💡First, I need to solve for the complex number ( z ) in the equation ( (1 - i)z = 2 + 3i ).To isolate ( z ), I'll divide both sides of the equation by ( 1 - i ):[z = frac{2 + 3i}{1 - i}]Next, to simplify the expression, I'll multiply the numerator and the denominator by the complex conjugate of the denominator, which is ( 1 + i ):[z = frac{(2 + 3i)(1 + i)}{(1 - i)(1 + i)}]Expanding both the numerator and the denominator:[text{Numerator: } (2 + 3i)(1 + i) = 2 cdot 1 + 2 cdot i + 3i cdot 1 + 3i cdot i = 2 + 2i + 3i + 3i^2]Since ( i^2 = -1 ):[2 + 2i + 3i - 3 = -1 + 5i][text{Denominator: } (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 2]So, the simplified form of ( z ) is:[z = frac{-1 + 5i}{2} = -frac{1}{2} + frac{5}{2}i]The real part of ( z ) is ( -frac{1}{2} ) and the imaginary part is ( frac{5}{2} ). In the complex plane, a negative real part and a positive imaginary part place the point in the second quadrant.

💡First, I need to find the sum of the first 6 numbers and the sum of the last 6 numbers. The sum of the first 6 numbers is calculated by multiplying the average of these numbers by the count: 6 * 10.5 = 63.Similarly, the sum of the last 6 numbers is:6 * 11.4 = 68.4.Next, I'll find the total sum of all 11 numbers by multiplying the overall average by the total count:11 * 10.7 = 117.7.Now, I'll set up an equation that represents the total sum of the 11 numbers. This total sum is equal to the sum of the first 5 numbers, the sixth number (X), and the sum of the last 5 numbers. Since the sixth number is included in both the first 6 and the last 6 numbers, I need to account for this overlap by subtracting X once:117.7 = (63 - X) + X + (68.4 - X).Simplifying the equation:117.7 = 63 + 68.4 - X.Combining the constants:117.7 = 131.4 - X.Finally, solving for X:X = 131.4 - 117.7X = 13.7.Therefore, the sixth number is 13.7.

💡Alright, so I have this problem here about a new operation defined for real numbers a and b. The operation is given by a⊗b = ab² - b. The question is asking about the range of values for k such that the equation 1⊗x = k has two distinct real roots with respect to x. The options are A: k > -1/4, B: k < -1/4, C: k > -1/4 and k ≠ 0, and D: k ≥ -1/4 and k ≠ 0.Okay, let me try to break this down step by step. First, I need to understand what this new operation means. It's defined as a⊗b = ab² - b. So, for any two real numbers a and b, when I perform this operation, I multiply a by b squared and then subtract b from that product.Now, the equation given is 1⊗x = k. So, substituting a = 1 and b = x into the operation, we get 1⊗x = 1*x² - x. That simplifies to x² - x. So, the equation becomes x² - x = k.To find the values of k for which this equation has two distinct real roots, I need to analyze this quadratic equation. Let me rewrite it in standard quadratic form. So, moving k to the left side, we have x² - x - k = 0.Now, for a quadratic equation ax² + bx + c = 0, the discriminant D is given by D = b² - 4ac. The discriminant tells us about the nature of the roots. If D > 0, there are two distinct real roots; if D = 0, there's exactly one real root (a repeated root); and if D < 0, there are no real roots, just complex ones.In our equation, a = 1, b = -1, and c = -k. Plugging these into the discriminant formula, we get D = (-1)² - 4*(1)*(-k). Calculating that, (-1) squared is 1, and then 4*1*(-k) is -4k. So, D = 1 - (-4k) which simplifies to D = 1 + 4k.For the equation to have two distinct real roots, the discriminant must be greater than zero. So, we set up the inequality 1 + 4k > 0. Solving for k, we subtract 1 from both sides: 4k > -1. Then, divide both sides by 4: k > -1/4.Hmm, so that suggests that k must be greater than -1/4 for the