A cube is inscribed in a sphere. A right circular cylinder is placed within the cube touching all the vertical faces, and a right circular cone is inside the cylinder. Their heights are the same and the diameter of the cone is equal to that of the cylinder. Consider the following statements: 1) The surface area of the sphere is equal to square root of five times the curved surface area of the cone. 2) The surface area of the cube is equal to the curved surface area of the cylinder. Which of the above statements is or are correct?

Introduction / Context:
This question tests three dimensional geometry, relationships between standard solids, and the ability to compare surface areas using algebraic expressions. A cube is inscribed in a sphere, a right circular cylinder fits exactly inside the cube, and a right circular cone fits inside the cylinder. You must check two statements about ratios of surface areas. Such questions require careful visualisation and systematic formula based reasoning.

Given Data / Assumptions:

• A cube of side a is inscribed in a sphere.
• A right circular cylinder fits in the cube, touching all four vertical faces, so its height is a and its diameter is a.
• A right circular cone is inside the cylinder with the same height a and with diameter equal to that of the cylinder, so radius is a divided by 2.
• Statement 1: Surface area of the sphere equals square root of five multiplied by the curved surface area of the cone.
• Statement 2: Surface area of the cube equals the curved surface area of the cylinder.

Concept / Approach:
First, express each surface area in terms of the cube side a. For the sphere, you must find its radius using the space diagonal of the cube. For the cylinder and cone, use the given relationships between side, height, and diameter. Then compute the surface area formulas: for a sphere, 4 * pi * r^2; for a cylinder, curved area 2 * pi * r * h; and for a right circular cone, curved area pi * r * l, where l is the slant height computed using Pythagoras theorem. After that, compare ratios to see whether the given equalities can hold for any positive a.

Step-by-Step Solution:
Step 1: Let the cube have side length a. Its space diagonal is a * sqrt(3). The sphere is drawn around the cube, so the diameter of the sphere equals the space diagonal.Step 2: Radius of the sphere r equals a * sqrt(3) divided by 2, so the surface area of the sphere is 4 * pi * r^2 = 4 * pi * (a^2 * 3 / 4) = 3 * pi * a^2.Step 3: The cylinder fits inside the cube. Its height h equals a, and its diameter equals a, so its radius is a divided by 2. Curved surface area of the cylinder is 2 * pi * r * h = 2 * pi * (a / 2) * a = pi * a^2.Step 4: The cone sits inside the cylinder with the same height a and radius a divided by 2. Slant height l is sqrt(r^2 + h^2) = sqrt((a^2 / 4) + a^2) = a * sqrt(5 / 4) = a * sqrt(5) / 2.Step 5: Curved surface area of the cone equals pi * r * l = pi * (a / 2) * (a * sqrt(5) / 2) = pi * a^2 * sqrt(5) / 4.Step 6: Compare the surface area of the sphere and the curved surface area of the cone. The ratio is (3 * pi * a^2) divided by (pi * a^2 * sqrt(5) / 4) = 12 divided by sqrt(5), which is not equal to sqrt(5). So statement 1 is false.Step 7: Surface area of the cube is 6 * a^2. The curved surface area of the cylinder is pi * a^2. For these to be equal, pi would have to equal 6, which is impossible. So statement 2 is also false.

Verification / Alternative check:
You can verify statement 1 numerically by taking a convenient value, such as a equals 2. Then surface area of the sphere is 3 * pi * 4 = 12 * pi. Curved surface area of the cone is pi * 4 * sqrt(5) / 4 = pi * sqrt(5). The ratio 12 * pi divided by pi * sqrt(5) again gives 12 divided by sqrt(5), not square root of five. For statement 2, take a equals 1. The cube has surface area 6, while the cylinder has curved surface area pi, which is about 3.14, clearly not equal.

Why Other Options Are Wrong:
Option A and option B each assume that exactly one of the statements is true, which contradicts the algebraic comparisons above. Option C assumes that both equalities hold, which is mathematically impossible. Only option D, which states that neither 1 nor 2 is correct, matches the calculations.

Common Pitfalls:
Students often make mistakes by confusing diameters and radii, or by forgetting that the inscribed cube touches the sphere at its vertices, leading to a wrong expression for the sphere radius. Another common error is to use total surface area instead of curved surface area for cylinders and cones. Some candidates also try to estimate visually rather than computing formulas, which is very risky in precision based questions. Always draw a rough diagram, label all lengths carefully, write each formula, and then simplify algebraically before comparing any expressions.

Final Answer:
Both given statements about the surface areas are incorrect, so the right choice is Neither 1 nor 2.