Tag: science & technology

An n-dimensional cube (n-cube) has each vertex connected to n edges. There are 2^n vertices total. Each edge is counted twice (once from each endpoint), so total edges = n * 2^n / 2 = n * 2^(n-1). This gives option C.

In a complete graph with n vertices, each vertex connects to (n-1) others. Total connections counted this way = n*(n-1). Each edge is counted twice (once from each endpoint), so actual edges = n*(n-1)/2. Option A is correct.

Kuratowski's theorem provides the complete characterization of planar graphs. A graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (complete graph on 5 vertices) or K3,3 (complete bipartite graph with 3+3 vertices). These two graphs are the minimal non-planar graphs - any graph containing them as a subgraph cannot be drawn without edge crossings.

Euler's formula for connected planar graphs states v - e + f = 2, where v is vertices, e is edges, and f is faces. Rearranging gives f = e - v + 2. This fundamental relationship applies to all connected planar graphs drawn without edge crossings.

Euler's formula for connected planar graphs states v - e + f = 2, where v is vertices, e is edges, and f is faces. Rearranging gives f = e - v + 2. This fundamental relationship applies to all connected planar graphs drawn without edge crossings.

Euler's theorem states that a connected graph has an Euler cycle (closed trail using every edge exactly once) if and only if every vertex has even degree. This is because each time an Euler trail enters a vertex, it must also exit, requiring paired edges. Loops contribute 2 to the degree count.

Euler's theorem states that a connected graph has an Euler cycle (closed trail using every edge exactly once) if and only if every vertex has even degree. This is because each time an Euler trail enters a vertex, it must also exit, requiring paired edges. Loops contribute 2 to the degree count.

Vitamin K is essential for blood clotting as it helps the liver produce several clotting factors required for the coagulation cascade. Without adequate Vitamin K, blood cannot clot properly, leading to excessive bleeding. Vitamins B1, B2, and D serve different functions in the body and are not directly involved in clotting.

An average adult human has approximately 5-6 liters of blood, which represents about 7-8% of total body weight. This volume can vary slightly based on body size, gender, and overall health. 3-4 liters is too low, while 8-12 liters would be excessive.

The pancreas secretes insulin, a hormone produced by beta cells in the islets of Langerhans. Insulin is essential for regulating blood glucose levels by facilitating glucose uptake into cells. Bile juice is produced by the liver, and peptic juice comes from the stomach.