GATE Metallurgical 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 Metallurgical Engineering, not a summary of it.

1. 1

   Linear Algebra

   3 topics

   • Matrices and Determinants
   • Systems of linear equations
   • Eigen values and Eigen vectors
2. 2

   Calculus

   5 topics

   • Limit, Continuity and Differentiability
   • Partial derivatives
   • Maxima and minima
   • 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

   5 topics

   • Linear and non-linear first order ODEs
   • Higher order linear ODEs with constant coefficients
   • Cauchy’s and Euler’s equations
   • Laplace transforms
   • PDEs –Laplace, one dimensional heat and wave equations
5. 5

   Probability and Statistics

   7 topics

   • Definitions of probability and sampling theorems
   • Conditional probability
   • Mean, median, mode and standard deviation
   • Random variables
   • Poisson, normal and binomial distributions
   • Analysis of experimental data
   • Linear least squares method
6. 6

   Numerical Methods

   3 topics

   • Solutions of linear and non-linear algebraic equations
     • Bisection
     • Secant
     • Newton-Raphson methods
   • Integration by trapezoidal and Simpson’s rule
   • Single and multi-step methods for differential equations

Engineering Mathematics flashcards for GATE Metallurgical Engineering

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

1. What is 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 its rows/columns are linearly independent.

2. State the formula for the inverse of a non-singular matrix $A$ in terms of its adjugate.

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

3. For a $2\times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, give $\det(A)$ and $A^{-1}$.

   $\det(A) = ad - bc$, and $$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.$$

4. How does a determinant change under the elementary row operation of swapping two rows, and of multiplying a row by a scalar $k$?

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

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

   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 $\vec{b}$.

6. Using the rank of the coefficient matrix $A$ and augmented matrix $[A\,|\,b]$, state the consistency conditions for a system of linear equations in $n$ unknowns.

   If $\text{rank}(A) \neq \text{rank}([A|b])$: inconsistent (no solution). If $\text{rank}(A) = \text{rank}([A|b]) = n$: unique solution. If $\text{rank}(A) = \text{rank}([A|b]) < n$: infinitely many solutions.

7. For a homogeneous system $A\vec{x} = \vec{0}$, when does a non-trivial solution exist?

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

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

   A scalar $\lambda$ is an eigenvalue with eigenvector $\vec{x}\neq\vec{0}$ if $A\vec{x} = \lambda\vec{x}$. They are found from the characteristic equation $\det(A - \lambda I) = 0$.

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

   The sum of eigenvalues equals the trace: $\sum_i \lambda_i = \text{tr}(A)$. The product of eigenvalues equals the determinant: $\prod_i \lambda_i = \det(A)$.

10. What does the Cayley–Hamilton theorem state?

    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 $A$.

11. If $\lambda$ is an eigenvalue of $A$, what are the corresponding eigenvalues of $A^{-1}$, $A^k$, and $A + cI$?

    $A^{-1}$: $\frac{1}{\lambda}$; $A^k$: $\lambda^k$; $A + cI$: $\lambda + c$. The eigenvectors remain the same in each case.

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

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

13. State the three conditions for a function $f$ to be continuous at $x = a$.

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

14. What is the relationship between differentiability and continuity of a function?

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

15. State L'Hôpital's rule for the indeterminate form $\frac{0}{0}$.

    If $\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$ and the limit on the right exists, then $$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}.$$

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

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

17. State the condition for the equality of mixed second-order partial derivatives (Clairaut's/Schwarz's theorem).

    If $f_{xy}$ and $f_{yx}$ are continuous in a neighborhood of a point, then $f_{xy} = f_{yx}$ (order of partial differentiation does not matter).

18. For a function $f(x,y)$, state the second-derivative test to classify a critical point where $f_x = f_y = 0$.

    Let $D = f_{xx}f_{yy} - (f_{xy})^2$. If $D>0$ and $f_{xx}>0$: local minimum; $D>0$ and $f_{xx}<0$: local maximum; $D<0$: saddle point; $D=0$: test inconclusive.

19. Describe the method of Lagrange multipliers for finding extrema of $f$ subject to constraint $g=0$.

    Solve $\nabla f = \lambda \nabla g$ together with $g(x,y,\dots)=0$. The scalar $\lambda$ is the Lagrange multiplier; solutions give the constrained extrema.

20. State the first-derivative (necessary) condition for a local maximum or minimum of a single-variable function $f(x)$.

    At an interior extremum, $f'(x) = 0$ (a stationary/critical point). Classification uses the second derivative: $f''>0$ minimum, $f''<0$ maximum.

21. State the $n$-th term test (divergence test) for an infinite series $\sum a_n$.

    If $\lim_{n\to\infty} a_n \neq 0$ (or does not exist), the series $\sum a_n$ diverges. If $\lim a_n = 0$, the test is inconclusive.

See more Engineering Mathematics flashcards →

Planning Engineering Mathematics for GATE Metallurgical Engineering

Engineering Mathematics (GATE Metallurgical Engineering) FAQ