answer：Okay, so I have this problem here: If x divided by (11p) is a certain type of number, where x is a positive integer and p is a prime number, and the least value of x is 66. I need to figure out what type of number x divided by (11p) is. Hmm, let's break this down step by step.First, let's understand the components of the problem. We have x, which is a positive integer, and p, which is a prime number. The expression we're looking at is x divided by (11p), and we're told that the smallest x that makes this expression a certain type of number is 66. So, my goal is to figure out what type of number x/(11p) is, given these conditions.Alright, let's start by plugging in the smallest x, which is 66, into the expression. So, x/(11p) becomes 66/(11p). I can simplify this fraction by dividing both the numerator and the denominator by 11. That gives me 6/p. So now, the expression simplifies to 6 divided by p.Now, since p is a prime number, it can only be divided by 1 and itself. The prime numbers less than 6 are 2, 3, and 5. Let's consider each of these possibilities for p and see what happens to the expression 6/p.If p is 2, then 6/p is 6/2, which equals 3. That's an integer.If p is 3, then 6/p is 6/3, which equals 2. That's also an integer.If p is 5, then 6/p is 6/5, which is 1.2. That's a decimal, not an integer.So, when p is 2 or 3, the expression 6/p results in an integer, but when p is 5, it doesn't. Since we're looking for the smallest x that makes x/(11p) a certain type of number, and x is 66, it seems that the expression must be an integer. Otherwise, if p were 5, x would have to be larger to make 6/p an integer, but 66 is the smallest x given.Wait, let me think about that again. If p were 5, then 6/p is 1.2, which isn't an integer. But since x is 66, which is the smallest x that makes x/(11p) a certain type of number, it must be that x/(11p) is an integer. Otherwise, if it were a different type of number, like a fraction or a decimal, there might be a smaller x that satisfies the condition.So, if x/(11p) has to be an integer, then 6/p must be an integer. That means p has to be a divisor of 6. The divisors of 6 are 1, 2, 3, and 6. But p is a prime number, so p can only be 2 or 3. Therefore, p must be either 2 or 3.Let's verify this. If p is 2, then x/(11p) is 66/(11*2) = 66/22 = 3, which is an integer. If p is 3, then x/(11p) is 66/(11*3) = 66/33 = 2, which is also an integer. So, both cases result in integers.But what if p were a different prime number, like 5 or 7? Let's check. If p is 5, then x/(11p) would be 66/(11*5) = 66/55 = 1.2, which isn't an integer. Similarly, if p is 7, then x/(11p) is 66/(11*7) = 66/77 = 6/7, which is approximately 0.846, also not an integer.So, it seems that only when p is 2 or 3 does x/(11p) result in an integer. Since 66 is the smallest x that satisfies this condition, it must be that x/(11p) is an integer. Therefore, the type of number that x/(11p) is, given the least value of x is 66 and p is a prime number, is an integer.Wait, but let me make sure I'm not missing something. Is there a possibility that x/(11p) could be another type of number, like a whole number or a natural number? Well, integers include whole numbers and natural numbers, so if x/(11p) is an integer, it's also a whole number and a natural number. But since the problem specifies a "certain type of number," and given that 66 is the smallest x, it's most precise to say that x/(11p) is an integer.Also, considering that x is a positive integer and p is a prime number, there's no other type of number that fits the condition with x being as small as 66. If x were smaller, say 33, then x/(11p) would be 33/(11p) = 3/p. For this to be an integer, p would have to be 1 or 3, but p is a prime number, so p can't be 1. Therefore, p would have to be 3, making x/(11p) = 3/3 = 1, which is an integer. But since the problem states that the least value of x is 66, it implies that x can't be smaller than 66, so p must be 2 or 3, making x/(11p) an integer.In conclusion, after considering different values of p and verifying the results, it's clear that x/(11p) must be an integer when x is 66 and p is a prime number. Therefore, the type of number that x/(11p) is, given the least value of x is 66 and p is a prime number, is an integer.

