💡First, I need to identify the weight range that satisfies all the given opinions.Arun's opinion: greater than 66 kg but less than 72 kg.Brother's opinion: greater than 60 kg but less than 70 kg.Mother's opinion: not greater than 69 kg.Father's opinion: between 65 kg and 71 kg.Sister's opinion: more than 62 kg but not more than 68 kg.Next, I'll determine the overlapping range that meets all these criteria.The lowest weight that satisfies all opinions is the highest of the lower limits: max(66, 60, 65, 62) = 66 kg.The highest weight that satisfies all opinions is the lowest of the upper limits: min(72, 70, 69, 71, 68) = 68 kg.Therefore, Arun's weight must be greater than 66 kg and not more than 68 kg, which means the possible weights are 67 kg and 68 kg.To find the average of these probable weights, I'll add them together and divide by the number of values:Average weight = (67 + 68) / 2 = 135 / 2 = 67.5 kgThus, the average of the different probable weights of Arun is 67.5 kg.

💡Okay, so I have this problem where I need to find all possible pairs of (w, l) where w is the width and l is the length of a rectangle, and both are positive integers. The area of the rectangle has to be 18 square units. I need to figure out how to plot these pairs correctly.First, I know that the area of a rectangle is calculated by multiplying its width and length. So, the formula is:[ text{Area} = w times l ]Given that the area is 18, I can set up the equation:[ w times l = 18 ]Now, since both w and l are positive integers, I need to find all pairs of integers that multiply together to give 18. This means I need to list all the factors of 18.Let me start by listing the factors of 18. Factors are numbers that divide 18 without leaving a remainder.1. Start with 1: 18 divided by 1 is 18. So, (1, 18) is one pair.2. Next, 2: 18 divided by 2 is 9. So, (2, 9) is another pair.3. Then, 3: 18 divided by 3 is 6. So, (3, 6) is another pair.4. Now, 6: 18 divided by 6 is 3. So, (6, 3) is another pair.5. Next, 9: 18 divided by 9 is 2. So, (9, 2) is another pair.6. Finally, 18: 18 divided by 18 is 1. So, (18, 1) is the last pair.So, the pairs I have are (1, 18), (2, 9), (3, 6), (6, 3), (9, 2), and (18, 1).Wait, but I need to make sure that both width and length are positive integers. All these pairs consist of positive integers, so that's good.Now, I need to plot these pairs. Since w is the width and l is the length, I can plot them on a coordinate plane where the x-axis represents width (w) and the y-axis represents length (l).So, each pair (w, l) will correspond to a point on the graph. For example, (1, 18) will be a point where x=1 and y=18, (2, 9) will be at x=2 and y=9, and so on.Let me visualize this. If I plot these points, they should form a hyperbola because the product of w and l is constant (18). However, since we are only considering positive integers, we will have discrete points rather than a continuous curve.Looking at the options provided:A) (1, 18), (2, 9), (3, 6), (6, 3), (9, 2), (18, 1)B) (1, 18), (2, 7), (4, 4.5)C) (1, 18), (2, 8.5), (3, 6), (6, 2)D) All pairs with integer values as long as w ⋅ l = 18 but excluding (18, 1)Option A lists all the pairs I found earlier. Option B includes some non-integer values like (4, 4.5), which doesn't make sense because both width and length should be integers. Option C also includes a non-integer value (2, 8.5) and misses some pairs like (9, 2) and (18, 1). Option D suggests excluding (18, 1), but that pair is valid since both are integers and their product is 18.Therefore, the correct answer should include all the integer pairs without excluding any valid ones. So, option A is the right choice.

