sum of Fu. Fu = -4.2 - 8 + 0 = -12.2 Step 3: Calculate the mean using the assumed mean formula. x = A + ( Fu)/( F) × c Substitute A=13, Fu = -12.2, F = 23, and c=5: x = 13 + (-12.2)/(23) × 5 x = 13 + (-61)/(23) x = 13 - 2.65217... x ≈ 10.35 The completed table is shown above, and the mean is 10.35. That's 2 down. 3 left today — send the next one.