answer：First, I need to expand the left side of the equation ((x + 5)(x + n)). Multiplying the two binomials gives me (x^2 + (5 + n)x + 5n).Next, I compare this expanded form with the right side of the equation, which is (x^2 + mx - 5). By equating the coefficients of the corresponding terms, I can set up two equations: (5 + n = m) and (5n = -5).Solving the second equation, (5n = -5), I find that (n = -1). Substituting this value of (n) into the first equation, (5 + (-1) = m), I determine that (m = 4).Finally, to find the product (mn), I multiply (m = 4) by (n = -1), resulting in (mn = -4).

answer：First, I recognize that the average of 15 numbers is 40. To find the total sum of these numbers, I multiply the average by the number of values: 40 multiplied by 15 equals 600.Next, I need to determine how adding 15 to each number affects the total sum. Since there are 15 numbers, adding 15 to each number increases the total sum by 15 multiplied by 15, which is 225.Adding this increase to the original sum gives me the new total sum: 600 plus 225 equals 825.Finally, to find the new average, I divide the new total sum by the number of values: 825 divided by 15 equals 55.

answer：Okay, so I have this problem here with vectors and a function, and I need to find the probability that the function has a root. Hmm, let me try to break this down step by step.First, the problem gives me two vectors, **a** = (-2, 2) and **b** = (5, m). It also says that the magnitude of **a** + **b** is less than or equal to 5. I need to find the probability that the function f(x) = √3 cos x - sin x + m has a root. The options are given as fractions, so I think it's a probability question involving some range of m.Alright, let's start with the vectors. If I add vectors **a** and **b**, I get:**a** + **b** = (-2 + 5, 2 + m) = (3, 2 + m)The magnitude of this vector is given by the square root of the sum of the squares of its components. So,|**a** + **b**| = √(3² + (2 + m)²) = √(9 + (2 + m)²)The problem states that this magnitude is less than or equal to 5. So,√(9 + (2 + m)²) ≤ 5If I square both sides to eliminate the square root,9 + (2 + m)² ≤ 25Subtracting 9 from both sides,(2 + m)² ≤ 16Taking square roots on both sides,|2 + m| ≤ 4This means that 2 + m is between -4 and 4. So,-4 ≤ 2 + m ≤ 4Subtracting 2 from all parts,-6 ≤ m ≤ 2Okay, so m is between -6 and 2. That's the range of m based on the vector condition.Now, moving on to the function f(x) = √3 cos x - sin x + m. The problem asks for the probability that this function has a root. A root means that f(x) = 0 for some x. So, we need to find when √3 cos x - sin x + m = 0.Let me rewrite that equation:√3 cos x - sin x = -mHmm, this looks like a trigonometric equation. Maybe I can express the left side as a single sine or cosine function. I remember that expressions like a cos x + b sin x can be written as R cos(x - φ) or R sin(x + φ), where R is the amplitude and φ is the phase shift.Let me try that. The expression √3 cos x - sin x can be written in the form R cos(x + φ). Let's compute R and φ.The formula is:R = √(a² + b²) where a is the coefficient of cos x and b is the coefficient of sin x.Here, a = √3 and b = -1.So,R = √((√3)² + (-1)²) = √(3 + 1) = √4 = 2Okay, so R is 2. Now, to find φ, we can use:cos