answer：Alright, so I have this problem here: "If the distance from a point M on the parabola x = 4y² to the focus is 1/2, what is the x-coordinate of point M?" Hmm, okay. Let me try to figure this out step by step.First, I remember that the standard form of a parabola can help me identify its key features like the vertex, focus, and directrix. The given equation is x = 4y². I think this is a horizontal parabola because the x is by itself and the y is squared. In standard form, a horizontal parabola is usually written as (y - k)² = 4p(x - h), where (h, k) is the vertex, and p is the distance from the vertex to the focus.So, comparing x = 4y² to the standard form, it seems like the vertex is at (0, 0) because there are no shifts in the equation. That makes sense. Now, in the standard form, 4p would be equal to 4, right? Because in the given equation, it's 4y², so 4p = 4. Therefore, p = 1. Okay, so p is 1. That means the focus is p units away from the vertex along the axis of the parabola. Since it's a horizontal parabola opening to the right, the focus should be at (h + p, k), which is (0 + 1, 0) = (1, 0). Got that.Now, the problem says that the distance from point M on the parabola to the focus is 1/2. So, point M is somewhere on the parabola x = 4y², and the distance from M to the focus (1, 0) is 1/2. I need to find the x-coordinate of M.Let me denote point M as (x, y). Since it's on the parabola, it must satisfy the equation x = 4y². So, x = 4y². That's one equation.Now, the distance from M to the focus (1, 0) is given by the distance formula: √[(x - 1)² + (y - 0)²] = 1/2. So, √[(x - 1)² + y²] = 1/2. If I square both sides to eliminate the square root, I get (x - 1)² + y² = (1/2)² = 1/4.So now I have two equations:1. x = 4y²2. (x - 1)² + y² = 1/4I can substitute the first equation into the second to solve for y, and then find x.Substituting x = 4y² into the second equation:(4y² - 1)² + y² = 1/4Let me expand (4y² - 1)². That would be (4y²)² - 2*4y²*1 + 1² = 16y⁴ - 8y² + 1.So, substituting back:16y⁴ - 8y² + 1 + y² = 1/4Combine like terms:16y⁴ - 7y² + 1 = 1/4Subtract 1/4 from both sides to set the equation to zero:16y⁴ - 7y² + 1 - 1/4 = 0Simplify 1 - 1/4 = 3/4:16y⁴ - 7y² + 3/4 = 0Hmm, this is a quartic equation, but it's quadratic in terms of y². Let me set z = y², so the equation becomes:16z² - 7z + 3/4 = 0Now, I can solve this quadratic equation for z.Using the quadratic formula: z = [7 ± √(49 - 4*16*(3/4))]/(2*16)First, calculate the discriminant:√(49 - 4*16*(3/4)) = √(49 - 16*3) = √(49 - 48) = √1 = 1So, z = [7 ± 1]/32Therefore, two solutions:z = (7 + 1)/32 = 8/32 = 1/4z = (7 - 1)/32 = 6/32 = 3/16So, z = 1/4 or z = 3/16But z = y², so y² = 1/4 or y² = 3/16Therefore, y = ±1/2 or y = ±√(3)/4Now, let's find the corresponding x for each y using x = 4y².First, for y² = 1/4:x = 4*(1/4) = 1Second, for y² = 3/16:x = 4*(3/16) = 12/16 = 3/4So, the possible x-coordinates are 1 and 3/4.But wait, the distance from M to the focus is 1/2. Let's verify if both these points satisfy the distance condition.First, for x = 1:Point M is (1, y). Since x = 1, from the parabola equation, y² = 1/4, so y = ±1/2.So, points are (1, 1/2) and (1, -1/2).Distance from (1, 1/2) to focus (1, 0):√[(1 - 1)² + (1/2 - 0)²] = √[0 + 1/4] = √(1/4) = 1/2Similarly for (1, -1/2), same distance.Now, for x = 3/4:Point M is (3/4, y). From the parabola equation, y² = 3/16, so y = ±√(3)/4.So, points are (3/4, √3/4) and (3/4, -√3/4).Distance from (3/4, √3/4) to focus (1, 0):√[(3/4 - 1)² + (√3/4 - 0)²] = √[(-1/4)² + (√3/4)²] = √[1/16 + 3/16] = √[4/16] = √(1/4) = 1/2Same for (3/4, -√3/4).So, both x = 1 and x = 3/4 satisfy the distance condition. But wait, the problem asks for the x-coordinate of point M. So, are there two possible x-coordinates? Or is there something I missed?Looking back at the parabola equation x = 4y², it's symmetric about the x-axis, so for each y, there's a corresponding point with the same x but opposite y. So, both x = 1 and x = 3/4 are valid.But the problem says "the x-coordinate of point M." It doesn't specify if there are multiple points or just one. Maybe I need to consider the definition of a parabola.Wait, in a parabola, the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. The directrix of this parabola is x = -1, since for a horizontal parabola, the directrix is x = h - p, which is 0 - 1 = -1.So, if the distance from M to the focus is 1/2, then the distance from M to the directrix should also be 1/2.Let me check that.For point (1, 1/2), distance to directrix x = -1 is |1 - (-1)| = |2| = 2, which is not equal to 1/2. Wait, that doesn't make sense. Did I make a mistake?Wait, no. The definition says that the distance from a point on the parabola to the focus equals the distance from that point to the directrix. So, if the distance to the focus is 1/2, the distance to the directrix should also be 1/2.But for point (1, 1/2), distance to directrix is |1 - (-1)| = 2, which is not equal to 1/2. That contradicts the definition. So, something's wrong here.Wait, maybe I messed up the directrix. Let me recalculate the directrix.Given the standard form (y - k)² = 4p(x - h), the directrix is x = h - p.In our case, h = 0, k = 0, and 4p = 4, so p = 1. Therefore, directrix is x = 0 - 1 = -1. That's correct.But then, for point (1, 1/2), distance to directrix is |1 - (-1)| = 2, which is not equal to 1/2. That's a problem because according to the definition, they should be equal.Wait, but the distance from M to the focus is 1/2, so the distance from M to the directrix should also be 1/2. But for point (1, 1/2), it's 2, which is not equal. So, that point shouldn't be on the parabola? But it is on the parabola because x = 4y², 1 = 4*(1/2)² = 4*(1/4) = 1. So, it is on the parabola.This is confusing. Maybe I need to rethink.Wait, perhaps I made a mistake in calculating the distance to the directrix. The directrix is a vertical line x = -1, so the distance from a point (x, y) to the directrix is |x - (-1)| = |x + 1|.So, for point (1, 1/2), distance to directrix is |1 + 1| = 2, which is not equal to 1/2. But according to the definition, it should be equal to the distance to the focus, which is 1/2. So, that's a contradiction.Similarly, for point (3/4, √3/4), distance to directrix is |3/4 + 1| = |7/4| = 7/4, which is not equal to 1/2 either.Wait, that can't be. There must be a mistake in my reasoning.Let me go back. The standard form is (y - k)² = 4p(x - h). So, in our case, (y)^2 = 4*(1)*(x - 0), so 4p = 4, so p = 1. Therefore, the focus is at (h + p, k) = (1, 0), and the directrix is x = h - p = -1.So, that's correct.Now, for any point on the parabola, the distance to the focus equals the distance to the directrix.So, if the distance to the focus is 1/2, the distance to the directrix should also be 1/2.But when I calculated for x = 1, the distance to directrix was 2, not 1/2. So, that point shouldn't be on the parabola? But it is on the parabola.Wait, maybe I'm misunderstanding the definition. The definition says that for any point on the parabola, the distance to the focus equals the distance to the directrix. So, if I have a point on the parabola, these distances must be equal. But in this problem, it's given that the distance to the focus is 1/2, so the distance to the directrix must also be 1/2.Therefore, the point M must satisfy both being on the parabola and having distance 1/2 to the focus and directrix.But when I found x = 1 and x = 3/4, both points are on the parabola, but their distances to the directrix are 2 and 7/4, respectively, which are not equal to 1/2. So, that suggests that my earlier approach is flawed.Wait, maybe I need to set up the equation differently. Let's try again.Given that the distance from M(x, y) to the focus (1, 0) is 1/2, and M is on the parabola x = 4y².So, distance formula: √[(x - 1)² + y²] = 1/2Square both sides: (x - 1)² + y² = 1/4But since x = 4y², substitute:(4y² - 1)² + y² = 1/4Expand (4y² - 1)²: 16y⁴ - 8y² + 1So, 16y⁴ - 8y² + 1 + y² = 1/4Combine like terms: 16y⁴ - 7y² + 