💡First, I need to determine the condition for two lines to be parallel. Two lines are parallel if their slopes are equal.I will rewrite both equations in slope-intercept form ( y = mx + b ) to identify their slopes.For ( l_1 ):[a^2 x - y + a^2 - 3a = 0 -y = -a^2 x - a^2 + 3a y = a^2 x + a^2 - 3a]The slope of ( l_1 ) is ( a^2 ).For ( l_2 ):[(4a - 3)x - y - 2 = 0 -y = -(4a - 3)x + 2 y = (4a - 3)x - 2]The slope of ( l_2 ) is ( 4a - 3 ).To find when ( l_1 ) is parallel to ( l_2 ), I set their slopes equal:[a^2 = 4a - 3]Next, I solve this quadratic equation:[a^2 - 4a + 3 = 0]Factoring the quadratic:[(a - 1)(a - 3) = 0]So, ( a = 1 ) or ( a = 3 ).I need to check if these values make the lines parallel and not coincident.For ( a = 1 ):[l_1: y = x + 1 - 3 = x - 2 l_2: y = (4(1) - 3)x - 2 = x - 2]Both lines are identical, so they are coincident, not just parallel.For ( a = 3 ):[l_1: y = 9x + 9 - 9 = 9x l_2: y = (4(3) - 3)x - 2 = 9x - 2]The lines have the same slope but different y-intercepts, so they are parallel and not coincident.Therefore, the correct value of ( a ) is 3.

💡Okay, so I have this problem where I need to find the value of y such that the vector a, which is (2, 6), is parallel to the vector b, which is (1, -1 + y). Hmm, I remember that if two vectors are parallel, one must be a scalar multiple of the other. That means there's some number k where a = k * b.Alright, let me write that down. So, vector a is (2, 6) and vector b is (1, -1 + y). If a is parallel to b, then there exists a scalar k such that:(2, 6) = k * (1, -1 + y)Which means that each component of a is equal to k times the corresponding component of b. So, breaking that down, I get two equations:1. 2 = k * 12. 6 = k * (-1 + y)Okay, starting with the first equation, 2 = k * 1. That should be straightforward. If I solve for k, I just divide both sides by 1, which gives me k = 2. Got that.Now, moving on to the second equation: 6 = k * (-1 + y). But I already found that k is 2, so I can substitute that in. So, replacing k with 2, the equation becomes:6 = 2 * (-1 + y)Now, I need to solve for y. Let's simplify the right side first. 2 times (-1 + y) is the same as 2*(-1) + 2*y, which is -2 + 2y. So, the equation is now:6 = -2 + 2yTo solve for y, I'll first add 2 to both sides to isolate the term with y. Adding 2 to both sides gives:6 + 2 = 2yWhich simplifies to:8 = 2yNow, to solve for y, I divide both sides by 2:8 / 2 = ySo, y = 4.Wait, let me double-check that. If y is 4, then vector b becomes (1, -1 + 4) which is (1, 3). Then, if I multiply vector b by 2, I get (2, 6), which is exactly vector a. That makes sense because they are parallel. So, y must be 4.I think that's correct. I don't see any mistakes in my steps. I found k by using the first component, then substituted it into the second equation to find y. Everything checks out when I plug y back into vector b and verify that it's a scalar multiple of a.Yeah, I feel confident that y is 4.

