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 the questions from the image. 1. Find the value of x. Step 1: Convert the numbers from base x to base 10. The number "25" in base x is equivalent to 2x^1 + 5x^0 = 2x + 5 in base 10. The number "52" in base x is equivalent to 5x^1 + 2x^0 = 5x + 2 in base 10. Step 2: Set up the equation. According to the problem, twice the first number equals the second number: 2(2x + 5) = 5x + 2 Step 3: Solve for x. 4x + 10 = 5x + 2 Subtract 4x from both sides: 10 = x + 2 Subtract 2 from both sides: x = 8 The value of x is 8. 2. Evaluate each of the following without using tables or calculator, to four significant figures. Given sqrt(5) = 2.2361 and sqrt(7) = 2.6458. i) sqrt(72) - sqrt(7) Step 1: Analyze the expression. We need to evaluate sqrt(72) - sqrt(7). We can simplify sqrt(72) as sqrt(36 × 2) = 6sqrt(2). The expression becomes 6sqrt(2) - sqrt(7). However, the problem only provides values for sqrt(5) and sqrt(7), and explicitly states "without using tables or calculator". Since sqrt(2) is not provided, and we cannot use a calculator to find its value, this expression cannot be evaluated to four significant figures using the given information. This expression cannot be evaluated to four significant figures with the provided values for sqrt(5) and sqrt(7) only, as sqrt(2) is required and not given. ii) sqrt(5) + 2sqrt(5)sqrt(5) Step 1: Simplify the numerator. sqrt(5) + 2sqrt(5)sqrt(5) = 3sqrt(5)sqrt(5) Step 2: Simplify the fraction. 3sqrt(5)sqrt(5) = 3 To four significant figures, the value is 3.000. 3. Evaluate _2 sqrt((a^2 b)/(c^2)) Given _2 a = 3, _2 b = -2, and _2 c = 5. Step 1: Apply logarithm properties to expand the expression. _2 sqrt((a^2 b)/(c^2)) = _2 ((a^2 b)/(c^2))^(1)/(2) Using the power rule for logarithms, x^n = n x: = (1)/(2) _2 ((a^2 b)/(c^2)) Using the quotient rule, (x)/(y) = x - y: = (1)/(2) (_2 (a^2 b) - _2 (c^2)) Using the product rule, (xy) = x + y, and the power rule again: = (1)/(2) (2_2 a + _2 b - 2_2 c) Step 2: Substitute the given values. Substitute _2 a = 3, _2 b = -2, and _2 c = 5: = (1)/(2) (2(3) + (-2) - 2(5)) Step 3: Calculate the final value. = (1)/(2) (6 - 2 - 10) = (1)/(2) (4 - 10) = (1)/(2) (-6) = -3 The value of the expression is -3. 4. Find the mean of the numbers 5 + x_1, 5 + x_2, 5 + x_3, 5 + x_4. Given that the mean of x_1, x_2, x_3, x_4 is 7. Step 1: Find the sum of the first set of numbers. The mean of x_1, x_2, x_3, x_4 is 7. There are 4 numbers. (x_1 + x_2 + x_3 + x_4)/(4) = 7 x_1 + x_2 + x_3 + x_4 = 7 × 4 = 28 Step 2: Find the sum of the second set of numbers. The second set of numbers is 5 + x_1, 5 + x_2, 5 + x_3, 5 + x_4. Their sum is: (5 + x_1) + (5 + x_2) + (5 + x_3) + (5 + x_4) = (5 × 4) + (x_1 + x_2 + x_3 + x_4) = 20 + 28 = 48 Step 3: Calculate the mean of the second set of numbers. There are 4 numbers in the second set. Mean = Sum of numbersNumber of numbers = (48)/(4) = 12 The mean of the numbers is 12. 5. Calculate the mean age of the class. The frequency table is: | Age (years) | Frequency | |-------------|-----------| | 12 | 8 | | 13 | 10 | | 14 | 4 | | 15 | 2 | Step 1: Calculate the sum of (age × frequency). (f × x) = (12 × 8) + (13 × 10) + (14 × 4) + (15 × 2) = 96 + 130 + 56 + 30 = 312 Step 2: Calculate the total frequency. f = 8 + 10 + 4 + 2 = 24 Step 3: Calculate the mean age. Mean age = ( (f × x))/( f) = (312)/(24) = 13 The mean age of the class is 13 years. 6. Make u the subject of the relation fx = w((u^2 - v^2)/(2a)). Step 1: Isolate the term containing u^2. Multiply both sides by 2a: 2afx = w(u^2 - v^2) Divide both sides by w: (2afx)/(w) = u^2 - v^2 Step 2: Isolate u^2. Add v^2 to both sides: (2afx)/(w) + v^2 = u^2 Step 3: Solve for u. Take the square root of both sides: u = ± sqrt((2afx)/(w) + v^2) The subject of the relation is u = ± sqrt((2afx)/(w) + v^2). 7. Calculate the: i) Distance XZ ii) Bearing of Z from X Step 1: Determine the angle XYZ in triangle XYZ. Draw a diagram with North lines. The bearing of Y from X is 030^. The bearing of Z from Y is 078^. Draw a North line at Y. The angle between the North line at Y and the line YX (back towards X) is 30^ (alternate interior angles with the bearing from X to Y). The angle between the North line at Y and the line YZ is 78^. Both angles are measured clockwise from the North line at Y. Therefore, the interior angle XYZ = 78^ - 30^ = 48^. Step 2: Calculate the distance XZ using the Cosine Rule. In XYZ, we have XY = 10 km, YZ = 15 km, and XYZ = 48^. Using the Cosine Rule, XZ^2 = XY^2 + YZ^2 - 2(XY)(YZ)( XYZ): XZ^2 = 10^2 + 15^2 - 2(10)(15)(48^) XZ^2 = 100 + 225 - 300(48^) XZ^2 = 325 - 300(0.6691) XZ^2 = 325 - 200.73 XZ^2 = 124.27 XZ = sqrt(124.27) ≈ 11.1476 km Rounding to one decimal place, the distance XZ is 11.1 km. Step 3: Calculate angle YXZ using the Sine Rule. Using the Sine Rule, (( YXZ))/(YZ) = (( XYZ))/(XZ): (( YXZ))/(15) = ((48^))/(11.1476) ( YXZ) = (15 × (48^))/(11.1476) ( YXZ) = (15 × 0.7431)/(11.1476) ( YXZ) = (11.1465)/(11.1476) ≈ 0.9999 YXZ = (0.9999) ≈ 89.4^ Step 4: Calculate the bearing of Z from X. The bearing of Y from X is 030^. The bearing of Z from X is the bearing of Y from X plus the angle YXZ. Bearing of Z from X = 030^ + 89.4^ = 119.4^ The bearing of Z from X is 119.4^. What's next?