💡Okay, so I have this problem where I'm given three distinct real numbers a, b, and c. They satisfy the equation:[frac{a^3 - 12}{a} = frac{b^3 - 12}{b} = frac{c^3 - 12}{c}]And I need to find the value of (a^3 + b^3 + c^3).Hmm, let me try to parse this. All three fractions are equal to each other, so let's denote that common value as k. So, I can write:[frac{a^3 - 12}{a} = k, quad frac{b^3 - 12}{b} = k, quad frac{c^3 - 12}{c} = k]That means each of a, b, and c satisfies the equation:[frac{x^3 - 12}{x} = k]Let me rearrange this equation to make it a standard polynomial equation. Multiplying both sides by x (assuming x ≠ 0, which is fine because if x were 0, the original expression would be undefined), I get:[x^3 - 12 = kx]Which simplifies to:[x^3 - kx - 12 = 0]So, a, b, and c are roots of the cubic equation (x^3 - kx - 12 = 0). Since they are distinct, this cubic must have three distinct real roots.Now, I remember from Vieta's formulas that for a cubic equation (x^3 + px^2 + qx + r = 0), the sum of the roots is -p, the sum of the products of the roots two at a time is q, and the product of the roots is -r.In our case, the equation is (x^3 - kx - 12 = 0). So, comparing to the standard form, p is 0 (since there's no x² term), q is -k, and r is -12.Therefore, applying Vieta's formulas:1. The sum of the roots: (a + b + c = -p = 0)2. The sum of the products two at a time: (ab + bc + ac = q = -k)3. The product of the roots: (abc = -r = 12)So, we know that (a + b + c = 0), which might come in handy later.Now, the problem asks for (a^3 + b^3 + c^3). I remember there's a formula for the sum of cubes of roots in terms of the roots themselves and their products. Let me recall it.The formula is:[a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b + c)(ab + bc + ac) + 3abc]Wait, let me check that. Alternatively, I think another way to express (a^3 + b^3 + c^3) is using the identity:[a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b + c)(ab + bc + ac) + 3abc]Yes, that seems right. Let me verify with a simple case. Suppose a = 1, b = 1, c = 1. Then (a^3 + b^3 + c^3 = 3). Plugging into the formula:[(1 + 1 + 1)^3 - 3(1 + 1 + 1)(1*1 + 1*1 + 1*1) + 3*1*1*1 = 27 - 3*3*3 + 3 = 27 - 27 + 3 = 3]Okay, that works. So, the formula is correct.Given that, let's plug in the values we have from Vieta's formulas.We know that (a + b + c = 0), so the first term is 0³ = 0.The second term is -3*(a + b + c)*(ab + bc + ac). Since (a + b + c = 0), this entire term becomes 0.The third term is 3abc. We know from Vieta that (abc = 12), so this term is 3*12 = 36.Putting it all together:[a^3 + b^3 + c^3 = 0 - 0 + 36 = 36]So, the answer should be 36.Wait, let me make sure I didn't skip any steps or make any mistakes. Let me go through it again.We started by setting the common value of the fractions equal to k, leading to the cubic equation (x^3 - kx - 12 = 0). Then, using Vieta's formulas, we found that the sum of the roots is 0, the sum of the products two at a time is -k, and the product is 12.Then, to find (a^3 + b^3 + c^3), we used the identity which expanded to 0 - 0 + 36, giving 36.Alternatively, another approach is to consider that each root satisfies the equation (x^3 = kx + 12). So, for each root, (a^3 = ka + 12), (b^3 = kb + 12), and (c^3 = kc + 12). Adding these together:[a^3 + b^3 + c^3 = k(a + b + c) + 36]But since (a + b + c = 0), this simplifies to:[a^3 + b^3 + c^3 = 0 + 36 = 36]Yes, that also gives 36. So, both methods confirm that the answer is 36.I think that makes sense. The key was recognizing that a, b, c are roots of the same cubic equation, applying Vieta's formulas, and then using the identity for the sum of cubes. Alternatively, using the fact that each root satisfies the equation (x^3 = kx + 12) and summing them up also leads directly to the answer.So, I feel confident that the answer is 36.**Final Answer**boxed{36}

💡Okay, so I need to find the magnitude of the complex number 3/4 + 3i. Hmm, I remember that the magnitude of a complex number is like its distance from the origin on the complex plane. The formula for the magnitude is the square root of the sum of the squares of the real part and the imaginary part. Let me write that down to make sure I have it right. If I have a complex number a + bi, then its magnitude is |a + bi| = sqrt(a² + b²). So in this case, a is 3/4 and b is 3. Alright, so plugging those values into the formula, I get |3/4 + 3i| = sqrt((3/4)² + (3)²). Let me calculate each part step by step to avoid mistakes. First, (3/4) squared. That would be (3 squared) divided by (4 squared), so 9/16. Okay, that seems right. Now, the imaginary part is 3, so 3 squared is 9. So now I have sqrt(9/16 + 9). Hmm, I need to add these two fractions. But wait, 9 is a whole number, and 9/16 is a fraction. To add them, I should express 9 as a fraction with the same denominator as 9/16. The denominator here is 16, so 9 can be written as 144/16 because 9 times 16 is 144. So now, adding 9/16 + 144/16 gives me 153/16. So now the expression becomes sqrt(153/16). I can separate the square root of the numerator and the denominator, so that's sqrt(153)/sqrt(16). The square root of 16 is 4, so this simplifies to sqrt(153)/4. Wait, can I simplify sqrt(153) any further? Let me check. 