Mathematics

The total cost per share is Rs. 25 face value + Rs. 5 premium + Rs. 0.25 brokerage = Rs. 30.25. The dividend is 7.5% on the face value of Rs. 25, which is Rs. 1.875 per share. For 88 shares, income = 88 × 1.875 = Rs. 165.

The annual income from Rs. 2500 of 8% stock is calculated based on the face value of the stock, not the purchase price. The income is 8% of Rs. 2500, which equals Rs. 200. The stock price at 106 means the stock is bought at a premium (Rs. 2650), but the dividend income remains based on the face value. Therefore, the correct annual income derived from the stock is Rs. 200, which corresponds to option C.

The effective annual rate (EAR) is calculated using the formula EAR = (1 + nominal rate / number of compounding periods)^(number of compounding periods) - 1. For a nominal rate of 6% per annum compounded half yearly (2 compounding periods), EAR = (1 + 0.06/2)^2 - 1 = (1.03)^2 - 1 = 1.0609 - 1 = 0.0609 or 6.09%. Therefore, the correct effective annual rate is 6.09%, which corresponds to option D.

The first man earns simple interest: 6000 × 5% × 2 = Rs 600. The second man earns compound interest: 5000 × (1 + 8%)^2 - 5000 = 5000 × 1.1664 - 5000 = Rs 832. The difference is 832 - 600 = Rs 232, which matches option B.

The dividend rate is calculated as the income received divided by the total face value of the shares, expressed as a percentage. The man invested Rs. 1552 at a price of Rs. 97 per share, so the number of shares he bought is 1552 / 97 = 16 shares. Each share has a face value of Rs. 100, so the total face value is 16 × 100 = Rs. 1600. The income received is Rs. 128. Therefore, the dividend rate = (128 / 1600) × 100 = 8%. Hence, the correct dividend rate is 8%, which corresponds to option B.

The difference between compound interest (CI) and simple interest (SI) for 2 years at 8% per annum can be calculated using the formula: Difference = P × r², where r is the rate in decimal form. Here, the difference is Rs 786, and the rate is 8% or 0.08. So, 786 = P × (0.08)² = P × 0.0064. Solving for P gives P = 786 / 0.0064 = 122,812.5 approximately. Among the given options, Rs 120,000 (option C) is the closest to the calculated principal. Therefore, the correct answer is Rs 120,000.

The amount after 2 years is Rs. 2420 and after 3 years is Rs. 2662. The increase from the 2nd to the 3rd year is Rs. 2662 - Rs. 2420 = Rs. 242. This increase represents the interest for one year on Rs. 2420. The rate of interest is therefore (242 / 2420) × 100 = 10%. This matches option B. The compound interest formula confirms this since (2662 / 2420) = 1.1, so the rate is 10%. Hence, the correct answer is 10%.

The compound interest earned in the third year is Rs.1440 and in the second year is Rs.1200. The ratio of the third year's interest to the second year's interest is 1440/1200 = 1.2. This ratio equals (1 + r/100), where r is the rate of interest. Solving for r, we get 1 + r/100 = 1.2, so r/100 = 0.2, which means r = 20%. Therefore, the rate of interest is 20% per annum, which corresponds to option C.

The speed of the man 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 ¼ minutes = 11.25/60 = 0.1875 hours, Time with stream = 7 ½ minutes = 7.5/60 = 0.125 hours. Speed against stream = 0.75 / 0.1875 = 4 km/hr. Speed with stream = 0.75 / 0.125 = 6 km/hr. Speed in still water = (4 + 6) / 2 = 5 km/hr. Therefore, the correct answer is D (5 km/hr).

The simple interest (S.I.) for 2 years is Rs. 660, so the interest for 1 year is Rs. 330. Using the formula S.I. = (P × R × T)/100, and knowing the compound interest (C.I.) for 2 years is Rs. 696.30, the rate of interest is calculated as 10%.