Mathematics

The profit shares of partners A, B, and C are proportional to their capitals. The capitals are Rs. 8000, Rs. 10000, and Rs. 12000 respectively, so the ratio of their capitals (and hence profit shares) is 8:10:12. Given B's profit share is Rs. 1500, each share corresponds to Rs. 150 (1500 ÷ 10). Therefore, A's share is 8 × 150 = Rs. 1200 and C's share is 12 × 150 = Rs. 1800. The difference between A and C's profit shares is Rs. 1800 - Rs. 1200 = Rs. 600. Among the options given (Rs. 300, Rs. 400, Rs. 500, Rs. 600), the correct difference is Rs. 600, which corresponds to option D.

A receives 30% of the total profit as a managing partner, and the remaining 70% is divided among A, B, and C in the ratio of their capitals (5000:6000:4000). Let total profit be P. A's total profit = 0.3P + (5000/15000)*0.7P = 0.3P + (1/3)*0.7P = 0.3P + 0.2333P = 0.5333P. B's profit = (6000/15000)*0.7P = 0.28P. C's profit = (4000/15000)*0.7P = 0.1867P. Given A's profit is Rs.200 more than B's and C's combined profit: 0.5333P = 0.28P + 0.1867P + 200 => 0.5333P = 0.4667P + 200 => 0.0666P = 200 => P = 3000. Hence, total profit is Rs.3000.

Option B (246) is the only number composed entirely of even digits, while the others (358, 134, 862) contain at least one odd digit. Therefore, 246 does not belong to the group.

Option A (27) is the only number that is not a prime number, while options B (37), C (47), and D (67) are all prime numbers. Therefore, 27 does not belong to the group.

Option B (KOMQO) is the only group where the letters do not follow a consistent alphabetical pattern or symmetry like the others. The other options have letters that form a pattern or symmetry in their positions.

Since A and B invest Rs.6000 and Rs.8000 respectively for the same duration (one year), their profit sharing ratio is directly proportional to their investments, which is 6000:8000 or 3:4.

The profit share among partners is proportional to the product of the amount invested and the time for which it is invested. Let A's investment be x. Then B invests 2x after 6 months, so for 6 months, and C invests 3x after 8 months, so for 4 months. Calculating the weighted investments: A = x * 12 = 12x, B = 2x * 6 = 12x, C = 3x * 4 = 12x. The ratio of their investments multiplied by time is 1:1:1, so the profit is shared equally. The total profit is Rs.18,000, so A's share is one-third of Rs.18,000, which is Rs.6,000. Therefore, the correct answer is option C (Rs.6000).

A invests Rs. 2000 for 1 year, B invests Rs. 3000 for 2 years, and C invests Rs. 4000 for 2 years. The profit shares are proportional to the product of investment and time: A = 2000×1 = 2000, B = 3000×2 = 6000, C = 4000×2 = 8000. Total = 2000 + 6000 + 8000 = 16000. A's share of the profit = (2000/16000) × 3200 = Rs. 400. Therefore, A's share in the profit is Rs. 400, which corresponds to option D.

The angle of elevation θ can be found using tan θ = opposite/adjacent = height/shadow length = (2√3)/2 = √3. Therefore, θ = 60°.

Kamal invested Rs. 9000 for 12 months, so his capital-time product is 9000 × 12 = 108000. Sameer invested Rs. 8000 for 7 months, so his capital-time product is 8000 × 7 = 56000. The total capital-time product is 108000 + 56000 = 164000. Sameer's share of profit = (56000 / 164000) × 6970 ≈ Rs. 2380. However, this matches option B, but the calculation shows option B is correct. Rechecking: 56000/164000 = 0.34146; 0.34146 × 6970 = 2380. So option B is correct. The initial explanation was incorrect. Therefore, the correct answer is B.