Mathematics

To simplify 1095/1168, we find the greatest common divisor (GCD) of 1095 and 1168, which is 43. Dividing numerator and denominator by 43 gives 25/26, making option D correct.

Let the radius of the base be r. The volume of the cone is $\frac{1}{3}\pi r^{2}h$. Let the height of the cylinder be H. The volume of the cylinder is $\pi r^{2}H$. The total volume of the solid is the sum of the volumes of the cylinder and the cone: $\pi r^{2}H+\frac{1}{3}\pi r^{2}h$. According to the problem, this total volume is three times the volume of the cone, so: $$\pi r^{2}H+\frac{1}{3}\pi r^{2}h=3\times \frac{1}{3}\pi r^{2}h=\pi r^{2}h$$ Dividing both sides by $\pi r^2$ gives: $$H+\frac{1}{3}h=h$$ Solving for H: $$H=h-\frac{1}{3}h=\frac{2}{3}h$$ Therefore, the height of the cylinder is $\frac{2}{3}h$, which corresponds to option C.

The highest common factor (HCF) of 36 and 84 is 12, as 12 is the largest number that divides both 36 and 84 without leaving a remainder.

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.

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.

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 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.

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 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.

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.

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 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).

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 surface area of a sphere is proportional to the square of its diameter. Given the diameters are in ratio 1:2, the surface areas are in ratio 1²:2² = 1:4.

The volume of a cone is given by (1/3)πr²h. If the height is increased by 100%, it doubles, so the new volume is twice the original volume, meaning a 100% increase.

First, convert the flow speed from km/h to m/min: 3 km/h = 3000 m/60 min = 50 m/min. The cross-sectional area of the river is depth × width = 2 m × 45 m = 90 m². The volume flow per minute is area × speed = 90 m² × 50 m/min = 4500 m³/min. However, option A is 4500 m³, which matches this calculation. Therefore, correct answer is A, not D.

The volume of each cube is the cube of its edge length: 6³ = 216 cm³, 8³ = 512 cm³, and 10³ = 1000 cm³. The total volume after melting is 216 + 512 + 1000 = 1728 cm³. The edge of the new cube is the cube root of 1728, which is 12 cm.

First, convert the room dimensions to centimeters: length = 624 cm, width = 432 cm. The largest square tile size that can exactly cover the floor is the greatest common divisor (GCD) of 624 and 432, which is 48 cm. The number of tiles needed is (624/48) × (432/48) = 13 × 9 = 117 tiles. However, since 117 is option D, and the calculation confirms 117 tiles, the correct answer is D, not C. Therefore, the original correct_answer 'D' is correct. The explanation clarifies the method to find the least number of square tiles.

First, convert all dimensions to the same unit (cm). The wall volume is 2500 cm * 200 cm * 75 cm = 37,500,000 cm³. The brick volume is 20 cm * 10 cm * 7.5 cm = 1500 cm³. Dividing the wall volume by the brick volume gives 37,500,000 / 1500 = 25,000 bricks. However, this calculation assumes no mortar gaps. The options given suggest the closest correct answer is 25,000, which corresponds to option D. But since the calculation matches 25,000 exactly, the correct answer should be D, not A. Therefore, the original correct_answer 'D' is correct. The explanation clarifies the calculation steps.

The number of plants leaves a remainder of 5 when divided by 6, 8, 10, and 12, but is divisible by 13. The least common multiple (LCM) of 6, 8, 10, and 12 is 120. The number of plants is therefore 120k + 5. Checking multiples of 120 plus 5 for divisibility by 13, 845 (120*7 + 5) is divisible by 13, so the number of plants is 845.

The total surface area of a sphere is given by the formula 4πr². A hemisphere is half of a sphere, but its total surface area includes both the curved surface area and the base (a circle). The curved surface area of a hemisphere is half that of the sphere, i.e., 2πr², and the base area is πr². Therefore, the total surface area of a hemisphere is 3πr². The ratio of the total surface area of a sphere to that of a hemisphere is 4πr² : 3πr², which simplifies to 4:3. Hence, the correct ratio is 4:3, corresponding to option B.

Given the floor area (length × width) = 20 m², longer wall area (length × height) = 15 m², and shorter wall area (width × height) = 12 m², we can find the dimensions of the room. Let length = L, width = W, and height = H. From the equations: L × W = 20, L × H = 15, and W × H = 12. Multiplying all three gives (L × W × H)² = 20 × 15 × 12 = 3600, so L × W × H = 60 m³, which is the volume of the room. The question asks for the volume of the cube with the same volume as the room, so the cube's volume is also 60 m³. Therefore, the correct answer is option C (60 m³).

Given the LCM (495) and HCF (5) of two numbers and their sum (100), we can find the numbers using the relationships: product = HCF × LCM = 5 × 495 = 2475, and sum = 100. Let the numbers be a and b. Then a + b = 100 and ab = 2475. Using the formula for difference: (a - b)^2 = (a + b)^2 - 4ab = 100^2 - 4 × 2475 = 10000 - 9900 = 100, so a - b = 10. Therefore, the difference between the two numbers is 10, which corresponds to option A.

The area of the four walls of a rectangular room is calculated using the formula: Area = 2 × height × (length + breadth). Here, length = 15 m, breadth = 4 m, and height = 3 m. So, the area = 2 × 3 × (15 + 4) = 2 × 3 × 19 = 114 m². The cost of whitewashing is given as 50 paise per m², which is Rs. 0.5 per m². Therefore, the total cost = 114 × 0.5 = Rs. 57. Hence, the correct answer is Rs. 57, which corresponds to option B.

The numbers are in the ratio 1:2:3, so let the numbers be 12x, 24x, and 36x. Since their H.C.F is 12, the common factor x must be 1. Therefore, the numbers are 12, 24, and 36.