💡Okay, so I have this function f(x) = 2√3 cos²x + sin2x - √3, and I need to figure out which of the given conclusions A, B, C, D are correct. Let me take it step by step.First, I think it might help to simplify the function f(x). I remember there are some trigonometric identities that could be useful here. For example, the double-angle identity for cosine: cos²x = (1 + cos2x)/2. Let me try applying that.So, substituting cos²x with (1 + cos2x)/2, the function becomes:f(x) = 2√3 * (1 + cos2x)/2 + sin2x - √3Simplifying that, the 2 in the numerator and denominator cancels out:f(x) = √3(1 + cos2x) + sin2x - √3Now, distribute the √3:f(x) = √3 + √3 cos2x + sin2x - √3The √3 and -√3 cancel each other out:f(x) = √3 cos2x + sin2xHmm, that's simpler. Now, I have f(x) = √3 cos2x + sin2x. This looks like a combination of sine and cosine functions with the same argument, 2x. I think I can express this as a single sine or cosine function using the amplitude-phase form. The general formula is A cosθ + B sinθ = C sin(θ + φ), where C = √(A² + B²) and tanφ = A/B.Let me compute the amplitude C:C = √[(√3)² + (1)²] = √[3 + 1] = √4 = 2Okay, so the amplitude is 2. Now, let's find the phase shift φ:tanφ = (√3)/1 = √3So, φ = arctan(√3). I remember that tan(π/3) = √3, so φ = π/3.Therefore, f(x) can be written as:f(x) = 2 sin(2x + π/3)Wait, let me double-check that. The formula is A cosθ + B sinθ = C sin(θ + φ). So, expanding 2 sin(2x + π/3):2 sin(2x + π/3) = 2 [sin2x cos(π/3) + cos2x sin(π/3)]We know that cos(π/3) = 1/2 and sin(π/3) = √3/2, so:= 2 [sin2x*(1/2) + cos2x*(√3/2)]= 2*(1/2 sin2x + √3/2 cos2x)= sin2x + √3 cos2xWhich matches our simplified f(x). Great, so f(x) = 2 sin(2x + π/3).Now, let's analyze each option one by one.**Option A: The smallest positive period of f(x) is π.**Since f(x) is expressed as 2 sin(2x + π/3), the period of sin(kx + φ) is 2π/k. Here, k = 2, so the period is 2π/2 = π. So, yes, the smallest positive period is π. Therefore, option A is correct.**Option B: The graph of f(x) is symmetric about the line x = π/6.**Symmetry about a vertical line x = a means that f(a + h) = f(a - h) for any h. Let's test this with a = π/6.Compute f(π/6 + h) and f(π/6 - h):f(π/6 + h) = 2 sin[2(π/6 + h) + π/3] = 2 sin(π/3 + 2h + π/3) = 2 sin(2π/3 + 2h)f(π/6 - h) = 2 sin[2(π/6 - h) + π/3] = 2 sin(π/3 - 2h + π/3) = 2 sin(2π/3 - 2h)Now, let's see if these are equal:Is 2 sin(2π/3 + 2h) equal to 2 sin(2π/3 - 2h)?Using the identity sin(A + B) = sinA cosB + cosA sinB and sin(A - B) = sinA cosB - cosA sinB.So,sin(2π/3 + 2h) = sin(2π/3)cos(2h) + cos(2π/3)sin(2h)sin(2π/3 - 2h) = sin(2π/3)cos(2h) - cos(2π/3)sin(2h)Subtracting these two:sin(2π/3 + 2h) - sin(2π/3 - 2h) = 2 cos(2π/3) sin(2h)Since cos(2π/3) = -1/2, this becomes:= 2*(-1/2)*sin(2h) = -sin(2h)Which is not zero unless sin(2h) = 0. Therefore, f(π/6 + h) ≠ f(π/6 - h) in general. So, the function is not symmetric about x = π/6. Therefore, option B is incorrect.**Option C: One zero of f(x) is -π/6.**A zero of f(x) is a value x where f(x) = 0. Let's compute f(-π/6):f(-π/6) = 2 sin[2*(-π/6) + π/3] = 2 sin(-π/3 + π/3) = 2 sin(0) = 0So, yes, f(-π/6) = 0. Therefore, -π/6 is indeed a zero of f(x). Option C is correct.**Option D: The maximum value of f(x) is √3 + 1.**Since f(x) = 2 sin(2x + π/3), the maximum value of sin is 1, so the maximum value of f(x) is 2*1 = 2. √3 + 1 is approximately 1.732 + 1 = 2.732, which is greater than 2. Therefore, the maximum value is not √3 + 1. Option D is incorrect.So, summarizing:- A: Correct- B: Incorrect- C: Correct- D: IncorrectTherefore, the correct conclusions are A and C.**Final Answer**The correct options are boxed{A} and boxed{C}.