💡Okay, so I have this problem where I need to find the value of z in the compound C6H8Oz, and I know that the molecular weight is 192. Alright, let's see. I remember that molecular weight is the sum of the atomic weights of all the atoms in the molecule. So, I need to calculate the atomic weights for carbon, hydrogen, and oxygen, and then see how many oxygens are needed to make the total molecular weight 192.First, let's break it down. The formula is C6H8Oz. That means there are 6 carbon atoms, 8 hydrogen atoms, and z oxygen atoms. I need to find z.I think the atomic weights are roughly: Carbon is about 12, Hydrogen is about 1, and Oxygen is about 16. Let me double-check that. Yeah, I think those are the approximate atomic weights.So, for carbon: 6 atoms times 12 g/mol equals 72 g/mol.For hydrogen: 8 atoms times 1 g/mol equals 8 g/mol.For oxygen: z atoms times 16 g/mol equals 16z g/mol.Now, the total molecular weight is the sum of these three: 72 + 8 + 16z.And the problem says that the total molecular weight is 192 g/mol. So, I can set up the equation:72 + 8 + 16z = 192Let me add 72 and 8 first. 72 plus 8 is 80. So, the equation simplifies to:80 + 16z = 192Now, I need to solve for z. I'll subtract 80 from both sides:16z = 192 - 80192 minus 80 is 112. So, now I have:16z = 112To find z, I'll divide both sides by 16:z = 112 / 16Let me do that division. 16 times 7 is 112, so z equals 7.Wait, does that make sense? Let me check my calculations again to be sure.Carbon: 6 * 12 = 72Hydrogen: 8 * 1 = 8Oxygen: 7 * 16 = 112Total: 72 + 8 + 112 = 192Yes, that adds up correctly. So, z is indeed 7.I think that's it. The value of z is 7.

💡Alright, let's tackle this problem step by step. It's a bit involved, but I'll take it one piece at a time.**Problem I2.1: Solving the simultaneous equations**We have the system of equations:[left{begin{array}{l}x - y - z = -1 y - x - z = -2 z - x - y = -4end{array} right.]Hmm, okay. Let's see. I need to find the values of (x), (y), and (z) that satisfy all three equations simultaneously.First, I notice that each equation has three variables, and they are all linear. Maybe I can use substitution or elimination. Let me try adding all three equations together to see if that simplifies things.Adding the left sides:[(x - y - z) + (y - x - z) + (z - x - y)]Simplify:[x - y - z + y - x - z + z - x - y]Combine like terms:[(x - x - x) + (-y + y - y) + (-z - z + z)]Which simplifies to:[- x - y - z]Now, adding the right sides:[-1 + (-2) + (-4) = -7]So, we have:[- x - y - z = -7]Multiply both sides by -1:[x + y + z = 7]Okay, that's a useful equation. Now, let's see if I can find another equation to relate these variables.Looking back at the original equations:1. (x - y - z = -1)2. (y - x - z = -2)3. (z - x - y = -4)Let me try to express each equation in terms of (x + y + z).From equation 1:[x - y - z = -1]But (x + y + z = 7), so maybe I can add these two equations.Adding equation 1 and (x + y + z = 7):[(x - y - z) + (x + y + z) = -1 + 7]Simplify:[2x = 6 implies x = 3]Great, found (x = 3). Now, let's find (y) and (z).From equation 2:[y - x - z = -2]We know (x = 3), so:[y - 3 - z = -2 implies y - z = 1]From equation 3:[z - x - y = -4]Again, (x = 3), so:[z - 3 - y = -4 implies z - y = -1]Now, we have two equations:1. (y - z = 1)2. (z - y = -1)These are actually the same equation. So, it seems we need another approach.Wait, we know (x + y + z = 7), and (x = 3), so:[3 + y + z = 7 implies y + z = 4]Now, we have:1. (y - z = 1)2. (y + z = 