1 = 1/4Subtract 1/4: 16y⁴ - 7y² + 3/4 = 0Let z = y²: 16z² - 7z + 3/4 = 0Multiply both sides by 4 to eliminate the fraction: 64z² - 28z + 3 = 0Now, use quadratic formula: z = [28 ± √(784 - 768)] / 128Calculate discriminant: √(784 - 768) = √16 = 4So, z = [28 ± 4]/128Therefore, z = (28 + 4)/128 = 32/128 = 1/4Or z = (28 - 4)/128 = 24/128 = 3/16So, y² = 1/4 or y² = 3/16Thus, y = ±1/2 or y = ±√3/4Then, x = 4y²:For y² = 1/4: x = 4*(1/4) = 1For y² = 3/16: x = 4*(3/16) = 3/4So, points are (1, ±1/2) and (3/4, ±√3/4)Now, check the distance to the directrix for these points.For (1, 1/2): distance to directrix x = -1 is |1 - (-1)| = 2But according to the definition, it should be equal to the distance to the focus, which is 1/2. So, 2 ≠ 1/2. Contradiction.Similarly, for (3/4, √3/4): distance to directrix is |3/4 - (-1)| = |7/4| = 7/4, which is not equal to 1/2.Wait, this is confusing. How come the points satisfy the distance to the focus but not to the directrix? That shouldn't happen because the definition of a parabola states that these distances are equal.I think I made a mistake in interpreting the problem. Maybe the point M is not on the parabola? But the problem says it is. Hmm.Wait, no, the problem says "the distance from a point M on the parabola x = 4y² to the focus is 1/2." So, M is on the parabola, and the distance to the focus is 1/2. Therefore, according to the definition, the distance to the directrix should also be 1/2.But when I calculated, it wasn't. So, perhaps there's a mistake in my calculations.Wait, let's recalculate the distance to the directrix for point (3/4, √3/4).Directrix is x = -1.Distance from (3/4, √3/4) to x = -1 is |3/4 - (-1)| = |3/4 + 1| = |7/4| = 7/4.But according to the definition, this should be equal to the distance to the focus, which is 1/2. So, 7/4 ≠ 1/2. Contradiction.Similarly, for (1, 1/2), distance to directrix is 2, which is not equal to 1/2.This suggests that there is no such point M on the parabola x = 4y² that is at a distance of 1/2 from the focus. But the problem states that such a point exists. So, I must have made a mistake.Wait, maybe I messed up the directrix. Let me double-check.Given the standard form (y - k)² = 4p(x - h), the directrix is x = h - p.In our case, h = 0, k = 0, and 4p = 4, so p = 1. Therefore, directrix is x = -1. That's correct.But then, for any point on the parabola, distance to focus equals distance to directrix.So, if I have a point M on the parabola, then distance to focus = distance to directrix.But in our case, we have distance to focus = 1/2, so distance to directrix should also be 1/2.But when I calculated, it wasn't. So, perhaps the points I found are not on the parabola? But they are, because x = 4y².Wait, maybe I need to consider that the distance to the directrix is |x + 1|, and set that equal to 1/2.So, |x + 1| = 1/2Therefore, x + 1 = 1/2 or x + 1 = -1/2So, x = -1/2 or x = -3/2But the parabola is x = 4y², which is x ≥ 0. So, x cannot be negative. Therefore, x = -1/2 and x = -3/2 are not on the parabola.This suggests that there is no point on the parabola x = 4y² that is at a distance of 1/2 from the focus, because the corresponding x would have to be negative, which is not possible.But the problem states that such a point exists. So, I must have made a mistake in my approach.Wait, maybe I need to consider that the distance from M to the focus is 1/2, and M is on the parabola. So, using the definition, distance to focus = distance to directrix = 1/2.But since the directrix is x = -1, the distance from M to directrix is |x + 1| = 1/2So, |x + 1| = 1/2Therefore, x + 1 = 1/2 or x + 1 = -1/2So, x = -1/2 or x = -3/2But x cannot be negative on the parabola x = 4y², so no solution.But the problem says there is such a point. So, perhaps the parabola is defined differently.Wait, maybe I misread the equation. The given parabola is x = 4y². Is that the standard form?Yes, it's a horizontal parabola opening to the right. Standard form is x = 4p y², so 4p = 4, so p = 1.Therefore, focus