Mathematics

Given the product of two numbers is 2028 and their HCF is 13, we can write the numbers as 13a and 13b where HCF(a,b) = 1. Then, 13a × 13b = 2028 ⇒ 169ab = 2028 ⇒ ab = 12. The number of pairs (a,b) with product 12 and HCF 1 are (1,12) and (3,4), so there are 2 such pairs. Hence, the number of such pairs is 2.

The curved surface area of a cone is given by the formula π × radius × slant height. Here, radius = 4 cm and slant height = 12 cm, so the curved surface area = π × 4 × 12 = 48 π cm², which corresponds to option C.

The large cube has an edge length of 1 meter, which is equivalent to 10 decimeters (dm) since 1 m = 10 dm. The smaller cubes each have an edge length of 2 dm. To find how many smaller cubes fit along one edge of the large cube, divide 10 dm by 2 dm, which equals 5. Since the cubes fit perfectly along each dimension, the total number of smaller cubes that can be cut out is 5 × 5 × 5 = 125. Therefore, the correct answer is D (125).

The volume of one sphere with diameter 6 cm (radius 3 cm) is calculated as V = (4/3)πr³ = (4/3)π(3)³ = 36π cm³. The volume of the cylinder with height 45 cm and diameter 4 cm (radius 2 cm) is V = πr²h = π(2)²(45) = 180π cm³. The number of spheres that can be moulded to form the cylinder is the volume of the cylinder divided by the volume of one sphere: 180π / 36π = 5. Therefore, 5 spheres can be moulded to form the cylinder, making option C the correct answer.

The box is open, so it has five faces. The outer dimensions are 52 cm × 40 cm × 29 cm, and the thickness is 2 cm, so the inner dimensions are (52-4) cm × (40-4) cm × (29-2) cm = 48 cm × 36 cm × 27 cm. The volume of metal used is the difference between the outer and inner volumes: (52×40×29) - (48×36×27) = 60240 - 46656 = 13584 cm³. Since 1 cm³ weighs 0.5 g, total weight = 13584 × 0.5 = 6792 g = 6.792 kg, which is closest to option D (6.832 kg). Therefore, the correct answer is D, not B as originally stated.

The surface area of a cube is given by 6a² = 726, so a² = 121 and a = 11 m. The volume is a³ = 11³ = 1331 m³, which corresponds to option B.

Given the ratio of two numbers is 3:4 and their HCF is 4, the numbers can be expressed as 3×4=12 and 4×4=16. The LCM of two numbers is given by (Product of the numbers) / HCF. So, LCM = (12 × 16) / 4 = 48. Therefore, the correct LCM is 48, which corresponds to option D.

The volumes of the three cubes are 3³ = 27 dm³, 4³ = 64 dm³, and 5³ = 125 dm³. Their total volume is 27 + 64 + 125 = 216 dm³. The edge of the new cube is the cube root of 216, which is 6 dm.

For two co-prime numbers, their H.C.F is 1. The product of two numbers equals the product of their H.C.F and L.C.M. Since H.C.F is 1, L.C.M equals the product, which is 117.

The volume of a cone is proportional to the product of the base area and the height. Given the perimeters of the bases are in ratio 3:4, the radii are in ratio 3:4 (since perimeter = 2πr). The base areas ratio is therefore (3^2):(4^2) = 9:16. The heights ratio is 1:2. Thus, the volume ratio is (base area ratio) × (height ratio) = (9:16) × (1:2) = 9:32.