💡Okay, so I have this problem with vectors a and b. It says that the magnitude of vector a is 1, which is good to know. Then there's this equation: (a - b) · (a + b) equals 1/2. Hmm, I remember that when you have a dot product like (a - b) · (a + b), it's similar to the difference of squares in algebra. So maybe I can expand that.Let me try expanding it: (a - b) · (a + b) = a · a + a · b - b · a - b · b. Since dot product is commutative, a · b is the same as b · a, so those middle terms cancel out. So it simplifies to a · a - b · b. Which is |a|² - |b|². Given that |a| is 1, so |a|² is 1. Therefore, the equation becomes 1 - |b|² = 1/2. Solving for |b|², I subtract 1/2 from both sides: 1 - 1/2 = |b|², so |b|² is 1/2. Taking the square root, |b| is sqrt(1/2), which is sqrt(2)/2. Okay, so the magnitude of vector b is sqrt(2)/2.Now, part (1) says that a · b is 1/2, and we need to find the angle between a and b. I remember that the dot product formula is a · b = |a||b|cos(theta), where theta is the angle between them. Plugging in the values we have: 1/2 = |a||b|cos(theta). We know |a| is 1 and |b| is sqrt(2)/2, so substituting those in: 1/2 = (1)(sqrt(2)/2)cos(theta). Let me write that out: 1/2 = (sqrt(2)/2)cos(theta). To solve for cos(theta), I can multiply both sides by 2/sqrt(2). So cos(theta) = (1/2) * (2/sqrt(2)) = 1/sqrt(2). I know that cos(pi/4) is 1/sqrt(2), so theta must be pi/4 radians. That seems right because pi/4 is 45 degrees, and the cosine of 45 degrees is indeed 1/sqrt(2). So the angle between vectors a and b is pi/4.Moving on to part (2), we need to find the magnitude of a - 2b. So |a - 2b|. I remember that the magnitude squared of a vector is equal to the dot product of the vector with itself. So |a - 2b|² = (a - 2b) · (a - 2b). Let me expand that.Expanding the dot product: a · a - 2a · 2b + (2b) · (2b). Wait, no, that's not quite right. Let me do it step by step. (a - 2b) · (a - 2b) = a · a - 2a · 2b + 4b · b. Wait, no, that's not correct either. Let me recall the formula: (u - v) · (u - v) = |u|² - 2u · v + |v|². So in this case, u is a and v is 2b. So it's |a|² - 2a · (2b) + |2b|².Wait, that might not be the right way to think about it. Let me actually distribute it properly. (a - 2b) · (a - 2b) = a · a - a · 2b - 2b · a + 2b · 2b. Simplifying each term: a · a is |a|², which is 1. Then, -a · 2b is -2(a · b), which we know is -2*(1/2) = -1. Similarly, -2b · a is also -2(a · b) = -1. Then, 2b · 2b is 4(b · b) which is 4|b|². We know |b|² is 1/2, so 4*(1/2) is 2.Putting it all together: 1 -1 -1 + 2. Let's compute that: 1 -1 is 0, 0 -1 is -1, -1 +2 is 1. So |a - 2b|² is 1. Therefore, |a - 2b| is sqrt(1) which is 1.Wait, that seems too straightforward. Let me double-check. Starting from |a - 2b|² = |a|² + |2b|² - 4(a · b). Because (a - 2b) · (a - 2b) = |a|² + |2b|² - 4(a · b). Calculating each term: |a|² is 1, |2b|² is 4|b|² which is 4*(1/2) = 2. Then, 4(a · b) is 4*(1/2) = 2. So putting it all together: 1 + 2 - 2 = 1. So yes, |a - 2b|² is 1, so |a - 2b| is 1. That checks out.I think that makes sense. So the magnitude of a - 2b is 1. It's interesting because even though we scaled vector b by 2, the magnitude ended up being the same as vector a. Maybe because of the specific angle and magnitudes involved.So, in summary, for part (1), the angle between a and b is pi/4, and for part (2), the magnitude of a - 2b is 1.**Final Answer**(1) The angle between vectors overrightarrow{a} and overrightarrow{b} is boxed{dfrac{pi}{4}}.(2) The value of |overrightarrow{a} - 2overrightarrow{b}| is boxed{1}.

