GATE Marine Engineering Engineering Mathematics Syllabus

Engineering Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Engineering Mathematics in GATE Marine Engineering, not a summary of it.

1. 1

   Determinants and Matrices

   2 topics

   • Determinants
   • Matrices
2. 2

   Systems of Linear Equations

   1 topic

   • Linear Equations
3. 3

   Eigen Values and Eigen Vectors

   2 topics

   • Eigen Values
   • Eigen Vectors
4. 4

   Functions

   4 topics

   • Gradient
   • Divergence
   • Curl
   • Chain Rules
5. 5

   Partial Derivatives

   1 topic

   • Directional Derivatives
6. 6

   Definite and Indefinite Integrals

   3 topics

   • Line Integrals
   • Surface Integrals
   • Volume Integrals
7. 7

   Theorems of Stokes, Gauss, and Green

   overview

   Examined as a single unit within Engineering Mathematics — no further topic split in the official outline.

8. 8

   Ordinary and Partial Differential Equations

   5 topics

   • Linear Equations
   • Non-linear Equations
   • First Order Equations
   • Higher Order Equations
   • Partial Differential Equations
     • Separation of Variables
9. 9

   Laplace Transformation

   overview

   Examined as a single unit within Engineering Mathematics — no further topic split in the official outline.

10. 10

    Analytical Functions of Complex Variables

    overview

    Examined as a single unit within Engineering Mathematics — no further topic split in the official outline.

11. 11

    Fourier Series

    overview

    Examined as a single unit within Engineering Mathematics — no further topic split in the official outline.

12. 12

    Numerical Methods for Differentiation and Integration

    overview

    Examined as a single unit within Engineering Mathematics — no further topic split in the official outline.

13. 13

    Complex Analysis

    overview

    Examined as a single unit within Engineering Mathematics — no further topic split in the official outline.

14. 14

    Probability and Statistics

    overview

    Examined as a single unit within Engineering Mathematics — no further topic split in the official outline.

Engineering Mathematics flashcards for GATE Marine Engineering

21 of 51 cards from the Engineering Mathematics deck — real questions with worked answers.

1. What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

   $$\det = ad - bc$$

2. How does the determinant change when two rows (or two columns) of a matrix are interchanged?

   The determinant changes sign (is multiplied by $-1$). Interchanging two rows or two columns reverses the sign of the determinant.

3. What is the value of the determinant if any two rows (or columns) of a square matrix are identical or proportional?

   The determinant is $0$. A matrix with two identical or proportional rows/columns is singular.

4. If $A$ is an $n \times n$ matrix and $k$ is a scalar, how is $\det(kA)$ related to $\det(A)$?

   $$\det(kA) = k^{n}\det(A)$$ because each of the $n$ rows contributes a factor of $k$.

5. State the multiplicative property of determinants for two square matrices $A$ and $B$ of the same order.

   $$\det(AB) = \det(A)\,\det(B)$$

6. How is the determinant of a triangular matrix (upper or lower) computed?

   It equals the product of the diagonal entries: $\det = a_{11}a_{22}\cdots a_{nn}$.

7. What is the relationship between $\det(A)$ and $\det(A^{T})$?

   $$\det(A^{T}) = \det(A)$$ The determinant is unchanged by transposition.

8. For an invertible matrix $A$, what is $\det(A^{-1})$?

   $$\det(A^{-1}) = \frac{1}{\det(A)}$$ valid only when $\det(A) \neq 0$.

9. What is the formula for the inverse of a non-singular matrix $A$ using the adjugate?

   $$A^{-1} = \frac{1}{\det(A)}\,\operatorname{adj}(A)$$ where $\operatorname{adj}(A)$ is the transpose of the cofactor matrix.

10. Define the rank of a matrix.

    The rank is the order of the largest non-zero minor, equivalently the number of linearly independent rows (or columns), equivalently the number of non-zero rows in row-echelon form.

11. What condition on a square matrix $A$ makes it singular (non-invertible)?

    $A$ is singular if and only if $\det(A) = 0$, equivalently its rank is less than its order, equivalently $0$ is an eigenvalue.

12. Define a symmetric matrix and a skew-symmetric matrix.

    Symmetric: $A^{T} = A$ (so $a_{ij} = a_{ji}$). Skew-symmetric: $A^{T} = -A$ (so $a_{ij} = -a_{ji}$ and all diagonal entries are $0$).

13. What defines an orthogonal matrix, and what is its key determinant property?

    A matrix $A$ is orthogonal if $A^{T}A = AA^{T} = I$, i.e. $A^{-1} = A^{T}$. Its determinant is $\det(A) = \pm 1$.

14. State Cramer's rule for solving a system $AX = B$ of $n$ linear equations.

    If $\det(A) \neq 0$, then $x_{i} = \dfrac{\det(A_{i})}{\det(A)}$, where $A_{i}$ is $A$ with its $i$-th column replaced by $B$.

15. For the homogeneous system $AX = 0$, when does a non-trivial (non-zero) solution exist?

    A non-trivial solution exists if and only if $\det(A) = 0$ (the coefficient matrix is singular, i.e. rank $< n$).

16. Using rank, state the consistency conditions for the linear system $AX = B$ with $n$ unknowns.

    Let $r = \operatorname{rank}(A)$ and $r' = \operatorname{rank}([A\,|\,B])$. If $r \neq r'$: inconsistent (no solution). If $r = r' = n$: unique solution. If $r = r' < n$: infinitely many solutions.

17. Define an eigenvalue and eigenvector of a square matrix $A$.

    A non-zero vector $\vec{x}$ is an eigenvector with eigenvalue $\lambda$ if $$A\vec{x} = \lambda\vec{x}.$$

18. What is the characteristic equation used to find eigenvalues of $A$?

    $$\det(A - \lambda I) = 0$$ Its roots $\lambda$ are the eigenvalues.

19. What does the sum of the eigenvalues of a matrix $A$ equal?

    The sum of eigenvalues equals the trace of $A$ (sum of diagonal elements): $\sum \lambda_{i} = \operatorname{tr}(A)$.

20. What does the product of the eigenvalues of a matrix $A$ equal?

    The product of eigenvalues equals the determinant: $\prod \lambda_{i} = \det(A)$.

21. State the Cayley-Hamilton theorem.

    Every square matrix satisfies its own characteristic equation. If the characteristic polynomial is $p(\lambda) = 0$, then $p(A) = 0$ (the zero matrix).

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