equation to have two distinct real roots. Looking back at the options, option A is k > -1/4, which matches what I just found.But wait, let me double-check if there's any other condition I might have missed. The problem mentions "two distinct real roots with respect to x." So, I need to ensure that the quadratic equation doesn't have any repeated roots or complex roots. Since the discriminant is positive when k > -1/4, that should be sufficient.Is there any reason why k couldn't be equal to -1/4? If k = -1/4, then the discriminant D would be 1 + 4*(-1/4) = 1 - 1 = 0. So, in that case, there would be exactly one real root (a repeated root). Since the problem specifies two distinct real roots, k cannot be equal to -1/4. Therefore, k must be strictly greater than -1/4.Looking at the options again, option A is k > -1/4, which fits. Option C says k > -1/4 and k ≠ 0, but why would k ≠ 0 matter? Let me think. If k = 0, then the equation becomes x² - x = 0, which factors to x(x - 1) = 0, giving roots at x = 0 and x = 1. These are two distinct real roots, so k = 0 is acceptable. Therefore, option C is unnecessarily excluding k = 0, which is actually a valid value. Similarly, option D says k ≥ -1/4 and k ≠ 0, but again, k = 0 is fine, so D is also incorrect.Option B is k < -1/4, which would make the discriminant negative, leading to no real roots, so that's not what we want. Therefore, the correct answer should be option A: k > -1/4.Wait, but just to make sure, let me test k = -1/4. If k = -1/4, then the equation is x² - x + 1/4 = 0. Let's compute the discriminant: D = (-1)² - 4*(1)*(1/4) = 1 - 1 = 0. So, it has a repeated root at x = [1 ± sqrt(0)]/2 = 1/2. So, only one real root, which confirms that k cannot be equal to -1/4.What about k = 0? As I mentioned earlier, the equation becomes x² - x = 0, which factors to x(x - 1) = 0, giving x = 0 and x = 1, which are two distinct real roots. So, k = 0 is acceptable, meaning that option C and D, which exclude k = 0, are incorrect.Therefore, the correct range is k > -1/4, which is option A.Just to recap, the steps were:1. Substitute the operation into the equation to get x² - x = k.2. Rewrite it as x² - x - k = 0.3. Compute the discriminant D = 1 + 4k.4. Set D > 0 to find k > -1/4.5. Verify that k = 0 is acceptable and k = -1/4 is not.6. Conclude that option A is correct.I think that covers everything. I don't see any mistakes in my reasoning, so I feel confident that option A is the right answer.**Final Answer**boxed{A}

💡Alright, so I've got this problem here about finding the minimum value of α + β + γ for a number A, which is factored into primes as 2^α × 3^β × 5^γ. The conditions are that half of A is a perfect square, one-third of A is a perfect cube, and one-fifth of A is a fifth power. Hmm, okay, let's break this down step by step.First, let's understand what each condition implies about the exponents α, β, and γ.Starting with the first condition: half of A is a perfect square. So, A/2 should be a perfect square. If A is 2^α × 3^β × 5^γ, then A/2 would be 2^(α-1) × 3^β × 5^γ. For this to be a perfect square, all the exponents in its prime factorization must be even. That means:- α - 1 must be even, so α must be odd.- β must be even.- γ must be even.Okay, so from the first condition, α is odd, and both β and γ are even.Moving on to the second condition: one-third of A is a perfect cube. So, A/3 should be a perfect cube. That would be 2^α × 3^(β-1) × 5^γ. For this to be a perfect cube, all exponents must be multiples of 3. Therefore:- α must be a multiple of 3.- β - 1 must be a multiple of 3, so β ≡ 1 mod 3.