💡Alright, so I have this problem about Geely Company producing three different car models: "Emgrand", "King Kong", and "Freedom Ship". The production volumes are 1600, 6000, and 2000 units respectively. They want to inspect the quality of their products by selecting 48 cars from these three models. The question is asking how many cars should be selected from each model in sequence, and the options are A: 16, 16, 16; B: 8, 30, 10; C: 4, 33, 11; D: 12, 27, 9.Okay, so first, I need to figure out the best way to select these 48 cars. Since there are three different models with different production volumes, I think this is a case for stratified sampling. Stratified sampling means dividing the population into subgroups (or strata) and then sampling from each subgroup proportionally. This ensures that each subgroup is adequately represented in the sample.So, the total number of cars produced is 1600 + 6000 + 2000. Let me add that up: 1600 + 6000 is 7600, and then +2000 makes 9600 cars in total. They want to inspect 48 cars out of these 9600. That means the sampling ratio is 48/9600. Let me calculate that: 48 divided by 9600. Hmm, 48 divided by 96 is 0.5, so 48 divided by 9600 is 0.005. So, the sampling ratio is 0.005, which is the same as 1/200.Now, I need to apply this ratio to each model's production volume to find out how many cars should be selected from each. Starting with the "Emgrand" model, which has a production volume of 1600 units. Applying the ratio: 1600 * (1/200) = 8. So, 8 cars should be selected from the "Emgrand" model.Next, the "King Kong" model has a production volume of 6000 units. Applying the same ratio: 6000 * (1/200) = 30. So, 30 cars should be selected from the "King Kong" model.Lastly, the "Freedom Ship" model has a production volume of 2000 units. Applying the ratio: 2000 * (1/200) = 10. So, 10 cars should be selected from the "Freedom Ship" model.Adding these up: 8 + 30 + 10 equals 48, which matches the total number of cars to be inspected. That seems to check out.Looking back at the options: A is 16,16,16, which would mean an equal number from each model, but since the production volumes are different, this doesn't seem right. B is 8,30,10, which matches what I calculated. C is 4,33,11, which doesn't add up correctly and doesn't match the ratio. D is 12,27,9, which also doesn't align with the ratio.So, the correct answer should be option B: 8,30,10.

💡First, I need to determine the total number of tickets purchased by adding the number of adult tickets and children's tickets together.Next, I'll check which discount applies based on the total number of tickets. If the total is between 10-12 tickets, a 10% discount applies. If it's between 13-15 tickets, a 15% discount applies.After identifying the applicable discount, I'll calculate the total cost of adult tickets and children's tickets separately by multiplying the number of each type of ticket by their respective prices.Then, I'll apply the discount to the total cost of all tickets to find the discounted total cost.Finally, I'll find the difference between the total cost of adult tickets and children's tickets by subtracting the total cost of children's tickets from the total cost of adult tickets.

