Area and Perimeter

Original area = 25,000. After 5% length increase and 10% breadth increase, new area = 25000 × 1.05 × 1.10 = 25000 × 1.155 = 28,875. The statement claims 28,775, which is incorrect. 25,000 × 1.155 = 28,875, not 28,775.

Let sides be 12x and 12y where x,y are coprime. Perimeter 2(12x + 12y) = 384, so x + y = 16. Coprime pairs (x,y) for 16 are (1,15), (3,13), (5,11), (7,9). There are 4 distinct pairs, representing 4 distinct rectangles.

For perimeter 258, if length = l and breadth = b, then 2(l+b) = 258, so l+b = 129. For integer dimensions, l can be 1 to 128, giving 128 ordered pairs. Since l>l+b, b>0 gives l<129, and we need l >= b for distinct rectangles. Counting gives 32 values of l from 65 to 96, and 32 values where l < b, totaling 64.

A regular heptagon has 7 equal sides. If its perimeter is 84 cm, each side measures 12 cm. However, the question only states that 'a length of a side' of the square equals 'a length of a side' of the heptagon - this is grammatically ambiguous. It could mean any one side length, not necessarily all sides being equal. Without clarity on whether the square's side equals the heptagon's side length (12 cm), the area cannot be definitively determined.

The negative square root of 36 is -6. Thus, t = -6. Substituting into y = 2t + 28 gives y = 2(-6) + 28 = 16. Area = (x+y)(x-y) = x^2 - y^2. Substituting y=16 gives x^2 - 256.

The goat is tied to a corner, so it can graze in a quarter-circle (90° sector) of radius 7m. Area of full circle = πr² = π×49. Quarter of this = π×49/4 = 38.48 ≈ 38.5 sq.m. The rope doesn't reach the full 12m side, so only the quarter-circle area matters.

Area of park = 2400. Area of lawn = 2109. Area of roads = 2400 - 2109 = 291. Let width be w. Road area = 60w + 40w - w^2 = 291. This gives w^2 - 100w + 291 = 0. Solving for w, (w-3)(w-97)=0. Since w cannot be 97m in a 40m wide park, w = 3m.

Let the side be s. The new area is (s+3)^2 and the old area is s^2. (s+3)^2 - s^2 = 33. Expanding gives s^2 + 6s + 9 - s^2 = 33, so 6s = 24, and s = 4. Distractors like 6 or 2 result in different area increases.

A regular hexagon has 6 equal sides. If each side is 22 cm, the perimeter is 6 × 22 = 132 cm. This correctly uses the centimeter unit.

A square has 4 equal sides. If the perimeter is 56 meters, each side is 56 ÷ 4 = 14 meters. Option A would be double the correct side length.

The area of a square is given by side × side (s²), not length × breadth. Length × breadth is the formula for a rectangle. Since a square has all equal sides, the correct formula is side². The statement is indeed false, though the question has a typo ('Area a' should be 'Area of a').

The original area is L × W. After changes, new area = 1.1L × 0.9W = 0.99LW. This is a 1% decrease overall. The 10% increase and 10% decrease don't cancel out exactly - they multiply to 0.99, giving a net loss.