💡Okay, so I have this problem about arranging four people—A, B, C, and D—to be on duty from May 1 to May 4. Each person is assigned to one day, so it's like a permutation problem. But there are some constraints. A can only be on duty on May 1 or May 4, and A and B cannot be on duty on adjacent days. Hmm, okay, let me break this down step by step.First, let's understand the basic setup. We have four days: May 1, 2, 3, and 4. Each day needs one person, and each person is assigned to exactly one day. So, without any constraints, the number of possible arrangements would just be 4 factorial, which is 4! = 24. But we have some constraints here, so the number will be less than 24.The first constraint is that A can only be on duty on May 1 or May 4. So, A has two possible choices: either May 1 or May 4. Let's consider these two cases separately.**Case 1: A is on duty on May 1.**If A is on May 1, then we need to assign B, C, and D to May 2, 3, and 4. But there's another constraint: A and B cannot be on adjacent days. Since A is on May 1, B cannot be on May 2 because that's adjacent to May 1. So, B can only be on May 3 or May 4.Let's break this down further.- **Subcase 1a: B is on May 3.** Then, we have to assign C and D to May 2 and May 4. There are 2 people and 2 days, so that's 2! = 2 ways.- **Subcase 1b: B is on May 4.** Then, we have to assign C and D to May 2 and May 3. Again, that's 2! = 2 ways.So, for Case 1, the total number of arrangements is 2 + 2 = 4.**Case 2: A is on duty on May 4.**Similarly, if A is on May 4, then we need to assign B, C, and D to May 1, 2, and 3. Again, A and B cannot be on adjacent days. Since A is on May 4, B cannot be on May 3 because that's adjacent to May 4. So, B can only be on May 1 or May 2.Let's break this down as well.- **Subcase 2a: B is on May 1.** Then, we have to assign C and D to May 2 and May 3. That's 2! = 2 ways.- **Subcase 2b: B is on May 2.** Then, we have to assign C and D to May 1 and May 3. Again, that's 2! = 2 ways.So, for Case 2, the total number of arrangements is also 2 + 2 = 4.**Total Arrangements:**Adding up the arrangements from both cases, we get 4 (from Case 1) + 4 (from Case 2) = 8.Wait, let me double-check to make sure I haven't missed anything. Sometimes in combinatorial problems, it's easy to overlook a case or miscount.In Case 1, A is fixed on May 1, B can't be on May 2, so B has two choices: May 3 or May 4. For each choice of B, the remaining two people can be arranged in 2 ways. So, 2 * 2 = 4. That seems right.In Case 2, A is fixed on May 4, B can't be on May 3, so B has two choices: May 1 or May 2. For each choice of B, the remaining two people can be arranged in 2 ways. So, again, 2 * 2 = 4. That also seems correct.Adding them together gives 8 total arrangements. I don't see any overlaps or missed cases. So, I think 8 is the correct answer.**Final Answer**The number of possible arrangements is boxed{8}.

💡Okay, so I have this problem here about a cone. The front view of the cone is an equilateral triangle with a side length of 4. I need to find the surface area of the cone, and the options are 4π, 8π, 12π, or 16π. Hmm, let me think about how to approach this.First, I remember that when you look at a cone from the front, you essentially see a triangle. In this case, it's an equilateral triangle. An equilateral triangle has all sides equal, so each side is 4 units long. That means the height of the triangle, which is also the height of the cone, can be calculated using the properties of an equilateral triangle.Wait, actually, in the front view, the triangle's base is the diameter of the cone's base, right? So if the side length of the equilateral triangle is 4, that should correspond to the slant height of the cone. Hmm, no, wait. Let me clarify.In the front view, the triangle you see is formed by the height of the cone and the diameter of the base. But if it's an equilateral triangle, all sides are equal, so the two sides (which are the slant heights of the cone) and the base (which is the diameter) are all 4 units. So that means the diameter of the base of the cone is 4, so the radius is half of that, which is 2.Okay, so radius r = 2. Now, the slant height l of the cone is equal to the side length of the triangle, which is 4. So l = 4.Now, the surface area of a cone is given by the formula πr(r + l). So plugging in the values, we have π*2*(2 + 4) = π*2*6 = 12π. So that would be option C.Wait, let me double-check. The surface area formula is πr² + πrl, which is the same as πr(r + l). So yes, with r=2 and l=4, it's π*2*(2 + 4) = 12π. That seems right.But just to make sure, let me think about the height of the cone. Since the triangle is equilateral, all sides are 4, so the height of the triangle can be calculated using the formula for the height of an equilateral triangle, which is (√3/2)*side. So that would be (√3/2)*4 = 2√3. But wait, is that the height of the cone?Hmm, in the front view, the height of the triangle is the height of the cone. So the height h of the cone is 2√3. But do I need that for the surface area? No, because surface area only requires the radius and the slant height. So I think my initial calculation is correct.Alternatively, if I had to find the lateral surface area, it would be πrl, which is π*2*4=8π, but the total surface area includes the base, which is πr²=π*4=4π. So total surface area is 8π + 4π=12π. Yep, that's consistent.So I think the answer is 12π, which is option C.