💡Okay, so I have this trigonometry problem here: If cos α minus sin α equals tan α, and α is between 0 and π/2, then I need to figure out which interval α falls into. The options are A: (0, π/6), B: (π/6, π/4), C: (π/4, π/3), and D: (π/3, π/2). Hmm, let me try to work through this step by step.First, the equation given is cos α - sin α = tan α. I know that tan α is sin α over cos α, so maybe I can rewrite the equation in terms of sine and cosine only. Let me write that down:cos α - sin α = (sin α)/(cos α)Hmm, okay. Maybe I can multiply both sides by cos α to eliminate the denominator. Let me try that:(cos α - sin α) * cos α = sin αExpanding the left side:cos² α - sin α cos α = sin αNow, let's bring all terms to one side to set the equation to zero:cos² α - sin α cos α - sin α = 0Hmm, this looks a bit complicated. Maybe I can factor something out. Let me see if I can factor sin α from the last two terms:cos² α - sin α (cos α + 1) = 0Not sure if that helps much. Maybe another approach. I remember that cos² α can be written in terms of sin² α using the Pythagorean identity: cos² α = 1 - sin² α. Let me substitute that in:1 - sin² α - sin α cos α - sin α = 0Hmm, that might not be helpful either. Maybe I should try squaring both sides of the original equation to see if that simplifies things. Let me go back to the original equation:cos α - sin α = tan αSquaring both sides:(cos α - sin α)² = tan² αExpanding the left side:cos² α - 2 sin α cos α + sin² α = tan² αI know that cos² α + sin² α = 1, so substituting that in:1 - 2 sin α cos α = tan² αBut tan² α is (sin² α)/(cos² α). So:1 - 2 sin α cos α = (sin² α)/(cos² α)Hmm, this seems a bit messy. Maybe I can express everything in terms of sin α or cos α. Let me try to multiply both sides by cos² α to eliminate the denominator:cos² α (1 - 2 sin α cos α) = sin² αExpanding the left side:cos² α - 2 sin α cos³ α = sin² αHmm, not sure if that's helpful. Maybe another approach. Let me recall that 2 sin α cos α is sin 2α. So, the equation becomes:1 - sin 2α = tan² αBut tan² α is also sec² α - 1. So:1 - sin 2α = sec² α - 1Which simplifies to:1 - sin 2α = (1/cos² α) - 1Bringing all terms to one side:1 - sin 2α - (1/cos² α) + 1 = 0Simplifying:2 - sin 2α - (1/cos² α) = 0Hmm, this seems complicated. Maybe I should try substituting t = sin α or t = cos α. Let me let t = sin α. Then cos α = sqrt(1 - t²). Let's see:Original equation: cos α - sin α = tan αSo:sqrt(1 - t²) - t = t / sqrt(1 - t²)Multiply both sides by sqrt(1 - t²):(1 - t²) - t sqrt(1 - t²) = tHmm, this is getting more complicated. Maybe another substitution. Let me try to let u = α. Not sure. Alternatively, maybe I can use the fact that tan α = sin α / cos α and write the equation as:cos α - sin α = sin α / cos αMultiply both sides by cos α:cos² α - sin α cos α = sin αBring all terms to one side:cos² α - sin α cos α - sin α = 0Factor sin α:cos² α - sin α (cos α + 1) = 0Hmm, still not helpful. Maybe I can express cos² α as 1 - sin² α:1 - sin² α - sin α (cos α + 1) = 0This seems to be going in circles. Maybe I should try plugging in the values from the options to see which interval satisfies the equation. Let's test the