153 divided by 9 is 17, so 153 is 9 times 17. Therefore, sqrt(153) is sqrt(9*17), which is sqrt(9)*sqrt(17) = 3*sqrt(17). So substituting back, sqrt(153)/4 becomes 3*sqrt(17)/4. Let me double-check my steps to make sure I didn't make any mistakes. Starting with the formula, calculating each part, converting 9 to 144/16, adding to get 153/16, taking the square root, and simplifying sqrt(153) to 3*sqrt(17). Everything seems to add up correctly. So, the magnitude of the complex number 3/4 + 3i is 3*sqrt(17)/4.

💡Okay, so I have this problem about tiling an 8x8 chessboard with two opposite corners missing using 2x1 dominoes. Hmm, I've heard about domino tiling problems before, but I'm not exactly sure how to approach this one. Let me think through it step by step.First, I know that a standard 8x8 chessboard has 64 squares. If we remove two opposite corners, that leaves us with 62 squares. Now, each domino covers exactly two squares, so theoretically, if 62 is divisible by 2, it should be possible to tile the board. 62 divided by 2 is 31, so we would need 31 dominoes. That makes sense mathematically, but I remember there's more to it because of the way the dominoes cover the board.I recall that a chessboard is colored in an alternating black and white pattern. Each domino, no matter how you place it, will always cover one black square and one white square. So, for the tiling to be possible, the number of black and white squares must be equal. Let me check that.In a standard chessboard, there are 32 black squares and 32 white squares. If we remove two opposite corners, what happens? Well, the corners of a chessboard are all the same color. For example, the top-left corner is usually black, and the bottom-right corner is also black. So, removing two black squares means we now have 30 black squares and still 32 white squares.Wait a minute, that's a problem. If we have 30 black squares and 32 white squares, the numbers aren't equal anymore. Each domino needs to cover one black and one white square. So, if we have more white squares than black ones, we can't pair them all up. There will always be two white squares left uncovered because we don't have enough black squares to match them with.Let me visualize this. Imagine the chessboard with the two opposite corners missing. The missing squares are both black, so the remaining squares have an unequal number of black and white squares. If I try to place dominoes, I'll quickly run out of black squares to pair with the white ones. Eventually, I'll end up with some white squares that can't be covered because there are no adjacent black squares left.Is there another way to look at this? Maybe using graph theory or something? I think about the chessboard as a graph where each square is a vertex, and edges connect adjacent squares. Tiling with dominoes is like finding a perfect matching in this graph. But for a perfect matching to exist, certain conditions must be met, like the number of vertices being even and the graph being connected in a way that allows pairing.In this case, the graph is still connected because the two missing squares are corners, and the rest of the board remains connected. However, the issue is the imbalance in the number of black and white squares, which breaks the condition for a perfect matching. Each domino corresponds to an edge connecting a black and a white square, so without an equal number of both, it's impossible to cover all squares.Maybe I can try a smaller example to see if this holds. Let's say I have a 2x2 chessboard and remove two opposite corners. That leaves me with two squares of the same color. Can I tile that with dominoes? No, because a domino needs to cover two squares, but they have to be adjacent and of different colors. In this case, the two remaining squares are the same color and not adjacent, so it's impossible.Extending this idea to the 8x8 board, the same principle applies. Even though the board is much larger, the removal of two squares of the same color creates an imbalance that makes tiling impossible.I wonder if there's a way to rearrange the dominoes or if some special configuration could work. But no, because each domino inherently requires one black and one white square, and with the imbalance, it's impossible to cover all squares without leaving some uncovered.So, after thinking through the coloring argument and considering smaller examples, I'm pretty confident that it's impossible to tile the modified chessboard with 2x1 dominoes.