at (1, 0), directrix x = -1.So, according to this, there is no point on the parabola with distance 1/2 to the focus, because that would require x to be negative, which is impossible.But the problem says there is such a point. So, perhaps I made a mistake in interpreting the distance.Wait, maybe the distance from M to the focus is 1/2, but M is not necessarily on the parabola? No, the problem says M is on the parabola.Wait, perhaps I need to consider that the distance from M to the focus is 1/2, and M is on the parabola, but the definition says that the distance to the focus equals the distance to the directrix. So, if distance to focus is 1/2, distance to directrix is also 1/2.But as we saw, this leads to x = -1/2 or x = -3/2, which are not on the parabola.Therefore, there is no such point M on the parabola x = 4y² that is at a distance of 1/2 from the focus.But the problem says there is. So, perhaps I made a mistake in the standard form.Wait, let me double-check the standard form.For a horizontal parabola, the standard form is (y - k)² = 4p(x - h), where p is the distance from vertex to focus.Given x = 4y², we can write it as y² = (1/4)x, so 4p = 1/4, so p = 1/16.Wait, wait, that's different from what I thought earlier.Wait, hold on. If the standard form is (y - k)² = 4p(x - h), then comparing to y² = (1/4)x, we have 4p = 1/4, so p = 1/16.Therefore, the focus is at (h + p, k) = (0 + 1/16, 0) = (1/16, 0)And the directrix is x = h - p = 0 - 1/16 = -1/16Ah, so I made a mistake earlier. I thought 4p = 4, but actually, 4p = 1/4, so p = 1/16.That changes everything.So, the focus is at (1/16, 0), and the directrix is x = -1/16.Now, let's redo the problem with this correct information.Given that the distance from M(x, y) to the focus (1/16, 0) is 1/2.So, distance formula: √[(x - 1/16)² + y²] = 1/2Square both sides: (x - 1/16)² + y² = 1/4But M is on the parabola x = 4y², so x = 4y².Substitute x = 4y² into the equation:(4y² - 1/16)² + y² = 1/4Let me expand (4y² - 1/16)²:= (4y²)² - 2*(4y²)*(1/16) + (1/16)²= 16y⁴ - (8y²)/16 + 1/256Simplify:= 16y⁴ - (1/2)y² + 1/256So, the equation becomes:16y⁴ - (1/2)y² + 1/256 + y² = 1/4Combine like terms:16y⁴ + ( -1/2 + 1 )y² + 1/256 = 1/4Simplify:16y⁴ + (1/2)y² + 1/256 = 1/4Subtract 1/4 from both sides:16y⁴ + (1/2)y² + 1/256 - 1/4 = 0Calculate 1/256 - 1/4:Convert to common denominator, which is 256:1/256 - 64/256 = -63/256So, the equation is:16y⁴ + (1/2)y² - 63/256 = 0Multiply through by 256 to eliminate denominators:256*16y⁴ + 256*(1/2)y² - 63 = 0Calculate:256*16 = 4096256*(1/2) = 128So, 4096y⁴ + 128y² - 63 = 0Let z = y², so equation becomes:4096z² + 128z - 63 = 0This is a quadratic in z. Let's solve it using the quadratic formula:z = [-128 ± √(128² - 4*4096*(-63))]/(2*4096)Calculate discriminant:128² = 163844*4096*63 = 4*4096*63First, 4*4096 = 1638416384*63: Let's calculate 16384*60 = 983040, and 16384*3 = 49152, so total is 983040 + 49152 = 1,032,192So, discriminant is √(16384 + 1,032,192) = √(1,048,576)Wait, 16384 + 1,032,192 = 1,048,576√1,048,576 = 1024So, z = [-128 ± 1024]/(8192)First, z = (-128 + 1024)/8192 = 896/8192 = 896 ÷ 896 = 1/9.140625? Wait, let me calculate.Wait, 896 ÷ 8192: 8192 ÷ 896 = 9.140625, so 896/8192 = 1/9.140625 ≈ 0.109375But let's do it properly:896 ÷ 8192 = (896 ÷ 896)/(8192 ÷ 896) = 1/9.140625But 896 = 64*14, 8192 = 64*128So, 896/8192 = 14/128 = 7/64Similarly, z = (-128 - 1024)/8192 = (-1152)/8192 = -1152/8192 = -9/64But z = y² cannot be negative, so discard z = -9/64Therefore, z = 7/64So, y² = 7/64Thus, y = ±√(7)/8Now, find x:x = 4y² = 4*(7/64) = 28/64 = 7/16So, the x-coordinate of point M is 7/16.Let me verify this.Point M is (7/16, ±√7/8)Distance to focus (1/16, 0):√[(7/16 - 1/16)² + (√7/8 - 0)²] = √[(6/16)² + (7/64)] = √[(9/64) + (7/64)] = √(16/64) = √(1/4) = 1/2Good, that matches.Also, distance to directrix x = -1/16:|7/16 - (-1/16)| = |8/16| = 1/2Perfect, so both distances are equal to 1/2, as per the definition of a parabola.Therefore, the x-coordinate of point M is 7/16.

