Mathematics

The difference between compound interest (CI) and simple interest (SI) for 2 years at the same rate R% on principal P is given by the formula: CI - SI = P * (R/100)^2. Given the difference is Rs. 72 and P = 5000, we have 72 = 5000 * (R/100)^2. Simplifying, (R/100)^2 = 72/5000 = 0.0144, so R/100 = 0.12, which means R = 12%. Therefore, the rate of interest is 12%. The original answer given as option D (8%) is incorrect; the correct answer is option B (12%).

The speed of the man in still water is the average of his speeds against and along the current. Given speed against current = 3.5 kmph and speed in still water = 5 kmph, the speed along the current = 2 × 5 - 3.5 = 6 kmph.

Using the compound interest formula A = P(1 + r)^t, we check values: for 2 years, A = 800 × (1.05)^2 = 800 × 1.1025 = 882. Hence, the amount becomes Rs. 882 in 2 years.

The man's speed in still water is the average of his speed against the stream and with the stream. Given: Distance = 0.75 km. Time against stream = 11 minutes = 11/60 hours. Time down stream = 7 minutes = 7/60 hours. Speed against stream = 0.75 / (11/60) = 4.09 km/hr. Speed down stream = 0.75 / (7/60) = 6.43 km/hr. Speed in still water = (4.09 + 6.43) / 2 = 5.26 km/hr, which rounds to 5 km/hr. Therefore, the correct answer is D (5 km/hr).

The sum after 3 years is Rs.6690 and after 6 years is Rs.10035. Using compound interest, the amount after 6 years is the amount after 3 years multiplied by (1 + r)^3. So, (1 + r)^3 = 10035 / 6690 = 1.5. Therefore, the principal P = 6690 / (1.5) = Rs.4460. However, this calculation is incorrect because (1 + r)^3 = 10035 / 6690 = 1.5, so (1 + r) = 1.5^(1/3) ≈ 1.1447. Then, P = 6690 / (1.1447)^3 = 6690 / 1.5 = Rs.4460. This matches option C, so the correct answer is Rs.4460.

The effective annual rate (EAR) is calculated using the formula EAR = (1 + nominal rate/m)^m - 1, where m is the number of compounding periods per year. For a nominal rate of 6% per annum compounded half-yearly (m=2), EAR = (1 + 0.06/2)^2 - 1 = (1.03)^2 - 1 = 1.0609 - 1 = 0.0609 or 6.09%. This matches option D exactly, so the correct answer is 6.09%.

The simple interest (SI) on Rs. 1200 for one year at 10% per annum is calculated as SI = (1200 × 10 × 1)/100 = Rs. 120. The compound interest (CI) compounded half-yearly means the interest rate per half year is 10%/2 = 5%, and there are 2 compounding periods in one year. CI = 1200 × (1 + 0.05)^2 - 1200 = 1200 × 1.1025 - 1200 = Rs. 123. The difference between compound interest and simple interest is Rs. 123 - Rs. 120 = Rs. 3. Therefore, the correct answer is Rs. 3, which corresponds to option B.

The downstream speed is calculated as 100 km / 10 hours = 10 km/h, and the upstream speed is 75 km / 15 hours = 5 km/h. The speed of the stream is half the difference between downstream and upstream speeds: (10 - 5) / 2 = 2.5 km/h. This matches option D (2 (1/2) km/h). Therefore, the correct answer is D, not B.

Since A runs 1 2/3 (which is 5/3) times as fast as B, the ratio of their speeds is 5:3. If A gives B a start of 80 m, let the total distance to the winning post be x meters. B runs (x - 80) meters while A runs x meters. Since time is distance/speed and they finish together, x/5 = (x - 80)/3. Solving gives x = 270 m.

The speed downstream is the sum of the speed in still water and the speed of the stream, so 15 kmph + 5 kmph = 20 kmph.