💡Okay, so I have this problem here where two lines, l₁ and l₂, intersect at a point P. The equations of the lines are given in terms of a parameter m, which is a real number. I need to find the range of values for the distance d from point P to the line x + y = 0. The options are given as A, B, C, D with different intervals.First, I should probably write down the equations of the lines again to make sure I have them correctly:l₁: mx - y - 3m + 1 = 0l₂: x + my - 3m - 1 = 0So, both lines have the parameter m in them. Since m is a real number, these lines can change their slopes and positions depending on the value of m. The point P is where these two lines intersect, so P depends on m as well.I need to find the distance from P to the line x + y = 0. The distance from a point (x₀, y₀) to the line ax + by + c = 0 is given by the formula:d = |ax₀ + by₀ + c| / sqrt(a² + b²)In this case, the line is x + y = 0, which can be rewritten as x + y + 0 = 0. So, a = 1, b = 1, c = 0. Therefore, the distance from P(x₀, y₀) to this line is:d = |x₀ + y₀| / sqrt(2)So, my goal is to find the range of d as m varies over all real numbers.To find the coordinates of point P, I need to solve the system of equations given by l₁ and l₂.Let me write them again:1) mx - y - 3m + 1 = 02) x + my - 3m - 1 = 0I can solve this system for x and y in terms of m.From equation 1, I can express y in terms of x:y = mx - 3m + 1Now, substitute this expression for y into equation 2:x + m*(mx - 3m + 1) - 3m - 1 = 0Let me expand this:x + m²x - 3m² + m - 3m - 1 = 0Combine like terms:x + m²x - 3m² - 2m - 1 = 0Factor x:x*(1 + m²) - 3m² - 2m - 1 = 0Now, solve for x:x = (3m² + 2m + 1) / (1 + m²)Okay, so x is expressed in terms of m. Now, let's find y using the expression from equation 1:y = mx - 3m + 1Substitute x:y = m*(3m² + 2m + 1)/(1 + m²) - 3m + 1Let me compute this step by step.First, compute m*(3m² + 2m + 1):= 3m³ + 2m² + mThen, divide by (1 + m²):= (3m³ + 2m² + m) / (1 + m²)Now, subtract 3m:= (3m³ + 2m² + m)/(1 + m²) - 3mTo combine these, I need a common denominator:= (3m³ + 2m² + m - 3m*(1 + m²)) / (1 + m²)Compute 3m*(1 + m²):= 3m + 3m³Subtract this from the numerator:= 3m³ + 2m² + m - 3m - 3m³Simplify:3m³ - 3m³ = 02m² remainsm - 3m = -2mSo, numerator becomes:2m² - 2mTherefore, y = (2m² - 2m)/(1 + m²)Simplify numerator:= 2m(m - 1)/(1 + m²)So, now we have expressions for x and y in terms of m:x = (3m² + 2m + 1)/(1 + m²)y = (2m² - 2m)/(1 + m²)Now, I need to compute x + y because the distance formula involves |x + y|.Compute x + y:= (3m² + 2m + 1)/(1 + m²) + (2m² - 2m)/(1 + m²)Since denominators are the same, add numerators:= [3m² + 2m + 1 + 2m² - 2m] / (1 + m²)Simplify numerator:3m² + 2m² = 5m²2m - 2m = 01 remainsSo, numerator is 5m² + 1Therefore, x + y = (5m² + 1)/(1 + m²)So, the distance d is |x + y| / sqrt(2) = |(5m² + 1)/(1 + m²)| / sqrt(2)Since 