Counterexamples disprove all three: for (i), n = 4 gives 4(16) + 1 = 65 = 5 × 13; for (ii), n = 10 gives 121 = 11²; for (iii), n = 3 gives 67? No, n = 2 gives ...
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The cylinder fitting inside a cube of edge r has diameter r, so base radius is r/2 and height is r. Its volume is π × (r/2)² × r = (πr³)/4 cm³. Class 9 Chapter 14 Math of Space Surface Area ...
Given r₁ : r₂ = 2 : 3 and h₁ : h₂ = 3 : 2. The volume ratio is (r₁/r₂)²(h₁/h₂) = (4/9)(3/2) = 2 : 3. The curved surface area ratio is (r₁/r₂)(h₁/h₂) = (2/3)(3/2) = 1 : 1. Class ...
Let cylinder A have radius r and height h; cylinder B has radius 2r and height h/2. Curved surface area ratio is 2πrh : 2π(2r)(h/2) = 1 : 1. Volume ratio is πr²h : π(2r)²(h/2) = 1 : 2. CBSE Class ...
The surface area condition requires lw + wh + hl = 50 for integers l ≥ w ≥ h ≥ 1. Systematic testing gives exactly two integer solutions: dimensions 8 cm × 3 cm × 2 cm and 12 cm ...