φ = a/R = √3 / 2sin φ = b/R = -1 / 2So, cos φ = √3/2 and sin φ = -1/2. That means φ is in the fourth quadrant. The angle whose cosine is √3/2 and sine is -1/2 is -30 degrees or 330 degrees, which is equivalent to -π/6 radians.So, φ = -π/6.Therefore, √3 cos x - sin x can be written as 2 cos(x - (-π/6)) = 2 cos(x + π/6).Wait, actually, let me double-check that. If I have a cos x + b sin x = R cos(x - φ), then:a = R cos φb = R sin φBut in our case, it's √3 cos x - sin x, so b is negative. So, maybe it's better to write it as 2 cos(x + π/6). Let me verify:2 cos(x + π/6) = 2 [cos x cos π/6 - sin x sin π/6] = 2 [ (√3/2) cos x - (1/2) sin x ] = √3 cos x - sin xYes, that's correct. So, √3 cos x - sin x = 2 cos(x + π/6).So, going back to the equation:√3 cos x - sin x = -mWhich becomes:2 cos(x + π/6) = -mTherefore,cos(x + π/6) = -m / 2Now, for this equation to have a solution, the right-hand side must be between -1 and 1 because the cosine function only takes values in that interval. So,-1 ≤ -m / 2 ≤ 1Multiplying all parts by -2 (and remembering to reverse the inequalities when multiplying by a negative):2 ≥ m ≥ -2So,-2 ≤ m ≤ 2Wait, but earlier we found that m is between -6 and 2. So, the function f(x) has a root only when m is between -2 and 2. So, within the range of m from -6 to 2, the function has a root when m is between -2 and 2.Therefore, the length of the interval where the function has a root is from -2 to 2, which is 4 units. The total possible interval for m is from -6 to 2, which is 8 units.So, the probability is the ratio of the favorable interval to the total interval, which is 4/8 = 1/2.Wait, but let me make sure I didn't make a mistake here. The total interval is from -6 to 2, which is 8 units. The favorable interval is from -2 to 2, which is 4 units. So, the probability is 4/8 = 1/2. So, the probability is 1/2, which is option D.But hold on, let me think again. The function f(x) = √3 cos x - sin x + m has a root when m is between -2 and 2. But m itself is constrained between -6 and 2 because of the vector condition. So, the overlap is from -2 to 2, which is 4 units. The total possible m is from -6 to 2, which is 8 units. So, 4/8 is indeed 1/2.Wait, but another way to think about it is, the function f(x) has a root when m is between -2 and 2. So, the favorable range is 4, and the total possible range is 8, so 4/8 is 1/2. So, yeah, that makes sense.But just to double-check, maybe I should consider the function f(x) = √3 cos x - sin x + m. The maximum and minimum values of √3 cos x - sin x are 2 and -2, respectively, because we expressed it as 2 cos(x + π/6), which has an amplitude of 2. So, the function f(x) will have a root when m is between -2 and 2 because f(x) can vary between -2 + m and 2 + m. For f(x) to cross zero, m must be such that the minimum of f(x) is less than or equal to zero and the maximum is greater than or equal to zero.So, the minimum of f(x) is -2 + m, and the maximum is 2 + m. For f(x) to have a root, we need:-2 + m ≤ 0 and 2 + m ≥ 0Which simplifies to:m ≤ 2 and m ≥ -2So, again, m must be between -2 and 2.Therefore, the probability is the length of this interval divided by the total interval of m, which is from -6 to 2. So, 4/8 = 1/2.Alright, I think that's solid. So, the answer should be D: 1/2.