answer：Okay, so I need to find the derivative of the function f(x) = (2 + x)^2 - 3x and then evaluate it at x = 1. Hmm, let's break this down step by step. First, I remember that to find the derivative, I can use basic differentiation rules. The function has two parts: (2 + x)^2 and -3x. I think I can differentiate each part separately and then combine the results.Starting with the first part, (2 + x)^2. I recall that the derivative of a function like (u)^n is n*u^(n-1)*u', where u is a function of x and u' is its derivative. This is called the chain rule. So, in this case, u = 2 + x, and n = 2. Let me compute the derivative of (2 + x)^2. The outer function is u^2, and its derivative is 2u. The inner function is u = 2 + x, and its derivative is 1 because the derivative of 2 is 0 and the derivative of x is 1. So, applying the chain rule, the derivative of (2 + x)^2 is 2*(2 + x)*1, which simplifies to 2*(2 + x).Okay, that seems straightforward. Now, moving on to the second part of the function, which is -3x. I remember that the derivative of a linear term like ax is just a. So, the derivative of -3x should be -3.Putting it all together, the derivative of f(x) is the derivative of (2 + x)^2 minus the derivative of 3x. So, f'(x) = 2*(2 + x) - 3.Wait, let me make sure I did that correctly. The original function is (2 + x)^2 - 3x, so the derivative should be 2*(2 + x) - 3. Yeah, that looks right.Now, I need to simplify this expression to make it easier to evaluate at x = 1. Let's distribute the 2 in the first term: 2*(2 + x) = 4 + 2x. So, f'(x) = 4 + 2x - 3.Combining like terms, 4 - 3 is 1, so f'(x) simplifies to 2x + 1. That seems much simpler.Alright, now I need to find f'(1). That means I substitute x = 1 into the derivative. So, f'(1) = 2*(1) + 1.Calculating that, 2*1 is 2, and 2 + 1 is 3. So, f'(1) should be 3.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the beginning, f(x) = (2 + x)^2 - 3x. Differentiating term by term, the derivative of (2 + x)^2 is 2*(2 + x), and the derivative of -3x is -3. Combining these gives 2*(2 + x) - 3, which simplifies to 4 + 2x - 3, and then to 2x + 1. Substituting x = 1 gives 2*1 + 1 = 3. Yeah, that all checks out.I think I've got it right. The derivative at x = 1 is 3.