- γ must be a multiple of 3.Alright, so from the second condition, α is a multiple of 3, β is one more than a multiple of 3, and γ is a multiple of 3.Now, the third condition: one-fifth of A is a fifth power. So, A/5 should be a perfect fifth power, which is 2^α × 3^β × 5^(γ-1). For this to be a perfect fifth power, all exponents must be multiples of 5. Hence:- α must be a multiple of 5.- β must be a multiple of 5.- γ - 1 must be a multiple of 5, so γ ≡ 1 mod 5.Putting all these together, let's summarize the conditions for each exponent:For α:- From the first condition: α is odd.- From the second condition: α is a multiple of 3.- From the third condition: α is a multiple of 5.So, α needs to be the smallest number that is odd, a multiple of 3, and a multiple of 5. The least common multiple of 3 and 5 is 15, and 15 is odd, so α = 15.For β:- From the first condition: β is even.- From the second condition: β ≡ 1 mod 3.- From the third condition: β is a multiple of 5.So, β needs to be the smallest number that is even, one more than a multiple of 3, and a multiple of 5. Let's list the multiples of 5 and see which one fits the other conditions.Multiples of 5: 5, 10, 15, 20, 25, 30, ...Check which of these are even: 10, 20, 30, ...Now, check which of these satisfy β ≡ 1 mod 3.10 mod 3 = 1, because 10 divided by 3 is 3 with a remainder of 1. So, 10 works.So, β = 10.For γ:- From the first condition: γ is even.- From the second condition: γ is a multiple of 3.- From the third condition: γ ≡ 1 mod 5.So, γ needs to be the smallest number that is even, a multiple of 3, and one more than a multiple of 5. Let's list numbers that are one more than multiples of 5 and check the other conditions.Numbers that are one more than multiples of 5: 6, 11, 16, 21, 26, 31, 36, ...From these, pick the ones that are even and multiples of 3.6: even, multiple of 3. 6 mod 5 = 1, so 6 is 5×1 +1. So, 6 works.So, γ = 6.Now, let's add them up: α + β + γ = 15 + 10 + 6 = 31.Wait, but looking back at the options, 31 is an option (D). But I recall that sometimes there might be smaller exponents that satisfy all conditions if we consider higher multiples. Let me double-check if there are smaller values for α, β, γ that satisfy all conditions.For α: It's the least common multiple of 3 and 5, which is 15. Since 15 is odd, it's the smallest possible.For β: We found 10, which is even, multiple of 5, and 10 mod 3 = 1. Is there a smaller number? The next smaller multiple of 5 is 5, but 5 is odd, so it doesn't satisfy the first condition. So, 10 is indeed the smallest.For γ: 6 is even, multiple of 3, and 6 mod 5 = 1. The next smaller number that is one more than a multiple of 5 is 1, but 1 isn't even or a multiple of 3. So, 6 is the smallest.Therefore, the minimal sum is 15 + 10 + 6 = 31.But wait, the options include 17 and 23, which are smaller than 31. Did I make a mistake somewhere?Let me re-examine the conditions.For α:- Must be odd, multiple of 3 and 5. So, 15 is correct.For β:- Must be even, β ≡1 mod 3, multiple of 5. 10 is correct.For γ:- Must be even, multiple of 3, γ ≡1 mod 5. 6 is correct.Hmm, maybe I missed a smaller γ? Let's see:Looking for γ such that γ is even, multiple of 3, and γ ≡1 mod 5.Numbers that are one more than multiples of 5: 6, 11, 16, 21, 26, 31, 36,...From these, even and multiple of 3: 6, 36,...6 is the smallest.So, γ=6 is correct.Wait, maybe I can find smaller exponents by considering higher exponents in a way that they satisfy multiple conditions? For example, maybe α can be smaller if it's a multiple of both 3 and 5, but still odd.But 15 is the smallest such number. 