Mathematics

The longest pole that can be kept in the room corresponds to the space diagonal of the cuboid formed by the room's dimensions. Using the formula for the space diagonal $d=\sqrt{l^2+b^2+h^2}$, we get $d=\sqrt{12^2+4^2+3^2}=\sqrt{144+16+9}=\sqrt{169}=13$ meters. Therefore, the longest pole is 13 meters long.

The wall volume is calculated as 10 m × 4 m × 0.5 m = 20 m³. Since 10% of the volume is mortar, the volume occupied by bricks is 90% of 20 m³, which is 18 m³. Each brick measures 25 cm × 15 cm × 8 cm, which converts to 0.25 m × 0.15 m × 0.08 m = 0.003 m³ per brick. The number of bricks required is the brick volume divided by the volume of one brick: 18 m³ ÷ 0.003 m³ = 6,000 bricks. This matches option C. The initial confusion in the question's explanation about wall thickness being 5 m instead of 0.5 m was incorrect. The correct thickness is 0.5 m as stated in the question, leading to the correct answer of 6,000 bricks (option C).

Let the three numbers be 3x, 4x, and 5x. Since 3, 4, and 5 have no common factors other than 1, the HCF of the three numbers is x. The LCM of 3, 4, and 5 is 60, so the LCM of the three numbers is 60x. Given that the LCM is 2400, we have 60x = 2400, which gives x = 40. Therefore, the HCF of the three numbers is 40.

The surface area of a rectangular box (cuboid) is calculated using the formula 2(lw + lh + wh), where l, w, and h are the length, width, and height respectively. For the box with dimensions 25 cm × 15 cm × 8 cm, the surface area is 2(25×15 + 25×8 + 15×8) = 2(375 + 200 + 120) = 2(695) = 1390 cm². Therefore, the area of cardboard needed to make the box is 1390 cm², which corresponds to option A.

The curved surface area of a cylinder is given by 2πrh. Since the heights are the same and the radii are in the ratio 1:2, the curved surface areas are in the ratio 1×h : 2×h, which simplifies to 1:2.

Let the three co-prime numbers be a, b, and c. Given a*b = 551 and b*c = 1073. Since the numbers are co-prime in pairs, b must be 1 (the only common factor). Then a = 551 and c = 1073. The sum is 551 + 1 + 1073 = 1625, which is not among the options, so this suggests b is not 1. Instead, factorize 551 = 19 * 29 and 1073 = 29 * 37. Since the numbers are co-prime in pairs, the common factor between the two products is 29, so b = 29, a = 19, and c = 37. Sum = 19 + 29 + 37 = 85, which corresponds to option C.

The volume of a cylinder is proportional to the square of its radius when the height is constant. Given the radii ratio 1:3, the volume ratio is 1²:3² = 1:9. Since the first cylinder's volume is 40 cc, the second's volume is 40 × 9 = 360 cc.

The question asks for the least number which, when doubled, is exactly divisible by 12, 18, 21, and 30. This means that twice the number must be a common multiple of these numbers. The least common multiple (LCM) of 12, 18, 21, and 30 is 1260. Therefore, 2 × number = 1260, so number = 1260 ÷ 2 = 630. Among the options given, 630 corresponds to option B, making it the correct answer.

The curved surface area of a sphere is given by 4πr². Given 64π = 4πr², dividing both sides by 4π gives r² = 16, so r = 4 cm. However, the options provided do not include 4 cm as the correct radius for 64π surface area. Rechecking the calculation: 4πr² = 64π implies r² = 16, so r = 4 cm, which matches option C. Therefore, the original correct_answer 'C' is correct. The explanation was missing and is now provided. Difficulty is set to easy as this is a straightforward calculation.

Given the ratio 6:5:4, let the dimensions be 6x, 5x, and 4x. The total surface area of a cuboid is 2(lb + bh + hl). Substituting, 2(6x*5x + 5x*4x + 6x*4x) = 33300, which simplifies to 2(30x^2 + 20x^2 + 24x^2) = 33300, or 2(74x^2) = 33300, so 148x^2 = 33300, giving x^2 = 225, and x = 15. Therefore, length = 6*15 = 90 cm, breadth = 5*15 = 75 cm, height = 4*15 = 60 cm, matching option D.