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 problems: 15. Evaluate 11001_two + 1111_two. Step 1: Align the binary numbers for addition. c 11001_two \\ + 01111_two \\ Step 2: Perform binary addition from right to left. 1 + 1 = 10_two (write 0, carry 1) 0 + 1 + 1 (carry) = 10_two (write 0, carry 1) 0 + 1 + 1 (carry) = 10_two (write 0, carry 1) 1 + 1 + 1 (carry) = 11_two (write 1, carry 1) 1 + 0 + 1 (carry) = 10_two (write 0, carry 1) The final carry is 1. c 11001_two \\ + 01111_two \\ 101000_two The correct option is B. The sum is 101000_two. 16. An aircraft flies from Q (25^ N, 38^ E) to R (72^ S, 38^ E). Calculate the distance QR along the line of longitude, correct to the nearest kilometre. (Take R = 6,400 km, = (22)/(7)) Step 1: Determine the angular difference in latitude. Since the points are in different hemispheres (North and South), add their latitudes. = 25^ + 72^ = 97^ Step 2: Convert the angular difference from degrees to radians. _rad = 97 × ()/(180) Step 3: Calculate the distance using the formula D = R × _rad, where R is the radius of the Earth. D = 6400 km × 97 × (22)/(7 × 180) D = 6400 × (97 × 22)/(1260) D = (6400 × 2134)/(1260) D = (13657600)/(1260) D ≈ 10839.365 km Step 4: Round the distance to the nearest kilometre. D ≈ 10839 km The correct option is A. The distance QR is 10839 km. 17. Find the gradient of the curve y = 2x^2 - 3x + 4 at the point x=2. Step 1: Find the first derivative of the curve equation to get the gradient function. y = 2x^2 - 3x + 4 (dy)/(dx) = (d)/(dx)(2x^2) - (d)/(dx)(3x) + (d)/(dx)(4) (dy)/(dx) = 4x - 3 Step 2: Substitute x=2 into the gradient function to find the gradient at that point. Gradient at x=2 = 4(2) - 3 = 8 - 3 = 5 The correct option is B. The gradient of the curve at x=2 is 5. 18. Evaluate _8 16. Assumption: The question "Evaluate log8 log16" is interpreted as _16 8 due to the available options. If it were _8 16, the answer would be (4)/(3), which is not an option. If it were _8(_10 16), the answer would not be a simple fraction. Step 1: Set the logarithm equal to a variable, say k. _16 8 = k Step 2: Convert the logarithmic equation to an exponential equation. 16^k = 8 Step 3: Express both sides of the equation with a common base, which is 2. (2^4)^k = 2^3 2^4k = 2^3 Step 4: Equate the exponents and solve for k. 4k = 3 k = (3)/(4) The correct option is A. The value is (3)/(4). Send me the next one 📸