answer：Alright, so I've got this problem here: I need to find all two-digit numbers whose sum of digits doesn't change when the number is multiplied by 2, 3, 4, 5, 6, 7, 8, or 9. Hmm, that sounds a bit tricky, but let's break it down step by step.First off, let's understand what it means for the sum of digits to remain the same after multiplication. For example, if I take a number like 18, and multiply it by 2, I get 36. The sum of the digits of 18 is 1 + 8 = 9, and the sum of the digits of 36 is 3 + 6 = 9. So, in this case, the sum remains the same. That's a good sign.But wait, I need this to hold true not just for multiplication by 2, but for all the way up to 9. So, I need to check if multiplying the number by each of these digits doesn't change the sum of its digits. That seems like a lot of checking, but maybe there's a pattern or a rule that can help me narrow it down.I remember something about the divisibility rule for 9, which states that if the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9. So, if the sum of the digits remains the same after multiplication, it might mean that the original number is divisible by 9. That could be a useful clue.Let me list out all two-digit numbers that are divisible by 9. They are: 18, 27, 36, 45, 54, 63, 72, 81, 90, and 99. Okay, so these are the candidates I need to check.Now, I'll go through each of these numbers and see if multiplying them by 2 through 9 keeps the sum of their digits the same.Starting with 18:- 18 × 2 = 36 → 3 + 6 = 9- 18 × 3 = 54 → 5 + 4 = 9- 18 × 4 = 72 → 7 + 2 = 9- 18 × 5 = 90 → 9 + 0 = 9- 18 × 6 = 108 → 1 + 0 + 8 = 9- 18 × 7 = 126 → 1 + 2 + 6 = 9- 18 × 8 = 144 → 1 + 4 + 4 = 9- 18 × 9 = 162 → 1 + 6 + 2 = 9Okay, so 18 seems to work for all multipliers. Let's move on to 27:- 27 × 2 = 54 → 5 + 4 = 9- 27 × 3 = 81 → 8 + 1 = 9- 27 × 4 = 108 → 1 + 0 + 8 = 9- 27 × 5 = 135 → 1 + 3 + 5 = 9- 27 × 6 = 162 → 1 + 6 + 2 = 9- 27 × 7 = 189 → 1 + 8 + 9 = 18 (Uh-oh, that's not 9) Alright, so 27 fails at multiplication by 7. I'll have to exclude 27.Next up is 36:- 36 × 2 = 72 → 7 + 2 = 9- 36 × 3 = 108 → 1 + 0 + 8 = 9- 36 × 4 = 144 → 1 + 4 + 4 = 9- 36 × 5 = 180 → 1 + 8 + 0 = 9- 36 × 6 = 216 → 2 + 1 + 6 = 9- 36 × 7 = 252 → 2 + 5 + 2 = 9- 36 × 8 = 288 → 2 + 8 + 8 = 18 (Not 9) So, 36 also fails at multiplication by 8. Out it goes.Moving on to 45:- 45 × 2 = 90 → 9 + 0 = 9- 45 × 3 = 135 → 1 + 3 + 5 = 9- 45 × 4 = 180 → 1 + 8 + 0 = 9- 45 × 5 = 225 → 2 + 2 + 5 = 9- 45 × 6 = 270 → 2 + 7 + 0 = 9- 45 × 7 = 315 → 3 + 1 + 5 = 9- 45 × 8 = 360 → 3 + 6 + 0 = 9- 45 × 9 = 405 → 4 + 0 + 5 = 9Great, 45 works for all multipliers. Let's check 54:- 54 × 2 = 108 → 1 + 0 + 8 = 9- 54 × 3 = 162 → 1 + 6 + 2 = 9- 54 × 4 = 216 → 2 + 1 + 6 = 9- 54 × 5 = 270 → 2 + 7 + 0 = 9- 54 × 6 = 324 → 3 + 2 + 4 = 9- 54 × 7 = 378 → 3 + 7 + 8 = 18 (Not 9) So, 54 fails at multiplication by 7. Next is 63:- 63 × 2 = 126 → 1 + 2 + 6 = 9- 63 × 3 = 189 → 1 + 8 + 9 = 18 (Not 9) 63 fails early on. Moving to 72:- 72 × 2 = 144 → 1 + 4 + 4 = 9- 72 × 3 = 216 → 2 + 1 + 6 = 9- 72 × 4 = 288 → 2 + 8 + 8 = 18 (Not 9) 72 fails at multiplication by 4. Next is 81:- 81 × 2 = 162 → 1 + 6 + 2 = 9- 81 × 3 = 243 → 2 + 4 + 3 = 9- 81 × 4 = 324 → 3 + 2 + 4 = 9- 81 × 5 = 405 → 4 + 0 + 5 = 9- 81 × 6 = 486 → 4 + 8 + 6 = 18 (Not 9) 81 fails at multiplication by 6. Finally, 90:- 90 × 2 = 180 → 1 + 8 + 0 = 9- 90 × 3 = 270 → 2 + 7 + 0 = 9- 90 × 4 = 360 → 3 + 6 + 0 = 9- 90 × 5 = 450 → 4 + 5 + 0 = 9- 90 × 6 = 540 → 5 + 4 + 0 = 9- 90 × 7 = 630 → 6 + 3 + 0 = 9- 90 × 8 = 720 → 7 + 2 + 0 = 9- 90 × 9 = 810 → 8 + 1 + 0 = 9Perfect, 90 works for all multipliers. Lastly, 99:- 99 × 2 = 198 → 1 + 9 + 8 = 18 (Not 9) 99 fails right away.So, after checking all the two-digit numbers divisible by 9, only 18, 45, and 90 satisfy the condition that their digit sum remains unchanged when multiplied by 2 through 9.