4)Let's add these two equations:[(y - z) + (y + z) = 1 + 4]Simplify:[2y = 5 implies y = frac{5}{2}]Now, substitute (y = frac{5}{2}) into (y + z = 4):[frac{5}{2} + z = 4 implies z = 4 - frac{5}{2} = frac{3}{2}]So, the solution is:[x = 3, quad y = frac{5}{2}, quad z = frac{3}{2}]**Problem I2.2: Calculating the remainder (beta)**We need to find the remainder when a number consisting of 10011 ones (i.e., (underbrace{111ldots111}_{10011 text{ 1's}})) is divided by (alpha). But wait, what is (alpha)? It wasn't defined in the problem. Hmm, maybe (alpha) is the solution from the previous problem? Let me check.From Problem I2.1, we found (x = 3), (y = frac{5}{2}), (z = frac{3}{2}). Maybe (alpha) is one of these? Or perhaps it's a typo and should be (alpha = x), which is 3.Assuming (alpha = 3), let's proceed.We need to find the remainder when (underbrace{111ldots111}_{10011 text{ 1's}}) is divided by 3.I remember that a number is divisible by 3 if the sum of its digits is divisible by 3. Since all digits are 1, the sum is 10011. Let's find the remainder when 10011 is divided by 3.Divide 10011 by 3:[10011 ÷ 3 = 3337 text{ with a remainder of } 0]Because 3 × 3337 = 10011.So, the sum of the digits is divisible by 3, which means the number itself is divisible by 3. Therefore, the remainder (beta) is 0.Wait, but in the original problem, it says "If (beta) is the remainder of (underbrace{111 cdots 111}_{10011 text{ 1's}} div alpha)", and I assumed (alpha = 3). If (alpha) is something else, this might change. But since (alpha) wasn't defined, I think it's safe to assume it's 3 based on the previous solution.**Problem I2.3: Calculating (gamma)**We need to find the remainder when (left[(beta - 2)^{100} + beta^{50} + (beta + 2)^{25}right]) is divided by 3.From the previous step, (beta = 0).Substitute (beta = 0):[left[(0 - 2)^{100} + 0^{50} + (0 + 2)^{25}right] = [(-2)^{100} + 0 + 2^{25}]]Calculate each term:- ((-2)^{100}): Since the exponent is even, this is (2^{100}).- (0^{50}): This is 0.- (2^{25}): This is a large number, but we only need the remainder modulo 3.Let's find each term modulo 3.First, (2 equiv -1 pmod{3}).So, (2^{100} equiv (-1)^{100} equiv 1 pmod{3}).Similarly, (2^{25} equiv (-1)^{25} equiv -1 pmod{3}).Therefore, the expression modulo 3 is:[1 + 0 + (-1) = 0 pmod{3}]So, the remainder (gamma) is 0.Wait, but in the original problem, it says "If (gamma) is the remainder...", and I got 0. Let me double-check.[(-2)^{100} = 2^{100} equiv 1 pmod{3}][0^{50} = 0][2^{25} equiv (-1)^{25} = -1 equiv 2 pmod{3}]Wait, hold on. (2^{25}) modulo 3: Since 2 ≡ -1 mod 3, so 2^25 ≡ (-1)^25 ≡ -1 ≡ 2 mod 3.So, the expression is:[1 + 0 + 2 = 3 equiv 0 pmod{3}]Yes, so (gamma = 0).**Problem I2.4: Solving the polynomial equation**We have the equation:[x^{4} + a x^{2} + b x + delta = 0]It has four real roots, three of which are (1), (gamma), and (gamma^{2}). We need to find (delta).From the previous step, (gamma = 0). So, the roots are (1), (0), and (0^{2} = 0). Wait, that means two of the roots are 0? So, the roots are (1), (0), (0), and another root, say (t).But wait, the problem says three of the roots are (1), (gamma), and (gamma^{2}). So, if (gamma = 0), then the roots are (1), (0), and (0). So, the fourth root is also 0? Or is it another distinct root?Wait, let me clarify. The polynomial is quartic, so it has four roots. Three of them are given: (1), (gamma), and (gamma^{2}). So, the fourth root is something else, say (t).But if (gamma = 0), then the roots are (1), (0), (0), and (t). So, two roots are 0, one is 1, and the fourth is (t).But let's use Vieta's formulas to find (delta).Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.For a quartic equation:[x^{4} + c x^{3} + d x^{2} + e x + f = 0]The sum of the roots is (-c), the sum of the products of the roots taken two at a time is (d), the sum of the products taken three at a time is (-e), and the product of all roots is (f).In our case, the equation is:[x^{4} + a x^{2} + b x + delta = 0]Notice that the coefficient of (x^{3}) is 0, so (c = 0). Therefore, the sum of the roots is 0.Let the roots be (1), (gamma), (gamma^{2}), and (t). So:[1 + gamma + gamma^{2} + t = 0]We know (gamma = 0), so:[1 + 0 + 0 + t = 0 implies t = -1]Now, the product of the roots is (delta) (since the constant term is (delta), and the leading coefficient is 1). So:[1 times gamma times gamma^{2} times t = delta]Substitute (gamma = 0) and (t = -1):[1 times 0 times 0 times (-1) = 0 = delta]Wait, that seems too straightforward. Let me double-check.Given the roots are (1), (0), (0), and (-1), their product is indeed (1 times 0 times 0 times (-1) = 0). So, (delta = 0).But let me make sure I didn't miss anything. The polynomial is:[(x - 1)(x - 0)(x - 0)(x + 1) = (x - 1)x^{2}(x + 1)]Multiply it out:[(x - 1)(x + 1) times x^{2} = (x^{2} - 1) times x^{2} = x^{4} - x^{2}]So, the polynomial is (x^{4} - x^{2}), which means (a = -1), (b = 0), and (delta = 0).Yes, that matches. So, (delta = 0).**Final Answer**boxed{0}

💡Okay, so I have this triangle ABC with sides a, b, c. There are two vectors given: vector m is (sin B - sin A, sqrt(3)a + c) and vector n is (sin C, a + b). It's also given that vector m is parallel to vector n. I need to find the value of angle B.First, I remember that if two vectors are parallel, one is a scalar multiple of the other. So, that means the components of vector m must be proportional to the components of vector n. So, I can set up a proportion:(sin B - sin A) / sin C = (sqrt(3)a + c) / (a + b)Hmm, okay. Now, I also recall the Law of Sines, which says that in any triangle, a/sin A = b/sin B = c/sin C. Maybe I can use that to relate the sides and the sines of the angles.Let me denote the common ratio as 2R, where R is the radius of the circumscribed circle. So, a = 2R sin A, b = 2R sin B, c = 2R sin C.Wait, but in the proportion above, I have sin B - sin A over sin C. Maybe I can express sin B - sin A in terms of sides. Since a = 2R sin A and b = 2R sin B, then sin A = a/(2R) and sin B = b/(2R). So, sin B - sin A = (b - a)/(2R).Similarly, sin C = c/(2R). So, the left side of the proportion becomes (b - a)/(2R) divided by c/(2R), which simplifies to (b - a)/c.So, now the proportion is (b - a)/c = (sqrt(3)a + c)/(a + b).Okay, so we have:(b - a)/c = (sqrt(3)a + c)/(a + b)Let me cross-multiply to eliminate the fractions:(b - a)(a + b) = c(sqrt(3)a + c)Expanding the left side: (b - a)(a + b) = b^2 - a^2So, b^2 - a^2 = c(sqrt(3)a + c)Let me write that as:b^2 - a^2 = sqrt(3) a c + c^2Hmm, let's rearrange the terms:b^2 - a^2 - sqrt(3) a c - c^2 = 0Wait, maybe I can write this as:b^2 = a^2 + c^2 + sqrt(3) a cHmm, that looks a bit like the Law of Cosines, which is:b^2 = a^2 + c^2 - 2ac cos BComparing the two equations:a^2 + c^2 + sqrt(3) a c = a^2 + c^2 - 2ac cos BSubtracting a^2 + c^2 from both sides:sqrt(3) a c = -2ac cos BDivide both sides by ac (assuming a and c are not zero, which they can't be in a triangle):sqrt(3) = -2 cos BSo, cos B = -sqrt(3)/2Now, I need to find the angle