💡Okay, so I have this problem where the arithmetic mean, geometric mean, and harmonic mean of three numbers x, y, and z are given as 10, 7, and 4 respectively. I need to find the value of x² + y² + z². Hmm, let me think about how to approach this.First, I remember that the arithmetic mean (AM) of three numbers is just the sum of the numbers divided by three. So, if the AM is 10, that means (x + y + z)/3 = 10. Therefore, multiplying both sides by 3, I get x + y + z = 30. That seems straightforward.Next, the geometric mean (GM) is given as 7. The GM of three numbers is the cube root of their product. So, the cube root of (xyz) equals 7. To find xyz, I need to cube both sides, which gives me xyz = 7³ = 343. Okay, so the product of the three numbers is 343.Now, the harmonic mean (HM) is 4. The HM of three numbers is a bit trickier. I recall that the harmonic mean is 3 divided by the sum of the reciprocals of the numbers. So, HM = 3 / [(1/x) + (1/y) + (1/z)] = 4. Therefore, (1/x) + (1/y) + (1/z) = 3/4. Hmm, how can I relate this to something I know? Well, if I have (1/x) + (1/y) + (1/z) = 3/4, I can write this as (xy + yz + zx)/(xyz) = 3/4. Since I already know that xyz = 343, I can substitute that in. So, (xy + yz + zx)/343 = 3/4. Multiplying both sides by 343, I get xy + yz + zx = (3/4)*343. Let me calculate that: 343 divided by 4 is 85.75, and 85.75 multiplied by 3 is 257.25. So, xy + yz + zx = 257.25. Wait, 257.25 seems a bit messy. Maybe I should write it as a fraction. 343 is 7³, which is 343. So, (3/4)*343 is 1029/4, which is 257.25. Yeah, that's correct. So, the sum of the products of the numbers two at a time is 1029/4.Now, I need to find x² + y² + z². I remember that there's a formula relating the square of the sum of the numbers to the sum of their squares and twice the sum of their products. The formula is:(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)I have x + y + z = 30, so (30)² = 900. And I have xy + yz + zx = 1029/4. So, plugging into the formula:900 = x² + y² + z² + 2*(1029/4)Let me compute 2*(1029/4). That's the same as 1029/2, which is 514.5. So, 900 = x² + y² + z² + 514.5To find x² + y² + z², I subtract 514.5 from both sides:x² + y² + z² = 900 - 514.5 = 385.5Hmm, 385.5 is a decimal. I wonder if I can express that as a fraction. 0.5 is 1/2, so 385.5 is 385 and 1/2, which is 771/2. So, x² + y² + z² = 771/2.Wait, let me double-check my calculations to make sure I didn't make a mistake. Starting from the beginning:1. AM: (x + y + z)/3 = 10 => x + y + z = 30. Correct.2. GM: (xyz)^(1/3) = 7 => xyz = 343. Correct.3. HM: 3 / [(1/x) + (1/y) + (1/z)] = 4 => (1/x) + (1/y) + (1/z) = 3/4. Then, (xy + yz + zx)/xyz = 3/4 => (xy + yz + zx) = (3/4)*343 = 257.25 or 1029/4. Correct.4. Then, (x + y + z)² = 900 = x² + y² + z² + 2*(1029/4) = x² + y² + z² + 514.5. So, x² + y² + z² = 900 - 514.5 = 385.5. Yes, that seems right.Alternatively, if I express everything in fractions to avoid decimals:(x + y + z)² = 9002*(xy + yz + zx) = 2*(1029/4) = 1029/2 = 514.5So, 900 - 514.5 = 385.5, which is 771/2. So, both ways, it's consistent.I think that makes sense. So, the value of x² + y² + z² is 385.5 or 771/2.**Final Answer**The value of (x^2 + y^2 + z^2) is boxed{385.5}.

