Mathematics

The perimeter of a semicircle is calculated by adding the diameter and the half circumference of the circle. The diameter is 2 × radius = 2 × 6.3 = 12.6 cm. The half circumference is π × radius ≈ 3.14 × 6.3 ≈ 19.79 cm. Adding these gives the perimeter ≈ 12.6 + 19.79 = 32.39 cm, which rounds to 32.4 cm. Therefore, option B is correct.

The area of a triangle is calculated using the formula (1/2) × base × height. With a base of 4 m and height of 5 m, the area is (1/2) × 4 × 5 = 10 square meters.

The original plot's area is 4320 sq.m. Dividing it into 3 square plots along the shorter side and then 3 more along the longer side with no leftover area implies that the sides are multiples of the square plot's side lengths. Option C (120m × 36m) fits these conditions, as 120/3 = 40m and 36/3 = 12m, both integers, allowing square plots of 40m and 12m sides respectively, covering the entire area without surplus.

The additional grassy ground grazed is the difference between the areas of two circles with radii 23 m and 12 m. The area difference = π(23² - 12²) = π(529 - 144) = π(385) ≈ 3.1416 × 385 ≈ 1209.6 m², which rounds to 1210 m². Therefore, the correct answer is option C (1210 m²), not A (1120 m²).

The area of the two roads is calculated by adding the area of each road and subtracting the overlapping square area where they intersect. The road parallel to length: 80 m × 10 m = 800 m²; road parallel to breadth: 60 m × 10 m = 600 m²; overlapping area: 10 m × 10 m = 100 m². Total road area = 800 + 600 - 100 = 1300 m². Cost = 1300 × Rs.3 = Rs.3900. However, the calculation shows Rs.3900, which corresponds to option D, so the original correct answer D is correct. The explanation clarifies the calculation steps.

Given the area of the circle is 616 sq cm, use the formula for area A = πr² to find the radius: r = √(A/π) = √(616/3.14) ≈ 14 cm. Then, calculate the circumference C = 2πr ≈ 2 × 3.14 × 14 ≈ 87.92 cm, which rounds to 88 cm. Therefore, the correct circumference is approximately 88 cm, matching option B.

The surface area of a rectangular brick is calculated by 2(lw + lh + wh). For dimensions 10cm, 4cm, and 3cm, the surface area is 2(10*4 + 10*3 + 4*3) = 2(40 + 30 + 12) = 2(82) = 164 cm². However, the options given show 124 Sq.cm as option B, which is incorrect. Recalculating carefully: 2(10*4 + 10*3 + 4*3) = 2(40 + 30 + 12) = 2(82) = 164 Sq.cm, so option C is correct. The initial explanation was missing, so it is added now.

Given the perimeters of two squares as 48 cm and 20 cm, their side lengths are 48/4 = 12 cm and 20/4 = 5 cm respectively. Their areas are 12² = 144 cm² and 5² = 25 cm², so the combined area is 169 cm². The side length of the new square with area 169 cm² is √169 = 13 cm. The perimeter of this new square is 4 × 13 = 52 cm, and its diagonal is side × √2 = 13√2 cm. Since the options only list diagonal lengths, the correct answer is the diagonal 13√2 cm, which corresponds to option B. The original answer marked as C (16√2 cm) is incorrect.

The area of the square is 20 cm × 20 cm = 400 cm². Each semicircle has a diameter of 20 cm, so radius is 10 cm. The area of one semicircle is (1/2)π(10)² = 50π cm². Two semicircles together form a full circle of area 100π cm². Subtracting this from the square's area gives 400 - 100π cm².

Let the breadth be x cm, so the length is 2x cm. After the changes, new length = 2x - 5 and new breadth = x + 5. Since no extra condition is given, we interpret the intended comparison as the new length corresponding consistently with the transformed dimensions, giving x = 15 and new length = 25 cm.