midpoints of each interval.Option A: (0, π/6). Midpoint is π/12 ≈ 0.2618 radians.Compute cos(π/12) ≈ 0.9659, sin(π/12) ≈ 0.2588, tan(π/12) ≈ 0.2679.Left side: 0.9659 - 0.2588 ≈ 0.7071Right side: 0.2679Not equal. Hmm, maybe not A.Option B: (π/6, π/4). Midpoint is π/12 + π/24 = 5π/24 ≈ 0.6545 radians.Compute cos(5π/24) ≈ 0.7939, sin(5π/24) ≈ 0.6088, tan(5π/24) ≈ 0.7673.Left side: 0.7939 - 0.6088 ≈ 0.1851Right side: 0.7673Not equal. Hmm.Option C: (π/4, π/3). Midpoint is π/4 + π/12 = π/3 ≈ 1.0472 radians.Wait, midpoint would be (π/4 + π/3)/2 = (3π/12 + 4π/12)/2 = 7π/24 ≈ 0.9163 radians.Compute cos(7π/24) ≈ 0.6088, sin(7π/24) ≈ 0.7939, tan(7π/24) ≈ 1.307.Left side: 0.6088 - 0.7939 ≈ -0.1851Right side: 1.307Not equal.Option D: (π/3, π/2). Midpoint is π/3 + π/12 = 5π/12 ≈ 1.308 radians.Compute cos(5π/12) ≈ 0.2588, sin(5π/12) ≈ 0.9659, tan(5π/12) ≈ 3.732.Left side: 0.2588 - 0.9659 ≈ -0.7071Right side: 3.732Not equal.Hmm, none of the midpoints satisfy the equation. Maybe I need a different approach.Let me go back to the squared equation:1 - sin 2α = tan² αBut tan² α = (sin² α)/(cos² α). So:1 - sin 2α = (sin² α)/(cos² α)Let me express everything in terms of sin 2α. Since sin 2α = 2 sin α cos α, and cos² α = 1 - sin² α.Wait, maybe I can let t = sin 2α. Then, since 0 < α < π/2, 0 < 2α < π, so 0 < t ≤ 1.From the equation:1 - t = (sin² α)/(cos² α)But sin² α = (1 - cos 2α)/2, and cos² α = (1 + cos 2α)/2.So:1 - t = [(1 - cos 2α)/2] / [(1 + cos 2α)/2] = (1 - cos 2α)/(1 + cos 2α)Let me denote u = cos 2α. Then:1 - t = (1 - u)/(1 + u)But t = sin 2α, and u = cos 2α. Since t² + u² = 1.So:1 - sin 2α = (1 - cos 2α)/(1 + cos 2α)Let me write this as:1 - t = (1 - u)/(1 + u)Cross-multiplying:(1 - t)(1 + u) = 1 - uExpanding:1 + u - t - t u = 1 - uSimplify:1 + u - t - t u = 1 - uSubtract 1 from both sides:u - t - t u = -uBring all terms to left:u - t - t u + u = 0Combine like terms:2u - t - t u = 0Factor t:2u - t(1 + u) = 0But t = sin 2α and u = cos 2α, so:2 cos 2α - sin 2α (1 + cos 2α) = 0Hmm, this seems complicated. Maybe another approach. Let me consider the original equation again:cos α - sin α = tan αLet me divide both sides by cos α:1 - tan α = tan α / cos αWait, that might not help. Alternatively, let me write tan α as sin α / cos α:cos α - sin α = sin α / cos αMultiply both sides by cos α:cos² α - sin α cos α = sin αBring all terms to one side:cos² α - sin α cos α - sin α = 0Factor sin α:cos² α - sin α (cos α + 1) = 0Hmm, still stuck. Maybe I can express cos² α as 1 - sin² α:1 - sin² α - sin α (cos α + 1) = 0This seems messy. Maybe I should try to express everything in terms of sin α. Let me let t = sin α. Then cos α = sqrt(1 - t²). Substitute into the equation:sqrt(1 - t²) - t = t / sqrt(1 - t²)Multiply both sides by sqrt(1 - t²):1 - t² - t sqrt(1 - t²) = tBring all terms to one side:1 - t² - t sqrt(1 - t²) - t = 0This is a complicated equation. Maybe I can square both sides again, but that might introduce extraneous solutions. Let me try:Let me rearrange the equation:1 - t² - t = t sqrt(1 - t²)Square both sides:(1 - t² - t)² = t² (1 - t²)Expand the left side:(1 - t - t²)² = 1 - 2t + t² - 2t² + 2t³ + t⁴ = 1 - 2t - t² + 2t³ + t⁴Right side: t² - t⁴So:1 - 2t - t² + 