answer：First, I need to consider the structure of the license plate, which is four characters long. The first and last characters must be letters, while the two middle characters can be either letters or digits.Since two adjacent characters must be identical, the second character must be the same as the first character, and the fourth character must be the same as the third character.There are 26 possible choices for the first character (A-Z). Because the second character must match the first, there is only 1 choice for the second character.For the third character, there are 36 possible choices (26 letters + 10 digits). The fourth character must match the third, so there is only 1 choice for the fourth character.To find the total number of valid license plates, I multiply the number of choices for each position:26 (first character) × 1 (second character) × 36 (third character) × 1 (fourth character) = 26 × 36 = 936.Therefore, there are 936 possible ways to create such a license plate.

answer：Alright, so I have this problem about a parabola. Let me try to understand it step by step. First, part (I) says: "Given a parabola C with its vertex at the origin, symmetric about the x-axis, and passing through point P(1, 2). Find the standard equation of the parabola C and the equation of its directrix."Okay, so the parabola has its vertex at (0,0), it's symmetric about the x-axis, which means it opens either to the right or to the left. Since it passes through the point (1,2), which is in the first quadrant, it must open to the right. I remember that the standard form of a parabola that opens to the right with vertex at the origin is ( y^2 = 4px ), where p is the distance from the vertex to the focus. The directrix is the line ( x = -p ).So, we need to find p. Since the parabola passes through (1,2), we can substitute x=1 and y=2 into the equation to solve for p.Let me write that out:( (2)^2 = 4p(1) )( 4 = 4p )Divide both sides by 4:( p = 1 )Wait, hold on. If p=1, then the equation would be ( y^2 = 4x ), and the directrix would be ( x = -1 ). Hmm, that seems right. Let me double-check.If p=1, then the focus is at (1,0), and the directrix is x=-1. The parabola ( y^2 = 4x ) does pass through (1,2) because ( 2^2 = 4*1 ), which is 4=4. So that's correct.Wait, but in the initial problem statement, it says the parabola is symmetric about the x-axis. So, yes, that makes sense because the standard form ( y^2 = 4px ) is symmetric about the x-axis.So, part (I) seems done. The standard equation is ( y^2 = 4x ), and the directrix is ( x = -1 ).Now, moving on to part (II): "Given a line l that does not pass through point P and has a slope of 1, and intersects the parabola C at points A and B. If the circle with AB as its diameter passes through point P, find the equation of line l."Alright, so line l has a slope of 1, so its equation is ( y = x + b ), where b is the y-intercept. Since it doesn't pass through P(1,2), substituting x=1 and y=2 into the line equation should not satisfy it. So, ( 2 neq 1 + b ), which implies ( b neq 1 ).This line intersects the parabola ( y^2 = 4x ) at points A and B. Let's find the points of intersection.Substitute ( y = x + b ) into ( y^2 = 4x ):( (x + b)^2 = 4x )Expanding the left side:( x^2 + 2bx + b^2 = 4x )Bring all terms to one side:( x^2 + 2bx + b^2 - 4x = 0 )Combine like terms:( x^2 + (2b - 4)x + b^2 = 0 )This is a quadratic in x. Let's denote this as:( x^2 + (2b - 4)x + b^2 = 0 )Let me denote this quadratic equation as ( x^2 + (2b - 4)x + b^2 = 0 ). Let's find its discriminant to ensure it has two real solutions (since the line intersects the parabola at two points A and B).The discriminant D is:( D = (2b - 4)^2 - 4 * 1 * b^2 )Calculate that:( D = 4b^2 - 16b + 16 - 4b^2 )Simplify:( D = -16b + 16 )For the quadratic to have two real solutions, D must be positive:( -16b + 16 > 0 )( -16b > -16 )Divide both sides by -16 (inequality sign flips):( b < 1 )So, b must be less than 1. Also, earlier we saw that b ≠ 1 because the line doesn't pass through P(1,2). So, b < 1 is acceptable.Now, let's denote the roots of the quadratic as x1 and x2, which correspond to the x-coordinates of points A and B. From quadratic theory, we know that:Sum of roots: ( x1 + x2 = -(2b - 4) = 4 - 2b )Product of roots: ( x1 * x2 = b^2 )Similarly, since y = x + b, the y-coordinates of A and B are y1 = x1 + b and y2 = x2 + b.So, sum of y-coordinates: ( y1 + y2 = (x1 + b) + (x2 + b) = (x1 + x2) + 2b = (4 - 2b) + 2b = 4 )Product of y-coordinates: ( y1 * y2 = (x1 + b)(x2 + b) = x1x2 + b(x1 + x2) + b^2 = b^2 + b(4 - 2b) + b^2 = b^2 + 4b - 2b^2 + b^2 = 4b )So, we have:( x1 + x2 = 4 - 2b )( x1x2 = b^2 )( y1 + y2 = 4 )( y1y2 = 4b )Now, the circle with AB as its diameter passes through point P(1,2). Let's recall that if a circle has diameter AB, then any point P on the circle satisfies the condition that angle APB is a right angle. In vector terms, this means that vectors PA and PB are perpendicular, so their dot product is zero.So, let's express vectors PA and PB.Point A is (x1, y1), Point B is (x2, y2), and Point P is (1,2).Vector PA = (x1 - 1, y1 - 2)Vector PB = (x2 - 1, y2 - 2)Their dot product is:( (x1 - 1)(x2 - 1) + (y1 - 2)(y2 - 2) = 0 )Let me compute each part step by step.First, compute ( (x1 - 1)(x2 - 1) ):( x1x2 - x1 - x2 + 1 )We know x1x2 = b^2, and x1 + x2 = 4 - 2b, so:( b^2 - (4 - 2b) + 1 = b^2 - 4 + 2b + 1 = b^2 + 2b - 3 )Next, compute ( (y1 - 2)(y2 - 2) ):( y1y2 - 2y1 - 2y2 + 4 )We know y1y2 = 4b, and y1 + y2 = 4, so:( 4b - 2*4 + 4 = 4b - 8 + 4 = 4b - 4 )Now, add both parts together:( (b^2 + 2b - 3) + (4b - 4) = 0 )Simplify:( b^2 + 2b - 3 + 4b - 4 = 0 )Combine like terms:( b^2 + 6b - 7 = 0 )So, we have the quadratic equation:( b^2 + 6b - 7 = 0 )Let me solve for b using the quadratic formula:( b = frac{-6 pm sqrt{36 + 28}}{2} = frac{-6 pm sqrt{64}}{2} = frac{-6 pm 8}{2} )So, two solutions:1. ( b = frac{-6 + 8}{2} = frac{2}{2} = 1 )2. ( b = frac{-6 - 8}{2} = frac{-14}{2} = -7 )But earlier, we found that b must be less than 1, and also, the line shouldn't pass through P(1,2), which would happen if b=1 because substituting x=1, y=2 into y = x + 1 gives 2 = 2, which is true. So, b=1 is invalid because the line would pass through P, which is not allowed.Therefore, the only valid solution is b = -7.So, the equation of line l is:( y = x - 7 )But it's often written in standard form, so let's rearrange:( y = x - 7 ) can be written as ( x - y - 7 = 0 )Let me double-check this result.Given b = -7, the line is y = x - 7.Let's find points A and B by solving the intersection with the parabola ( y^2 = 4x ).Substitute y = x - 7 into ( y^2 = 4x ):( (x - 7)^2 = 4x )Expand:( x^2 - 14x + 49 = 4x )Bring all terms to one side:( x^2 - 18x + 49 = 0 )Compute discriminant:( D = (-18)^2 - 4*1*49 = 324 - 196 = 128 )Since D is positive, there are two real roots, which is correct.Now, let's find the points A and B.Using quadratic formula:( x = frac{18 pm sqrt{128}}{2} = frac{18 pm 8sqrt{2}}{2} = 9 pm 4sqrt{2} )So, x1 = 9 + 4√2, x2 = 9 - 4√2Corresponding y1 = x1 - 7 = 2 + 4√2, y2 = x2 - 7 = 2 - 4√2So, points A(9 + 4√2, 2 + 4√2) and B(9 - 4√2, 2 - 4√2)Now, let's check if the circle with AB as diameter passes through P(1,2).The circle equation can be written as:( (x - x1)(x - x2) + (y - y1)(y - y2) = 0 )But since AB is the diameter, the equation can also be written using the midpoint as center and radius as half the distance between A and B.But maybe a simpler way is to use the condition that PA is perpendicular to PB.Alternatively, since we already used that condition to find b, and we arrived at b = -7, which satisfies all the constraints, I think this is correct.Therefore, the equation of line l is ( x - y - 7 = 0 ).**Final Answer**(I) The standard equation of parabola ( C ) is (boxed{y^2 = 4x}), and the equation of its directrix is (boxed{x = -1}).(II) The equation of line ( l ) is (boxed{x - y - 7 = 0}).