5m² + 1 and 1 + m² are always positive (as squares are non-negative and we're adding positive constants), the absolute value is redundant here.So, d = (5m² + 1)/(1 + m²) / sqrt(2) = (5m² + 1)/(sqrt(2)*(1 + m²))Now, I need to find the range of d as m varies over all real numbers.Let me denote t = m². Since m is real, t ≥ 0.So, d becomes:d = (5t + 1)/(sqrt(2)*(1 + t))So, now, I can consider d as a function of t:d(t) = (5t + 1)/(sqrt(2)*(1 + t)), t ≥ 0I need to find the range of d(t) for t ≥ 0.To find the range, I can analyze the function d(t). Let's see if it's increasing or decreasing, find its maximum and minimum.First, let's compute the derivative of d(t) with respect to t to find critical points.But before that, maybe simplify the expression:d(t) = (5t + 1)/(sqrt(2)*(1 + t)) = [5t + 1]/[sqrt(2)(t + 1)]Let me write it as:d(t) = (5t + 1)/(sqrt(2)(t + 1))Let me compute the derivative d'(t):Let me denote numerator as N = 5t + 1, denominator as D = sqrt(2)(t + 1)Then, d(t) = N/DSo, derivative d'(t) = (N’ D - N D’) / D²Compute N’ = 5D = sqrt(2)(t + 1), so D’ = sqrt(2)Thus,d'(t) = [5 * sqrt(2)(t + 1) - (5t + 1)*sqrt(2)] / [sqrt(2)(t + 1)]²Factor sqrt(2) in numerator:= sqrt(2)[5(t + 1) - (5t + 1)] / [2(t + 1)²]Compute inside the brackets:5(t + 1) = 5t + 5Subtract (5t + 1):5t + 5 - 5t - 1 = 4So, numerator becomes sqrt(2)*4Denominator is 2(t + 1)²Thus,d'(t) = (4 sqrt(2)) / [2(t + 1)²] = (2 sqrt(2))/(t + 1)²Since t ≥ 0, denominator is always positive, and numerator is positive. Therefore, d'(t) > 0 for all t ≥ 0.This means that d(t) is strictly increasing for t ≥ 0.Therefore, the minimum value occurs at t = 0, and as t approaches infinity, d(t) approaches its limit.Compute d(0):d(0) = (5*0 + 1)/(sqrt(2)*(0 + 1)) = 1/sqrt(2) = sqrt(2)/2 ≈ 0.707Wait, but looking at the options, the minimum is sqrt(2) ≈ 1.414, which is larger than sqrt(2)/2. Hmm, maybe I made a mistake.Wait, let me check my calculations again.Wait, in the expression for d(t):d(t) = (5t + 1)/(sqrt(2)*(1 + t))Wait, when t = 0:d(0) = (0 + 1)/(sqrt(2)*(1 + 0)) = 1/sqrt(2) ≈ 0.707But in the options, the minimum is sqrt(2). So, that suggests that perhaps my approach is missing something.Wait, perhaps I made a mistake in computing x + y.Let me double-check the earlier steps.We had:From equation 1: y = mx - 3m + 1Substituted into equation 2:x + m*(mx - 3m + 1) - 3m - 1 = 0Which became:x + m²x - 3m² + m - 3m - 1 = 0Then:x(1 + m²) - 3m² - 2m - 1 = 0So, x = (3m² + 2m + 1)/(1 + m²)Then, y = mx - 3m + 1So, y = m*(3m² + 2m + 1)/(1 + m²) - 3m + 1Compute numerator:3m³ + 2m² + m - 3m*(1 + m²) + 1*(1 + m²)Wait, perhaps I made a mistake in the substitution.Wait, let me re-express y:y = [m*(3m² + 2m + 1) - 3m*(1 + m²) + (1 + m²)] / (1 + m²)Wait, no, that's not correct. Let me re-express y correctly.Wait, y = mx - 3m + 1But x = (3m² + 2m + 1)/(1 + m²)So, y = m*(3m² + 2m + 1)/(1 + m²) - 3m + 1To combine these terms, we need a common denominator.So, let me write -3m as -3m*(1 + m²)/(1 + m²) and 1 as (1 + m²)/(1 + m²)So,y = [m*(3m² + 2m + 1) - 3m*(1 + m²) + (1 + m²)] / (1 + m²)Now, compute numerator:First term: m*(3m² + 2m + 1) = 3m³ + 2m² + mSecond term: -3m*(1 + m²) = -3m - 3m³Third term: (1 + m²) = 1 + m²Now, add them all together:3m³ + 2m² + m - 3m - 3m³ + 1 + m²Combine like terms:3m³ - 3m³ = 02m² + m² = 3m²m - 3m = -2mAnd +1So, numerator is 3m² - 2m + 1Therefore, y = (3m² - 2m + 1)/(1 + m²)Wait, earlier I had y = (2m² - 2m)/(1 + m²). That was incorrect. So, I made a mistake there.So, correct expressions:x = (3m² + 2m + 1)/(1 + m²)y = (3m² - 2m + 1)/(1 + m²)Now, compute x + y:x + y = [3m² + 2m + 1 + 3m² - 2m + 1]/(1 + m²)Simplify numerator:3m² + 3m² = 6m²2m - 2m = 01 + 1 = 2So, numerator is 6m² + 2Thus, x + y = (6m² + 2)/(1 + m²) = 2*(3m² + 1)/(1 + m²)Therefore, the distance d is |x + y| / sqrt(2) = (6m² + 2)/(sqrt(2)*(1 + m²)) = [2*(3m² + 1)]/[sqrt(2)*(1 + m²)] = (3m² + 1)/(sqrt(2)/2*(1 + m²))Wait, let me compute it correctly:d = (6m² + 2)/(sqrt(2)*(1 + m²)) = [2*(3m² + 1)]/[sqrt(2)*(1 + m²)] = (3m² + 1)/(sqrt(2)/2*(1 + m²))Wait, no, let me factor 2 in numerator:= 2*(3m² + 1)/(sqrt(2)*(1 + m²)) = (2/sqrt(2))*(3m² + 1)/(1 + m²) = sqrt(2)*(3m² + 1)/(1 + m²)So, d = sqrt(2)*(3m² + 1)/(1 + m²)Now, let me write this as:d(m) = sqrt(2)*(3m² + 1)/(1 + m²)Now, let me analyze this function.Let me set t = m², t ≥ 0.So, d(t) = sqrt(2)*(3t + 1)/(1 + t)Now, I need to find the range of d(t) for t ≥ 0.Let me compute the limit as t approaches infinity:lim(t→∞) d(t) = sqrt(2)*(3t)/(t) = 3 sqrt(2)At t = 0:d(0) = sqrt(2)*(0 + 1)/(1 + 0) = sqrt(2)Now, let's check if the function is increasing or decreasing.Compute derivative of d(t):d(t) = sqrt(2)*(3t + 1)/(1 + t)Let me write it as:d(t) = sqrt(2)*(3t + 1)/(t + 1)Compute derivative:d'(t) = sqrt(2)*[(3)(t + 1) - (3t + 1)(1)]/(t + 1)^2Simplify numerator:3(t + 1) - (3t + 1) = 3t + 3 - 3t - 1 = 2So, d'(t) = sqrt(2)*2/(t + 1)^2Since t ≥ 0, denominator is positive, so d'(t) > 0 for all t ≥ 0.This means that d(t) is strictly increasing for t ≥ 0.Therefore, the minimum value of d(t) is at t = 0, which is sqrt(2), and as t approaches infinity, d(t) approaches 3 sqrt(2).Therefore, the range of d is [sqrt(2), 3 sqrt(2)].Looking back at the options:A: [sqrt(2), 3 sqrt(2)]B: [sqrt(3), 2 sqrt(3)]C: [sqrt(3), 3 sqrt(3)]D: [sqrt(2), 2 sqrt(2)]So, the correct answer is A.But wait, in my initial calculation, I had a mistake in computing y, which led me to an incorrect expression for x + y. After correcting that, I arrived at the correct expression for d(t), which is sqrt(2)*(3t + 1)/(1 + t), and found that it's increasing from sqrt(2) to 3 sqrt(2). So, the range is [sqrt(2), 3 sqrt(2)], which is option A.I think that's the correct answer.