answer：Alright, so I have this problem about an even function ( f(x) ) that's monotonic on the interval ([0, +infty)). It passes through two points: A at ((0, -1)) and B at ((3, 1)). I need to find the solution set for ( f(x) < 1 ).First, let me recall what an even function is. An even function satisfies ( f(-x) = f(x) ) for all ( x ) in its domain. This means the graph of the function is symmetric with respect to the y-axis. So, whatever behavior the function has on the right side of the y-axis (positive x-values), it mirrors on the left side (negative x-values).Next, the function is monotonic on ([0, +infty)). Monotonic means it's either entirely non-increasing or non-decreasing on that interval. Given that it passes through ((0, -1)) and ((3, 1)), let's see if it's increasing or decreasing.At ( x = 0 ), ( f(0) = -1 ), and at ( x = 3 ), ( f(3) = 1 ). Since ( -1 < 1 ) and moving from ( x = 0 ) to ( x = 3 ) increases the function's value, it seems like the function is increasing on ([0, +infty)).Since it's an even function, what does that mean for the left side of the y-axis? Well, because ( f(-x) = f(x) ), the function on the left side (negative x-values) will be a mirror image of the right side. So, if it's increasing on the right, it must be decreasing on the left. That makes sense because as ( x ) becomes more negative, the function's value would decrease from ( f(0) = -1 ) towards ( f(-3) = 1 ), but wait, that seems contradictory.Hold on, if ( f(3) = 1 ), then ( f(-3) = 1 ) as well. But since the function is increasing on the right, it goes from ( -1 ) at ( x = 0 ) up to ( 1 ) at ( x = 3 ). On the left side, because it's even, it should mirror this behavior. So, starting from ( x = 0 ) with ( f(0) = -1 ), as ( x ) becomes more negative, the function should also increase? Wait, that doesn't make sense because increasing on the right would imply decreasing on the left if it's symmetric.Let me think again. If ( f(x) ) is increasing on ([0, +infty)), then for ( x > 0 ), as ( x ) increases, ( f(x) ) increases. On the left side, since it's even, ( f(-x) = f(x) ). So, as ( x ) becomes more negative (i.e., ( -x ) increases), ( f(x) ) should also increase. But that would mean that on the left side, as ( x ) decreases (becomes more negative), ( f(x) ) increases. So, in terms of monotonicity, on the left side, the function is decreasing as ( x ) increases, but since ( x ) is negative, it's a bit confusing.Maybe it's better to consider the behavior separately on each side. On the right side (( x geq 0 )), the function is increasing, so as ( x ) increases, ( f(x) ) increases. On the left side (( x leq 0 )), since it's even, the function must be decreasing as ( x ) increases (because as ( x ) becomes less negative, moving towards zero, the function value decreases from ( f(-3) = 1 ) down to ( f(0) = -1 )).So, summarizing:- For ( x geq 0 ), ( f(x) ) is increasing.- For ( x leq 0 ), ( f(x) ) is decreasing.Now, the problem is to find where ( f(x) < 1 ). Let's break this down into two parts: ( x geq 0 ) and ( x < 0 ).Starting with ( x geq 0 ):We know ( f(3) = 1 ). Since the function is increasing on this interval, for ( x < 3 ), ( f(x) < f(3) = 1 ). So, for ( x ) in ([0, 3)), ( f(x) < 1 ).Now, for ( x < 0 ):Since the function is even, ( f(x) = f(-x) ). We need to find where ( f(x) < 1 ). Let's consider the corresponding positive ( x ) values. For ( x < 0 ), ( -x > 0 ). We already know that for ( -x < 3 ), ( f(-x) < 1 ). Therefore, ( f(x) = f(-x) < 1 ) when ( -x < 3 ), which implies ( x > -3 ).But since ( x < 0 ) in this case, combining both conditions, ( x ) must be greater than ( -3 ) and less than ( 0 ). So, ( x in (-3, 0) ).Putting both intervals together:- For ( x geq 0 ), ( x in [0, 3) )- For ( x < 0 ), ( x in (-3, 0) )Combining these, the solution set is ( (-3, 3) ).Wait, but let me double-check. At ( x = 3 ), ( f(3) = 1 ), so it's not included. Similarly, at ( x = -3 ), ( f(-3) = 1 ), so it's not included either. Therefore, the open interval ( (-3, 3) ) is correct.I think that's the solution. The function is less than 1 between ( -3 ) and ( 3 ), not including the endpoints because at those points, the function equals 1.**Final Answer**The solution set for ( f(x) < 1 ) is boxed{(-3, 3)}.

answer：First, I need to determine the combined speed of the escalator and the person walking on it. The escalator moves at 12 feet per second, and the person walks at 2 feet per second towards the top level. By adding these two speeds, I get the total effective speed.Next, I know that the person takes 15 seconds to cover the entire length of the escalator. Using the formula for distance, which is speed multiplied by time, I can calculate the length of the escalator by multiplying the effective speed by the time taken.Finally, by performing the multiplication, I will obtain the total length of the escalator.