answer:To find the speed of the current, we first need to determine the speed of the boat when it is rowing downstream. Since the man can row at 22 kmph in still water, we need to convert this speed to meters per second (m/s) to match the units of the distance covered downstream. 1 km = 1000 meters 1 hour = 3600 seconds So, 22 kmph in m/s is: 22 kmph * (1000 m / 1 km) * (1 hour / 3600 s) = 22 * 1000 / 3600 m/s 22 * 1000 / 3600 = 6.111 m/s (approximately) This is the speed of the boat in still water. Now, let's find the speed of the boat downstream. The man takes 11.519078473722104 seconds to cover 80 meters downstream. To find the speed downstream, we use the formula: Speed = Distance / Time Speed downstream = 80 meters / 11.519078473722104 seconds Speed downstream ≈ 6.944 m/s (approximately) Now, we know the speed of the boat in still water (6.111 m/s) and the speed of the boat downstream (6.944 m/s). The difference between these two speeds is the speed of the current. Speed of current = Speed downstream - Speed of boat in still water Speed of current ≈ 6.944 m/s - 6.111 m/s Speed of current ≈ 0.833 m/s To convert this speed back to kmph: Speed of current in kmph = 0.833 m/s * (3600 s / 1 hour) * (1 km / 1000 m) Speed of current in kmph ≈ 0.833 * 3600 / 1000 Speed of current in kmph ≈ 2.9988 kmph Therefore, the speed of the current is approximately boxed{2.9988} kmph.

answer:1. We start with the given sequence: [ c_{n} = (n+1) int_{0}^{1} x^{n} cos(pi x) , dx ] 2. To evaluate the integral, we use integration by parts. Let ( u = x^n ) and ( dv = cos(pi x) , dx ). Then, ( du = n x^{n-1} , dx ) and ( v = frac{sin(pi x)}{pi} ). Applying integration by parts: [ int_{0}^{1} x^n cos(pi x) , dx = left. frac{x^n sin(pi x)}{pi} right|_{0}^{1} - int_{0}^{1} frac{n x^{n-1} sin(pi x)}{pi} , dx ] Evaluating the boundary term: [ left. frac{x^n sin(pi x)}{pi} right|_{0}^{1} = frac{sin(pi)}{pi} - frac{0}{pi} = 0 ] Thus, we have: [ int_{0}^{1} x^n cos(pi x) , dx = -frac{n}{pi} int_{0}^{1} x^{n-1} sin(pi x) , dx ] 3. We now need to evaluate the integral involving (sin(pi x)). Using integration by parts again, let ( u = x^{n-1} ) and ( dv = sin(pi x) , dx ). Then, ( du = (n-1) x^{n-2} , dx ) and ( v = -frac{cos(pi x)}{pi} ). Applying integration by parts: [ int_{0}^{1} x^{n-1} sin(pi x) , dx = left. -frac{x^{n-1} cos(pi x)}{pi} right|_{0}^{1} + int_{0}^{1} frac{(n-1) x^{n-2} cos(pi x)}{pi} , dx ] Evaluating the boundary term: [ left. -frac{x^{n-1} cos(pi x)}{pi} right|_{0}^{1} = -frac{cos(pi)}{pi} + frac{1}{pi} = 0 ] Thus, we have: [ int_{0}^{1} x^{n-1} sin(pi x) , dx = frac{n-1}{pi} int_{0}^{1} x^{n-2} cos(pi x) , dx ] 4. Substituting back, we get: [ int_{0}^{1} x^n cos(pi x) , dx = -frac{n}{pi} cdot frac{n-1}{pi} int_{0}^{1} x^{n-2} cos(pi x) , dx ] This recursive relation suggests that: [ int_{0}^{1} x^n cos(pi x) , dx = (-1)^k frac{n(n-1)cdots(n-k+1)}{pi^k} int_{0}^{1} x^{n-k} cos(pi x) , dx ] As ( n to infty ), the integral term tends to zero, and we can approximate: [ c_n approx -1 ] 5. Given that ( lambda = -1 ), we rewrite the sequence: [ c_n + 1 = -frac{pi^2}{(n+2)(n+3)} c_{n+2} ] 6. We now need to find: [ lim_{n to infty} frac{c_{n+1} - lambda}{c_n - lambda} = lim_{n to infty} frac{c_{n+1} + 1}{c_n + 1} ] Using the recursive relation: [ frac{c_{n+1} + 1}{c_n + 1} = frac{-frac{pi^2}{(n+3)(n+4)} c_{n+3}}{-frac{pi^2}{(n+2)(n+3)} c_{n+2}} = frac{(n+2)(n+3)}{(n+3)(n+4)} cdot frac{c_{n+3}}{c_{n+2}} ] As ( n to infty ), this ratio approaches 1: [ lim_{n to infty} frac{c_{n+1} + 1}{c_n + 1} = 1 ] The final answer is (boxed{1})

answer:When y=0, we have x=0.5; When x=0, we have y=-1. Therefore, the coordinates of the intersection point between the line y=2x-1 and the x-axis are (0.5, 0), and the coordinates of the intersection point with the y-axis are (0, -1). So, the final answers are boxed{(0.5, 0)} for the intersection with the x-axis and boxed{(0, -1)} for the intersection with the y-axis.

answer:To solve this problem, we need to calculate the share of profit for each person based on their investment and the time they invested in the business. The profit is divided in the ratio of the product of the capital invested and the time for which it was invested. Suresh invested Rs. 18000 for 12 months. Rohan invested Rs. 12000, but he joined after 3 months, so he invested for 9 months. Sudhir invested Rs. 9000, and he joined after 4 months, so he invested for 8 months. Now, let's calculate the ratio of their investments: Suresh's investment ratio = Rs. 18000 * 12 months = 216000 Rohan's investment ratio = Rs. 12000 * 9 months = 108000 Sudhir's investment ratio = Rs. 9000 * 8 months = 72000 The total ratio of their investments = 216000 + 108000 + 72000 = 396000 Now, let's find the individual ratios: Suresh's ratio = 216000 / 396000 = 216 / 396 Rohan's ratio = 108000 / 396000 = 108 / 396 Sudhir's ratio = 72000 / 396000 = 72 / 396 We can simplify these ratios by dividing each by 36 (the greatest common divisor): Suresh's simplified ratio = 216 / 36 = 6 Rohan's simplified ratio = 108 / 36 = 3 Sudhir's simplified ratio = 72 / 36 = 2 Now, the ratio of their profits is 6:3:2. The difference between Rohan's and Sudhir's share is Rs. 352, which corresponds to 1 part in the ratio (since 3 - 2 = 1). So, 1 part = Rs. 352 The total ratio is 6 + 3 + 2 = 11 parts. Therefore, the total profit is 11 parts, and since 1 part is Rs. 352, the total profit is: Total profit = 11 parts * Rs. 352/part = Rs. 3872 So, the total profit at the end of the year was Rs. boxed{3872} .