UPSC NDA 2 2022 Mathematics Solved Paper

Concept:To check if a line is tangent to a sphere, compare the distance from the sphere's centre to the line with the sphere's radius.For the plane containing the centre, simply substitute the centre coordinates into the plane equation.Explanation:Given sphere: $x^2 + y^2 + z^2 - 4x - 6y - 8z - 16 = 0$.Complete squares: $(x-2)^2 + (y-3)^2 + (z-4)^2 = 45$.Thus centre $C = (2,3,4)$ and radius $r = \sqrt{45}$.Statement 1: The z-axis is the line $x=0,\ y=0$.Distance from $C$ to the z-axis is $\sqrt{(2-0)^2 + (3-0)^2} = \sqrt{13}$.Since $\sqrt{13} \neq \sqrt{45}$, the z-axis is not tangent to the sphere.Hence statement 1 is false.Statement 2: Check if centre lies on plane $x + y + z - 9 = 0$.Substitute $(2,3,4)$: $2 + 3 + 4 - 9 = 0$.Yes, the centre satisfies the equation, so the centre lies on the plane.Thus statement 2 is true.Answer:B. 2 only