Statements I & III are false:
According to the mutation theory of evolution given by Hugo de Vries, evolution can occur in a single generation due to mutations which are random (can occur any time), discontinuous (occur occasionally without prior precedent) and directionless (useful, harmful or have no effects at all on the organisms).
He believed that single step large mutations (saltations) can form entirely new species in a single generation.
Darwin on the other hand believed that small or minor variations accumulate over successive generations leading to the formation of new species. Therefore, Darwin believed that evolution is a gradual process.
Statement II is true:
The modern synthetic theory of evolution or Neo-Darwinism, is an improvement of Darwin’s theory of natural selection. It is a combination of ideas from Darwin's theory of natural selection, mutation theory and Mendel’s principles of heredity. It states that genetic variations can be introduced into the gene pool of a population by gene mutations, genetic recombination, change in chromosome number or structure, hybridisation or migration.