B where the cosine is -sqrt(3)/2. I remember that cos(30°) = sqrt(3)/2, so cos(150°) = -sqrt(3)/2 because cosine is negative in the second quadrant.150° is equal to 5π/6 radians.So, angle B is 5π/6.Let me double-check my steps to make sure I didn't make a mistake.1. Set up the proportion because vectors are parallel.2. Used Law of Sines to express sines in terms of sides.3. Simplified the proportion to (b - a)/c = (sqrt(3)a + c)/(a + b)4. Cross-multiplied and expanded to get b^2 - a^2 = sqrt(3) a c + c^25. Rearranged to b^2 = a^2 + c^2 + sqrt(3) a c6. Compared with Law of Cosines: b^2 = a^2 + c^2 - 2ac cos B7. Solved for cos B: sqrt(3) = -2 cos B => cos B = -sqrt(3)/28. Found angle B = 5π/6Everything seems to check out. So, I think the answer is 5π/6.**Final Answer**boxed{B}

💡Okay, so I have this system of equations with three variables x, y, and z. Each equation involves the sign function of the sum of the other two variables. The sign function can be +1, 0, or -1 depending on whether the input is positive, zero, or negative. Let me write down the equations again to make sure I have them correctly:1. ( x = 2018 - 2019 cdot text{sign}(y + z) )2. ( y = 2018 - 2019 cdot text{sign}(z + x) )3. ( z = 2018 - 2019 cdot text{sign}(x + y) )Hmm, so each variable is determined by the sign of the sum of the other two. Since the sign function can only take three values, each equation can only result in three possible values for x, y, or z. Let me figure out what those possible values are.For the first equation, ( x = 2018 - 2019 cdot text{sign}(y + z) ). Depending on the value of sign(y + z), x can be:- If sign(y + z) = +1, then x = 2018 - 2019 = -1- If sign(y + z) = 0, then x = 2018 - 0 = 2018- If sign(y + z) = -1, then x = 2018 + 2019 = 4037So x can be -1, 2018, or 4037. Similarly, y and z can also be -1, 2018, or 4037.Now, I need to find all possible triples (x, y, z) where each of x, y, z is one of these three values, and they satisfy all three equations. Let me consider the possible cases for each variable.First, let's assume that x = -1. Then, from the first equation, sign(y + z) must be +1, which means y + z > 0. So y + z must be positive. Similarly, if x = 2018, then sign(y + z) = 0, which means y + z = 0. If x = 4037, then sign(y + z) = -1, which means y + z < 0.Same logic applies to y and z. So, for each variable, depending on its value, the sum of the other two variables must be positive, zero, or negative.Now, let's try to see if we can find consistent values for x, y, z.Case 1: Suppose x = -1. Then y + z > 0.Similarly, let's suppose y = -1. Then z + x > 0. Since x = -1, z + (-1) > 0 => z > 1.But z can only be -1, 2018, or 4037. So z must be 2018 or 4037.If z = 2018, then from the third equation, sign(x + y) = sign(-1 + (-1)) = sign(-2) = -1. So z = 2018 - 2019*(-1) = 2018 + 2019 = 4037. But we assumed z = 2018, which leads to a contradiction.If z = 4037, then sign(x + y) = sign(-1 + (-1)) = -1. So z = 2018 - 2019*(-1) = 4037. That's consistent. So in this case, x = -1, y = -1, z = 4037.Let me check all equations:1. ( x = 2018 - 2019 cdot text{sign}(y + z) = 2018 - 2019 cdot text{sign}(-1 + 4037) = 2018 - 2019 cdot 1 = -1 ) ✓2. ( y = 2018 - 2019 cdot text{sign}(z + x) = 2018 - 2019 cdot text{sign}(4037 - 1) = 2018 - 2019 cdot 1 = -1 ) ✓3. ( z = 2018 - 2019 cdot text{sign}(x + y) = 2018 - 2019 cdot text{sign}(-1 -1) = 2018 - 2019 cdot (-1) = 4037 ) ✓So this is a valid solution: (-1, -1, 4037).Case 2: Let's see if there are other solutions where x, y, z take different values.Suppose x = -1, y = 2018. Then from the second equation, sign(z + x) = sign(z -1). Since y = 2018, sign(z + x) must be 0 because y = 2018 - 2019*sign(z + x). So 2018 = 2018 - 2019*sign(z + x) implies sign(z + x) = 0. Therefore, z + x = 0 => z = -x = 1. But z can only be -1, 2018, or 4037. 