• Java Exception Handling Keywords:
• throw: Keyword used to explicitly throw or generate an exception from any method or block of code (e.g., throw new ArithmeticException("Divide by zero");).
• throws: Keyword used in method signatures to declare the exceptions that the method might throw.
• try: Encloses the block of code to monitor for exceptions.
• catch: Handles the specific exception thrown by a try block.
• Conclusion: An exception is explicitly generated using throw (Option 3).

Correct option: (3) Throw

• Principles of Linear Discriminant Analysis (LDA):
• Statement (I): In a discriminant analysis, dependent variable is parametric and the independent variable is categorical. $\implies$ Incorrect.
• In discriminant analysis, the dependent variable is categorical (representing discrete group membership/classes) and the independent variables are metric/continuous (parametric). (The statement has them inverted).
• Statement (II): Discriminant analysis minimizes the variance within a class and maximizes it between class. $\implies$ Correct.
• Fisher's linear discriminant maximizes the ratio of between-class scatter matrix ($S_B$) to within-class scatter matrix ($S_W$):

• Conclusion: Statement (I) is incorrect but Statement (II) is correct (Option 4).

• Principal Component Analysis (PCA):
• Statement (I): Principal Component Analysis is a supervised learning method. $\implies$ Incorrect.
• PCA is an unsupervised learning method because it operates strictly on feature vectors without utilizing any target labels or class response variables.
• Statement (II): Principal Component Analysis is a dimension reduction method. $\implies$ Correct.
• PCA reduces the dimensionality of large multivariate datasets by finding orthogonal linear combinations (principal components) that maximize variance along successive axes.
• Conclusion: Statement (I) is incorrect but Statement (II) is correct (Option 4).

• Solving Transportation Problems to Optimality:
• Initial Basic Feasible Solution (IBFS) Methods:
• North-West Corner Cell Method: Quick allocation from top-left, ignores shipping costs.
• Vogel's Approximation Method (VAM): Heuristic penalty method providing an initial feasible solution.
• Optimality Determination Methods:
• Modified Distribution (MODI) Method (or $u-v$ method): Calculates dual variables $u_i + v_j = c_{ij}$ for basic cells and computes net evaluation $\Delta_{ij} = c_{ij} - (u_i + v_j)$ for non-basic cells to iteratively attain the minimum cost optimal solution.
• Stepping Stone Method: Evaluates closed loops to test optimality.
• Conclusion: Minimum cost solution is obtained by Modified distribution method (Option 4).

Correct option: (4) Modified distribution method