Mensuration

Square area = 1089 cm², so side = √1089 = 33 cm and perimeter = 132 cm. When bent into a circle, circumference = 132 cm, so 2πr = 132 and r = 66/π. Circle area = πr² = π × (66/π)² = 4356/π ≈ 1386 cm² (using π ≈ 22/7).

Sphere volume = (4/3)πr^3. Surface area = 4πr^2. Volume/Surface area = r/3 = 27. Solving: r = 81 cm. Option C is correct.

Sphere volume = (4/3)π(3)^3 = 36π. Cylindrical vessel volume increase = π(6)^2 × h = 36πh. Equating: 36π = 36πh, giving h = 1 cm. Option C is correct.

Lateral surface area = perimeter of base × slant height. For triangular base with each edge 8cm, perimeter = 3 × 8 = 24cm. LSA = 24 × slant height = 24 × 4 = 96 cm² would be correct, but options suggest a different calculation. The question has inconsistent data (slant height given as 4cm in English but 5cm in Hindi, and answer matches 4cm).

Volume of original sphere = (4/3)π*8³ = 2048π/3 cm³. Volume of each bullet = (4/3)π*0.2³ = 0.008π/3 cm³. Number of bullets = (2048π/3) / (0.008π/3) = 2048/0.008 = 64000. Option D is correct.

Square perimeter 114.8m gives side = 114.8 ÷ 4 = 28.7m. This is the circle's radius. Area = π × r² = 3.14 × 28.7² = 3.14 × 823.69 = 2586.39 sqm (approximately). The closest option is D (2568.16 sqm) but there's a calculation discrepancy. Using precise calculation: π × (28.7)² = π × 823.69 ≈ 2587.4 sqm. None of the given options match this result, so 'None of these' is correct.

Area of first circle = π × 5^2 = 25π. Area of second = π × 12^2 = 144π. Combined area = 169π. For the new circle: π × r^2 = 169π, so r^2 = 169 and r = 13. Diameter = 2 × 13 = 26 m. Option A is correct.

Circle diameter = 56 cm, so radius = 28 cm. Circumference = 2πr = 2 × (22/7) × 28 = 176 cm. Perimeter of square = 272 - 176 = 96 cm, so side = 96/4 = 24 cm. Area of circle = πr² = (22/7) × 28² = 2464 sq.cm. Area of square = 24² = 576 sq.cm. Sum = 2464 + 576 = 3040 sq.cm.

The largest circle inside a square has diameter equal to the square's side. For side 28 cm, radius = 14 cm. Area = πr² = (22/7) × 14² = (22/7) × 196 = 22 × 28 = 616 cm². Option C (784) would be the square's area, and option A (625) assumes π = 25, which is incorrect.

When a cone is cut at mid-height (12 cm), the upper portion forms a smaller similar cone with height 12 cm. By similar triangles, the radius at the cut is half of base radius = 3.5 cm. Volume = (1/3) × π × (3.5)² × 12 = 49π ≈ 154 cm³. Option A (169) is incorrect calculation, option C (1078) is the full cone volume, and option D (800) has no relation.

If radius is reduced by 50% (to 0.5r) and height increased by 60% (to 1.6h), new volume = π(0.5r)²(1.6h) = 0.4πr²h, which is 40% of original. This is a 60% decrease. Note: The English text says 'increased' but Hindi says 'कम' (decreased/less) - Hindi is correct.

For cylinder: CSA = 2πrh = 264, Volume = πr²h = 924. Dividing Volume by CSA: (πr²h)/(2πrh) = 924/264, which gives r/2 = 3.5, so r = 7. Then h = (924)/(π × 7²) = 3. Diameter:height = 2r:h = 14:3 = 7:3.

Area ratio 9:16 means radius² ratio is 9:16, so radius ratio is 3:4. Let radii be 3r and 4r. Difference in diameters = 2(4r-3r) = 2r = 4cm, so r = 2cm. Outer radius = 4r = 8cm. Area = π(8)² = 64π cm².

If radius increases by 1%, new radius = 1.01r. New area = π(1.01r)² = 1.0201πr². Increase = 1.0201 - 1 = 0.0201 = 2.01%. Alternatively: % increase in area ≈ 2 × (% increase in radius) for small changes, giving approximately 2%, but exact calculation shows 2.01%.

The circle circumscribing a rectangle has its diameter equal to the rectangle's diagonal. Using Pythagorean theorem: diagonal = √(16² + 12²) = √(256 + 144) = √400 = 20 cm. Radius = 10 cm, so area = πr² = π(10)² = 100π cm².