answer：Okay, so I have this problem here where I need to find the probability that the function ( f(x) = x^2 + 2ax + 2 ) has two distinct zeros when a fair die is thrown, and the number of dots facing up is ( a ). The options are A: ( frac{1}{3} ), B: ( frac{1}{2} ), C: ( frac{2}{3} ), D: ( frac{5}{6} ).First, I remember that for a quadratic function ( ax^2 + bx + c ), the discriminant ( Delta ) determines the nature of its roots. The discriminant is given by ( Delta = b^2 - 4ac ). If ( Delta > 0 ), the quadratic has two distinct real roots. If ( Delta = 0 ), it has exactly one real root, and if ( Delta < 0 ), it has no real roots.In this problem, the quadratic function is ( f(x) = x^2 + 2ax + 2 ). So, comparing this to the standard form ( ax^2 + bx + c ), here, the coefficient of ( x^2 ) is 1, the coefficient of ( x ) is ( 2a ), and the constant term is 2.So, let me calculate the discriminant for this function. The discriminant ( Delta ) would be:( Delta = (2a)^2 - 4 times 1 times 2 )Simplifying that:( Delta = 4a^2 - 8 )For the quadratic to have two distinct zeros, we need ( Delta > 0 ). So:( 4a^2 - 8 > 0 )Let me solve this inequality step by step. First, add 8 to both sides:( 4a^2 > 8 )Then, divide both sides by 4:( a^2 > 2 )Taking square roots on both sides, we get:( |a| > sqrt{2} )Since ( a ) is the number of dots facing up on a die, it can only take positive integer values from 1 to 6. So, ( a ) is an integer in the set {1, 2, 3, 4, 5, 6}.Given that ( |a| > sqrt{2} ), and since ( a ) is positive, this simplifies to:( a > sqrt{2} )Calculating ( sqrt{2} ) is approximately 1.414. Therefore, ( a ) must be greater than 1.414. Since ( a ) is an integer, the smallest integer greater than 1.414 is 2.So, the possible values of ( a ) that satisfy this condition are 2, 3, 4, 5, and 6. That's 5 values in total.Now, since a fair die has 6 faces, each with an equal probability of landing face up, the probability of each outcome is ( frac{1}{6} ).Therefore, the probability that ( a ) is 2, 3, 4, 5, or 6 is the sum of the probabilities of each of these individual outcomes. Since each has a probability of ( frac{1}{6} ), the total probability is:( 5 times frac{1}{6} = frac{5}{6} )So, the probability that the function ( f(x) = x^2 + 2ax + 2 ) has two distinct zeros is ( frac{5}{6} ).Let me just double-check my steps to make sure I didn't make a mistake. I calculated the discriminant correctly, set it greater than zero, solved for ( a ), considered the possible integer values of ( a ) on a die, and then computed the probability. It all seems to check out.Therefore, I'm confident that the correct answer is D: ( frac{5}{6} ).**Final Answer**boxed{D}

answer：First, identify the real and imaginary parts of the complex number ( z = 3 - i ).The real part is ( 3 ) and the imaginary part is ( -1 ).Next, apply the formula for the modulus of a complex number:[|z| = sqrt{(text{real part})^2 + (text{imaginary part})^2}]Substitute the identified values into the formula:[|z| = sqrt{3^2 + (-1)^2} = sqrt{9 + 1} = sqrt{10}]Therefore, the modulus of ( z ) is ( sqrt{10} ).

answer：Okay, so I have this problem here where I need to figure out which of the given numbers doesn't round to 72.36 when rounded to the nearest hundredth. Let me break this down step by step because I want to make sure I understand it correctly.First, rounding to the nearest hundredth means I need to look at the number up to the thousandth place to determine whether to round the hundredth place up or down. The hundredth place is the second digit after the decimal point, and the thousandth place is the third digit after the decimal point.Let me list out the options again to keep things clear:A) 72.361 B) 72.358 C) 72.3539999 D) 72.3601 E) 72.35999 Alright, let's tackle each one by one.**Option A: 72.361**Looking at this number, the hundredth place is 6, and the thousandth place is 1. Since 1 is less than 5, we don't round up the hundredth place. So, 72.361 rounded to the nearest hundredth is 72.36. That seems straightforward.**Option B: 72.358**Here, the hundredth place is 5, and the thousandth place is 8. Since 8 is greater than or equal to 5, we round up the hundredth place. Rounding 5 up gives us 6, so 72.358 becomes 72.36 when rounded to the nearest hundredth. That also makes sense.**Option C: 72.3539999**This one looks a bit trickier because there are multiple digits after the decimal. The hundredth place is 5, and the thousandth place is 3. Since 3 is less than 5, we don't round up the hundredth place. So, 72.3539999 should round down to 72.35. Wait, that's different from the others. It doesn't round to 72.36. Hmm, that's interesting.**Option D: 72.3601**The hundredth place is 6, and the thousandth place is 0. Since 0 is less than 5, we don't round up the hundredth place. So, 72.3601 remains 72.36 when rounded to the nearest hundredth. That's consistent with the others except for option C.**Option E: 72.35999**Here, the hundredth place is 5, and the thousandth place is 9. Since 9 is greater than or equal to 5, we round up the hundredth place. Rounding 5 up gives us 6, so 72.35999 becomes 72.36 when rounded to the nearest hundredth. That's the same as options A, B, D, and E.Wait a second, so option C is the only one that doesn't round to 72.36. It rounds to 72.35 instead. That must be the answer. Let me double-check to make sure I didn't make a mistake.Looking back at option C: 72.3539999. The hundredth place is 5, and the thousandth place is 3. Since 3 is less than 5, we keep the hundredth place as it is, which is 5. So, it rounds down to 72.35. Yep, that's correct.Just to be thorough, let's go through the others again quickly:- A: 72.361 → 72.36 (thousandth place is 1, round down)- B: 72.358 → 72.36 (thousandth place is 8, round up)- C: 72.3539999 → 72.35 (thousandth place is 3, round down)- D: 72.3601 → 72.36 (thousandth place is 0, round down)- E: 72.35999 → 72.36 (thousandth place is 9, round up)Yep, everything checks out. Option C is the only one that doesn't round to 72.36. I think I've got it.

