GATE Electrical 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 Electrical Engineering, not a summary of it.

1. 1

   Linear Algebra

   4 topics

   • Matrix Algebra
   • Systems of linear equations
   • Eigen values
   • Eigen vectors
2. 2

   Calculus

   16 topics

   • Mean value theorems
   • Theorems of integral calculus
   • Evaluation of definite and improper integrals
   • Partial Derivatives
   • Maxima and minima
   • Multiple integrals
   • Fourier series
   • Vector identities
   • Directional derivatives
   • Line integral
   • Surface integral
   • Volume integral
   • Stokes’s theorem
   • Gauss’s theorem
   • Divergence theorem
   • Green’s theorem
3. 3

   Differential equations

   8 topics

   • First order equations (linear and nonlinear)
   • Higher order linear differential equations with constant coefficients
   • Method of variation of parameters
   • Cauchy’s equation
   • Euler’s equation
   • Initial and boundary value problems
   • Partial Differential Equations
   • Method of separation of variables
4. 4

   Complex variables

   7 topics

   • Analytic functions
   • Cauchy’s integral theorem
   • Cauchy’s integral formula
   • Taylor series
   • Laurent series
   • Residue theorem
   • Solution integrals
5. 5

   Probability and Statistics

   11 topics

   • Sampling theorems
   • Conditional probability
   • Mean, Median, Mode
   • Standard Deviation
   • Random variables
   • Discrete and Continuous distributions
   • Poisson distribution
   • Normal distribution
   • Binomial distribution
   • Correlation analysis
   • Regression analysis

Engineering Mathematics flashcards for GATE Electrical Engineering

21 of 50 cards from the Engineering Mathematics deck — real questions with worked answers.

1. What is the rank of a matrix?

   The rank of a matrix is the maximum number of linearly independent rows (or columns), equivalently the order of the largest non-zero minor. For an $m \times n$ matrix, $\text{rank} \leq \min(m,n)$.

2. State the condition for a square matrix $A$ to be invertible (non-singular).

   $A$ is invertible if and only if $\det(A) \neq 0$. Equivalently, $A$ has full rank and all its eigenvalues are non-zero.

3. For a $2\times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, give the formula for $A^{-1}$.

   $$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, \quad ad-bc \neq 0.$$

4. How is the inverse of a matrix expressed using the adjugate?

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

5. State the defining equation for an eigenvalue $\lambda$ and eigenvector $\vec{x}$ of a matrix $A$.

   $$A\vec{x} = \lambda \vec{x}, \quad \vec{x} \neq \vec{0}.$$ Equivalently $(A - \lambda I)\vec{x} = \vec{0}$.

6. What is the characteristic equation used to find eigenvalues?

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

7. State the relationship between the trace of a matrix and its eigenvalues.

   The trace (sum of diagonal entries) equals the sum of the eigenvalues: $\text{tr}(A) = \sum_{i} \lambda_i$.

8. State the relationship between the determinant of a matrix and its eigenvalues.

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

9. State the Cayley–Hamilton theorem.

   Every square matrix satisfies its own characteristic equation. If $p(\lambda) = \det(A - \lambda I)$, then $p(A) = 0$.

10. What are the eigenvalues of a symmetric real matrix, and what property do its eigenvectors have?

    A real symmetric matrix has all real eigenvalues, and eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.

11. What are the eigenvalues of a triangular matrix?

    For a triangular (upper or lower) matrix, the eigenvalues are exactly the diagonal entries.

12. If $\lambda$ is an eigenvalue of $A$, what is the corresponding eigenvalue of $A^{-1}$ and of $A^k$?

    $A^{-1}$ has eigenvalue $\frac{1}{\lambda}$ and $A^k$ has eigenvalue $\lambda^k$, with the same eigenvector $\vec{x}$.

13. State Cramer's rule for solving a linear system $A\vec{x} = \vec{b}$.

    For $\det(A)\neq 0$, $x_i = \dfrac{\det(A_i)}{\det(A)}$, where $A_i$ is $A$ with its $i$-th column replaced by $\vec{b}$.

14. State the consistency condition for the linear system $A\vec{x} = \vec{b}$ in terms of rank.

    The system is consistent if and only if $\text{rank}(A) = \text{rank}([A\,|\,\vec{b}])$. A unique solution exists when this common rank equals the number of unknowns $n$; infinitely many when it is less than $n$.

15. When does a homogeneous system $A\vec{x} = \vec{0}$ have a non-trivial solution?

    A non-trivial (non-zero) solution exists if and only if $\det(A) = 0$, i.e. $\text{rank}(A) < n$ (number of unknowns).

16. 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 $f'(c)=0$.

17. 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 $$f'(c) = \frac{f(b)-f(a)}{b-a}.$$

18. State Cauchy's Mean Value Theorem.

    If $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x)\neq 0$, then there exists $c\in(a,b)$ with $$\frac{f'(c)}{g'(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}.$$

19. State the First Fundamental Theorem of Calculus.

    If $F(x) = \int_a^x f(t)\,dt$ with $f$ continuous, then $F'(x) = f(x)$.

20. State the Second Fundamental Theorem of Calculus.

    If $F$ is an antiderivative of a continuous $f$ on $[a,b]$, then $$\int_a^b f(x)\,dx = F(b) - F(a).$$

21. State the Mean Value Theorem for integrals.

    If $f$ is continuous on $[a,b]$, there exists $c\in[a,b]$ such that $$\int_a^b f(x)\,dx = f(c)\,(b-a).$$

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