💡First, I need to evaluate the expression (sqrt{(-2)^{2}} + (sqrt{2})^{2}).For the first term, (sqrt{(-2)^{2}}), I calculate the square of (-2), which is (4). Taking the square root of (4) gives me (2).Next, for the second term, ((sqrt{2})^{2}), squaring the square root of (2) simply returns (2).Adding these two results together, (2 + 2), gives me the final answer of (4).

💡Okay, so I need to figure out the domain of this function f(x) = sqrt(4 - x²)/x. Hmm, the domain is basically all the real numbers x for which the function is defined, right? So, I need to make sure that both the numerator and the denominator are valid.First, let's look at the numerator, which is sqrt(4 - x²). I remember that the expression inside a square root has to be greater than or equal to zero because you can't take the square root of a negative number in the set of real numbers. So, I need to solve the inequality 4 - x² ≥ 0.Let me write that down: 4 - x² ≥ 0. To solve this, I can rearrange it to x² ≤ 4. Taking the square root of both sides, I get |x| ≤ 2. Which means x is between -2 and 2, inclusive. So, x ∈ [-2, 2]. That's the first condition.Now, looking at the denominator, which is x. Since we can't divide by zero, x cannot be zero. So, x ≠ 0. That's the second condition.Putting these two conditions together, the domain of f(x) is all real numbers x such that x is between -2 and 2, but x cannot be zero. So, in interval notation, that would be [-2, 0) union (0, 2]. Wait, let me double-check. If x is between -2 and 2, excluding zero, then yes, that makes sense. For example, if x is 1, then sqrt(4 - 1) = sqrt(3), which is fine, and 1 is not zero, so the function is defined. If x is -1, sqrt(4 - 1) is still sqrt(3), and -1 is not zero, so that's also fine. But if x is 0, the denominator becomes zero, which is undefined. Similarly, if x is greater than 2 or less than -2, the expression inside the square root becomes negative, which isn't allowed in real numbers.So, I think I've got it right. The domain is all real numbers from -2 to 2, excluding zero. So, in interval notation, that's two intervals: from -2 to 0, not including 0, and from 0 to 2, not including 0. So, [-2, 0) union (0, 2].I guess another way to write it is x ∈ [-2, 2] but x ≠ 0. But in interval notation, it's clearer to write it as two separate intervals.Just to make sure, let me test a value in each interval. Let's pick x = -2: sqrt(4 - (-2)^2) = sqrt(0) = 0, and denominator is -2, so f(-2) = 0/-2 = 0, which is defined. Similarly, x = 2: sqrt(4 - 4) = 0, so f(2) = 0/2 = 0, which is also defined.What about x = -1: sqrt(4 - 1) = sqrt(3), and denominator is -1, so f(-1) = sqrt(3)/-1 = -sqrt(3), which is defined. Similarly, x = 1: sqrt(3)/1 = sqrt(3), which is fine.If I pick x = 0, the function is undefined because of division by zero. If I pick x = 3, sqrt(4 - 9) = sqrt(-5), which isn't a real number, so that's excluded. Similarly, x = -3 would give the same issue.So, yeah, I think I'm confident that the domain is [-2, 0) union (0, 2].