💡Alright, so I have this problem here with a function f(x) = A cos(x/4 + π/6). They told me that when x is π/3, f(x) is √2. I need to find A first. Hmm, okay, let's see.So, plugging x = π/3 into the function, I get f(π/3) = A cos((π/3)/4 + π/6). Let me compute the argument inside the cosine first. (π/3)/4 is π/12, and adding π/6 to that. Wait, π/6 is the same as 2π/12, right? So π/12 + 2π/12 is 3π/12, which simplifies to π/4. So, f(π/3) = A cos(π/4). I know that cos(π/4) is √2/2, so this becomes A*(√2/2) = √2. To solve for A, I can set up the equation: A*(√2/2) = √2. If I divide both sides by √2, I get A/2 = 1, so A must be 2. Okay, that wasn't too bad. So, part (1) is done, A is 2.Now, moving on to part (2). They give me two more equations involving f evaluated at different points, and I need to find cos(α + β). Let me write down what's given.First, f(4α + 4π/3) = -30/17. Since I already found A is 2, the function is f(x) = 2 cos(x/4 + π/6). So, plugging in 4α + 4π/3 into f(x), I get:f(4α + 4π/3) = 2 cos((4α + 4π/3)/4 + π/6). Let me simplify the argument inside the cosine. Dividing 4α by 4 gives α, and 4π/3 divided by 4 is π/3. So, the argument becomes α + π/3 + π/6. Adding π/3 and π/6 together: π/3 is 2π/6, so 2π/6 + π/6 is 3π/6, which is π/2. So, the argument is α + π/2. Therefore, f(4α + 4π/3) = 2 cos(α + π/2). I remember that cos(θ + π/2) is equal to -sin θ. So, this becomes 2*(-sin α) = -2 sin α. They told me this equals -30/17. So, -2 sin α = -30/17. Dividing both sides by -2, I get sin α = 15/17.Okay, so sin α is 15/17. Since α is between 0 and π/2, which is the first quadrant, all trigonometric functions are positive here. So, cos α should be positive as well. I can find cos α using the Pythagorean identity: sin² α + cos² α = 1. So, cos² α = 1 - sin² α = 1 - (225/289) = (289 - 225)/289 = 64/289. Taking the square root, cos α = 8/17.Alright, moving on to the second equation: f(4β - 2π/3) = 8/5. Again, using the function f(x) = 2 cos(x/4 + π/6). Plugging in 4β - 2π/3, we get:f(4β - 2π/3) = 2 cos((4β - 2π/3)/4 + π/6). Let's simplify the argument inside the cosine. Dividing 4β by 4 gives β, and -2π/3 divided by 4 is -π/6. So, the argument becomes β - π/6 + π/6. The -π/6 and +π/6 cancel each other out, leaving just β. So, f(4β - 2π/3) = 2 cos β. They told me this equals 8/5. Therefore, 2 cos β = 8/5. Dividing both sides by 2, I get cos β = 4/5.Again, since β is in [0, π/2], sin β will be positive. Using the Pythagorean identity, sin² β + cos² β = 1. So, sin² β = 1 - cos² β = 1 - 16/25 = 9/25. Taking the square root, sin β = 3/5.Now, I need to find cos(α + β). I remember the cosine addition formula: cos(α + β) = cos α cos β - sin α sin β. Plugging in the values I found:cos α = 8/17, cos β = 4/5, sin α = 15/17, sin β = 3/5.So, cos(α + β) = (8/17)(4/5) - (15/17)(3/5). Let me compute each term:First term: (8/17)(4/5) = 32/85.Second term: (15/17)(3/5) = 45/85.Subtracting the second term from the first: 32/85 - 45/85 = (32 - 45)/85 = (-13)/85.So, cos(α + β) is -13/85.Wait, let me double-check my calculations to make sure I didn't make any mistakes.For the first part, f(π/3) = 2 cos(π/4) = 2*(√2/2) = √2. That checks out.For the second part, f(4α + 4π/3) = 2 cos(α + π/2) = -2 sin α = -30/17, so sin α = 15/17. Then, cos α = 8/17. That seems correct.For the third part, f(4β - 2π/3) = 2 cos β = 8/5, so cos β = 4/5. Then, sin β = 3/5. That also seems correct.Applying the cosine addition formula: (8/17)(4/5) = 32/85, (15/17)(3/5) = 45/85. 32 - 45 is -13, so -13/85. That seems right.Hmm, okay, I think I did everything correctly. So, the final answers are A = 2 and cos(α + β) = -13/85.**Final Answer**(1) boxed{2}(2) boxed{-dfrac{13}{85}}

