💡Okay, so I have this problem here with four options, A through D, and I need to figure out which ones are true. Let me take them one by one.Starting with **Option A**: It says cos²15° minus sin²15° equals √3 over 2. Hmm, I remember there's a trigonometric identity that relates cos²θ minus sin²θ to something else. Isn't that the double-angle formula for cosine? Yeah, cos(2θ) equals cos²θ minus sin²θ. So if I apply that here, cos²15° minus sin²15° should be equal to cos(30°). And I know that cos(30°) is √3 over 2. So that means Option A is true. Got it.Moving on to **Option B**: It says (1/2)sin40° plus (√3/2)cos40° equals sin70°. Hmm, this looks like a combination of sine and cosine terms. Maybe I can use the sine addition formula here. The formula is sin(A + B) equals sinA cosB plus cosA sinB. If I compare that to what's given, it seems like the coefficients 1/2 and √3/2 might correspond to sine and cosine of some angle. Let me think, 1/2 is cos60° and √3/2 is sin60°. So maybe this expression is sin40°cos60° + cos40°sin60°, which would be sin(40° + 60°) = sin100°. But the right side of the equation is sin70°, not sin100°. Wait, sin100° is equal to sin80°, because sine is positive in the second quadrant and sin(180° - θ) equals sinθ. So sin100° is sin80°, which is not equal to sin70°. Therefore, Option B is false.Next is **Option C**: It says sin(π/8)cos(π/8) equals √2 over 4. I recall that there's a double-angle identity for sine that says sin2θ equals 2sinθcosθ. So if I let θ be π/8, then sin(π/4) equals 2sin(π/8)cos(π/8). Therefore, sin(π/8)cos(π/8) equals sin(π/4) divided by 2. I know that sin(π/4) is √2 over 2, so dividing that by 2 gives √2 over 4. That matches the right side of the equation, so Option C is true.Finally, **Option D**: It says tan15° equals 2 minus √3. I remember that tan(45° - 30°) can be calculated using the tangent subtraction formula: tan(A - B) equals (tanA - tanB)/(1 + tanA tanB). Let's apply that with A as 45° and B as 30°. So tan45° is 1, and tan30° is 1/√3. Plugging those in, we get (1 - 1/√3)/(1 + 1*1/√3). Let's simplify numerator and denominator. Multiply numerator and denominator by √3 to rationalize: (√3 - 1)/(√3 + 1). To simplify further, multiply numerator and denominator by (√3 - 1): [(√3 - 1)^2]/[(√3)^2 - 1^2] = (3 - 2√3 + 1)/(3 - 1) = (4 - 2√3)/2 = 2 - √3. So tan15° indeed equals 2 - √3. Therefore, Option D is true.So, summarizing:- Option A is true.- Option B is false.- Option C is true.- Option D is true.So the correct options are A, C, and D.

💡First, I need to find the mean of the given numbers. I'll add all the numbers together and then divide by the total number of integers, which is 5.Next, I'll calculate the mean by summing up the numbers:x + (x + 2) + (x + 4) + (x + 7) + (x + 32) = 5x + 45Then, I'll divide this sum by 5 to get the mean:Mean = (5x + 45) / 5 = x + 9Now, I'll determine the median. Since there are 5 numbers, the median is the third number when the numbers are arranged in ascending order. The numbers in order are:x, x + 2, x + 4, x + 7, x + 32So, the median is x + 4.Finally, I'll find the difference between the mean and the median:Difference = Mean - Median = (x + 9) - (x + 4) = 5Therefore, the mean is 5 greater than the median.

💡First, I recognize that the number of diagonals (D) that can be drawn from one vertex of a polygon is given by the formula D = n - 3, where n is the number of vertices.Given that there are six diagonals drawn from one vertex, I set up the equation 6 = n - 3.To solve for n, I add 3 to both sides of the equation: 6 + 3 = n.This simplifies to n = 9.Therefore, the polygon has 9 vertices.