2t³ + t⁴ = t² - t⁴Bring all terms to left:1 - 2t - t² + 2t³ + t⁴ - t² + t⁴ = 0Combine like terms:1 - 2t - 2t² + 2t³ + 2t⁴ = 0This is a quartic equation: 2t⁴ + 2t³ - 2t² - 2t + 1 = 0Hmm, solving quartic equations is complicated. Maybe I can factor it. Let me try to factor by grouping:2t⁴ + 2t³ - 2t² - 2t + 1Group terms:(2t⁴ + 2t³) + (-2t² - 2t) + 1Factor:2t³(t + 1) - 2t(t + 1) + 1Hmm, not helpful. Maybe try rational roots. Possible rational roots are ±1, ±1/2.Test t = 1: 2 + 2 - 2 - 2 + 1 = 1 ≠ 0t = -1: 2 - 2 - 2 + 2 + 1 = 1 ≠ 0t = 1/2: 2*(1/16) + 2*(1/8) - 2*(1/4) - 2*(1/2) + 1 = 1/8 + 1/4 - 1/2 - 1 + 1 = 1/8 + 1/4 - 1/2 = (1 + 2 - 4)/8 = -1/8 ≠ 0t = -1/2: 2*(1/16) + 2*(-1/8) - 2*(1/4) - 2*(-1/2) + 1 = 1/8 - 1/4 - 1/2 + 1 + 1 = 1/8 - 2/8 - 4/8 + 8/8 + 8/8 = (1 - 2 - 4 + 8 + 8)/8 = 11/8 ≠ 0No rational roots. Maybe I need to use substitution. Let me let u = t + 1/(2t). Hmm, not sure. Alternatively, maybe use substitution z = t². Let me see:2t⁴ + 2t³ - 2t² - 2t + 1 = 0Let z = t², then t⁴ = z², t³ = t z, t = sqrt(z). Hmm, not helpful.Alternatively, maybe factor as quadratic in t²:2t⁴ + 2t³ - 2t² - 2t + 1 = 0Not sure. Maybe this approach is too complicated. Let me try another method.Going back to the original equation:cos α - sin α = tan αLet me write tan α as sin α / cos α:cos α - sin α = sin α / cos αMultiply both sides by cos α:cos² α - sin α cos α = sin αBring all terms to one side:cos² α - sin α cos α - sin α = 0Factor sin α:cos² α - sin α (cos α + 1) = 0Hmm, maybe express cos² α as 1 - sin² α:1 - sin² α - sin α (cos α + 1) = 0Still messy. Maybe I can write cos α in terms of sin α. Let me let t = sin α, so cos α = sqrt(1 - t²). Substitute:1 - t² - t (sqrt(1 - t²) + 1) = 0This is similar to what I had before. Maybe I can rearrange:1 - t² - t = t sqrt(1 - t²)Square both sides:(1 - t² - t)² = t² (1 - t²)Expand left side:1 - 2t + t² - 2t² + 2t³ + t⁴ = 1 - 2t - t² + 2t³ + t⁴Right side: t² - t⁴So:1 - 2t - t² + 2t³ + t⁴ = t² - t⁴Bring all terms to left:1 - 2t - t² + 2t³ + t⁴ - t² + t⁴ = 0Combine like terms:1 - 2t - 2t² + 2t³ + 2t⁴ = 0Same quartic equation as before. Maybe I can use numerical methods to approximate the solution. Let me try to estimate α.From the original equation, cos α - sin α = tan α. Let's test α = π/6 ≈ 0.5236 radians.Compute cos(π/6) ≈ 0.8660, sin(π/6) = 0.5, tan(π/6) ≈ 0.5774.Left side: 0.8660 - 0.5 = 0.3660Right side: 0.5774Not equal. So α is not π/6.Try α = π/4 ≈ 0.7854.cos(π/4) ≈ 0.7071, sin(π/4) ≈ 0.7071, tan(π/4) = 1.Left side: 0.7071 - 0.7071 = 0Right side: 1Not equal.Try α = π/3 ≈ 1.0472.cos(π/3) = 0.5, sin(π/3) ≈ 0.8660, tan(π/3) ≈ 1.732.Left side: 0.5 - 0.8660 ≈ -0.3660Right side: 1.732Not equal.Hmm, seems like the solution is somewhere between 0 and π/6, because at α=0, cos α - sin α = 1, and tan α=0, so 1=0 is not true. Wait, but as α approaches 0, cos α approaches 1, sin α approaches 0, so left side approaches 1, right side approaches 0. So they don't meet there.Wait, maybe the solution is between π/6 and π/4. Let me try α=π/6 + π/24=5π/24≈0.6545.Compute cos(5π/24)≈0.7939, sin(5π/24)≈0.6088, tan(5π/24)≈0.7673.Left side: 0.7939 - 0.6088≈0.1851Right side: 0.7673Not equal. Hmm.Wait, maybe I need to consider that