💡To determine the range of values for ( a ) such that the fraction ( frac{2}{a+1} ) is meaningful, we need to ensure that the denominator is not zero. 1. **Condition for the Denominator**: The denominator ( a + 1 ) must not be zero. 2. **Solving the Inequality**: [ a + 1 neq 0 ] Subtracting 1 from both sides gives: [ a neq -1 ]Therefore, ( a ) must not equal (-1) for the fraction to be meaningful.

💡First, I need to find the exterior angles corresponding to each of the given interior angles of the triangle. An exterior angle is calculated by subtracting the interior angle from 180 degrees.For the first interior angle of 45 degrees:180 degrees - 45 degrees = 135 degreesFor the second interior angle of 75 degrees:180 degrees - 75 degrees = 105 degreesFor the third interior angle of 60 degrees:180 degrees - 60 degrees = 120 degreesNext, I will add up all the exterior angles to find their total sum:135 degrees + 105 degrees + 120 degrees = 360 degreesFinally, to find the mean value of the exterior angles, I will divide the total sum by the number of exterior angles, which is 3:360 degrees / 3 = 120 degreesTherefore, the mean value of the measures of the three exterior angles is 120 degrees.

💡First, I need to understand the structure of the club and the changes that occur each year. The club starts with 21 members, which includes 3 chairpersons and 18 active members.Each year, the 3 chairpersons step down, leaving only the active members. Each active member then recruits 3 new members. This means that the number of active members grows by a factor of 4 each year (since each active member becomes 4 members: themselves plus 3 new recruits).After the recruitment, 3 new chairpersons are added from outside the club. This process repeats every year.To find the number of members after 4 years, I'll calculate the number of active members after each year and then add the 3 new chairpersons at the end.Starting with 18 active members:- After 1 year: 18 * 4 = 72 active members- After 2 years: 72 * 4 = 288 active members- After 3 years: 288 * 4 = 1,152 active members- After 4 years: 1,152 * 4 = 4,608 active membersFinally, adding the 3 new chairpersons:4,608 + 3 = 4,611 members

💡Okay, so I have this function f(x) = e^x - a ln x - a. There are two parts to the problem. Let me tackle them one by one.Starting with part (I): When a = e, find the equation of the tangent line to the curve y = f(x) at the point (1, f(1)).Alright, so first, I need to find f(1) when a = e. Let me plug in x = 1 and a = e into the function.f(1) = e^1 - e ln(1) - e. Hmm, ln(1) is 0, so that term disappears. So f(1) = e - 0 - e = 0. So the point is (1, 0).Now, to find the equation of the tangent line, I need the derivative f'(x) at x = 1. Let's compute f'(x).f(x) = e^x - a ln x - a, so f'(x) is the derivative of e^x, which is e^x, minus the derivative of a ln x, which is a/x, and the derivative of -a is 0. So f'(x) = e^x - a/x.Since a = e, f'(x) becomes e^x - e/x. Now evaluate this at x = 1.f'(1) = e^1 - e/1 = e - e = 0. So the slope of the tangent line at x = 1 is 0.Therefore, the tangent line is a horizontal line passing through (1, 0). The equation is y = 0.Wait, that seems straightforward. Let me double-check. f(1) is indeed 0, and f'(1) is 0, so yes, the tangent line is y = 0. Okay, part (I) seems done.Moving on to part (II): Prove that for all a in (0, e), f(x) has a minimum value in the interval (a/e, 1), and the minimum value is greater than 0.Hmm, okay. So I need to show two things: first, that f(x) attains a minimum in that interval, and second, that this minimum is positive.Let me think about how to approach this. Since f(x) is differentiable, I can use calculus. I'll find the critical points by setting f'(x) = 0 and then analyze the behavior.First, let's write down f'(x) again. It's e^x - a/x.So, set f'(x) = 0: e^x - a/x = 0. So e^x = a/x. That's the equation we need to solve for x in the interval (a/e, 1).I need to show that there exists a solution x0 in (a/e, 1) such that e^{x0} = a/x0. Then, since f'(x) changes sign from negative to positive around x0 (since f''(x) is positive, as e^x grows exponentially and a/x decreases), x0 is a minimum.Wait, let me make sure. Let's check the sign of f'(x) at the endpoints of the interval.Compute f'(a/e): e^{a/e} - a/(a/e) = e^{a/e} - e. Since a is in (0, e), a/e is in (0, 1). So e^{a/e} is less than e^1 = e because a/e < 1. Hence, e^{a/e} - e is negative. So f'(a/e) < 0.Now, compute f'(1): e^1 - a/1 = e - a. Since a is in (0, e), e - a is positive. So f'(1) > 0.Therefore, by the Intermediate Value Theorem, since f'(x) is continuous (as a composition of continuous functions), and it goes from negative at x = a/e to positive at x = 1, there must be some x0 in (a/e, 1) where f'(x0) = 0. So that's the critical point.Now, to confirm it's a minimum, let's check the second derivative or the sign changes.Alternatively, since f'(x) is increasing because f''(x) = e^x + a/x^2, which is always positive for x > 0. So f'(x) is strictly increasing. Therefore, the critical point x0 is the only one, and since f'(x) changes from negative to positive, it's a minimum.So f(x) has a minimum at x0 in (a/e, 1). Now, I need to show that f(x0) > 0.Let me compute f(x0). Since at x0, e^{x0} = a/x0, so f(x0) = e^{x0} - a ln x0 - a = (a/x0) - a ln x0 - a.Factor out a: f(x0) = a(1/x0 - ln x0 - 1).So I need to show that 1/x0 - ln x0 - 1 > 0.Let me define g(x) = 1/x - ln x - 1. I need to show that g(x0) > 0.Compute g(x) in the interval (a/e, 1). Let's analyze g(x).First, compute g'(x): derivative of 1/x is -1/x^2, derivative of -ln x is -1/x, and derivative of -1 is 0. So g'(x) = -1/x^2 - 1/x = - (1 + x)/x^2.Since x > 0, g'(x) is negative because both numerator and denominator are positive, but with a negative sign. So g(x) is decreasing on (0, ∞).Therefore, on the interval (a/e, 1), g(x) is decreasing. So its minimum on this interval is at x = 1.Compute g(1): 1/1 - ln 1 - 1 = 1 - 0 - 1 = 0.So since g(x) is decreasing, for x in (a/e, 1), g(x) > g(1) = 0. Therefore, g(x0) > 0.Hence, f(x0) = a * g(x0) > 0, since a > 0.Therefore, the minimum value f(x0) is greater than 0.Wait, let me recap to make sure I didn't skip any steps.1. Showed that f'(x) = e^x - a/x.2. Evaluated f'(a/e) < 0 and f'(1) > 0, so by IVT, there exists x0 in (a/e, 1) where f'(x0) = 0.3. Since f''(x) > 0, f'(x) is increasing, so x0 is a minimum.4. Expressed f(x0) in terms of a and x0, substituted e^{x0} = a/x0.5. Defined g(x) = 1/x - ln x - 1, showed it's decreasing, so g(x0) > g(1) = 0.6. Therefore, f(x0) = a * g(x0) > 0.Yes, that seems solid. I think that covers the proof.So, summarizing:(I) The tangent line at (1, f(1)) when a = e is y = 0.(II) For all a in (0, e), f(x) has a minimum in (a/e, 1), and this minimum is positive.I think that's all. I don't see any gaps in the logic, but let me think if there's another way someone might approach part (II). Maybe using convexity or something else, but the way I did it seems straightforward.Another thought: Could I have used the fact that f(x) is convex? Since f''(x) = e^x + a/x^2 > 0, f is convex, so any critical point is a global minimum. But wait, the interval is (a/e, 1), so it's a local minimum within that interval. But since f is convex, it's also the global minimum on that interval, which is fine.Alternatively, maybe using inequalities like AM-GM or something, but I think the calculus approach is more straightforward here.Alright, I think I'm confident with the solution.**Final Answer**(I) The equation of the tangent line is boxed{y=0}.(II) The minimum value of the function is greater than boxed{0}.