answer：First, I observe that the expression has multiple layers of nested parentheses, which suggests that I should simplify it step by step, starting from the innermost parentheses and working my way outward.I begin by simplifying the innermost expression, which is (2 - y). This gives me a value that I can substitute back into the expression.Next, I substitute this simplified value back into the expression and continue simplifying each subsequent layer of parentheses. Each step involves basic arithmetic operations, such as subtraction and addition, which I perform carefully to ensure accuracy.As I work through each layer, I keep track of the changes in the expression, making sure that each substitution and simplification is correct. This step-by-step approach helps me avoid mistakes and ensures that I fully understand each transformation of the expression.Finally, after simplifying all layers of parentheses, I arrive at the fully simplified form of the original expression.

answer：Okay, so I've got this problem here: The sum of two numbers is 36, and their difference is 12. I need to find the greater of the two numbers. Hmm, let's see how I can approach this.Alright, first, I know that when dealing with two numbers and their sum and difference, it's often helpful to set up a system of equations. That way, I can solve for both numbers step by step. So, let me define the two numbers. Let's say the two numbers are x and y, and without loss of generality, I can assume that x is the larger number and y is the smaller one. That makes sense because the problem is asking for the greater number, so it's good to have that defined upfront.So, if x is the larger number and y is the smaller one, their sum is 36. That gives me the first equation: x + y = 36. Got that down. Now, their difference is 12. Since x is larger, subtracting y from x should give me 12. So, the second equation is x - y = 12. Perfect, now I have a system of two equations:1. x + y = 362. x - y = 12Now, I need to solve this system to find the values of x and y. There are a couple of methods to solve systems of equations, like substitution or elimination. I think elimination might be straightforward here because if I add the two equations together, the y terms will cancel out, leaving me with an equation in terms of x only. Let's try that.Adding equation 1 and equation 2:(x + y) + (x - y) = 36 + 12Simplifying the left side:x + y + x - y = 2xAnd the right side:36 + 12 = 48So, 2x = 48. To solve for x, I just divide both sides by 2:2x / 2 = 48 / 2x = 24Alright, so x is 24. Now, I need to find y. I can substitute x back into one of the original equations. Let's use the first equation, x + y = 36.Substituting x = 24:24 + y = 36To solve for y, subtract 24 from both sides:y = 36 - 24y = 12So, y is 12. Let me double-check my work to make sure I didn't make any mistakes. If x is 24 and y is 12, then their sum should be 36, and their difference should be 12.Checking the sum: 24 + 12 = 36. Yep, that's correct.Checking the difference: 24 - 12 = 12. That also checks out.So, everything seems to add up. Therefore, the greater of the two numbers is 24.Wait, just to make sure I didn't make any calculation errors, let me go through the steps again.Starting with the two equations:1. x + y = 362. x - y = 12Adding them:(x + y) + (x - y) = 36 + 12Simplifies to:2x = 48Divide both sides by 2:x = 24Substitute back into equation 1:24 + y = 36Subtract 24:y = 12Yep, same result. So, I'm confident that x is indeed 24 and y is 12. Therefore, the greater number is 24.I think that's solid. I don't see any mistakes in my reasoning or calculations. It's always good to verify, especially when dealing with simple equations because it's easy to mix up signs or make arithmetic errors. But in this case, everything seems consistent.Another way to think about it is by considering the average. If the sum of two numbers is 36, their average is 18. The difference is 12, which means one number is 6 more than the average, and the other is 6 less. So, 18 + 6 = 24 and 18 - 6 = 12. That also gives me the same result, which reinforces that 24 is the correct greater number.So, whether I use the system of equations approach or think about it in terms of averages and deviations, I arrive at the same conclusion. That gives me even more confidence that 24 is the right answer.Just to recap, the steps were:1. Define variables: x (greater number) and y (smaller number).2. Set up equations based on the given sum and difference.3. Use elimination to solve for x.4. Substitute back to find y.5. Verify the solution by plugging the values back into the original problem.6. Consider an alternative method (averages) to confirm the result.This thorough approach helps ensure that I haven't overlooked anything and that my answer is accurate. It's especially useful when dealing with problems that seem straightforward but can sometimes have hidden complexities or require careful attention to detail.In summary, by systematically setting up and solving the equations, as well as verifying through an alternative method, I've determined that the greater of the two numbers is 24.

answer：First, I need to calculate the price of the snowboard after the initial 40% discount on Friday. The original price is 200, and a 40% discount means the snowboard is sold for 60% of its original price.Next, I apply the 25% discount on Monday to the already discounted price from Friday. This means the snowboard is now sold for 75% of the Friday price.By multiplying these percentages with the respective prices, I can determine the final price of the snowboard on Monday.