answer：Okay, so I have this problem about a sequence where the sum of the first n terms is S_n. They give me that a_1 is 1, and also that 2S_n equals a_{n+1} for any positive integer n. I need to find a general formula for a_n.Alright, let me break this down. First, S_n is the sum of the first n terms of the sequence {a_n}. So, S_n = a_1 + a_2 + ... + a_n. They tell me that 2S_n = a_{n+1}. That means each term a_{n+1} is twice the sum of all the previous terms up to n.Given that a_1 is 1, let me try to compute the next few terms to see if I can spot a pattern.Starting with n=1: 2S_1 = a_2. Since S_1 is just a_1, which is 1, so 2*1 = a_2. Therefore, a_2 = 2.Now, n=2: 2S_2 = a_3. S_2 is a_1 + a_2 = 1 + 2 = 3. So, 2*3 = a_3, which means a_3 = 6.Next, n=3: 2S_3 = a_4. S_3 is a_1 + a_2 + a_3 = 1 + 2 + 6 = 9. So, 2*9 = a_4, which gives a_4 = 18.Continuing, n=4: 2S_4 = a_5. S_4 is 1 + 2 + 6 + 18 = 27. Thus, 2*27 = a_5, so a_5 = 54.Hmm, let me list out the terms I have so far:a_1 = 1a_2 = 2a_3 = 6a_4 = 18a_5 = 54Looking at these, I notice that starting from a_2, each term seems to be three times the previous term. Let me check:a_2 = 2a_3 = 6 = 3*2a_4 = 18 = 3*6a_5 = 54 = 3*18Yes, that seems consistent. So, starting from a_2, the sequence is geometric with a common ratio of 3.But a_1 is 1, which doesn't fit into this pattern. So, maybe the sequence is a combination of a first term and then a geometric sequence starting from the second term.So, perhaps the general formula for a_n is:a_n = 1 when n=1,and for n ≥ 2, a_n = 2 * 3^{n-2}Let me test this formula to see if it works.For n=2: 2 * 3^{0} = 2*1 = 2, which matches a_2.For n=3: 2 * 3^{1} = 6, which matches a_3.For n=4: 2 * 3^{2} = 18, which matches a_4.For n=5: 2 * 3^{3} = 54, which matches a_5.Good, so this seems to hold.But let me make sure this is consistent with the given condition 2S_n = a_{n+1}.Let me compute S_n using the formula for a_n and see if 2S_n indeed equals a_{n+1}.First, S_n is the sum from k=1 to n of a_k.Given that a_1 = 1, and for k ≥ 2, a_k = 2 * 3^{k-2}.So, S_n = a_1 + sum from k=2 to n of a_k= 1 + sum from k=2 to n of 2 * 3^{k-2}Let me adjust the index for the sum to make it easier. Let m = k - 2, so when k=2, m=0, and when k=n, m = n - 2.So, sum from k=2 to n of 2 * 3^{k-2} = sum from m=0 to n-2 of 2 * 3^{m}This is a geometric series with first term 2*3^0 = 2, common ratio 3, and number of terms (n - 2 + 1) = n - 1.The sum of a geometric series is given by S = a1*(r^{n} - 1)/(r - 1), where a1 is the first term, r is the common ratio, and n is the number of terms.So, sum from m=0 to n-2 of 2 * 3^{m} = 2*(3^{n-1} - 1)/(3 - 1) = 2*(3^{n-1} - 1)/2 = 3^{n-1} - 1Therefore, S_n = 1 + (3^{n-1} - 1) = 3^{n-1}So, S_n = 3^{n-1}Now, let's compute 2S_n: 2*3^{n-1}According to the given condition, 2S_n = a_{n+1}So, a_{n+1} = 2*3^{n-1}Which implies that a_n = 2*3^{n-2} for n ≥ 2Which is exactly what I found earlier.Therefore, the general term is:a_n = 1 when n=1,and a_n = 2*3^{n-2} when n ≥ 2.So, the answer is a piecewise function:a_n = { 1, if n=1; 2*3^{n-2}, if n ≥ 2 }I think that's the solution.**Final Answer**The general term of the sequence is boxed{begin{cases}1, & n=1 2times3^{n-2}, & ngeqslant 2end{cases}}.

answer：First, I need to determine the principal amount (P) that was invested at a certain simple interest rate (R%) for 9 years. The problem states that if the interest rate had been 5% higher, the interest earned would have been ₹1350 more.I'll start by setting up the equations for the simple interest at the original rate and the increased rate. The simple interest formula is:[ text{Simple Interest (SI)} = frac{text{Principal (P)} times text{Rate (R)} times text{Time (T)}}{100} ]For the original rate (R%), the interest (I₁) over 9 years is:[ I₁ = frac{P times R times 9}{100} ]For the increased rate (R + 5)%, the interest (I₂) over the same period is:[ I₂ = frac{P times (R + 5) times 9}{100} ]According to the problem, the difference between the two interests is ₹1350:[ I₂ - I₁ = 1350 ]Substituting the expressions for I₁ and I₂:[ frac{P times (R + 5) times 9}{100} - frac{P times R times 9}{100} = 1350 ]Simplifying the equation:[ frac{P times 9 times 5}{100} = 1350 ][ frac{45P}{100} = 1350 ]Solving for P:[ 45P = 1350 times 100 ][ 45P = 135000 ][ P = frac{135000}{45} ][ P = 3000 ]Therefore, the principal amount is ₹3000.

