Mathematics

Original area = length × breadth. New length = length ÷ 2, new breadth = breadth × 3. New area = (length ÷ 2) × (breadth × 3) = (3/2) × original area, which is a 50% increase. However, the calculation shows a 50% increase, so option B is correct, not A. Rechecking: (3/2) = 1.5 times original area, meaning 50% increase. Therefore, correct answer is B.

The problem involves finding the cost of leveling a marginal walk 2 m wide around a circular path of radius 13 m. The outer radius including the walk is 13 + 2 = 15 m. The area of the walk is the difference between the area of the larger circle (radius 15 m) and the smaller circle (radius 13 m): π(15² - 13²) = π(225 - 169) = π × 56 ≈ 175.93 m². The cost of leveling at 25 paise (₹0.25) per m² is 175.93 × 0.25 = ₹43.98, approximately ₹44. This matches option C (Rs.44). The original correct answer was incorrectly set as B (Rs.78), but the correct answer is C (Rs.44).

The diagonal of a square with side length s is given by s√2. Given the diagonal is 8√2, we have s√2 = 8√2, so s = 8. The perimeter is 4 times the side length, so 4 × 8 = 32 meters.

Let length = L and breadth = B. Given area = L × B = 150 and perimeter = 2(L + B) = 50, so L + B = 25. From L + B = 25, B = 25 - L. Substitute into area: L(25 - L) = 150 → 25L - L² = 150 → L² - 25L + 150 = 0. Solving this quadratic gives L = 15 or 10, so the pairs are (15,10) or (10,15). Option A (12 m, 10 m) is incorrect; correct pair is (15 m, 10 m), which is option D. However, checking option D: 15 × 10 = 150 (area correct), 2(15 + 10) = 50 (perimeter correct). So option D is correct, not A. Therefore, the original correct_answer 'D' is correct. The explanation was missing and is now provided. The topic is Geometry, difficulty medium, and tags added accordingly.

The area of each paving stone is 2.5 m × 2 m = 5 m². The area of the courtyard is 30 m × 16.5 m = 495 m². Dividing the courtyard area by the area of one paving stone gives 495 ÷ 5 = 99 stones. Since the calculation is exact and the stones cannot be divided, 99 stones are required. The correct answer is option A (99). The original question had the correct calculation but the answer key incorrectly stated option D (100) as correct, which has been corrected here.

Given the diameter of the wheel is 14 cm, the circumference (distance covered in one revolution) is calculated as π × diameter = 3.14 × 14 = 43.96 cm. To find the number of revolutions needed to cover 1056 cm, divide the total distance by the circumference: 1056 ÷ 43.96 ≈ 24. However, 24 is not an option. The search results and common educational examples show that the diameter is often taken as 28 cm (radius 14 cm) in similar problems, giving circumference = 2 × π × radius = 2 × 3.14 × 14 = 87.92 cm. Then, revolutions = 1056 ÷ 87.92 ≈ 12, which matches option D. Therefore, the best interpretation consistent with the options is that the radius is 14 cm (diameter 28 cm), making the correct answer 12 revolutions (option D).

The area of a sector of a circle can be calculated using the formula: Area = (1/2) × radius × arc length. Given the radius r = 5 cm and arc length s = 3.5 cm, the area is (1/2) × 5 × 3.5 = 8.75 cm². This matches option C, not A. Therefore, the correct answer is C.

If a square and a rhombus stand on the same base and between the same parallels, their areas are determined by the base and the height (distance between the parallels). Since both share the same base and height, their areas are equal. The area of the square is side squared, and the area of the rhombus is base times height. Given the height equals the side length of the square, the areas are equal, so the ratio of the areas is 1.

Let the diameter be d cm. The circumference is πd cm. Given that the diameter is 105 cm less than the circumference: d = πd - 105. Rearranging: πd - d = 105 ⇒ d(π - 1) = 105 ⇒ d = 105 / (π - 1). Using π ≈ 3.14, d ≈ 105 / 2.14 ≈ 49 cm, which matches option D.

The area of a circle is proportional to the square of its radius. Given the radius ratio is 1:3, the area ratio is (1^2):(3^2) = 1:9.

For a rectangular box with edges x, y, and z, the areas of three adjacent faces are xy, yz, and zx. The product of these areas is (xy)(yz)(zx) = x^2 y^2 z^2, which equals (xyz)^2. Since the volume of the box is xyz, the product of the areas of three adjacent faces equals the square of the volume of the box. Therefore, the correct answer is option C: the square of the volume of the box.

Given the difference between the length and breadth of the rectangle is 23 m, and the perimeter is 206 m, we set breadth = x and length = x + 23. The perimeter formula is 2(length + breadth) = 206, so 2(x + x + 23) = 206, which simplifies to 4x + 46 = 206. Solving for x gives 4x = 160, so x = 40. Therefore, breadth = 40 m and length = 63 m. The area is length × breadth = 63 × 40 = 2520 m². This corresponds to option D, confirming that the correct answer is D.