💡Okay, so I have this geometry problem here. It says that angle 1 is 100 degrees, angle 2 is 60 degrees, and angle 3 is 90 degrees. I need to find angle 4. Hmm, let me visualize the figure they're talking about. Since it's not provided, I'll assume it's a quadrilateral with four angles labeled 1, 2, 3, and 4. Alright, in a quadrilateral, the sum of all interior angles should be 360 degrees. That's a basic property I remember. So if I can figure out the sum of the given angles and subtract that from 360, I should get angle 4. Let me write that down:Angle 1 + Angle 2 + Angle 3 + Angle 4 = 360 degreesPlugging in the given values:100 degrees + 60 degrees + 90 degrees + Angle 4 = 360 degreesLet me add those numbers up. 100 plus 60 is 160, and 160 plus 90 is 250. So now the equation looks like:250 degrees + Angle 4 = 360 degreesTo find Angle 4, I subtract 250 degrees from both sides:Angle 4 = 360 degrees - 250 degreesAngle 4 = 110 degreesWait, that seems straightforward. But let me double-check. Maybe the figure isn't a simple quadrilateral? Could it be a triangle or something else? The problem mentions angles 1, 2, 3, and 4, which suggests four angles, so a quadrilateral makes sense. Alternatively, if it's a polygon with more sides, the sum of the interior angles would be different. But since only four angles are mentioned, I think quadrilateral is the right assumption. Another thought: could these angles be exterior angles instead of interior? If they were exterior angles, the sum would be 360 degrees regardless of the polygon. But the problem doesn't specify, so I think interior angles are more likely, especially since angle 3 is 90 degrees, which is common in quadrilaterals like rectangles or squares. Wait, but if angle 3 is 90 degrees, that could also be part of a right angle in a triangle. Maybe the figure is a triangle with an extension, making it a quadrilateral? Hmm, but without the figure, it's hard to be certain. Still, given the information, the most logical approach is to treat it as a quadrilateral and use the sum of interior angles. So, 360 degrees minus the sum of the given angles gives me angle 4 as 110 degrees. I think that's the answer. It makes sense, and I don't see any immediate errors in my reasoning. Maybe I should look up the sum of interior angles formula again to confirm. Yeah, for any n-sided polygon, the sum is (n-2)*180 degrees. For a quadrilateral, n=4, so (4-2)*180 = 360 degrees. Yep, that checks out. So, I'm confident that angle 4 is 110 degrees.

💡First, I need to evaluate the expression inside the parentheses: (6 div 2 times 3).Following the order of operations, I should perform the division first: (6 div 2 = 3).Next, I multiply the result by 3: (3 times 3 = 9).Now, I substitute this back into the original expression: (120 div 9).Finally, I perform the division: (120 div 9 = 13.overline{3}).