answer：Okay, so I have this problem about a function f(x) that's even and periodic with period 4. It's defined as f(x) = x + 2 on the interval [0, 2]. I need to find the distance between the closest two points where the line y = 4 intersects the graph of f(x). Hmm, let me try to break this down step by step.First, since f(x) is an even function, that means f(-x) = f(x). So whatever the graph looks like on the right side of the y-axis, it's mirrored on the left side. That's helpful because I can focus on understanding the function for x ≥ 0 and then just mirror it for x < 0.Next, the function is periodic with period 4. That means every 4 units along the x-axis, the function repeats itself. So if I can figure out what the function looks like between x = 0 and x = 4, I can extend that pattern indefinitely in both directions.Now, on the interval [0, 2], f(x) is given as x + 2. So that's a straight line starting at (0, 2) and going up to (2, 4). Since the function is even, on the interval [-2, 0], it should be a mirror image. So for x between -2 and 0, f(x) would be -x + 2. That makes sense because when x is negative, -x is positive, so it's like reflecting the right side over the y-axis.But what about between x = 2 and x = 4? Since the function is periodic with period 4, the behavior from x = 2 to x = 4 should be the same as from x = -2 to x = 0, right? Wait, no, that's not quite right. Let me think. The function is periodic, so f(x + 4) = f(x). So if I shift the graph of f(x) 4 units to the left or right, it should look the same.So, if I already have f(x) defined on [0, 2] as x + 2, and on [-2, 0] as -x + 2, then for x between 2 and 4, I can use the periodicity. Let me see: if x is in [2, 4], then x - 4 is in [-2, 0]. So f(x) = f(x - 4). But f(x - 4) is defined as - (x - 4) + 2 because x - 4 is in [-2, 0]. So f(x) = -x + 4 + 2 = -x + 6? Wait, that doesn't seem right because when x = 2, f(2) should be 4, and if I plug x = 2 into -x + 6, I get 4, which is correct. But when x = 4, f(4) should be equal to f(0), which is 2. Plugging x = 4 into -x + 6 gives 2, which is also correct. So actually, on [2, 4], f(x) is -x + 6. Hmm, that seems a bit more complicated than I thought.Wait, maybe I can approach it differently. Since the function is periodic with period 4, the graph from x = 0 to x = 4 should repeat every 4 units. So if I can sketch the graph from x = 0 to x = 4, I can then just copy that pattern to the left and right.From x = 0 to x = 2, it's a straight line from (0, 2) to (2, 4). Then, from x = 2 to x = 4, since the function is periodic, it should mirror the part from x = -2 to x = 0, but shifted by 4 units. Wait, no, because f(x + 4) = f(x), so the graph from x = 2 to x = 4 is the same as from x = -2 to x = 0. But from x = -2 to x = 0, f(x) is -x + 2. So shifting that 4 units to the right, the function from x = 2 to x = 4 would be f(x) = -(x - 4) + 2 = -x + 6. So that checks out.So, putting it all together, the function f(x) is defined as:- For x in [-2, 0]: f(x) = -x + 2- For x in [0, 2]: f(x) = x + 2- For x in [2, 4]: f(x) = -x + 6And this pattern repeats every 4 units.Now, I need to find where y = 4 intersects this function. So I need to solve f(x) = 4 for x.Let's consider each interval separately.1. For x in [-2, 0]: f(x) = -x + 2 = 4 Solving for x: -x + 2 = 4 => -x = 2 => x = -2 So x = -2 is a solution.2. For x in [0, 2]: f(x) = x + 2 = 4 Solving for x: x + 2 = 4 => x = 2 So x = 2 is a solution.3. For x in [2, 4]: f(x) = -x + 6 = 4 Solving for x: -x + 6 = 4 => -x = -2 => x = 2 Wait, x = 2 is already in the previous interval. So in [2, 4], the solution is x = 2, but that's the boundary point between [0, 2] and [2, 4]. So actually, x = 2 is a solution in both intervals, but it's the same point.But wait, let me check the function at x = 2. From the left (x approaching 2 from below), f(x) approaches 4. From the right (x approaching 2 from above), f(x) also approaches 4. So x = 2 is indeed a single point where f(x) = 4.Now, considering the periodicity, the function repeats every 4 units. So the solutions will also repeat every 4 units. So the solutions are x = -2 + 4k and x = 2 + 4k for any integer k.So the points where f(x) = 4 are at x = ..., -6, -2, 2, 6, 10, ... and so on.Now, the problem asks for the distance between the closest two points of intersection. So looking at the solutions, the closest points are x = -2 and x = 2. The distance between them is |2 - (-2)| = 4 units.Wait, but let me think again. Is there any closer intersection point? For example, between x = 2 and x = 6, the next solution is x = 6, which is 4 units away from x = 2. Similarly, between x = -6 and x = -2, it's also 4 units. So yes, the closest two points are 4 units apart.But just to make sure, let me visualize the graph. From x = -2 to x = 0, it's a line going from ( -2, 4) down to (0, 2). Then from x = 0 to x = 2, it goes up to (2, 4). Then from x = 2 to x = 4, it goes back down to (4, 2). So the peaks at x = -2, 2, 6, etc., all at y = 4. So the distance between each peak is 4 units. So the closest two points are 4 units apart.Therefore, the distance between the closest two points where y = 4 intersects f(x) is 4.