💡First, I need to determine how many job applications Carly sent to companies in her state. Let's denote this number as ( x ).According to the problem, Carly sent twice as many applications to companies in other states. Therefore, the number of applications sent to other states is ( 2x ).The total number of applications sent is the sum of the applications sent to companies in her state and other states, which is given as 600. This can be expressed as:[x + 2x = 600]Combining like terms, we get:[3x = 600]To find the value of ( x ), we divide both sides of the equation by 3:[x = frac{600}{3} = 200]Therefore, Carly sent 200 job applications to companies in her state.

💡Okay, so I have this problem here from the 2014 Yangzhou Final Exam. It says: If ( a_1 < a_2 ) and ( b_1 < b_2 ), then what is the size relationship between ( a_1b_1 + a_2b_2 ) and ( a_1b_2 + a_2b_1 )? Hmm, interesting. I need to figure out whether ( a_1b_1 + a_2b_2 ) is greater than, less than, or equal to ( a_1b_2 + a_2b_1 ).Let me start by writing down what I know. We have two pairs of numbers: ( a_1 < a_2 ) and ( b_1 < b_2 ). So, ( a_1 ) is less than ( a_2 ), and ( b_1 ) is less than ( b_2 ). I need to compare two expressions: one is ( a_1b_1 + a_2b_2 ) and the other is ( a_1b_2 + a_2b_1 ).Maybe I can subtract one expression from the other to see which one is larger. Let me try that. So, let's compute ( (a_1b_1 + a_2b_2) - (a_1b_2 + a_2b_1) ). Simplifying this, I get:( a_1b_1 + a_2b_2 - a_1b_2 - a_2b_1 ).Hmm, let's factor this expression. I notice that both ( a_1 ) and ( a_2 ) are multiplied by ( b_1 ) and ( b_2 ). Maybe I can factor out ( a_1 ) and ( a_2 ) or ( b_1 ) and ( b_2 ). Let me try factoring ( a_1 ) and ( a_2 ):( a_1(b_1 - b_2) + a_2(b_2 - b_1) ).Wait, that simplifies to:( a_1(b_1 - b_2) - a_2(b_1 - b_2) ).Now, I can factor out ( (b_1 - b_2) ):( (a_1 - a_2)(b_1 - b_2) ).Oh, that's interesting. So, the difference between the two expressions is ( (a_1 - a_2)(b_1 - b_2) ). Now, since ( a_1 < a_2 ), that means ( a_1 - a_2 ) is negative. Similarly, ( b_1 < b_2 ) implies ( b_1 - b_2 ) is also negative. So, multiplying two negative numbers gives a positive result. Therefore, ( (a_1 - a_2)(b_1 - b_2) > 0 ).This tells me that ( (a_1b_1 + a_2b_2) - (a_1b_2 + a_2b_1) > 0 ), which means ( a_1b_1 + a_2b_2 > a_1b_2 + a_2b_1 ).Wait, let me double-check that. If I have ( a_1 < a_2 ) and ( b_1 < b_2 ), then arranging the smaller numbers together and the larger numbers together gives a larger sum. That makes sense because when you pair the smaller ( a ) with the smaller ( b ) and the larger ( a ) with the larger ( b ), you get a bigger total than if you mix them. It's like the rearrangement inequality, where the sum is maximized when both sequences are similarly ordered.So, yeah, I think my reasoning is correct. The difference is positive, so ( a_1b_1 + a_2b_2 ) is greater than ( a_1b_2 + a_2b_1 ).**Final Answer**The size relationship is boxed{a_1b_1 + a_2b_2 > a_1b_2 + a_2b_1}.

💡First, calculate the distance A has traveled in 5 hours at a speed of 10 km/h:[text{Distance}_A = 10 , text{km/h} times 5 , text{hours} = 50 , text{km}]Next, determine the relative speed of B with respect to A:[text{Relative speed} = 20 , text{km/h} - 10 , text{km/h} = 10 , text{km/h}]Calculate the time it takes for B to catch up to A:[text{Time} = frac{text{Distance}_A}{text{Relative speed}} = frac{50 , text{km}}{10 , text{km/h}} = 5 , text{hours}]Finally, find the distance B travels in 5 hours:[text{Distance}_B = 20 , text{km/h} times 5 , text{hours} = 100 , text{km}]