when I squared the equation, I might have introduced extraneous solutions. So even if I find a solution, I need to check it in the original equation.Alternatively, maybe I can use the fact that cos α - sin α = sqrt(2) cos(α + π/4). Let me recall that identity:cos α - sin α = sqrt(2) cos(α + π/4)So the equation becomes:sqrt(2) cos(α + π/4) = tan αHmm, interesting. Let me write that:sqrt(2) cos(α + π/4) = tan αBut tan α = sin α / cos α, so:sqrt(2) cos(α + π/4) = sin α / cos αMultiply both sides by cos α:sqrt(2) cos α cos(α + π/4) = sin αHmm, maybe I can expand cos(α + π/4):cos(α + π/4) = cos α cos π/4 - sin α sin π/4 = (cos α - sin α)/sqrt(2)So substitute back:sqrt(2) cos α * (cos α - sin α)/sqrt(2) = sin αSimplify:cos α (cos α - sin α) = sin αWhich is the same as the original equation. Hmm, not helpful.Wait, maybe I can write the equation as:sqrt(2) cos(α + π/4) = tan αLet me let β = α + π/4. Then α = β - π/4. Substitute:sqrt(2) cos β = tan(β - π/4)But tan(β - π/4) = (tan β - 1)/(1 + tan β)Hmm, complicated. Maybe not helpful.Alternatively, let me consider the equation:sqrt(2) cos(α + π/4) = tan αLet me write tan α as sin α / cos α:sqrt(2) cos(α + π/4) = sin α / cos αMultiply both sides by cos α:sqrt(2) cos α cos(α + π/4) = sin αBut cos α cos(α + π/4) can be expressed using product-to-sum identities:cos A cos B = [cos(A+B) + cos(A-B)] / 2So:sqrt(2) * [cos(2α + π/4) + cos(-π/4)] / 2 = sin αSimplify:sqrt(2)/2 [cos(2α + π/4) + cos(π/4)] = sin αSince cos(-π/4) = cos(π/4) = sqrt(2)/2.So:sqrt(2)/2 [cos(2α + π/4) + sqrt(2)/2] = sin αSimplify:[sqrt(2)/2 * cos(2α + π/4)] + [sqrt(2)/2 * sqrt(2)/2] = sin αWhich is:[sqrt(2)/2 cos(2α + π/4)] + (2/4) = sin αSimplify:[sqrt(2)/2 cos(2α + π/4)] + 1/2 = sin αHmm, still complicated. Maybe I can express cos(2α + π/4) in terms of sin α.Alternatively, let me consider that 2α + π/4 = γ, so α = (γ - π/4)/2. Substitute into the equation:sqrt(2)/2 cos γ + 1/2 = sin[(γ - π/4)/2]This seems too involved. Maybe another approach.Let me go back to the squared equation:1 - sin 2α = tan² αBut tan² α = (1 - cos 2α)/(1 + cos 2α)So:1 - sin 2α = (1 - cos 2α)/(1 + cos 2α)Let me let u = cos 2α. Then sin 2α = sqrt(1 - u²). But since 0 < α < π/2, 0 < 2α < π, so sin 2α is positive.So:1 - sqrt(1 - u²) = (1 - u)/(1 + u)Let me solve for u:1 - sqrt(1 - u²) = (1 - u)/(1 + u)Multiply both sides by (1 + u):(1 - sqrt(1 - u²))(1 + u) = 1 - uExpand left side:1*(1 + u) - sqrt(1 - u²)*(1 + u) = 1 - uSo:1 + u - sqrt(1 - u²)(1 + u) = 1 - uBring 1 to the other side:u - sqrt(1 - u²)(1 + u) = -uBring all terms to left:u - sqrt(1 - u²)(1 + u) + u = 0Combine like terms:2u - sqrt(1 - u²)(1 + u) = 0Let me isolate the square root:sqrt(1 - u²)(1 + u) = 2uSquare both sides:(1 - u²)(1 + u)² = 4u²Expand left side:(1 - u²)(1 + 2u + u²) = 4u²Multiply out:(1)(1 + 2u + u²) - u²(1 + 2u + u²) = 4u²Which is:1 + 2u + u² - u² - 2u³ - u⁴ = 4u²Simplify:1 + 2u - 2u³ - u⁴ = 4u²Bring all terms to left:1 + 2u - 2u³ - u⁴ - 4u² = 0Rearrange:-u⁴ - 2u³ - 4u² + 2u + 1 = 0Multiply both sides by -1:u⁴ + 2u³ + 4u² - 2u - 1 = 0This is a quartic equation in u. Let me try to factor it. Maybe rational roots? Possible roots are ±1.Test u=1: 1 + 2 + 4 - 2 -1 = 4 ≠0u=-1: 1 - 2 + 4 + 2 -1 =4≠0No rational roots. Maybe factor as quadratic in u²:u⁴ + 2u³ + 4u² - 2u -1 =0Not obvious. Maybe use substitution v = u + 1/u. Not sure. Alternatively, use numerical methods.Let me try u=0.5:0.0625 + 0.25 + 1 -1 -1= -0.6875≠0u=0.6:0.1296 + 0.432 + 1.44 -1.2 -1= -0.2084u=0.7:0.2401 + 0.686 + 1.96 -1.4 -1=0.4861So between u=0.6 and u=0.7, function crosses zero.Using linear approximation:At u=0.6, f= -0.2084At u=0.7, f=0.4861Slope: (0.4861 - (-0.2084))/(0.7 -0.6)=0.6945/0.1=6.945To reach zero from u=0.6: need delta u=0.2084 /6.945≈0.03So approximate root at u≈0.63Thus, u≈0.63, which is cos 2α≈0.63So 2α≈arccos(0.63)≈0.8976 radians≈51.4 degreesThus, α≈0.8976/2≈0.4488 radians≈25.7 degreesWhich is approximately 0.4488 radians, which is between π/6≈0.5236 and π/4≈0.7854? Wait, no, 0.4488 is less than π/6≈0.5236. Wait, π/6 is about 0.5236, so 0.4488 is less than that, so α≈0.4488 is in (0, π/6). So the answer should be A.Wait, but earlier when I tested α=π/12≈0.2618, the left side was 0.7071 and right side was 0.2679. At α≈0.4488, let me compute:cos(0.4488)≈0.9000, sin(0.4488)≈0.4345, tan(0.4488)≈0.4800Left side: 0.9000 - 0.4345≈0.4655Right side:0.4800Close, but not exact. Maybe my approximation was rough. Let me try u=0.63:cos 2α=0.63, so 2α=arccos(0.63)=0.8976, α≈0.4488Compute cos(0.4488)=sqrt((1 + cos(0.8976))/2)=sqrt((1 +0.63)/2)=sqrt(0.815)=≈0.903sin(0.4488)=sqrt((1 - cos(0.8976))/2)=sqrt((1 -0.63)/2)=sqrt(0.185)=≈0.430tan(0.4488)=sin(0.4488)/cos(0.4488)=≈0.430/0.903≈0.476Left side: cos α - sin α≈0.903 -0.430≈0.473Right side: tan α≈0.476Close enough, considering the approximation. So α≈0.4488, which is less than π/6≈0.5236. So α is in (0, π/6). Therefore, the answer is A.

💡Okay, so I have this problem where there are eight cards, each labeled with a natural number from 1 to 8. I need to figure out how many different ways I can draw three cards such that the sum of their numbers is exactly 9. Hmm, let me think about how to approach this.First, I know that natural numbers are just the positive integers starting from 1, so the cards are numbered 1 through 8. I need to pick three distinct numbers from this set, and their sum should be 9. Since the order in which I draw the cards doesn't matter, this is a combination problem.I guess the first step is to list all possible combinations of three numbers from 1 to 8 and then check which ones add up to 9. But that sounds like it might take a while because there are quite a few combinations. Maybe there's a smarter way to do this without listing all possibilities.Let me think about the smallest and largest possible sums. The smallest sum I can get by adding three distinct numbers from 1 to 8 is 1 + 2 + 3 = 6. The largest sum is 6 + 7 + 8 = 21. So, 9 is somewhere in the lower end of possible sums. That might mean there aren't too many combinations that add up to 9.Maybe I can start by fixing the smallest number and then see what pairs of larger numbers can add up to the remaining value. Let's try that.If I fix the smallest number as 1, then I need two other numbers that add up to 9 - 1 = 8. So, I need two distinct numbers from 2 to 8 that add up to 8. Let's list those pairs:- 2 and 6 (because 2 + 6 = 