GATE E&C 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 E&C Engineering, not a summary of it.

1. 1

   Linear Algebra

   7 topics

   • Vector space
   • Basis
   • Linear dependence and independence
   • Matrix algebra
   • Eigen values and eigen vectors
   • Rank
   • Solution of linear equations- existence and uniqueness
2. 2

   Calculus

   8 topics

   • Mean value theorems
   • Theorems of integral calculus
   • Evaluation of definite and improper integrals
   • Partial derivatives
   • Maxima and minima
   • Multiple integrals
   • Line, surface and volume integrals
   • Taylor series
3. 3

   Differential Equations

   8 topics

   • First order equations (linear and nonlinear)
   • Higher order linear differential equations
   • Cauchy's and Euler's equations
   • Methods of solution using variation of parameters
   • Complementary function and particular integral
   • Partial differential equations
   • Variable separable method
   • Initial and boundary value problems
4. 4

   Vector Analysis

   4 topics

   • Vectors in plane and space
   • Vector operations
   • Gradient, divergence and curl
   • Gauss's, Green's and Stokes’ theorems
5. 5

   Complex Analysis

   8 topics

   • Analytic functions
   • Cauchy’s integral theorem
   • Cauchy’s integral formula
   • Sequences
   • Series
   • Convergence tests
   • Taylor and Laurent series
   • Residue theorem
6. 6

   Probability and Statistics

   11 topics

   • Mean
   • Median
   • Mode
   • Standard deviation
   • Combinatorial probability
   • Probability distributions
   • Binomial distribution
   • Poisson distribution
   • Exponential distribution
   • Normal distribution
   • Joint and conditional probability

Engineering Mathematics flashcards for GATE E&C Engineering

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

1. What is a vector space over a field $F$?

   A set $V$ with vector addition and scalar multiplication satisfying closure, associativity and commutativity of addition, an additive identity $\vec{0}$ and inverses, plus distributive and associative laws for scalars: $a(\vec{u}+\vec{v})=a\vec{u}+a\vec{v}$, $(a+b)\vec{v}=a\vec{v}+b\vec{v}$, $(ab)\vec{v}=a(b\vec{v})$, and $1\vec{v}=\vec{v}$.

2. What conditions make a subset $W\subseteq V$ a subspace?

   $W$ is nonempty (contains $\vec{0}$) and is closed under addition and scalar multiplication: for all $\vec{u},\vec{v}\in W$ and scalars $a,b$, $a\vec{u}+b\vec{v}\in W$.

3. Define a basis of a vector space.

   A basis is a set of vectors that is both linearly independent and spans the whole space. Every vector then has a unique representation as a linear combination of basis vectors, and the number of basis vectors equals the dimension.

4. When are vectors $\vec{v}_1,\dots,\vec{v}_n$ linearly independent?

   They are linearly independent if $c_1\vec{v}_1+\cdots+c_n\vec{v}_n=\vec{0}$ forces $c_1=c_2=\cdots=c_n=0$. If a nontrivial solution exists, they are linearly dependent.

5. How can you test linear independence of $n$ vectors in $\mathbb{R}^n$ using a determinant?

   Form the matrix $A$ with the vectors as columns. They are linearly independent if and only if $\det(A)\neq 0$ (equivalently $\operatorname{rank}(A)=n$).

6. State the relation between dimension, rank and nullity (rank-nullity theorem).

   For a linear map $T:V\to W$ (or matrix $A$ with $n$ columns), $\dim(V)=\operatorname{rank}(T)+\operatorname{nullity}(T)$, i.e. $n=\operatorname{rank}(A)+\dim(\ker A)$.

7. Define the rank of a matrix.

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

8. What is the condition $AB$ must satisfy for matrix multiplication, and is it commutative?

   If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is defined and is $m\times p$ (inner dimensions must match). Matrix multiplication is associative and distributive but in general not commutative: $AB\neq BA$.

9. State key transpose and inverse identities for matrices.

   $(AB)^{T}=B^{T}A^{T}$, $(A^{T})^{T}=A$, $(AB)^{-1}=B^{-1}A^{-1}$, and $(A^{-1})^{T}=(A^{T})^{-1}$, valid when the inverses exist.

10. When is a square matrix $A$ invertible (non-singular)?

    $A$ is invertible iff $\det(A)\neq 0$, equivalently full rank, columns linearly independent, $0$ is not an eigenvalue, and $A\vec{x}=\vec{0}$ has only the trivial solution.

11. How is the inverse of a non-singular matrix computed via the adjugate?

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

12. Define eigenvalues and eigenvectors of a square matrix $A$.

    A nonzero vector $\vec{x}$ is an eigenvector with eigenvalue $\lambda$ if $A\vec{x}=\lambda\vec{x}$. The eigenvalues are roots of the characteristic equation $\det(A-\lambda I)=0$.

13. State the relationship between eigenvalues and the trace and determinant of $A$.

    For an $n\times n$ matrix, $\sum_{i}\lambda_i=\operatorname{trace}(A)$ and $\prod_{i}\lambda_i=\det(A)$.

14. State the Cayley–Hamilton theorem.

    Every square matrix satisfies its own characteristic equation. If $p(\lambda)=\det(A-\lambda I)$, then $p(A)=0$; this can be used to compute $A^{-1}$ and powers of $A$.

15. What are the eigenvalues of $A^{-1}$, $A^{k}$ and $A^{T}$ in terms of those of $A$?

    If $\lambda$ is an eigenvalue of $A$, then $\frac{1}{\lambda}$ is an eigenvalue of $A^{-1}$, $\lambda^{k}$ of $A^{k}$, and $A^{T}$ has the same eigenvalues $\lambda$ as $A$ (eigenvectors may differ).

16. What special property do eigenvalues of real symmetric matrices have?

    A real symmetric matrix has all real eigenvalues, and eigenvectors corresponding to distinct eigenvalues are orthogonal; it is always diagonalizable by an orthogonal matrix.

17. For the system $A\vec{x}=\vec{b}$, state the consistency condition in terms of rank.

    The system is consistent iff $\operatorname{rank}(A)=\operatorname{rank}([A\,|\,\vec{b}])$. If ranks differ, the system is inconsistent (no solution).

18. Given a consistent system $A\vec{x}=\vec{b}$ with $n$ unknowns, when is the solution unique vs. infinitely many?

    If $\operatorname{rank}(A)=\operatorname{rank}([A|\vec{b}])=n$, the solution is unique. If the common rank $r<n$, there are infinitely many solutions with $n-r$ free parameters.

19. For a homogeneous system $A\vec{x}=\vec{0}$, when do nontrivial solutions exist?

    It always has the trivial solution $\vec{x}=\vec{0}$. Nontrivial solutions exist iff $\operatorname{rank}(A)<n$ (number of unknowns), i.e. $\det(A)=0$ for a square $A$.

20. State Rolle's theorem.

    If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c\in(a,b)$ with $f'(c)=0$.

21. State Lagrange's mean value theorem.

    If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in(a,b)$ such that $f'(c)=\dfrac{f(b)-f(a)}{b-a}$.

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