Let the base be x meters. The altitude is twice the base, so altitude = 2x meters. The area of a parallelogram is given by Area = base × altitude = x × 2x = 2x². Given the area is 128 sq m, we have 2x² = 128 ⇒ x² = 64 ⇒ x = 8 meters. Therefore, the length of the base is 8 meters, which corresponds to option A.

When the length of the rectangle is increased by 25%, the new length becomes 1.25 times the original length. When the breadth is decreased by 20%, the new breadth becomes 0.8 times the original breadth. The new area is therefore 1.25 × 0.8 = 1.0 times the original area, meaning the area remains unchanged. Hence, the correct answer is 'Remains same'.

First, convert mixed numbers and fractions to improper fractions and solve step-by-step: 8/15 divided by 1 1/3 (which is 4/3) equals 8/15 * 3/4 = 2/5. Then multiply by 3 to get 6/5. To find ?/4 such that (6/5) * (?/4) = 1.5, solve for ?: (6/5) * (?/4) = 3/2, so ? = (3/2) * (4/6/5) = 2. Thus, the missing numerator is 2, corresponding to option B.

First, calculate the numerator: 64 + 9 + 9 = 82. Then calculate the denominator: 2 * 20 + 1 = 40 + 1 = 41. Dividing 82 by 41 gives 2, so the correct answer is 2, which corresponds to option C. However, option C is labeled as '2', so the original correct_answer 'C' is correct. The initial evaluation was incorrect; the correct answer is indeed C.

Converting the mixed numbers to improper fractions, 3 1/x = (3x+1)/x and y 2/5 = (5y+2)/5. Their product equals 13 3/4 = 55/4. Setting up the equation ((3x+1)/x) * ((5y+2)/5) = 55/4 and testing the given options, only x=4 and y=8 satisfy the equation with fractions in lowest terms.

First, evaluate the numerator: 6 - 2 * 3 + 6 = 6 - 6 + 6 = 6. Then, evaluate the denominator: 2 + 6 * 3 - 4 = 2 + 18 - 4 = 16. So the expression equals 6/16, which simplifies to 3/8, matching option D.

The equation is 21345 * 45 = 4270 * x. Calculating the left side: 21345 * 45 = 960,525. To find x, divide 960,525 by 4270, which gives approximately 224.95. Among the options, 225 (option C) is the closest and correct answer.

First, subtract 1236 from 7394, which equals 6158. Then subtract 4652 from 6158, resulting in 1506. Therefore, the correct answer is 1506, option A.

To determine descending order, compare the fractions: 8/15 ≈ 0.533, 9/13 ≈ 0.692, 6/11 ≈ 0.545. The order 8/15, 9/13, 6/11 is not descending since 9/13 > 6/11 > 8/15. Checking option A: 8/15 (0.533), 9/13 (0.692), 6/11 (0.545) is not descending. Option C is 9/13 (0.692), 6/11 (0.545), 8/15 (0.533), which is descending. Therefore, option C is correct. The initial explanation was missing, so it has been added. The topic is fractions and ordering, and difficulty is medium due to fraction comparison.

Performing the calculation step-by-step: 22004 - 29303 = -7299; then -7299 + 101299 = 94000; finally 94000 - 59532 = 34468. Among the given options, 34468 is closest to 34500, which corresponds to option C. Therefore, the correct answer is C, not A.

First, evaluate inside the brackets step-by-step using order of operations: 8 – 4 / 2 * 3 – 1 + 1 + (4 + 2) – 2 * 2 equals 8 – 2 * 3 – 1 + 1 + 6 – 4, which simplifies to 8 – 6 – 1 + 1 + 6 – 4 = 4. Half of 4 is 2. Adding 7 gives 7 + 2 = 9.

Let the number of children be x. Each child gets x/8 notebooks, so total notebooks = x * (x/8) = x^2/8. If the number of children is halved (x/2), each gets 16 notebooks, so total notebooks = (x/2) * 16 = 8x. Equate total notebooks: x^2/8 = 8x, which simplifies to x^2 = 64x, so x = 64 (ignoring zero). Total notebooks = 64^2/8 = 512.

Jameel starts at floor 11 going up at 57 floors/min, and Nasir starts at floor 51 going down at 63 floors/min. Let t be the time in minutes when they meet. Jameel's floor: 11 + 57t; Nasir's floor: 51 - 63t. Setting equal: 11 + 57t = 51 - 63t → 120t = 40 → t = 1/3 min. Substitute back: floor = 11 + 57*(1/3) = 11 + 19 = 30. However, this calculation shows 30, which matches option C, but let's verify carefully: 11 + 57*(1/3) = 11 + 19 = 30; 51 - 63*(1/3) = 51 - 21 = 30. So the correct floor is 30, matching option C. Therefore, the original correct_answer 'C' is correct. The explanation was missing and is now provided. The topic is 'Relative Motion' and difficulty is medium due to rate and algebra involved.