| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | A. 50 + 50i Explanation: (i + 1)(5 – 5i)(5 + 5i) = (i + 1)(25 + 25) = (i + 1)(50) = 50i + 50 = 50 + 50i. |
A |
| deepseekr1 | The given expression is (i + 1)(5 – 5i)(5 + 5i). First, recognize that (5 – 5i) and (5 + 5i) are conjugates. Multiplying them: \[ (5 - 5i)(5 + 5i) = 5^2 - (5i)^2 = 25 - (-25) = 50. \] Now, the expression simplifies to: \[ (i + 1) \cdot 50 = 50(i + 1) = 50i + 50 = 50 + 50i. \] The result matches **A. 50 + 50i**. **Answer:** A |
A |
| gpto1 | A | A |
| GPT-4o | To solve the question, we need to simplify the expression \((i + 1)(5 - 5i)(5 + 5i)\). First, let's simplify \((5 - 5i)(5 + 5i)\): \[ (5 - 5i)(5 + 5i) = 5^2 - (5i)^2 = 25 - 25i^2 \] Since \(i^2 = -1\): \[ 25 - 25(-1) = 25 + 25 = 50 \] So, \((5 - 5i)(5 + 5i) = 50\). Next, we multiply this result by \((i + 1)\): \[ (i + 1) \cdot 50 = 50i + 50 \] So, the expression \((i + 1)(5 - 5i)(5 + 5i)\) simplifies to \(50 + 50i\). Therefore, the answer is: \[ A. 50 + 50i \] |
A |
| llama3.1-70B-Instruct | (i + 1)(5 – 5i)(5 + 5i) = (i + 1)(25 – 25i2) = (i + 1)(25 + 25) = (i + 1)(50) = 50 + 50i |
A |
| llama3.1-405B-Instruct | A |
A |
| DeepSeek-Coder-V2 | To solve the given expression \((i + 1)(5 - 5i)(5 + 5i)\), we will follow these steps: 1. **Simplify the product of the complex conjugates:** \[ (5 - 5i)(5 + 5i) \] Using the difference of squares formula \( (a - bi)(a + bi) = a^2 + b^2 \): \[ (5 - 5i)(5 + 5i) = 5^2 + (5i)^2 = 25 + 25(-1) = 25 - 25 = 25 \] 2. **Multiply the result by \((i + 1)\):** \[ (i + 1) \cdot 25 \] Distribute the 25: \[ 25(i + 1) = 25i + 25 \] Thus, the final expression is: \[ 25i + 25 \] Comparing this with the given options, the correct answer is: \[ \boxed{C} \] |
A |
| Qwen2-72B-Instruct | B. 50 – 50i | A |
| baichuan4 | B | A |
| doubao | B. First, calculate (5 - 5i)(5 + 5i) = 5^2 - (5i)^2 = 25 - 25i^2 = 25 - 25(-1) = 25 + 25 = 50. Then multiply by (i + 1): 50(i + 1) = 50i + 50 = 50 - 50i. |
A |
| minimax | To solve the given expression, we can use the difference of squares formula, which states that (a - b)(a + b) = a^2 - b^2. Here, we can consider (5 - 5i) and (5 + 5i) as a pair that fits the difference of squares formula, where a = 5 and b = 5i. So, (5 - 5i)(5 + 5i) = 5^2 - (5i)^2 = 25 - 25(-1) = 25 + 25 = 50. Now, we multiply this result by (i + 1): (i + 1)(50) = 50i + 50. Therefore, the correct answer is: A. 50 + 50i |
A |
| kimi | To solve this problem, we can use the difference of squares formula, which states that \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 5\) and \(b = 5i\), so we can simplify the expression \((5 - 5i)(5 + 5i)\) as follows: \[ (5 - 5i)(5 + 5i) = 5^2 - (5i)^2 = 25 - 25i^2 \] Since \(i^2 = -1\), we can substitute that in: \[ 25 - 25(-1) = 25 + 25 = 50 \] Now, we multiply this result by \(i + 1\): \[ (i + 1)(50) = 50i + 50 \] So the final answer is: \[ 50 + 50i \] Therefore, the correct answer is \(\boxed{\text{A}}\). |
A |
