Fundamental Concept

• Sets and Functions

Differential Equations

• Exact Differential Equations: Identifying exact equations and using the Integrating Factor method to solve non-exact ones.

• Second-Order LDE with Constant Coefficients: Solving for the Complementary Function (CF) and finding the Particular Integral (PI) (using variation of parameters or method of undetermined coefficients).

Calculus

• These topics require strong problem-solving skills, particularly with integration and ODEs.

• Mean Value Theorems (Rolle's & Lagrange's): Application of these theorems and understanding their geometric interpretation.

• Multivariable Maxima/Minima: Finding critical points and using the Second Derivative Test (Hessian matrix) to classify them.

• Change of Order of Integration: Mastering the process of changing the limits in double integrals (a very common question type).

Real Analysis

• Sequences and Series: This section tests your ability to handle rigorous proofs and application of convergence tests.

• Bolzano-Weierstrass Theorem: Understanding the existence of a convergent subsequence for any bounded sequence.

• Convergence Tests (Ratio & Root): Quick application of the Ratio Test for factorials/exponential terms and the Root Test for terms involving n-th powers.

• Absolute vs. Conditional Convergence: Distinguishing between the two and applying the Leibniz Test for alternating series.

  Power Series: Radius of Convergence: Calculating the radius and interval of convergence using the ratio or root test.

Linear Algebra

• This is often the most scoring section, focusing on computation and key theorems.

• Vector Spaces: Basis and Dimension: Testing for linear independence and finding the basis for the Null Space and Range Space.

• Rank-Nullity Theorem: Understanding the relationship: Rank(T)+Nullity(T)=Dimension (Domain).

  Eigenvalues, Eigenvectors & Diagonalization: Finding characteristic polynomials, properties of eigenvalues (trace, determinant), and conditions for Diagonalizability.

• Cayley-Hamilton Theorem: Using the theorem p(A)=0 for quick matrix calculations (e.g., finding Inverse of A).

• Systems of Linear Equations: Consistency conditions for Ax=b (Rank[A] = Rank[A|b]).