Mathematics

If the side of the original square is s, its area is s². The diagonal of the square is s√2, so the area of the square drawn on the diagonal is (s√2)² = 2s². Therefore, the ratio of the original square's area to the square on its diagonal is s² : 2s² = 1 : 2.

The circumference of the circular field is calculated as π × diameter = 3.14 × 28 = 87.92 meters, which is approximately 88 meters. The cost of fencing is then 88 meters × Rs.1.50 per meter = Rs.132. Hence, the correct answer is Rs.132, which corresponds to option B.

Let the breadth be x metres, then length = x + 20. The perimeter is 2(x + x + 20) = 4x + 40. The cost of fencing is Rs. 5300 at Rs. 26.50 per metre, so perimeter = 5300 / 26.50 = 200 metres. Thus, 4x + 40 = 200, giving 4x = 160 and x = 40. Therefore, length = 40 + 20 = 50 metres.

When the side of a square is increased by 25%, the new side length is 1.25 times the original. The area increases by (1.25)^2 - 1 = 1.5625 - 1 = 0.5625, or 56.25%.

The circumference of a circular wheel is given by the formula C = 2πr. For a wheel with radius r = 1.75 m, the circumference is C = 2 × π × 1.75 ≈ 11.0 m. To travel a distance of 1 km (1000 m), the number of revolutions the wheel makes is the total distance divided by the circumference: 1000 ÷ 11.0 ≈ 90.9 revolutions, which rounds to about 91 revolutions. Among the given options, 90 (option A) is the closest to this calculated value. Therefore, the correct answer is A.

The volume of earth dug from the drain equals the volume spread over the plot. Calculating the volume of the drain (outer rectangle minus inner rectangle times depth) and equating it to the volume increase on the plot (area times height increase) gives the drain depth as approximately 1.225 m, matching option B.

Let the breadth be x meters. Then the length is (4/3)x meters. The area is length × breadth = (4/3)x × x = (4/3)x² = 300. Solving for x² gives x² = 225, so x = 15 meters (breadth). Length = (4/3) × 15 = 20 meters. The difference between length and breadth is 20 - 15 = 5 meters. Since 5 meters is not among the given options (15m, 4m, 3m), the correct choice is 'None of these'.

The area of a rhombus is half the product of its diagonals. Here, area = (1/2) × 15 cm × 20 cm = 150 cm². The correct unit is square centimeters, not square meters.

The perimeter P of a rectangle is given by P = 2(length + breadth). Given P = 600 m and breadth = 100 m, we have 600 = 2(length + 100), so length + 100 = 300, thus length = 200 m. However, the correct calculation is length = 200 m, which corresponds to option C. Therefore, option C is correct, not D.

Since AB and AC are equal chords of length 6 cm in a circle of radius 5 cm, triangle ABC is isosceles with AB = AC = 6 cm. Using the circle's radius and chord length, the distance from the center to the chord can be found, and then applying the Law of Cosines or coordinate geometry, the length of chord BC is calculated as 12/5 cm.