Which of the following statements are true about the males in a colony of honey bees ? (i) They have 16 chromosomes per cell. (ii) They produce sperms by meiosis. (iii) They have a grandfather but no father. (iv) All males in the colony are haploid except one.

The correct option is (i) and (iii) only.

Let us break down the statements step-by-step to understand why they are true or false for male honey bees (drones) in a colony:

Analysis of Statement (i): They have 16 chromosomes per cell.
In honey bees, sex determination follows a haplodiploid system. Females (queens and worker bees) are diploid and possess 32 chromosomes. Male honey bees (drones) develop from unfertilized eggs through a process called parthenogenesis. Because they develop from unfertilized eggs, they are haploid and contain exactly half the number of chromosomes of a female.
Thus, the number of chromosomes in male honey bees is:
$$\frac{32}{2}=16$$
Therefore, statement (i) is true.

Analysis of Statement (ii): They produce sperms by meiosis.
Since male honey bees are already haploid (containing 16 chromosomes, or n), they cannot undergo meiosis (reduction division) to produce sperm cells, as meiosis would halve the chromosome count to an unviable number. Instead, they produce sperm cells via mitosis (equational division) so that the sperm also retains 16 chromosomes.
Therefore, statement (ii) is false.

Analysis of Statement (iii): They have a grandfather but no father.
Because males develop from unfertilized eggs laid by the queen bee, they have no biological father. However, the queen bee (their mother) developed from a fertilized egg, meaning she had both a mother (the grandmother) and a father (the grandfather). Consequently, a drone inherits genetic material from its mother's father, meaning it has a grandfather but no father.
Therefore, statement (iii) is true.

Analysis of Statement (iv): All males in the colony are haploid except one.
In a honey bee colony, all drones are produced parthenogenetically from unfertilized eggs and are strictly haploid. There is no special diploid male in the colony (the queen and workers are female).
Therefore, statement (iv) is false.

Since only statements (i) and (iii) are correct, the correct choice is (i) and (iii) only.