1 is not among these, so this case is impossible.Similarly, if x = -1 and y = 4037, then from the second equation, sign(z + x) = sign(z -1). Since y = 4037, which is 2018 - 2019*sign(z + x). So 4037 = 2018 - 2019*sign(z + x). Thus, 2019*sign(z + x) = 2018 - 4037 = -2019. Therefore, sign(z + x) = -1. So z + x < 0. Since x = -1, z -1 < 0 => z < 1. But z can be -1, 2018, or 4037. So z must be -1.So z = -1. Then, from the third equation, sign(x + y) = sign(-1 + 4037) = sign(4036) = +1. So z = 2018 - 2019*1 = -1. That's consistent. So we have x = -1, y = 4037, z = -1.Let me check all equations:1. ( x = 2018 - 2019 cdot text{sign}(y + z) = 2018 - 2019 cdot text{sign}(4037 -1) = 2018 - 2019 cdot 1 = -1 ) ✓2. ( y = 2018 - 2019 cdot text{sign}(z + x) = 2018 - 2019 cdot text{sign}(-1 -1) = 2018 - 2019 cdot (-1) = 4037 ) ✓3. ( z = 2018 - 2019 cdot text{sign}(x + y) = 2018 - 2019 cdot text{sign}(-1 + 4037) = 2018 - 2019 cdot 1 = -1 ) ✓So this is another valid solution: (-1, 4037, -1).Similarly, if I consider x = 4037, then sign(y + z) = -1, so y + z < 0. Let's suppose y = -1. Then from the second equation, sign(z + x) = sign(z + 4037). Since y = -1, we have -1 = 2018 - 2019*sign(z + 4037). Thus, 2019*sign(z + 4037) = 2018 - (-1) = 2019. Therefore, sign(z + 4037) = 1. So z + 4037 > 0 => z > -4037. But z can be -1, 2018, or 4037. So z can be any of these. Let's check each possibility.If z = -1, then from the third equation, sign(x + y) = sign(4037 -1) = sign(4036) = +1. So z = 2018 - 2019*1 = -1. That's consistent. So we have x = 4037, y = -1, z = -1.Checking all equations:1. ( x = 2018 - 2019 cdot text{sign}(y + z) = 2018 - 2019 cdot text{sign}(-1 -1) = 2018 - 2019 cdot (-1) = 4037 ) ✓2. ( y = 2018 - 2019 cdot text{sign}(z + x) = 2018 - 2019 cdot text{sign}(-1 + 4037) = 2018 - 2019 cdot 1 = -1 ) ✓3. ( z = 2018 - 2019 cdot text{sign}(x + y) = 2018 - 2019 cdot text{sign}(4037 -1) = 2018 - 2019 cdot 1 = -1 ) ✓So this is another valid solution: (4037, -1, -1).Are there any other solutions? Let's check if x, y, z can be 2018.Suppose x = 2018. Then sign(y + z) = 0, so y + z = 0. So y = -z.Similarly, from the second equation, y = 2018 - 2019*sign(z + x). Since x = 2018, z + 2018. So sign(z + 2018). If y = 2018, then 2018 = 2018 - 2019*sign(z + 2018). Thus, sign(z + 2018) = 0 => z + 2018 = 0 => z = -2018. But z can only be -1, 2018, or 4037. So z = -2018 is not allowed. Therefore, no solution where x = 2018.Similarly, if y = 2018, then sign(z + x) = 0 => z + x = 0. So z = -x. But x can be -1, 2018, or 4037. If x = -1, z = 1, which is not allowed. If x = 2018, z = -2018, not allowed. If x = 4037, z = -4037, not allowed. So no solution where y = 2018.Same for z = 2018. From the third equation, sign(x + y) = 0 => x + y = 0. So x = -y. But x and y can only be -1, 2018, or 4037. So x = -y implies y = -x. If x = -1, y = 1, not allowed. If x = 2018, y = -2018, not allowed. If x = 4037, y = -4037, not allowed. So no solution where z = 2018.Therefore, the only solutions are the three permutations where two variables are -1 and the third is 4037.So the valid triples are:1. (-1, -1, 4037)2. (-1, 4037, -1)3. (4037, -1, -1)Thus, there are exactly 3 solutions.