💡Okay, so I have this problem about a hyperbola, and I need to find the range of possible values for its eccentricity given a certain condition. Let me try to break this down step by step.First, the hyperbola is given by the equation (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), where (a > 0) and (b > 0). I remember that for hyperbolas, the standard form is similar to this, and the foci are located at ((pm c, 0)), where (c^2 = a^2 + b^2). The eccentricity (e) is defined as (e = frac{c}{a}), so (e > 1) for hyperbolas.Points (F) and (E) are the left and right foci, respectively. So, (F) is at ((-c, 0)) and (E) is at ((c, 0)).Next, the problem mentions a line passing through point (F) and perpendicular to the x-axis. Since it's perpendicular to the x-axis, it must be a vertical line. A vertical line passing through (F) would have the equation (x = -c). This line intersects the hyperbola at points (A) and (B). So, I need to find the coordinates of points (A) and (B) by plugging (x = -c) into the hyperbola equation.Let me do that:[frac{(-c)^2}{a^2} - frac{y^2}{b^2} = 1][frac{c^2}{a^2} - frac{y^2}{b^2} = 1]Since (c^2 = a^2 + b^2), substitute that in:[frac{a^2 + b^2}{a^2} - frac{y^2}{b^2} = 1]Simplify:[1 + frac{b^2}{a^2} - frac{y^2}{b^2} = 1]Subtract 1 from both sides:[frac{b^2}{a^2} - frac{y^2}{b^2} = 0]Multiply both sides by (a^2 b^2) to eliminate denominators:[b^4 - a^2 y^2 = 0]Rearrange:[a^2 y^2 = b^4]Divide both sides by (a^2):[y^2 = frac{b^4}{a^2}]Take square roots:[y = pm frac{b^2}{a}]So, the points (A) and (B) are ((-c, frac{b^2}{a})) and ((-c, -frac{b^2}{a})), respectively.Now, I need to compute the vectors (overrightarrow{EA}) and (overrightarrow{EB}) and then find their dot product. The condition given is that this dot product is greater than 0.First, let's find the coordinates of vectors (overrightarrow{EA}) and (overrightarrow{EB}).Point (E) is at ((c, 0)). Point (A) is at ((-c, frac{b^2}{a})), so vector (overrightarrow{EA}) is:[overrightarrow{EA} = A - E = (-c - c, frac{b^2}{a} - 0) = (-2c, frac{b^2}{a})]Similarly, point (B) is at ((-c, -frac{b^2}{a})), so vector (overrightarrow{EB}) is:[overrightarrow{EB} = B - E = (-c - c, -frac{b^2}{a} - 0) = (-2c, -frac{b^2}{a})]Now, compute the dot product (overrightarrow{EA} cdot overrightarrow{EB}):[overrightarrow{EA} cdot overrightarrow{EB} = (-2c)(-2c) + left(frac{b^2}{a}right)left(-frac{b^2}{a}right)][= 4c^2 - frac{b^4}{a^2}]The condition is that this dot product is greater than 0:[4c^2 - frac{b^4}{a^2} > 0][4c^2 > frac{b^4}{a^2}]Multiply both sides by (a^2) (since (a > 0), this doesn't change the inequality direction):[4c^2 a^2 > b^4]But I know that (c^2 = a^2 + b^2), so let's substitute that in:[4(a^2 + b^2) a^2 > b^4]Expand the left side:[4a^4 + 4a^2 b^2 > b^4]Bring all terms to the left side:[4a^4 + 4a^2 b^2 - b^4 > 0]Hmm, this looks a bit complicated. Maybe I can express everything in terms of (e), the eccentricity, since (e = frac{c}{a}) and (c^2 = a^2 + b^2). Let's express (b^2) in terms of (a^2) and (e):From (c^2 = a^2 + b^2), we have (b^2 = c^2 - a^2 = a^2(e^2 - 1)).So, let's substitute (b^2 = a^2(e^2 - 1)) into the inequality:First, compute (b^4):[b^4 = (a^2(e^2 - 1))^2 = a^4(e^2 - 1)^2]Now, substitute into the inequality:[4a^4 + 4a^2 cdot a^2(e^2 - 1) - a^4(e^2 - 1)^2 > 0]Simplify each term:First term: (4a^4)Second term: (4a^4(e^2 - 1))Third term: (-a^4(e^2 - 1)^2)So, the inequality becomes:[4a^4 + 4a^4(e^2 - 