answer：To find the largest whole number ( x ) such that eleven times the number is less than 150, we start by setting up the inequality:[ 11x < 150 ]Next, we solve for ( x ) by dividing both sides of the inequality by 11:[ x < frac{150}{11} ]Calculating the division:[ x < 13.6363 ]Since ( x ) must be a whole number, we take the largest integer less than 13.6363, which is 13. To verify, we check:- For ( x = 13 ): [ 11 times 13 = 143 ] Since 143 is less than 150, ( x = 13 ) satisfies the inequality.- For ( x = 14 ): [ 11 times 14 = 154 ] Since 154 is not less than 150, ( x = 14 ) does not satisfy the inequality.Thus, the largest whole number ( x ) that satisfies the condition is 13.

answer：Okay, so I have this problem where I need to find the equation of a tangent line from point P(1, 3) to the circle C: x² + y² = 10. The options are given as A, B, C, D. Hmm, let me think about how to approach this.First, I remember that a tangent line to a circle is perpendicular to the radius at the point of tangency. So, if I can find the point where the tangent touches the circle, I can find the slope of the radius and then determine the slope of the tangent line.Wait, but I don't know the point of tangency yet. Maybe I can use the fact that point P is outside the circle, and the tangent lines from P to the circle will satisfy certain conditions.Let me check if point P is inside, on, or outside the circle. The circle has equation x² + y² = 10, so plugging in P(1, 3):1² + 3² = 1 + 9 = 10.Oh, so point P is actually on the circle! That means the tangent line at point P is the line we're looking for. That simplifies things because I don't have to find another point of tangency.Since P is on the circle, the tangent line at P is perpendicular to the radius OP, where O is the center of the circle. The center of the circle is at (0, 0) because the equation is x² + y² = 10.So, the radius OP goes from (0, 0) to (1, 3). Let me find the slope of OP. The slope m_OP is (3 - 0)/(1 - 0) = 3/1 = 3.Since the tangent line is perpendicular to OP, its slope m_tangent should satisfy m_OP * m_tangent = -1. So, m_tangent = -1/m_OP = -1/3.Now that I have the slope of the tangent line and a point it passes through (1, 3), I can write the equation using point-slope form:y - y1 = m(x - x1)Plugging in the values:y - 3 = (-1/3)(x - 1)Let me simplify this equation:Multiply both sides by 3 to eliminate the fraction:3(y - 3) = -1(x - 1)Expanding both sides:3y - 9 = -x + 1Bring all terms to one side:x + 3y - 10 = 0So, the equation of the tangent line is x + 3y - 10 = 0, which is option A.Wait, let me double-check. If I substitute point P(1, 3) into the equation:1 + 3*3 - 10 = 1 + 9 - 10 = 0, which works. Also, the slope is -1/3, which is correct because it's perpendicular to the radius with slope 3.I think that's solid. So, the answer should be A.**Final Answer**boxed{A}

answer：First, I need to determine Shekar's total marks based on his average. Since the average is 71 and there are 5 subjects, the total marks would be 71 multiplied by 5, which equals 355.Next, I'll add up the marks from the known subjects: Science (65), Social Studies (82), English (47), and Biology (85). Adding these together gives a sum of 279.To find the marks in Mathematics, I'll subtract the sum of the known marks from the total marks. So, 355 minus 279 equals 76.Therefore, Shekar's marks in Mathematics are 76.

