List of top Mathematics Questions on introduction to three dimensional geometry

Let Q be the cube with the set of vertices {(x₁, x₂, x₃) ∈ R³: x₁, x₂, x₃ ∈ {0,1}}. Let F be the set of all twelve lines containing the diagonals of the six faces of cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in S. For lines l₁ and l₂, let d(l₁,l₂) denote the shortest distance between them. Then the maximum value of d(l₁,l₂) as l₁ varies over f and l₂ varies over S, is

Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Then the inradius of the triangle ABC is

Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Let a be the area of the triangle ABC. Then the value of (64a)² is

The equation of conic is $19x^2 + 15y^2 = 285$. A concentric circle with radius 4 units is given then angle of common tangent made by minor axis of ellipse is

Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Let a be the area of the triangle ABC. Then the value of (64a)² is

Let A₁, A₂, A₃, A₄,........, A₈ be the vertices of the regular octagons that lie on the circle of radius 2. Let p be a point on the circle and let PA_i denote the distance between the point P and Ai for i = 1,2,3,....,8. If P varies over the circle, then the maximum value of the product is PA₁.Pa₂..........PA₈, is

Let C₁ be the circle of radius 1 with the center at the origin. Let C₂ be the circle of radius r with center at the point A = (4,1), where 1 < r < 3 . Two distinct common tangents PQ and ST of C₁ and C₂ are drawn. The tangent PQ touches C₁ at P and C₂ at Q. The tangent ST touches C₁ at S and C₂ at T. Midpoints of the line segments PQ and ST are joined to form a line that meets the x-axis at a point B. If AB = √5, then the value of r² is :

Let A₁, A₂, A₃, A₄,........, A₈ be the vertices of the regular octagons that lie on the circle of radius 2. Let p be a point on the circle and let PA_i denote the distance between the point P and Ai for i = 1,2,3,....,8. If P varies over the circle, then the maximum value of the product is PA₁.PA₂..........PA₈, is