1) - a^4(e^2 - 1)^2 > 0]Factor out (a^4) (since (a > 0), we can divide both sides by (a^4) without changing the inequality):[4 + 4(e^2 - 1) - (e^2 - 1)^2 > 0]Let me simplify this expression step by step.First, expand (4(e^2 - 1)):[4e^2 - 4]So, the expression becomes:[4 + (4e^2 - 4) - (e^2 - 1)^2 > 0]Simplify (4 + (4e^2 - 4)):[4 + 4e^2 - 4 = 4e^2]So now, the inequality is:[4e^2 - (e^2 - 1)^2 > 0]Let me expand ((e^2 - 1)^2):[(e^2 - 1)^2 = e^4 - 2e^2 + 1]Substitute back into the inequality:[4e^2 - (e^4 - 2e^2 + 1) > 0][4e^2 - e^4 + 2e^2 - 1 > 0]Combine like terms:[(4e^2 + 2e^2) - e^4 - 1 > 0][6e^2 - e^4 - 1 > 0]Rearrange terms:[-e^4 + 6e^2 - 1 > 0]Multiply both sides by -1 (remember to reverse the inequality sign):[e^4 - 6e^2 + 1 < 0]So, we have the quartic inequality:[e^4 - 6e^2 + 1 < 0]This looks like a quadratic in terms of (e^2). Let me set (u = e^2), so the inequality becomes:[u^2 - 6u + 1 < 0]Now, solve this quadratic inequality. First, find the roots of the quadratic equation (u^2 - 6u + 1 = 0).Using the quadratic formula:[u = frac{6 pm sqrt{36 - 4}}{2} = frac{6 pm sqrt{32}}{2} = frac{6 pm 4sqrt{2}}{2} = 3 pm 2sqrt{2}]So, the roots are (u = 3 + 2sqrt{2}) and (u = 3 - 2sqrt{2}).Since (u = e^2) and (e > 1), (u > 1). Now, (3 - 2sqrt{2}) is approximately (3 - 2.828 = 0.172), which is less than 1, so we can ignore that root because (u > 1).The quadratic (u^2 - 6u + 1) opens upwards (since the coefficient of (u^2) is positive). Therefore, the inequality (u^2 - 6u + 1 < 0) holds between the roots. However, since one root is less than 1 and the other is greater than 1, the interval where the inequality holds for (u > 1) is between (u = 1) and (u = 3 + 2sqrt{2}).But wait, let's double-check. The quadratic is negative between its two roots. Since one root is less than 1 and the other is greater than 1, the interval where the quadratic is negative and (u > 1) is from (u = 1) to (u = 3 + 2sqrt{2}).Therefore, (1 < u < 3 + 2sqrt{2}), which translates back to (e) as:[1 < e^2 < 3 + 2sqrt{2}]Taking square roots:[1 < e < sqrt{3 + 2sqrt{2}}]Now, let's compute (sqrt{3 + 2sqrt{2}}). Notice that (sqrt{3 + 2sqrt{2}} = sqrt{2} + 1), because:[(sqrt{2} + 1)^2 = 2 + 2sqrt{2} + 1 = 3 + 2sqrt{2}]Yes, that's correct. So, (sqrt{3 + 2sqrt{2}} = 1 + sqrt{2}).Therefore, the range for (e) is:[1 < e < 1 + sqrt{2}]But wait, looking back at the options given:A: ((sqrt{2}, +∞)) B: ((1, sqrt{2}+1)) C: ((2, +∞)) D: ((1, 2))So, the range (1 < e < 1 + sqrt{2}) is option B. But hold on, in the initial problem, the hyperbola is given as (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), which is a standard hyperbola opening left and right, so (e > 1), which is consistent.But let me double-check my steps because the answer I got is option B, but the initial thought process mentioned option D. Maybe I made a mistake somewhere.Wait, going back to the dot product:I had:[overrightarrow{EA} cdot overrightarrow{EB} = 4c^2 - frac{b^4}{a^2} > 0]Then, substituting (c^2 = a^2 + b^2), I arrived at:[4(a^2 + b^2) a^2 > b^4]Which led to:[4a^4 + 4a^2 b^2 > b^4]Then, substituting (b^2 = a^2(e^2 - 1)), I got:[4a^4 + 4a^4(e^2 - 1) > a^4(e^2 - 1)^2]Dividing by (a^4):[4 + 4(e^2 - 1) > (e^2 - 1)^2]Which simplifies to:[4 + 4e^2 - 4 > e^4 - 2e^2 + 1][4e^2 > e^4 - 2e^2 + 1][0 > e^4 - 6e^2 + 1]Which is the same as:[e^4 - 6e^2 + 1 < 0]So, that part seems correct. Then, solving for (u = e^2), got the roots (3 pm 2sqrt{2}), and concluded that (1 < e < 1 + sqrt{2}), which is approximately (1 < e < 2.414).Looking back at the options, B is ((1, sqrt{2} + 1)), which is exactly what I got. However, in the initial thought process, the assistant concluded option D: ((1, 2)). So, which one is correct?Wait, maybe I made a mistake in interpreting the condition. The problem states that (overrightarrow{EA} cdot overrightarrow{EB} > 0). I computed the dot product as (4c^2 - frac{b^4}{a^2}), which is correct.But let me think about the geometric interpretation. The dot product being positive implies that the angle between vectors (overrightarrow{EA}) and (overrightarrow{EB}) is acute. Alternatively, it could also mean that the points (A) and (B) lie on the same side relative to point (E), but in this case, since (A) and (B) are symmetric with respect to the x-axis, the vectors (overrightarrow{EA}) and (overrightarrow{EB}) are symmetric with respect to the x-axis as well.Wait, actually, in this case, since both (A) and (B) are on the vertical line (x = -c), which is to the left of the origin, and (E) is at ((c, 0)), which is to the right. So, both vectors (overrightarrow{EA}) and (overrightarrow{EB}) point towards the left and upwards/downwards. The dot product being positive means that the angle between them is acute, which geometrically makes sense if the points (A) and (B) are not too far from (E).But according to my algebra, the condition leads to (e < 1 + sqrt{2}), which is approximately 2.414, so the range is ((1, 1 + sqrt{2})), which is option B.However, the initial thought process mentioned that the angle (angle AEF) must be less than 45°, leading to (e < 2). Maybe there's another way to approach this problem, considering the triangle (AEF).Let me try that approach.In triangle (AEF), point (A) is at ((-c, frac{b^2}{a})), point (E) is at ((c, 0)), and point (F) is at ((-c, 0)). So, triangle (AEF) has vertices at ((-c, frac{b^2}{a})), ((c, 0)), and ((-c, 0)).The angle at (E) is (angle AEF). The condition that (overrightarrow{EA} cdot overrightarrow{EB} > 0) implies that the angle between (overrightarrow{EA}) and (overrightarrow{EB}) is acute. However, since (B) is the reflection of (A) over the x-axis, the angle between (overrightarrow{EA}) and (overrightarrow{EB}) is twice the angle between (overrightarrow{EA}) and the x-axis.Alternatively, perhaps considering the triangle (AEF), the angle at (E) is related to the vectors (overrightarrow{EA}) and (overrightarrow{EF}).Wait, maybe the initial thought process was considering the angle between (overrightarrow{EA}) and (overrightarrow{EF}), but in reality, the dot product is between (overrightarrow{EA}) and (overrightarrow{EB}).Alternatively, perhaps using vectors, the dot product condition can be interpreted as:[overrightarrow{EA} cdot overrightarrow{EB} > 0]Which, as I computed earlier, leads to (4c^2 > frac{b^4}{a^2}).But let's see if there's another way to express this condition.Alternatively, since points (A) and (B) are symmetric with respect to the x-axis, the vectors (overrightarrow{EA}) and (overrightarrow{EB}) are symmetric with respect to the x-axis. Therefore, their dot product can be seen as the product of their magnitudes times the cosine of the angle between them.But since they are symmetric, the angle between them is twice the angle each makes with the x-axis.But perhaps it's more straightforward to stick with the algebraic approach.Given that I arrived at (e < 1 + sqrt{2}), which is approximately 2.414, and the options include B: ((1, sqrt{2} + 1)), which is exactly this interval, I think that should be the correct answer.However, the initial thought process concluded option D: ((1, 2)). Maybe there's a miscalculation there.Wait, let me check the initial thought process again.The assistant wrote:"To satisfy (overrightarrow {EA} cdot overrightarrow {EB} > 0), it is sufficient for (angle AEB) to be acute, which means that (angle AEF) must be less than (45°). In triangle (triangle AEF), we have [tan angle AEF = frac {|AE|}{|EF|} = frac {frac {b^{2}}{a}}{a+c} < 1.]This inequality leads to (c^2 - ac - 2a^2 < 0). Dividing both sides by (a^2), we get (e^2 - e - 2 < 0). Given that (e > 1), it follows that the range of the eccentricity (e) is ((1, 2)). Therefore, the correct answer is (boxed{D})."Hmm, so the assistant used a different approach, considering the tangent of angle (AEF) and setting it less than 1, leading to (e < 2). But in my approach, I arrived at (e < 1 + sqrt{2}), which is approximately 2.414, which is larger than 2.So, which approach is correct?Let me analyze the assistant's approach.They considered triangle (AEF), with points (A), (E), and (F). They said that (angle AEB) being acute implies that (angle AEF < 45°). Wait, but (angle AEB) is the angle at (E) between vectors (EA) and (EB), which are symmetric with respect to the x-axis. So, the angle between them is actually (2theta), where (theta) is the angle between (EA) and the x-axis.If (angle AEB) is acute, then (2theta < 90°), so (theta < 45°). Therefore, the angle between (EA) and the x-axis is less than (45°), which would mean that the slope of (EA) is less than 1 in magnitude.But in the assistant's approach, they considered (tan angle AEF = frac{|AE|}{|EF|}). Wait, let me visualize triangle (AEF).Point (A) is at ((-c, frac{b^2}{a})), point (E) is at ((c, 0)), and point (F) is at ((-c, 0)). So, triangle (AEF) has sides:- (AE): from (E) to (A), which is the vector ((-2c, frac{b^2}{a}))- (EF): from (E) to (F), which is the vector ((-2c, 0))- (AF): from (A) to (F), which is the vector ((0, frac{b^2}{a}))So, in triangle (AEF), the angle at (E) is (angle AEF), which is the angle between vectors (EA) and (EF). The assistant claims that (tan angle AEF = frac{|AE|}{|EF|}), but that's not correct.Actually, in triangle (AEF), the sides are:- (EA): length (sqrt{(2c)^2 + left(frac{b^2}{a}right)^2})- (EF): length (2c)- (AF): length (frac{b^2}{a})So, using the definition of tangent in triangle (AEF), if we consider the right triangle formed by dropping a perpendicular from (A) to (EF), but actually, triangle (AEF) is not a right triangle.Wait, perhaps the assistant made a mistake in assuming that (tan angle AEF = frac{|AE|}{|EF|}). That formula is not generally true unless the triangle is right-angled at (F), which it is not.Therefore, the assistant's approach might be flawed because they incorrectly applied the tangent formula.In reality, to find (tan angle AEF), we can use the coordinates.Point (E) is at ((c, 0)), point (A) is at ((-c, frac{b^2}{a})), and point (F) is at ((-c, 0)).So, vector (EA) is ((-2c, frac{b^2}{a})), and vector (EF) is ((-2c, 0)).The angle between vectors (EA) and (EF) is (angle AEF). To find (tan angle AEF), we can use the formula:[tan theta = frac{|text{opposite}|}{|text{adjacent}|}]In this case, the opposite side relative to angle (AEF) is the vertical component of