💡Okay, so I have this problem here: If z equals 4 plus 3i, then what is the dot z divided by the absolute value of z? And the options are A: 1, B: -1, C: 4/5 plus 3/5i, and D: 4/5 minus 3/5i.Hmm, first, I need to figure out what the dot over z means. I think in complex numbers, the dot over z usually represents the complex conjugate of z. So, if z is 4 plus 3i, then the conjugate, which is dot z, should be 4 minus 3i. Let me confirm that. Yeah, the complex conjugate of a complex number a plus bi is a minus bi. So, that makes sense.Next, I need to find the absolute value of z, which is the modulus of z. The modulus of a complex number a plus bi is the square root of (a squared plus b squared). So, for z equals 4 plus 3i, the modulus should be the square root of (4 squared plus 3 squared). Let me calculate that: 4 squared is 16, 3 squared is 9, so 16 plus 9 is 25, and the square root of 25 is 5. So, the modulus of z is 5.Now, the problem is asking for the complex conjugate of z divided by the modulus of z. So, that would be (4 minus 3i) divided by 5. Let me write that out: (4 - 3i)/5. If I separate the real and imaginary parts, that would be 4/5 minus (3/5)i.Looking at the options, option D is 4/5 minus 3/5i, which matches what I just calculated. So, I think the answer should be D.Wait, let me double-check my steps to make sure I didn't make a mistake. First, complex conjugate: yes, 4 minus 3i. Modulus: square root of (16 + 9) is 5, correct. Dividing each part by 5: 4/5 and -3/5i, yes. So, yeah, D is the right answer.I don't think I made any mistakes here. It seems straightforward once I remember what the complex conjugate is and how to find the modulus. Maybe I should also recall why the modulus is calculated that way. It's because the modulus represents the distance from the origin to the point (a, b) in the complex plane, which is a right triangle with sides a and b, so the distance is the hypotenuse, hence the square root of (a squared plus b squared). That makes sense.Also, just to make sure, if I had messed up the complex conjugate, I might have gotten a different answer. For example, if I had left it as 4 plus 3i, then dividing by 5 would give me 4/5 plus 3/5i, which is option C. But since the question has a dot over z, which I believe is the conjugate, I have to change the sign of the imaginary part, so it's 4 minus 3i, leading to option D.Another way to think about it is that dividing a complex number by its modulus gives a complex number of unit modulus, meaning it lies on the unit circle in the complex plane. So, in this case, (4 - 3i)/5 should have a modulus of 1. Let me check that: modulus squared would be (4/5)^2 plus (-3/5)^2, which is 16/25 plus 9/25, which is 25/25, which is 1. So, that confirms that the modulus is indeed 1, which is consistent with dividing by the modulus of z.So, all the steps check out, and I feel confident that D is the correct answer.**Final Answer**boxed{D}

💡Okay, so I have this problem here about a normal distribution. The mean is 14.5 and the standard deviation is 1.5. The question is asking for the value that's exactly 2 standard deviations less than the mean. Hmm, let me think about how to approach this.First, I remember that in a normal distribution, the mean is the center, and the standard deviation tells us how spread out the data is. So, if I want to find a value that's 2 standard deviations below the mean, I need to subtract twice the standard deviation from the mean.Let me write that down to visualize it better. The mean (μ) is 14.5, and the standard deviation (σ) is 1.5. So, 2 standard deviations would be 2 times 1.5, which is 3. Now, to find the value that's 2 standard deviations less than the mean, I subtract this 3 from the mean of 14.5.Wait, let me double-check that. If I have a mean of 14.5 and I subtract 3, that should give me 11.5. Does that make sense? Yeah, because 2 standard deviations below the mean would be moving to the left on the distribution curve, which in this case is 11.5.I think that's it. So, the value exactly 2 standard deviations less than the mean is 11.5. To make sure I didn't make any mistakes, I can plug the numbers back in. Mean is 14.5, subtract 2 times 1.5 (which is 3), and I get 11.5. Yep, that seems right.I guess there's not much more to it. It's a straightforward calculation once you understand what subtracting standard deviations from the mean entails. I don't think I need to consider any other factors here because the problem is pretty direct. It's just applying the basic formula for finding values a certain number of standard deviations away from the mean.So, final answer: 11.5.