💡Alright, so I'm trying to figure out how often Jaclyn gets her interest payments from the debentures she bought. Let's see, she invested 10,000 and earns 9.5% per annum simple interest. The interest is paid at certain intervals, and she gets 237.50 each time. The debenture period is 18 months. Hmm, okay, so I need to find out the frequency of these interest payments.First, I remember that simple interest is calculated using the formula:[ text{Interest} = text{Principal} times text{Rate} times text{Time} ]Where:- Principal is the initial amount invested, which is 10,000.- Rate is the annual interest rate, which is 9.5%, or 0.095 in decimal.- Time is the duration in years. Since the period is 18 months, that's 1.5 years.So, plugging in the numbers:[ text{Interest} = 10,000 times 0.095 times 1.5 ]Let me calculate that:[ 10,000 times 0.095 = 950 ][ 950 times 1.5 = 1,425 ]So, the total interest earned over 18 months is 1,425.Now, Jaclyn receives 237.50 each time she gets paid. To find out how many payments she receives, I can divide the total interest by the amount she gets each payment:[ text{Number of payments} = frac{1,425}{237.50} ]Calculating that:[ frac{1,425}{237.50} = 6 ]So, she receives 6 payments over the 18-month period.To find out how often she gets paid, I can divide the total period by the number of payments:[ text{Frequency of payments in months} = frac{18 text{ months}}{6} = 3 text{ months} ]Therefore, the interest is paid every 3 months.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, calculating the total interest:[ 10,000 times 0.095 = 950 ][ 950 times 1.5 = 1,425 ]Yes, that seems correct.Then, dividing the total interest by the interest per payment:[ frac{1,425}{237.50} = 6 ]That also checks out.Finally, dividing the total period by the number of payments:[ frac{18}{6} = 3 ]So, the frequency is every 3 months.I think that makes sense. She's getting paid every quarter, which is a common payment interval for interest.

💡Okay, so I have this problem here about vectors. Let me read it again: Given vectors a = (-1, 2) and b = (1, x), if a is perpendicular to (a + 2b), then what is the value of the real number x? Hmm, alright, let me try to figure this out step by step.First, I remember that if two vectors are perpendicular, their dot product is zero. So, since a is perpendicular to (a + 2b), their dot product should be zero. That gives me an equation to solve for x. Cool, that seems straightforward.Let me write down what I know. Vector a is (-1, 2), and vector b is (1, x). I need to find x such that a is perpendicular to (a + 2b). So, first, maybe I should compute the vector (a + 2b). Let me do that.To add vectors, I add their corresponding components. So, a + 2b would be:a + 2b = (-1, 2) + 2*(1, x)First, I need to compute 2*(1, x). That would be (2*1, 2*x) = (2, 2x). Then, adding this to vector a:a + 2b = (-1 + 2, 2 + 2x) = (1, 2 + 2x)Okay, so the vector (a + 2b) is (1, 2 + 2x). Now, since a is perpendicular to this vector, their dot product should be zero. Let me compute the dot product of a and (a + 2b).The dot product of two vectors (a1, a2) and (b1, b2) is a1*b1 + a2*b2. So, applying that here:a · (a + 2b) = (-1)*1 + 2*(2 + 2x)Let me compute each part step by step. First, (-1)*1 is -1. Then, 2*(2 + 2x) is 4 + 4x. So, adding these together:-1 + 4 + 4x = ( -1 + 4 ) + 4x = 3 + 4xSince the dot product is zero, we have:3 + 4x = 0Now, solving for x:4x = -3x = -3/4Wait, let me double-check my steps to make sure I didn't make a mistake. First, I computed 2b correctly as (2, 2x). Then, a + 2b is (-1 + 2, 2 + 2x) which is (1, 2 + 2x). Then, the dot product with a is (-1)*1 + 2*(2 + 2x). That gives -1 + 4 + 4x, which simplifies to 3 + 4x. Setting that equal to zero gives x = -3/4. Hmm, seems correct.Just to be thorough, let me substitute x = -3/4 back into the vectors and check if they are indeed perpendicular.First, vector b becomes (1, -3/4). Then, 2b is (2, -3/2). Adding a to 2b:a + 2b = (-1, 2) + (2, -3/2) = (1, 2 - 3/2) = (1, 1/2)Now, compute the dot product of a and (a + 2b):a · (a + 2b) = (-1)*1 + 2*(1/2) = -1 + 1 = 0Perfect, the dot product is zero, which confirms that they are indeed perpendicular. So, x = -3/4 is the correct answer.I think that's all. I didn't encounter any issues in the process, and my checks confirm the solution. So, I'm confident that x is -3/4.**Final Answer**The value of ( x ) is boxed{-dfrac{3}{4}}.