vector (EA), which is (frac{b^2}{a}), and the adjacent side is the horizontal component, which is (2c).Wait, actually, in the triangle, the sides adjacent and opposite to angle (AEF) would be:- Opposite side: (AF = frac{b^2}{a})- Adjacent side: (EF = 2c)But actually, in triangle (AEF), angle (AEF) is at point (E), so the sides are:- Opposite side: (AF = frac{b^2}{a})- Adjacent side: (EF = 2c)- Hypotenuse: (EA = sqrt{(2c)^2 + left(frac{b^2}{a}right)^2})Therefore, (tan angle AEF = frac{AF}{EF} = frac{frac{b^2}{a}}{2c}).So, the assistant wrote (tan angle AEF = frac{frac{b^2}{a}}{a + c}), but that seems incorrect. It should be (frac{frac{b^2}{a}}{2c}).Therefore, the correct expression is:[tan angle AEF = frac{frac{b^2}{a}}{2c}]The assistant mistakenly used (a + c) in the denominator instead of (2c). That was an error.Given that, let's correct the assistant's approach.They said that (tan angle AEF < 1), which would imply:[frac{frac{b^2}{a}}{2c} < 1][frac{b^2}{a} < 2c]Multiply both sides by (a):[b^2 < 2ac]But (b^2 = c^2 - a^2), so substitute:[c^2 - a^2 < 2ac]Rearrange:[c^2 - 2ac - a^2 < 0]Divide by (a^2):[left(frac{c}{a}right)^2 - 2left(frac{c}{a}right) - 1 < 0]Let (e = frac{c}{a}), so:[e^2 - 2e - 1 < 0]Solve the quadratic inequality (e^2 - 2e - 1 < 0).Find the roots:[e = frac{2 pm sqrt{4 + 4}}{2} = frac{2 pm sqrt{8}}{2} = frac{2 pm 2sqrt{2}}{2} = 1 pm sqrt{2}]So, the roots are (1 + sqrt{2}) and (1 - sqrt{2}). Since (e > 1), we consider (e < 1 + sqrt{2}).Thus, the inequality (e^2 - 2e - 1 < 0) holds for (1 - sqrt{2} < e < 1 + sqrt{2}). But since (e > 1), the valid interval is (1 < e < 1 + sqrt{2}), which is the same result as before.Therefore, the correct range is ((1, 1 + sqrt{2})), which is option B.So, the initial thought process had a mistake in the calculation of (tan angle AEF), leading to an incorrect conclusion of option D. The correct answer, after properly analyzing the dot product and considering the geometric interpretation, is option B.**Final Answer**boxed{B}

💡Okay, so I have this problem where I'm given that cos x equals zero, and then cos(x + z) equals -1/2. I need to find the smallest positive value of z in radians. Hmm, let me think about how to approach this.First, I remember that cos x equals zero at specific points on the unit circle. Specifically, cosine is zero at π/2, 3π/2, 5π/2, and so on. So, in general, x can be written as π/2 plus any multiple of π. So, x = π/2 + mπ, where m is any integer. That makes sense because cosine has a period of 2π, but since it's zero every π radians, that's why it's mπ.Now, moving on to the second part: cos(x + z) equals -1/2. I know that cosine equals -1/2 at specific angles too. Let me recall, cosine is -1/2 at 2π/3 and 4π/3 in the interval [0, 2π). So, in general, x + z can be written as 2πn ± 2π/3, where n is any integer. That's because cosine has a period of 2π, so adding any multiple of 2π will give the same value.So, putting it together, I have x = π/2 + mπ and x + z = 2πn ± 2π/3. I can substitute the expression for x into the second equation to solve for z.Substituting, we get:π/2 + mπ + z = 2πn ± 2π/3Now, let's solve for z:z = 2πn ± 2π/3 - π/2 - mπHmm, so z is expressed in terms of integers m and n. I need to find the smallest positive z. So, I should choose values of m and n such that z is positive and as small as possible.Let me simplify the expression for z:z = (2n - m)π ± 2π/3 - π/2To make this simpler, let's consider the