answer：Okay, so I have this integral to evaluate: it's written as -∫ from 1 to 2 of x dx. Hmm, let me make sure I understand the notation correctly. The integral is from 1 to 2, and the integrand is x. There's also a negative sign in front of the integral, so I need to remember that when I compute the result.First, I recall that the integral of x with respect to x is a standard integral. The antiderivative of x is (1/2)x², right? Because when you take the derivative of (1/2)x², you get x. So, that seems correct.Now, since this is a definite integral from 1 to 2, I need to evaluate the antiderivative at the upper limit, which is 2, and then subtract the value of the antiderivative at the lower limit, which is 1. So, putting that together, it would be (1/2)(2)² minus (1/2)(1)².Let me compute that step by step. First, (1/2)(2)²: 2 squared is 4, and half of 4 is 2. Then, (1/2)(1)²: 1 squared is 1, and half of 1 is 0.5. So, subtracting these, I get 2 minus 0.5, which is 1.5. But wait, there's a negative sign in front of the integral. So, I need to multiply this result by -1. That would give me -1.5, which is the same as -3/2. Let me double-check my steps to make sure I didn't make a mistake. The antiderivative of x is indeed (1/2)x². Evaluating at 2 gives (1/2)(4) = 2, and at 1 gives (1/2)(1) = 0.5. Subtracting 0.5 from 2 gives 1.5. Multiplying by -1 gives -1.5 or -3/2. That seems right.I wonder if there's another way to approach this problem. Maybe using geometry? The integral of x from 1 to 2 represents the area under the curve y = x from x = 1 to x = 2. That area is a trapezoid, right? The bases are at x=1 and x=2, with lengths 1 and 2 respectively, and the height is 1 unit (from x=1 to x=2). The area of a trapezoid is (base1 + base2)/2 * height. So, that would be (1 + 2)/2 * 1 = 1.5. Again, with the negative sign, it becomes -1.5 or -3/2. Hmm, both methods give me the same result, so I feel more confident now that -3/2 is the correct answer.Just to be thorough, let me consider if I interpreted the integral correctly. The integral is written as -∫ from 1 to 2 of x dx. So, the limits are 1 and 2, and the integrand is x. The negative sign is outside the integral, so it's multiplying the entire integral. If I had misread it as integrating from 2 to 1, that would change the sign as well, but since it's from 1 to 2, I think my original calculation is correct.Another thing to check is if I applied the Fundamental Theorem of Calculus properly. The theorem states that the definite integral from a to b of f(x) dx is F(b) - F(a), where F is the antiderivative of f. I used F(2) - F(1), which is correct. So, (1/2)(2)² - (1/2)(1)² is indeed 2 - 0.5 = 1.5, and then multiplied by -1 gives -1.5.I think I've covered all the bases here. I evaluated the integral using the antiderivative, checked it with a geometric interpretation, and made sure I applied the theorem correctly. The negative sign was crucial, and without it, the answer would have been positive 1.5. But since it's there, the result is negative.So, putting it all together, the value of the integral is -3/2. Looking at the options given:A: -3/2B: 3/2C: -1D: 1The correct answer is A: -3/2.