answer：Alright, so I have this problem about a construction team greening a 1500-meter road. They have two plans, Plan A and Plan B, each requiring different numbers of type A and type B flowers, and each costing a certain amount per meter. The first part asks me to find the cost per stem of type A and type B flowers. The second part is about minimizing the total cost of the project given a constraint on how much of the road must be greened using Plan B compared to Plan A.Okay, starting with part (1). I need to find the cost per stem of type A and type B flowers. Let me denote the cost per stem of type A as x yuan and type B as y yuan. From the problem, Plan A requires 2 stems of type A and 3 stems of type B for 1 meter, costing 22 yuan. So, I can write the equation: 2x + 3y = 22.Similarly, Plan B requires 1 stem of type A and 5 stems of type B for 1 meter, costing 25 yuan. So, the equation is: x + 5y = 25.Now, I have a system of two equations:1. 2x + 3y = 222. x + 5y = 25I need to solve this system to find x and y. Let me think about the best way to solve this. Maybe substitution or elimination. Let's try elimination.If I multiply the second equation by 2, I get:2x + 10y = 50Now, subtract the first equation from this new equation:(2x + 10y) - (2x + 3y) = 50 - 22Simplify:2x + 10y - 2x - 3y = 28Which simplifies to:7y = 28So, y = 4.Now that I have y, I can plug it back into one of the original equations to find x. Let's use the second equation:x + 5y = 25Substitute y = 4:x + 5*4 = 25x + 20 = 25x = 5So, the cost per stem of type A flowers is 5 yuan, and type B is 4 yuan. That seems straightforward.Moving on to part (2). I need to find the total cost of the project when the total length of road greened according to Plan A is how many meters to minimize the total cost. Also, find the minimum total cost.Let me denote the length of road greened according to Plan A as 'a' meters. Then, the length greened according to Plan B would be (1500 - a) meters.The constraint given is that the total length greened according to Plan B should not be less than twice the total length greened according to Plan A. So:1500 - a ≥ 2aLet me solve this inequality:1500 ≥ 3aa ≤ 500So, the maximum length that can be greened using Plan A is 500 meters.Now, I need to express the total cost in terms of 'a'. Since each meter of Plan A costs 22 yuan and each meter of Plan B costs 25 yuan, the total cost 'w' would be:w = 22a + 25(1500 - a)Simplify this:w = 22a + 37500 - 25aw = 37500 - 3aSo, the total cost decreases as 'a' increases. Therefore, to minimize the total cost, I need to maximize 'a' within the given constraint.From the earlier result, a ≤ 500. So, the maximum value of 'a' is 500 meters.Plugging this back into the total cost equation:w = 37500 - 3*500 = 37500 - 1500 = 36000 yuanTherefore, when the total length greened according to Plan A is 500 meters, the total cost is minimized, and the minimum total cost is 36000 yuan.Wait, let me double-check my calculations to make sure I didn't make any mistakes.For part (1), solving the system of equations:2x + 3y = 22x + 5y = 25I multiplied the second equation by 2 to get 2x + 10y = 50, then subtracted the first equation:2x + 10y - 2x - 3y = 50 - 22Which gives 7y = 28, so y = 4. Then x = 5. That seems correct.For part (2), setting up the total cost:w = 22a + 25(1500 - a) = 22a + 37500 -25a = 37500 -3aSince the coefficient of 'a' is negative, increasing 'a' decreases the total cost. Therefore, to minimize 'w', set 'a' as large as possible, which is 500 meters. Plugging back in, 37500 - 3*500 = 37500 - 1500 = 36000. That seems correct.I think my answers are correct.

