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Both are true.

1. If in 3NF, a relation will not be in BCNF if and only if

  • $A \to B$, where $B$ is a prime attribute and $A$ is not a super key and $B \nsubseteq A$.

But, if there is only one candidate key, there is no possibility of a $B$ prime attribute other than the FD becoming trivial. So, the relation in 3NF must be in BCNF. 

2. R may or may not be in BCNF - Even with one candidate key, we can have partial FD (if the key is composite) and hence the relation need not be even in 2NF.

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Given that table has only one C.K.,

if R is in 3NF, then for F.D. X->Y

either X is S.K (which is required for BCNF)

or Y shld have prime attributes, if this is so then then we get more than one C.Keys

Hence, R is in BCNF
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Statement 1$\Rightarrow$$\mathbf{If\: R\: is\: in\: 3NF,\: It \: is\; also\; in\: BCNF}$

$\Rightarrow$If it is in 3NF,then it is no doubt in 2NF$\Rightarrow$It does not have partial dependency.

$\Rightarrow$It won't contain Partial Dependency i.e

A functional dependency X → Y is a partial dependency if some attribute A ε X can be removed from X and the dependency still holds; that is, for some A ε X, (X – {A}) → Y.

  relation Prime Attribute$\rightarrow$Non Prime Attribute.so it won't contain Prime Attribute$\rightarrow$Prime Attribute.It may contain Prime Attribute$\rightarrow$Prime Attribute.

$\Rightarrow$It won't contain Transitive dependency i.e it won't contain relation Non Key$\rightarrow$Non key.But it may contain relation Prime Attribute$\rightarrow$Prime Attribute.

Now BCNF says that if it has Functional dependency X$\rightarrow$Y,X must be a superkey.

But in Relation Prime Attribute$\rightarrow$Prime Attribute,prime Attribute is not a super key,Makes Statement1 False .

Statement 2$\Rightarrow$R may be in BCNF

$\Rightarrow$If Relation contains Prime Attribute$\rightarrow$Prime Attribute,then it is not in BCNF.

if it won't contain this relation ,it will be in BCNF,so it may/may not be in BCNF.

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for 1st question:

AB->CD  D->A     ..................(AB is candidate key)D->A in 3NF but not in BCNF

for 2nd question,

A->B   B->C ......................(A is candidate key) B->C not in BCNF.
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