15 is 3×5, and it's odd. The next one would be 45, which is larger, so 15 is indeed minimal.Similarly, for β, 10 is the smallest multiple of 5 that is even and satisfies β ≡1 mod 3.And γ=6 is the smallest that fits all conditions.So, I think 31 is the correct answer. Maybe the options are trying to trick me into thinking it's smaller, but I don't see a way to get a smaller sum without violating one of the conditions.Wait, let me check the conditions again for each exponent:For α:- A/2 is a perfect square: α-1 must be even → α is odd.- A/3 is a perfect cube: α must be multiple of 3.- A/5 is a perfect fifth power: α must be multiple of 5.So, α must be the least common multiple of 3 and 5, which is 15, and it's odd. Correct.For β:- A/2 is a perfect square: β must be even.- A/3 is a perfect cube: β-1 must be multiple of 3.- A/5 is a perfect fifth power: β must be multiple of 5.So, β must be even, β ≡1 mod 3, and multiple of 5. The smallest such β is 10, as 10 is even, 10-1=9 which is multiple of 3, and 10 is multiple of 5. Correct.For γ:- A/2 is a perfect square: γ must be even.- A/3 is a perfect cube: γ must be multiple of 3.- A/5 is a perfect fifth power: γ-1 must be multiple of 5.So, γ must be even, multiple of 3, and γ ≡1 mod 5. The smallest such γ is 6, as 6 is even, multiple of 3, and 6-1=5 which is multiple of 5. Correct.Therefore, I think 31 is indeed the minimal sum. Maybe the options are trying to test if I consider exponents starting from 1 instead of 0, but since α, β, γ are natural numbers, they start from 1. But in this case, the minimal exponents are 15, 10, and 6, summing to 31.So, I think the answer is D. 31.

💡Let's break down the problem step by step.1. **Define Variables:** - Let the first integer be ( x ). - Let the second integer be ( y ).2. **Set Up the Equation:** - Tara writes down ( x ) three times: ( 3x ). - She writes down ( y ) four times: ( 4y ). - She adds double the first integer: ( 2x ). - The total sum is 160: [ 3x + 4y + 2x = 160 ] - Simplify the equation: [ 5x + 4y = 160 ]3. **Determine the Values of ( x ) and ( y ):** - We know one of the numbers is 12. We need to check both possibilities: - **Case 1:** ( x = 12 ) [ 5(12) + 4y = 160 ] [ 60 + 4y = 160 ] [ 4y = 100 ] [ y = 25 ] - **Case 2:** ( y = 12 ) [ 5x + 4(12) = 160 ] [ 5x + 48 = 160 ] [ 5x = 112 ] [ x = 22.4 ] Since ( x ) must be an integer, this case is invalid.4. **Conclusion:** - The other number is ( y = 25 ).Thus, the final answer is (boxed{textbf{(C) }25}).

💡Okay, so I need to figure out what the final price of the shirt is that Corna wants to buy. Let me read the problem again to make sure I understand all the details.The store buys the shirt for 20, and they charge 30% for profit. That means they add 30% to the cost price to set the selling price. But then, the shirt is on sale for 50% off the selling price. So, I need to calculate the final price after applying the discount.First, I think I need to find out what the selling price is before the discount. The store buys it for 20 and adds a 30% profit. So, 30% of 20 is how much? Let me calculate that.30% of 20 is 0.30 times 20, which is 6. So, the profit is 6. That means the selling price before any discount is 20 plus 6, which is 26.Okay, so the regular selling price is 26. Now, the shirt is on sale for 50% off the selling price. So, I need to find out what 50% of 26 is and then subtract that from 26 to get the final price.50% of 26 is 0.50 times 26, which is 13. So, the discount is 13. Subtracting that from the selling price, 26 minus 13 is 13.Wait, so the final price is 13? That seems pretty straightforward. But let me double-check my calculations to make sure I didn't make any mistakes.First, calculating the profit: 30% of 20 is indeed 6. Adding that to the cost price gives 26. Then, calculating 50% of 26 is 13, and subtracting that gives 13. Yeah, that seems correct.I think I got it. The final price Corna has to pay for the shirt is 13.