answer：First, let's define the cost price of the TV as ( x ).The store increases the selling price by 40%, so the new selling price becomes ( x times 1.40 ).Next, a 20% discount is applied to this increased price, which means the customer pays 80% of the increased price. Therefore, the final selling price after the discount is ( x times 1.40 times 0.80 ).The profit is calculated by subtracting the cost price from the final selling price. According to the problem, this profit is 360. So, the equation becomes:[x times 1.40 times 0.80 - x = 360]Simplifying the left side:[x times 1.12 - x = 360][0.12x = 360]Solving for ( x ):[x = frac{360}{0.12} = 3000]Therefore, the cost price of the TV is 3000.

answer：Okay, so I have this math problem here: (3 + 3 + 5) divided by 2 minus 1 divided by 2. Hmm, let me try to figure this out step by step. First, I think I should handle what's inside the parentheses because that's usually the first thing to do in math problems. So, 3 plus 3 is 6, and then 6 plus 5 is 11. Okay, so the expression simplifies to 11 divided by 2 minus 1 divided by 2.Now, I have 11/2 minus 1/2. I remember that when you subtract fractions with the same denominator, you can just subtract the numerators. So, 11 minus 1 is 10, and the denominator stays the same, which is 2. So, that gives me 10/2.Wait, is that right? Let me double-check. If I have 11 divided by 2, that's like 5.5, and 1 divided by 2 is 0.5. So, 5.5 minus 0.5 should be 5. And 10 divided by 2 is also 5. Okay, that matches up. So, it looks like I did that correctly.But just to make sure I'm not missing anything, let me go through it again. Starting with the original expression: (3 + 3 + 5) ÷ 2 - 1 ÷ 2. Adding inside the parentheses first: 3 + 3 is 6, plus 5 is 11. So, 11 ÷ 2 is 5.5, and 1 ÷ 2 is 0.5. Subtracting those gives me 5.5 - 0.5, which is 5. Yep, that's consistent.I guess another way to look at it is by combining the two divisions since they have the same denominator. So, instead of doing 11/2 minus 1/2 separately, I can subtract the numerators first: 11 - 1 is 10, and then divide by 2, which is 5. That seems simpler and quicker.Is there another method to solve this? Maybe using fractions throughout instead of decimals? Let's see. 11/2 minus 1/2. Since the denominators are the same, subtract the numerators: 11 - 1 = 10, so 10/2, which simplifies to 5. Yep, same result.I think I'm confident that the answer is 5. It makes sense both when I do it step by step and when I check it using different methods. I don't see any mistakes in my calculations, so I think I'm good.

