GATE Computer Science (CS) 2019 Solved Papers

$\mathit{X}\left(\mathit{P}\mathit{Q}\mathit{R}\mathit{S}\right)\left\{\mathit{Q}\mathit{R}\to \mathit{S},\mathit{R}\to \mathit{P},\mathit{S}\to \mathit{Q}\right\}$
decomposed into
$\begin{array}{cc}\mathit{Y}\left(\mathit{P}\mathit{R}\right)& \mathit{Z}\left(\mathit{Q}\mathit{R}\mathit{S}\right)\\ \left\{\mathit{R}\to \mathit{P}\right\}& \left\{\mathit{Q}\mathit{R}\to \mathit{S},\mathit{S}\to \mathit{Q}\right\}\end{array}$
Candidate key: R Candidate key : QR, RS
Relation Y in BCNF Relation Z in 3NF but not BCNF
Common attribute between Y and Zrelations is R which is key for relation Y.
So that given decomposition is losslessjoin decomposition.
R → P in Y
$\begin{array}{c}\mathit{Q}\mathit{R}\to \mathit{S}\\ \mathit{S}\to \mathit{Q}\end{array}\}$ are in Z and dependency preserving decomposition. Hence , C is the correct answer.