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: Question 99: The phrase "against all odds" means despite very unfavorable circumstances or great difficulty. A. easily - contradicts "against all odds" B. despite difficulties - aligns with "against all odds" C. with much help - does not directly explain "against all odds" D. without any problems - contradicts "against all odds" The option that best completes the gap is B. despite difficulties. Question 100: The phrase "called off" means cancelled. Given "due to unforeseen circumstances," a meeting would most logically be cancelled. A. off - forms the phrasal verb "called off" (cancelled) B. up - forms "called up" (summoned, telephoned) C. for - forms "called for" (required, demanded) D. in - forms "called in" (summoned to a place, asked for help) The option that best completes the gap is A. off. Mathematics Questions: 1. Convert 11011_2 to base 10. To convert a binary number to base 10, we multiply each digit by the corresponding power of 2 and sum the results. 11011_2 = (1 × 2^4) + (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0) = (1 × 16) + (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 16 + 8 + 0 + 2 + 1 = 27_10 The converted number is 27. 2. Convert 58 base 10 to base 2. To convert a base 10 number to base 2, we repeatedly divide the number by 2 and record the remainders. The binary number is formed by reading the remainders from bottom to top. 58 ÷ 2 = 29 R 0 29 ÷ 2 = 14 R 1 14 ÷ 2 = 7 R 0 7 ÷ 2 = 3 R 1 3 ÷ 2 = 1 R 1 1 ÷ 2 = 0 R 1 Reading the remainders from bottom to top gives 111010_2. The converted number is 111010_2. 3. Add 1011_2 and 111_2. We perform binary addition, carrying over when the sum is 2 or more. ` 1011_2 + 0111_2 ------- 10010_2 ` Step-by-step: Rightmost column: 1 + 1 = 10_2. Write down 0, carry over 1. Second column: 1 + 1 + carry 1 = 11_2. Write down 1, carry over 1. Third column: 0 + 1 + carry 1 = 10_2. Write down 0, carry over 1. Leftmost column: 1 + 0 + carry 1 = 10_2. Write down 0, carry over 1. The sum is 10010_2. 4. Subtract 1001_2 from 1110_2. We perform binary subtraction, borrowing from the left when needed. ` 1110_2 1001_2 ------- 0101_2 ` Step-by-step: Rightmost column: 0 - 1. We need to borrow. The 1 in the 2^1 position becomes 0, and the 0 in the 2^0 position becomes 10_2 (which is 2 in base 10). So, 10_2 - 1_2 = 1_2. Second column: The original 1 became 0 due to borrowing. So, 0 - 0 = 0. Third column: 1 - 0 = 1. Leftmost column: 1 - 1 = 0. The difference is 101_2. 5. Multiply 11_2 by 101_2. We perform binary multiplication similar to decimal multiplication. ` 101_2 x 11_2 ------- 101 (101_2 * 1_2) + 1010 (101_2 * 1_2, shifted one position left) ------- 1111_2 ` Step-by-step: Multiply 101_2 by the rightmost digit of 11_2 (which is 1_2): 101_2 × 1_2 = 101_2. Multiply 101_2 by the leftmost digit of 11_2 (which is 1_2), and shift the result one position to the left: 101_2 × 1_2 = 101_2, shifted becomes 1010_2. Add the partial products: 101_2 + 1010_2 = 1111_2. The product is 1111_2.