💡Alright, so I've got this problem here: If ( k ) is any integer, and ( 99^3 - 99 ) is divisible by ( k ), then ( k ) cannot be which of the following options? The options are A: 50, B: 100, C: 98, and D: 97.Hmm, okay. Let me think about how to approach this. I remember that when dealing with expressions like ( a^3 - a ), there's a factoring technique that can be applied. Maybe I can factor ( 99^3 - 99 ) to see what its prime factors are, and then check which of the given options isn't a factor.So, starting with ( 99^3 - 99 ). I notice that both terms have a common factor of 99, so I can factor that out first:( 99^3 - 99 = 99(99^2 - 1) )Now, looking at ( 99^2 - 1 ), that's a difference of squares, right? Because ( 99^2 ) is ( (99)^2 ) and 1 is ( (1)^2 ). So, the difference of squares formula is ( a^2 - b^2 = (a - b)(a + b) ). Applying that here:( 99^2 - 1 = (99 - 1)(99 + 1) = 98 times 100 )So, putting it all together:( 99^3 - 99 = 99 times 98 times 100 )Okay, so now I have the expression factored into three numbers: 99, 98, and 100. That means ( 99^3 - 99 ) is divisible by each of these factors, as well as any product of these factors.Now, looking at the options given:A: 50B: 100C: 98D: 97I need to see which of these numbers is not a divisor of ( 99^3 - 99 ).First, let's check option B: 100. Well, 100 is one of the factors we have, so ( 99^3 - 99 ) is definitely divisible by 100.Option C: 98. Similarly, 98 is also one of the factors, so it divides ( 99^3 - 99 ).Option A: 50. Hmm, 50 isn't directly one of the factors, but let's see if it's a factor of any of the existing factors. 50 is half of 100, right? So, since 100 is a factor, and 50 divides 100, then 50 must also be a factor of ( 99^3 - 99 ). So, 50 divides the expression as well.Now, option D: 97. 97 is a prime number, I believe. Let me check if 97 is a factor of ( 99^3 - 99 ). Since 97 is prime, it can only be a factor if it divides one of the existing factors: 99, 98, or 100.Let's check each:- 99 divided by 97: 97 goes into 99 once with a remainder of 2, so no.- 98 divided by 97: Similarly, 97 goes into 98 once with a remainder of 1, so no.- 100 divided by 97: 97 goes into 100 once with a remainder of 3, so no.Since 97 doesn't divide any of the factors 99, 98, or 100, it can't be a factor of ( 99^3 - 99 ). Therefore, ( k ) cannot be 97.Wait, but just to be thorough, maybe I should compute ( 99^3 - 99 ) and see if 97 divides it. Let me calculate:First, ( 99^3 ) is 99 * 99 * 99. Let's compute that step by step.99 * 99 is 9801. Then, 9801 * 99. Let's compute that:9801 * 99 = 9801 * (100 - 1) = 9801 * 100 - 9801 * 1 = 980100 - 9801 = 970299.So, ( 99^3 = 970299 ). Then, ( 99^3 - 99 = 970299 - 99 = 970200 ).Now, let's check if 97 divides 970200.Dividing 970200 by 97:First, 97 * 10000 = 970000. So, 970200 - 970000 = 200.Now, 200 divided by 97 is approximately 2 with a remainder of 6. So, 97 * 10002 = 970000 + 97*2 = 970000 + 194 = 970194.Subtracting that from 970200: 970200 - 970194 = 6.So, the remainder is 6, meaning 97 does not divide 970200 exactly. Therefore, 97 is not a divisor of ( 99^3 - 99 ).Therefore, the answer should be D: 97.

💡First, let's denote the remainder when ( x ) is divided by 9 as ( r ). Therefore, we can express ( x ) as:[x = 9k + r]where ( k ) is some integer and ( 0 leq r < 9 ).Next, we know that when ( 4x ) is divided by 9, the remainder is 2. This gives us the equation:[4x = 9m + 2]where ( m ) is some integer.Substituting the expression for ( x ) into the equation for ( 4x ), we get:[4(9k + r) = 9m + 2][36k + 4r = 9m + 2]Since ( 36k ) is a multiple of 9, we can ignore it for the purpose of finding the remainder. This simplifies our equation to:[4r equiv 2 pmod{9}]Now, we need to find a value of ( r ) that satisfies this congruence. We can test values of ( r ) from 0 to 8:- If ( r = 0 ), then ( 4r = 0 ), which does not equal 2.- If ( r = 1 ), then ( 4r = 4 ), which does not equal 2.- If ( r = 2 ), then ( 4r = 8 ), which does not equal 2.- If ( r = 3 ), then ( 4r = 12 ), and ( 12 mod 9 = 3 ), which does not equal 2.- If ( r = 4 ), then ( 4r = 16 ), and ( 16 mod 9 = 7 ), which does not equal 2.- If ( r = 5 ), then ( 4r = 20 ), and ( 20 mod 9 = 2 ), which equals 2.Therefore, the remainder when ( x ) is divided by 9 is ( 5 ).