| wenxinyiyan | B Explanation: To solve this problem, we need to multiply the given complex numbers. Let's do it step by step. Step 1: Multiply the first two complex numbers. (i + 1)(5 – 5i) = i(5) + i(-5i) + 1(5) + 1(-5i) = 5i - 5i^2 + 5 - 5i Since i^2 = -1, we can simplify the expression as: = 5i + 5 + 5 - 5i = 10 + 5i (This is the intermediate result) Step 2: Multiply the intermediate result with the third complex number. (10 + 5i)(5 + 5i) = 10(5) + 10(5i) + 5i(5) + 5i(5i) = 50 + 50i + 25i + 25i^2 Again, using i^2 = -1, we get: = 50 + 50i + 25i - 25 = 25 + 75i (This is the intermediate result) Oops! It seems like I made a mistake in Step 1. Let's correct it and redo the calculations. Step 1 (Corrected): Multiply the first two complex numbers. (i + 1)(5 – 5i) = i(5) + i(-5i) + 1(5) + 1(-5i) = 5i - 5i^2 + 5 - 5i Since i^2 = -1, we can simplify the expression as: = 5i + 5 + 5 - 5i = 10 (Corrected intermediate result) Step 2 (Redo): Multiply the corrected intermediate result with the third complex number. 10(5 + 5i) = 10(5) + 10(5i) = 50 + 50i However, we can further simplify this by multiplying it with the conjugate of the second complex number's denominator, which is actually (5 - 5i) in this case, but since we already multiplied it, we just need to multiply by the last part (5 + 5i). This will give us: 10(5 + 5i) / (5 + 5i)(5 - 5i) (We multiply by the conjugate to eliminate the imaginary part in the denominator) = 10(5 + 5i) / (25 - 25i^2) = 10(5 + 5i) / (25 + 25) (Since i^2 = -1) = 10(5 + 5i) / 50 = (5 + 5i) / 5 = 1 + i (This is the corrected intermediate result after dividing) Now, let's multiply this by the original first complex number (i + 1): (i + 1)(1 + i) = i(1) + i(i) + 1(1) + 1(i) = i + i^2 + 1 + i = i - 1 + 1 + i (Since i^2 = -1) = 2i (This is the final corrected intermediate result) Now, we multiply this by 50 (the result of multiplying the denominators): 2i * 50 = 100i Wait, this is still not correct. It seems like we went off track. Let's go back to the original expression and solve it in a different way. (i + 1)(5 – 5i)(5 + 5i) First, let's use the fact that (a - b)(a + b) = a^2 - b^2 to simplify the last two factors: (5 - 5i)(5 + 5i) = 5^2 - (5i)^2 = 25 - 25i^2 = 25 + 25 (Since i^2 = -1) = 50 Now, we multiply this by the first factor (i + 1): (i + 1) * 50 = 50i + 50 However, this is still not one of the options. It seems like there might be a typo in the question or the options |
A |
| zhipu | To solve the expression \((i + 1)(5 - 5i)(5 + 5i)\), let's break it down step by step. First, consider the product \((5 - 5i)(5 + 5i)\). This is a difference of squares: \[ (5 - 5i)(5 + 5i) = 5^2 - (5i)^2 \] We know that \(5^2 = 25\) and \((5i)^2 = 25i^2\). Since \(i^2 = -1\), we have: \[ (5i)^2 = 25(-1) = -25 \] So, the expression simplifies to: \[ 5^2 - (5i)^2 = 25 - (-25) = 25 + 25 = 50 \] Now, we need to multiply this result by \((i + 1)\): \[ (i + 1) \cdot 50 \] Distribute 50: \[ 50i + 50 \] So, the final result is: \[ 50 + 50i \] Therefore, the correct answer is: A. 50 + 50i |
A |
| gpto1 | A | A |
| deepseekv3 | To solve the expression \((i + 1)(5 - 5i)(5 + 5i)\), we proceed step by step. 1. **Simplify \((5 - 5i)(5 + 5i)\)**: This is a product of a complex number and its conjugate, so we use the formula \((a - bi)(a + bi) = a^2 + b^2\): \[ (5 - 5i)(5 + 5i) = 5^2 + 5^2 = 25 + 25 = 50. \] 2. **Multiply the result by \((i + 1)\)**: Now we have: \[ (i + 1) \cdot 50 = 50i + 50. \] 3. **Rearrange the terms**: \[ 50i + 50 = 50 + 50i. \] Thus, the correct answer is: **A. 50 + 50i** |
A |