💡First, I need to determine the length of the long side of the rectangular farm. I know the area of the farm is 1200 m² and the short side is 30 m. Using the formula for the area of a rectangle, which is Area = Length × Width, I can solve for the length:[text{Length} = frac{text{Area}}{text{Width}} = frac{1200 , text{m}^2}{30 , text{m}} = 40 , text{m}]Now that I have the length of the long side, I can find the length of the diagonal using the Pythagorean theorem. The diagonal of a rectangle forms a right triangle with the length and width:[text{Diagonal}^2 = text{Length}^2 + text{Width}^2][text{Diagonal}^2 = 40^2 + 30^2 = 1600 + 900 = 2500][text{Diagonal} = sqrt{2500} = 50 , text{m}]With the lengths of the long side, short side, and diagonal determined, I can now calculate the total cost of fencing. The costs per meter are as follows:- Long side: Rs. 16 per meter- Short side: Rs. 14 per meter- Diagonal: Rs. 18 per meterCalculating the cost for each side:[text{Cost for long side} = 16 , text{Rs/m} times 40 , text{m} = 640 , text{Rs}][text{Cost for short side} = 14 , text{Rs/m} times 30 , text{m} = 420 , text{Rs}][text{Cost for diagonal} = 18 , text{Rs/m} times 50 , text{m} = 900 , text{Rs}]Adding these costs together gives the total cost:[text{Total Cost} = 640 , text{Rs} + 420 , text{Rs} + 900 , text{Rs} = 1960 , text{Rs}]

💡Okay, so I have this problem where f(x) is an odd function defined on all real numbers. When x is greater than 0, f(x) is given as log base 2 of x minus 2x. I need to figure out what f(x) is when x is less than 0, and choose the correct option from A to D.First, I remember that an odd function has the property that f(-x) = -f(x). That means if I know the function for positive x, I can find it for negative x by using this relationship. So, for x < 0, I can write f(x) as -f(-x). Since -x will be positive when x is negative, I can use the given expression for f(-x).Let me write that down step by step. If x is negative, then -x is positive. So, f(-x) is log base 2 of (-x) minus 2 times (-x). That simplifies to log base 2 of (-x) plus 2x. But since f(x) is equal to -f(-x), I need to take the negative of that expression.So, f(x) = - [log base 2 of (-x) + 2x]. Distributing the negative sign, that becomes -log base 2 of (-x) minus 2x. Hmm, let me check that again. If f(-x) is log base 2 of (-x) minus 2*(-x), which is log base 2 of (-x) plus 2x, then f(x) is -f(-x), so it's -log base 2 of (-x) minus 2x.Looking at the options, option C is -log base 2 of (-x) minus 2x, which matches what I got. Option D is similar but has a plus instead of a minus before the 2x, so that's not right. Option A and B have log base 2 of x instead of (-x), which doesn't make sense because when x is negative, log base 2 of x isn't defined in real numbers. So, those can't be correct either.Wait, let me make sure I didn't make a mistake in the signs. If f(-x) = log base 2 of (-x) - 2*(-x), that's log base 2 of (-x) + 2x. Then f(x) = -f(-x) = -log base 2 of (-x) - 2x. Yeah, that seems correct. So, I think option C is the right answer.I also want to visualize this. If f(x) is odd, its graph is symmetric about the origin. So, for positive x, it's log base 2 of x minus 2x, which is a logarithmic curve minus a linear term. For negative x, it should be the mirror image across the origin, which would involve negative logs and negative linear terms. That aligns with option C.Just to double-check, if I plug in a negative value, say x = -1, into option C, I get -log base 2 of (-(-1)) - 2*(-1) = -log base 2 of 1 + 2 = -0 + 2 = 2. If I use the odd function property, f(-1) should be -f(1). f(1) is log base 2 of 1 - 2*1 = 0 - 2 = -2. So, f(-1) should be 2, which matches what I got from option C. That confirms it's correct.I think I've got it. The answer is option C.

