GATE Mining 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 Mining Engineering, not a summary of it.

1. 1

   Linear Algebra

   5 topics

   • Matrices and Determinants
   • Inverse and Rank of Matrix
   • Systems of Linear Equations
   • Eigen values and Eigen vectors
   • Cayley-Hamilton Theorem
2. 2

   Calculus

   9 topics

   • Limit, Continuity and Differentiability
   • Partial Derivatives
   • Mean Value Theorems
   • Indeterminate Forms and L’Hospital’s Rule
   • Maxima and Minima
   • Taylor’s Theorem
   • Sequences and Series
   • Test for Convergence
   • Fourier Series
3. 3

   Vector Calculus

   4 topics

   • Gradient
   • Divergence and Curl
   • Line, Surface and Volume Integrals
   • Stokes, Gauss and Green’s Theorems
4. 4

   Differential Equations

   3 topics

   • Linear and Non-linear First Order ODEs
   • Higher Order Linear ODEs with Constant Coefficients
   • Cauchy’s and Euler’s Equations
5. 5

   Probability and Statistics

   4 topics

   • Measures of Central Tendency and Dispersion
   • Hypothesis Testing
   • Binomial, Poisson, Exponential and Normal Distributions
   • Correlation and Regression Analysis
6. 6

   Numerical Methods

   4 topics

   • Solutions of Linear Algebraic Equations
   • Interpolation
   • Integration of Trapezoidal and Simpson’s Rule
   • Single and Multi-step Methods for Differential Equations

Engineering Mathematics flashcards for GATE Mining 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 $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

   $\det(A) = ad - bc$.

2. State three key properties of determinants regarding row operations.

   1) Swapping two rows multiplies the determinant by $-1$. 2) Multiplying a row by scalar $k$ multiplies the determinant by $k$. 3) Adding a multiple of one row to another leaves the determinant unchanged.

3. For an $n\times n$ matrix, how do $\det(AB)$, $\det(A^{T})$ and $\det(kA)$ relate to $\det(A)$?

   $\det(AB)=\det(A)\det(B)$, $\det(A^{T})=\det(A)$, and $\det(kA)=k^{n}\det(A)$.

4. What is the formula for the inverse of a matrix $A$ in terms of its adjugate?

   $A^{-1} = \dfrac{1}{\det(A)}\,\operatorname{adj}(A)$, valid when $\det(A)\neq 0$.

5. When does the inverse of a square matrix $A$ exist?

   $A^{-1}$ exists if and only if $A$ is non-singular, i.e. $\det(A)\neq 0$ (equivalently, $A$ has full rank).

6. Define the rank of a matrix.

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

7. For a system $A\vec{x}=\vec{b}$, state the consistency condition using rank.

   The system is consistent iff $\operatorname{rank}(A) = \operatorname{rank}([A\,|\,\vec{b}])$. If this common rank equals the number of unknowns $n$, the solution is unique; if it is less than $n$, there are infinitely many solutions.

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

   When $\operatorname{rank}(A) < n$ (number of unknowns), equivalently when $\det(A)=0$ for a square coefficient matrix.

9. By Cramer's rule, what is $x_{i}$ for the system $A\vec{x}=\vec{b}$?

   $x_{i} = \dfrac{\det(A_{i})}{\det(A)}$, where $A_{i}$ is $A$ with its $i$-th column replaced by $\vec{b}$, valid when $\det(A)\neq 0$.

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

    A scalar $\lambda$ is an eigenvalue and non-zero vector $\vec{x}$ an eigenvector if $A\vec{x}=\lambda\vec{x}$. They satisfy the characteristic equation $\det(A-\lambda I)=0$.

11. What is the relationship between the trace/determinant of $A$ and its eigenvalues?

    The sum of eigenvalues equals the trace: $\sum \lambda_{i} = \operatorname{tr}(A)$, and the product of eigenvalues equals the determinant: $\prod \lambda_{i} = \det(A)$.

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

    $A^{-1}$ has eigenvalue $\dfrac{1}{\lambda}$; $A^{k}$ has eigenvalue $\lambda^{k}$; $A^{T}$ has the same eigenvalue $\lambda$ (same eigenvectors are not guaranteed).

13. State key facts about eigenvalues of symmetric and skew-symmetric real matrices.

    A real symmetric matrix has all real eigenvalues with orthogonal eigenvectors; a real skew-symmetric matrix has eigenvalues that are zero or purely imaginary.

14. State the Cayley-Hamilton theorem.

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

15. How can the Cayley-Hamilton theorem be used to find $A^{-1}$?

    From the characteristic equation, express the identity in terms of powers of $A$ and solve for $A^{-1}$. E.g. for a $2\times 2$ matrix with $A^{2}-(\operatorname{tr}A)A+\det(A)I=0$, one gets $A^{-1}=\dfrac{1}{\det(A)}\big((\operatorname{tr}A)I - A\big)$.

16. State the formal $\epsilon$-$\delta$ definition of $\lim_{x\to a} f(x)=L$.

    For every $\epsilon>0$ there exists $\delta>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\epsilon$.

17. What are the three conditions for $f$ to be continuous at $x=a$?

    1) $f(a)$ is defined; 2) $\lim_{x\to a} f(x)$ exists; 3) $\lim_{x\to a} f(x) = f(a)$.

18. What is the relationship between differentiability and continuity?

    If $f$ is differentiable at a point, it is continuous there. The converse is false: continuity does not imply differentiability (e.g. $f(x)=|x|$ at $x=0$).

19. Define the partial derivative $\dfrac{\partial f}{\partial x}$ of $f(x,y)$.

    $\dfrac{\partial f}{\partial x} = \lim_{h\to 0} \dfrac{f(x+h,y)-f(x,y)}{h}$, the derivative with respect to $x$ holding $y$ constant.

20. State Clairaut's (Schwarz's) theorem on mixed partial derivatives.

    If the mixed second partials are continuous, then $\dfrac{\partial^{2} f}{\partial x\,\partial y} = \dfrac{\partial^{2} f}{\partial y\,\partial x}$.

21. State the total differential of $z=f(x,y)$.

    $dz = \dfrac{\partial f}{\partial x}\,dx + \dfrac{\partial f}{\partial y}\,dy$.

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