answer：First, I recognize the formula for converting Celsius to Fahrenheit: ( f = frac{9}{5}c + k ), where ( k ) is the constant we need to find.Next, I use the given information that an increase of 30 degrees Fahrenheit corresponds to an increase of 16.666666666666668 degrees Celsius. This relationship confirms the proportionality between Fahrenheit and Celsius scales but doesn't directly provide the constant.To find the constant, I rely on a known temperature reference point. The freezing point of water is a reliable reference: 0 degrees Celsius corresponds to 32 degrees Fahrenheit.Substituting these values into the formula:[32 = frac{9}{5} times 0 + k]Simplifying this equation gives:[32 = k]Therefore, the constant in the conversion formula is 32.

answer：Alright, let's tackle this problem step by step. So, we have a factory that produces a product with a factory price of 50 yuan per item and a cost of 25 yuan per item. For each product made, they discharge an average of 0.5 cubic meters of wastewater. They have two plans for treating this wastewater, and we need to figure out the profit relationships for each plan and determine which one is more profitable when producing 6000 items per month.First, let's understand what profit means here. Profit is calculated as Total Revenue minus Total Expenses. Total Revenue would be the money made from selling the products, and Total Expenses would include the cost of producing the products plus the cost of treating the wastewater.For both plans, we need to calculate the profit function y in terms of the number of products produced, x.**Plan 1:**- The factory treats the wastewater themselves.- They use 2 yuan of materials per cubic meter of wastewater.- There's also a monthly wear and tear cost of 30,000 yuan for the equipment.So, for each product, they discharge 0.5 cubic meters of wastewater. Therefore, for x products, the total wastewater generated is 0.5x cubic meters.The cost for treating this wastewater would be 2 yuan per cubic meter, so that's 2 * 0.5x = x yuan.Additionally, there's a fixed cost of 30,000 yuan per month for the equipment.So, the total expenses for Plan 1 are:- Cost of producing x products: 25x yuan- Wastewater treatment cost: x yuan- Fixed equipment cost: 30,000 yuanTotal Expenses = 25x + x + 30,000 = 26x + 30,000Total Revenue is simply the selling price times the number of products sold, which is 50x yuan.Therefore, Profit (y) for Plan 1 is:y = Total Revenue - Total Expensesy = 50x - (26x + 30,000)y = 50x - 26x - 30,000y = 24x - 30,000**Plan 2:**- The factory discharges the wastewater to a sewage treatment plant.- They pay 14 yuan per cubic meter treated.Again, for x products, the total wastewater is 0.5x cubic meters.The cost for treating this wastewater is 14 yuan per cubic meter, so that's 14 * 0.5x = 7x yuan.There are no additional fixed costs mentioned for Plan 2, only the variable cost based on the amount of wastewater.So, the total expenses for Plan 2 are:- Cost of producing x products: 25x yuan- Wastewater treatment cost: 7x yuanTotal Expenses = 25x + 7x = 32xTotal Revenue remains the same at 50x yuan.Therefore, Profit (y) for Plan 2 is:y = Total Revenue - Total Expensesy = 50x - 32xy = 18xNow, for part (2), we need to determine which plan is more profitable when producing 6000 products per month.Let's plug x = 6000 into both profit functions.**For Plan 1:**y = 24x - 30,000y = 24 * 6000 - 30,000y = 144,000 - 30,000y = 114,000 yuan**For Plan 2:**y = 18xy = 18 * 6000y = 108,000 yuanComparing the two, 114,000 yuan (Plan 1) is greater than 108,000 yuan (Plan 2). Therefore, when producing 6000 products per month, Plan 1 yields more profit for the factory.