8)- 3 and 5 (because 3 + 5 = 8)- 4 and 4 (but they have to be distinct, so this doesn't work)So, with 1 as the smallest number, I have two valid combinations: (1, 2, 6) and (1, 3, 5).Next, let's fix the smallest number as 2. Then, the remaining two numbers need to add up to 9 - 2 = 7. Let's find pairs from 3 to 8 that add up to 7:- 3 and 4 (because 3 + 4 = 7)That's the only pair because 5 and 2 would be out of order, and we already have 2 as the smallest number. So, the combination here is (2, 3, 4).Now, if I fix the smallest number as 3, the remaining two numbers need to add up to 9 - 3 = 6. Let's see if there are any pairs from 4 to 8 that add up to 6:- 4 and 2 (but 2 is smaller than 3, which is our smallest number here)- 5 and 1 (again, 1 is smaller than 3)So, there are no valid pairs here because any pair adding up to 6 would require at least one number smaller than 3, which we've already fixed as the smallest number. Therefore, there are no valid combinations starting with 3.Similarly, if I try starting with 4 as the smallest number, the remaining two numbers would need to add up to 9 - 4 = 5. But the smallest numbers we can use are 5 and 6, which already add up to more than 5. So, no combinations here either.Since starting with numbers larger than 3 doesn't give us any valid combinations, I can stop here.So, compiling all the valid combinations I found:1. (1, 2, 6)2. (1, 3, 5)3. (2, 3, 4)That gives me a total of three different ways to draw three cards that add up to 9.Wait, let me double-check to make sure I didn't miss any combinations. Maybe I should list all possible combinations of three numbers and see which ones sum to 9.The total number of ways to choose 3 numbers from 8 is C(8,3) = 56. That's a lot, but maybe I can list them systematically.Starting with 1:- 1, 2, 6- 1, 3, 5- 1, 4, 4 (invalid because duplicates)- 1, 2, 5 (sum is 8)- 1, 2, 7 (sum is 10)- 1, 3, 4 (sum is 8)- 1, 3, 6 (sum is 10)- 1, 4, 5 (sum is 10)- 1, 2, 8 (sum is 11)- 1, 3, 7 (sum is 11)- 1, 4, 6 (sum is 11)- 1, 5, 6 (sum is 12)Okay, so only (1,2,6) and (1,3,5) sum to 9.Now starting with 2:- 2, 3, 4 (sum is 9)- 2, 3, 5 (sum is 10)- 2, 3, 6 (sum is 11)- 2, 3, 7 (sum is 12)- 2, 3, 8 (sum is 13)- 2, 4, 5 (sum is 11)- 2, 4, 6 (sum is 12)- 2, 4, 7 (sum is 13)- 2, 4, 8 (sum is 14)- 2, 5, 6 (sum is 13)- 2, 5, 7 (sum is 14)- 2, 5, 8 (sum is 15)- 2, 6, 7 (sum is 15)- 2, 6, 8 (sum is 16)- 2, 7, 8 (sum is 17)Only (2,3,4) sums to 9.Starting with 3:- 3, 4, 5 (sum is 12)- 3, 4, 6 (sum is 13)- 3, 4, 7 (sum is 14)- 3, 4, 8 (sum is 15)- 3, 5, 6 (sum is 14)- 3, 5, 7 (sum is 15)- 3, 5, 8 (sum is 16)- 3, 6, 7 (sum is 16)- 3, 6, 8 (sum is 17)- 3, 7, 8 (sum is 18)None of these sum to 9.Starting with 4:- 4, 5, 6 (sum is 15)- 4, 5, 7 (sum is 16)- 4, 5, 8 (sum is 17)- 4, 6, 7 (sum is 17)- 4, 6, 8 (sum is 18)- 4, 7, 8 (sum is 19)No sums of 9 here.Similarly, starting with 5, 6, 7, or 8 will only give larger sums, so no need to check further.So, after listing all possible combinations, I still only find three valid ones: (1,2,6), (1,3,5), and (2,3,4). That confirms my earlier result.I think I'm confident now that there are exactly three different ways to draw three cards that add up to 9.