answer：Okay, so I need to find the domain of the function f(x) = 1 / [(x - 3) + (x - 9)]. Hmm, let me think about this step by step. First, I remember that the domain of a function refers to all the possible input values (x-values) for which the function is defined. Since this function is a rational function (a fraction), the main thing I need to worry about is the denominator. Specifically, the denominator cannot be zero because division by zero is undefined in mathematics. So, my goal is to find all the x-values that do not make the denominator zero.Alright, let's start by simplifying the denominator to make things easier. The denominator is (x - 3) + (x - 9). I can combine like terms here. Let's see, x minus 3 plus x minus 9. Combining the x terms: x + x is 2x. Combining the constants: -3 - 9 is -12. So, the denominator simplifies to 2x - 12. That makes the function f(x) = 1 / (2x - 12).Now, I need to find when the denominator equals zero because those are the values of x that are not allowed. So, let's set 2x - 12 equal to zero and solve for x.2x - 12 = 0Adding 12 to both sides:2x = 12Dividing both sides by 2:x = 6So, x equals 6 makes the denominator zero, which means the function is undefined at x = 6. Therefore, the domain of f(x) is all real numbers except x = 6.But how do I express this in interval notation? Interval notation is a way to describe sets of real numbers using intervals. Since x can be any real number except 6, I need to split the real number line into two intervals: one from negative infinity up to 6, and the other from 6 to positive infinity. In interval notation, we use parentheses to indicate that the endpoint is not included in the interval.So, the interval from negative infinity to 6 is written as (-∞, 6). The interval from 6 to positive infinity is written as (6, ∞). To combine these two intervals, we use the union symbol ∪.Putting it all together, the domain of f(x) in interval notation is (-∞, 6) ∪ (6, ∞).Wait, let me double-check to make sure I didn't make any mistakes. I simplified the denominator correctly: (x - 3) + (x - 9) becomes 2x - 12. Solving 2x - 12 = 0 gives x = 6. So, x cannot be 6. Therefore, the domain is all real numbers except 6, which in interval notation is indeed (-∞, 6) ∪ (6, ∞). Yeah, that seems right.I think I've got it. The key steps were simplifying the denominator, solving for when it equals zero, and then expressing the domain in interval notation by excluding that problematic x-value. It's important to remember that in interval notation, parentheses are used for exclusion, and since 6 is excluded, we use parentheses around it in both intervals.Just to visualize, if I were to graph this function, there would be a vertical asymptote at x = 6 because the function approaches infinity or negative infinity near that point. So, the graph would have two separate branches on either side of x = 6, which aligns with the domain I found.Another way to think about it is that the function f(x) = 1 / (2x - 12) is a hyperbola. Hyperbolas have asymptotes where the function cannot be defined, and in this case, the vertical asymptote is at x = 6. Therefore, the domain excludes x = 6, confirming my earlier conclusion.I guess it's also worth noting that if the denominator had more terms or was a more complex expression, I might need to factor it or use other methods to find where it equals zero. But in this case, it was straightforward since it simplified to a linear expression.So, to summarize my thought process: I started by simplifying the denominator, found the value that makes it zero, and then expressed the domain in interval notation by excluding that value. This gave me the final answer of (-∞, 6) ∪ (6, ∞).I think I'm confident with this solution. It's always good to go through the steps methodically, especially when dealing with domains and asymptotes, to avoid missing any critical points. By breaking down the problem into smaller parts—simplifying, solving for zero, and then expressing the domain—I was able to tackle each component effectively.One last check: if I plug in x = 6 into the original function, the denominator becomes (6 - 3) + (6 - 9) = 3 + (-3) = 0, which confirms that f(6) is undefined. Testing another value, say x = 0, gives f(0) = 1 / (0 - 3 + 0 - 9) = 1 / (-12), which is defined. Similarly, x = 7 gives f(7) = 1 / (14 - 12) = 1/2, which is also defined. This further solidifies that x = 6 is the only value excluded from the domain.In conclusion, by carefully analyzing the denominator and determining the values that make it zero, I was able to identify the domain of the function in interval notation. This approach ensures that I account for all restrictions on the input values, providing a clear and accurate solution.