💡Okay, so I have this problem about food hygiene inspections, and there are two parts. Let me try to figure them out step by step.First, the problem states that the pass rates for food types A and B are 90% and 80%, respectively. That means the probability of a food item being qualified is 0.9 for type A and 0.8 for type B. Therefore, the probability of an item being unqualified would be 1 minus the pass rate. So, for type A, the unqualified probability is 0.1, and for type B, it's 0.2.Now, moving on to part (1). It says that three students gather and each randomly selects one item from both types of food. We need to find the probability that exactly one student gets two items that are both unqualified.Hmm, okay. So each student is selecting one item from type A and one from type B. So, for each student, the probability that both items are unqualified would be the product of the individual probabilities, right? So that's 0.1 (for type A) multiplied by 0.2 (for type B), which is 0.02. So, each student has a 2% chance of getting both items unqualified.Now, since we have three students, and we want exactly one of them to have both items unqualified. This sounds like a binomial probability problem. The binomial formula is:P(k successes in n trials) = C(n, k) * (p)^k * (1-p)^(n-k)Where C(n, k) is the combination of n things taken k at a time.In this case, n is 3 (three students), k is 1 (exactly one student), p is 0.02 (probability of success, which here is getting both items unqualified).So, plugging in the numbers:C(3, 1) = 3p^k = (0.02)^1 = 0.02(1-p)^(n-k) = (0.98)^(3-1) = (0.98)^2 ≈ 0.9604So, the probability is 3 * 0.02 * 0.9604 ≈ 3 * 0.02 * 0.9604Calculating that:First, 0.02 * 0.9604 ≈ 0.019208Then, 3 * 0.019208 ≈ 0.057624So, approximately 0.0576, or 5.76%.Wait, that seems low, but considering that the chance of both items being unqualified is only 2%, it's not too surprising that the chance of exactly one out of three students having that is around 5.76%.Okay, moving on to part (2). A consumer randomly purchases one item from each of the two types of food, and ξ represents the number of unqualified food items purchased. We need to find the expected value of ξ.So, ξ can be 0, 1, or 2. Let's figure out the probabilities for each case.First, ξ=0 means both items are qualified. The probability of that is the product of both being qualified. So, for type A, it's 0.9, and for type B, it's 0.8. So, 0.9 * 0.8 = 0.72.Next, ξ=1 means either the type A is unqualified and type B is qualified, or type A is qualified and type B is unqualified. So, those are two separate cases.The probability for type A unqualified and type B qualified is 0.1 * 0.8 = 0.08.The probability for type A qualified and type B unqualified is 0.9 * 0.2 = 0.18.So, adding those together, 0.08 + 0.18 = 0.26.Lastly, ξ=2 means both items are unqualified. We already calculated that earlier as 0.02.To double-check, the total probabilities should add up to 1. So, 0.72 + 0.26 + 0.02 = 1.00. Perfect, that checks out.Now, the expected value E[ξ] is calculated as the sum of each outcome multiplied by its probability.So, E[ξ] = 0 * P(ξ=0) + 1 * P(ξ=1) + 2 * P(ξ=2)Plugging in the numbers:E[ξ] = 0 * 0.72 + 1 * 0.26 + 2 * 0.02Calculating each term:0 * 0.72 = 01 * 0.26 = 0.262 * 0.02 = 0.04Adding them up: 0 + 0.26 + 0.04 = 0.30So, the expected value is 0.30.Wait, let me think if there's another way to calculate the expected value without listing all the probabilities. Since ξ is the number of unqualified items, which can be seen as the sum of two Bernoulli random variables: one for type A and one for type B.So, if we let ξ_A be 1 if type A is unqualified, 0 otherwise, and ξ_B be 1 if type B is unqualified, 0 otherwise. Then, ξ = ξ_A + ξ_B.The expected value of ξ is E[ξ_A] + E[ξ_B].E[ξ_A] is the probability that type A is unqualified, which is 0.1.E[ξ_B] is the probability that type B is unqualified, which is 0.2.So, E[ξ] = 0.1 + 0.2 = 0.3.Yes, that's another way to get the same result, which confirms that the expected value is indeed 0.30.So, summarizing:1) The probability that exactly one student gets two unqualified items is approximately 0.0576.2) The expected number of unqualified items purchased is 0.30.**Final Answer**(1) The probability is boxed{0.0576}.(2) The expected value is boxed{0.30}.

💡To evaluate the expression (98^3 + 3(98^2)(2) + 3(98)(2^2) + 2^3), I recognize that it resembles the expansion of a binomial raised to the third power. I recall the binomial expansion formula for ((x + y)^3), which is:[(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3]By comparing this with the given expression, I identify (x) as 98 and (y) as 2. Substituting these values into the binomial formula:[(98 + 2)^3 = 98^3 + 3(98^2)(2) + 3(98)(2^2) + 2^3]This simplifies to:[100^3 = 1,000,000]Therefore, the expression evaluates to 1,000,000.

💡Alright, so I've got this problem here with four options, A through D, each involving square roots and some operations like addition, subtraction, multiplication, and division. I need to figure out which one is correct. Let's take it step by step.Starting with option A: √8 + √2 = √10. Hmm, okay. I remember that when you're adding square roots, they need to have the same radicand, which is the number under the square root symbol. So, √8 and √2 have different radicands. Maybe I can simplify √8 to see if it can be expressed in terms of √2.√8 can be broken down into √(4*2), which is √4 * √2. Since √4 is 2, that simplifies to 2√2. So, √8 is 2√2. Now, if I add √2 to that, it's like 2√2 + √2. Since both terms have √2, I can combine them. 2√2 + √2 is 3√2. But the right side of the equation is √10, which is approximately 3.16, and 3√2 is about 4.24. These aren't equal, so option A is incorrect.Moving on to option B: √8 - √2 = √6. Again, let's simplify √8 to 2√2. So, 2√2 - √2 would be √2. But the right side is √6, which is approximately 2.45, and √2 is about 1.41. These aren't equal either, so option B is also incorrect.Next is option C: √8 * √2 = 4. Multiplying square roots is straightforward; you can multiply the numbers under the roots first. So, √8 * √2 is √(8*2) = √16. √16 is 4. That matches the right side of the equation, so option C seems correct.Finally, option D: √8 ÷ √2 = 4. Dividing square roots works similarly to multiplying; you can divide the numbers under the roots first. So, √8 ÷ √2 is √(8/2) = √4. √4 is 2, not 4. Therefore, option D is incorrect.After checking all four options, only option C is correct.