constants:±2π/3 - π/2Let me compute both possibilities:1. Using the positive sign:2π/3 - π/2 = (4π/6 - 3π/6) = π/62. Using the negative sign:-2π/3 - π/2 = (-4π/6 - 3π/6) = -7π/6So, the two possibilities for the constants are π/6 and -7π/6.Now, considering z = (2n - m)π + π/6 or z = (2n - m)π - 7π/6.I need z to be positive, so let's look at the first case:z = (2n - m)π + π/6To make this as small as possible, I can set (2n - m)π to be as small as possible. The smallest value occurs when 2n - m is zero, which gives z = π/6. But wait, is that possible?If 2n - m = 0, then m = 2n. So, for example, if n = 0, then m = 0. Plugging back into the equation:z = 0 + π/6 = π/6Is that a valid solution? Let me check.If x = π/2 + 0π = π/2, then x + z = π/2 + π/6 = (3π/6 + π/6) = 4π/6 = 2π/3. And cos(2π/3) is indeed -1/2. So, yes, that works.Now, let's check the second case:z = (2n - m)π - 7π/6Again, to make z positive, (2n - m)π must be greater than 7π/6. The smallest integer values for 2n - m that make this positive would be when 2n - m = 1, so:z = π - 7π/6 = (6π/6 - 7π/6) = -π/6That's negative, which we don't want. If we try 2n - m = 2:z = 2π - 7π/6 = (12π/6 - 7π/6) = 5π/6That's positive, but 5π/6 is larger than π/6, so π/6 is still the smaller positive value.Therefore, the smallest positive value of z is π/6 radians.

💡Okay, so I have this series to calculate: 1/4 minus 1/16 plus 1/64 minus 1/256 plus 1/1024. Hmm, let me see. First, I need to figure out what kind of series this is. It looks like each term is being multiplied by a common ratio to get to the next term. Let me check that.Starting with 1/4, if I multiply by -1/4, I get -1/16. Then, multiplying -1/16 by -1/4 gives me 1/64. Okay, that seems consistent. So, this is a geometric series with the first term a = 1/4 and a common ratio r = -1/4. Now, I remember that the sum of a geometric series can be found using the formula S = a(1 - r^n)/(1 - r), where n is the number of terms. Let me count how many terms there are here. Starting from 1/4, then -1/16, 1/64, -1/256, and 1/1024. That's five terms in total, so n = 5.Plugging the values into the formula: S = (1/4)(1 - (-1/4)^5)/(1 - (-1/4)). Let me compute (-1/4)^5 first. Since the exponent is odd, it'll be negative. So, (-1/4)^5 = -1/1024. Therefore, 1 - (-1/1024) becomes 1 + 1/1024, which is 1025/1024.Now, the denominator is 1 - (-1/4) = 1 + 1/4 = 5/4. So, putting it all together: S = (1/4)(1025/1024)/(5/4). Dividing by 5/4 is the same as multiplying by 4/5, so S = (1/4)(1025/1024)(4/5). The 4s cancel out, leaving S = 1025/(1024*5) = 1025/5120.Wait, can this fraction be simplified further? Let me see. 1025 divided by 5 is 205, and 5120 divided by 5 is 1024. So, S = 205/1024. That seems like the simplest form.Let me double-check my calculations to make sure I didn't make a mistake. Starting with the formula, plugging in the values, computing the exponents, simplifying step by step. It all seems to add up correctly. So, I think the sum of the series is 205/1024.

💡First, I observe the original expression ( x(x - 3) + (3 - x) ).I notice that ( (3 - x) ) can be rewritten as ( -(x - 3) ) to make the expression more uniform.This allows me to rewrite the expression as ( x(x - 3) - (x - 3) ).Next, I identify the common factor ( (x - 3) ) in both terms.I factor out ( (x - 3) ) from both terms, resulting in ( (x - 3)(x - 1) ).Therefore, the factorized form of the given expression is ( (x - 3)(x - 1) ).