Correct Option: B\n\nLet the Cost Price (CP) of the item be $C$.\n\nStep 1: Calculate the Marked Price (MP).\nThe shopkeeper marks up the item by $20\\%$ above its cost price.\n$\\text{MP} = C + 20\\% \\text{ of } C$\n$\\text{MP} = C + 0.20C$\n$\\text{MP} = 1.20C$\n\nStep 2: Calculate the Selling Price (SP) after the discount.\nHe offers a discount of $10\\%$ on the Marked Price.\n$\\text{SP} = \\text{MP} - 10\\% \\text{ of } \\text{MP}$\n$\\text{SP} = 1.20C - 0.10 \\times (1.20C)$\n$\\text{SP} = 1.20C - 0.12C$\n$\\text{SP} = 1.08C$\n\nStep 3: Equate the Selling Price to the given value and solve for Cost Price.\nGiven that the item is sold for $\\text{INR } 1080$.\nTherefore, $1.08C = 1080$\n$C = \\frac{1080}{1.08}$\nTo simplify the division, multiply the numerator and denominator by 100:\n$C = \\frac{108000}{108}$\n$C = 1000$\n\nThus, the initial cost price of the item was $\\text{INR } 1000$.