💡Okay, so I have this problem where I need to categorize a bunch of numbers into three different sets: rational numbers, irrational numbers, and positive real numbers. The numbers given are: 3/5, sqrt(9), pi, 3.14, -cbrt(27), 0, -5.12345..., and -sqrt(3). Hmm, let's see how I can approach this.First, I need to remember what rational and irrational numbers are. Rational numbers are numbers that can be expressed as a fraction of two integers, like a/b where b is not zero. They can also be terminating or repeating decimals. Irrational numbers, on the other hand, cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating. Got it.Now, let's look at each number one by one.1. **3/5**: This is already a fraction of two integers, so it's definitely a rational number. It's also positive because both numerator and denominator are positive.2. **sqrt(9)**: The square root of 9 is 3, which is an integer. Integers are rational numbers because they can be expressed as a fraction (like 3/1). So, sqrt(9) is rational and positive.3. **pi**: Pi is a well-known irrational number. Its decimal goes on forever without repeating. So, pi is irrational. It's also positive because pi is approximately 3.14159...4. **3.14**: This is a decimal number. It looks like it's a finite decimal, which means it can be expressed as a fraction. For example, 3.14 is the same as 314/100, which simplifies to 157/50. So, 3.14 is rational and positive.5. **-cbrt(27)**: The cube root of 27 is 3, so -cbrt(27) is -3. Negative integers are still rational numbers because they can be expressed as fractions (-3/1). So, this is rational but negative.6. **0**: Zero is a rational number because it can be expressed as 0/1. It's neither positive nor negative.7. **-5.12345...**: This is a decimal number with the ellipsis (...), which usually indicates that the decimals continue indefinitely. If it's non-repeating and non-terminating, then it's irrational. Since it's negative, it's also a negative real number.8. **-sqrt(3)**: The square root of 3 is an irrational number because it cannot be expressed as a simple fraction, and its decimal goes on forever without repeating. The negative sign makes it a negative irrational number.Now, let's categorize them:1. **Set of rational numbers**: These are numbers that can be expressed as fractions. From the list, 3/5, sqrt(9), 3.14, -cbrt(27), and 0 are rational.2. **Set of irrational numbers**: These cannot be expressed as fractions. From the list, pi, -5.12345..., and -sqrt(3) are irrational.3. **Set of positive real numbers**: These are numbers greater than zero. From the list, 3/5, sqrt(9), pi, and 3.14 are positive.Wait, let me double-check to make sure I didn't miss anything. - 3/5: Rational and positive.- sqrt(9): Rational and positive.- pi: Irrational and positive.- 3.14: Rational and positive.- -cbrt(27): Rational and negative.- 0: Rational, neither positive nor negative.- -5.12345...: Irrational and negative.- -sqrt(3): Irrational and negative.Yes, that seems correct. So, the sets should be:1. Rational numbers: {3/5, sqrt(9), 3.14, -cbrt(27), 0}2. Irrational numbers: {pi, -5.12345..., -sqrt(3)}3. Positive real numbers: {3/5, sqrt(9), pi, 3.14}I think that's all. I hope I didn't make any mistakes!

💡First, I need to find the residue of -1234 modulo 32. To do this, I'll start by determining how many times 32 fits into 1234.Dividing 1234 by 32 gives approximately 38.56. This means that 32 multiplied by 38 is 1216, which is the largest multiple of 32 that is less than 1234.Next, I'll calculate the difference between 1234 and 1216:1234 - 1216 = 18.Since I'm dealing with -1234, I'll consider the negative counterpart of this difference:-1234 = -1216 - 18.Now, to find the residue modulo 32, I'll add 32 to -18 to ensure the result is within the range of 0 to 31:-18 + 32 = 14.Therefore, the residue of -1234 modulo 32 is 14.

💡First, identify the number of pencils in each row, which is 8.Next, determine the number of rows Faye can make, which is 4.To find the total number of pencils, multiply the number of pencils per row by the number of rows: 8 * 4 = 32.Therefore, Faye has 32 pencils in total.