GATE Chemical 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 Chemical Engineering, not a summary of it.

1. 1

   Linear Algebra

   3 topics

   • Matrix algebra
   • Systems of linear equations
   • Eigen values and eigenvectors
2. 2

   Calculus

   14 topics

   • Functions of single variable
   • Limit, continuity and differentiability
   • Taylor series
   • Mean value theorems
   • Evaluation of definite and improper integrals
   • Partial derivatives
   • Total derivative
   • Maxima and minima
   • Gradient
   • Divergence and Curl
   • Vector identities
   • Directional derivatives
   • Line, Surface and Volume integrals
   • Stokes, Gauss and Green’s theorems
3. 3

   Differential equations

   6 topics

   • First order equations (linear and nonlinear)
   • Higher order linear differential equations with constant coefficients
   • Cauchy’s and Euler’s equations
   • Initial and boundary value problems
   • Laplace transforms
   • Solutions of one-dimensional heat and wave equations and Laplace equation
4. 4

   Complex variables

   3 topics

   • Complex number
   • Polar form of complex number
   • Triangle inequality
5. 5

   Probability and Statistics

   6 topics

   • Definitions of probability and sampling theorems
   • Conditional probability
   • Mean, median, mode and standard deviation
   • Random variables
   • Poisson, Normal and Binomial distributions
   • Linear regression analysis
6. 6

   Numerical Methods

   3 topics

   • Numerical solutions of linear and non-linear algebraic equations
   • Integration by trapezoidal and Simpson’s rule
   • Single and multi-step methods for numerical solution of differential equations

Engineering Mathematics flashcards for GATE Chemical Engineering

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

1. What is a symmetric matrix, and how is it defined?

   A square matrix $A$ is symmetric if $A^{T} = A$, i.e. $a_{ij} = a_{ji}$ for all $i,j$.

2. Define a skew-symmetric matrix and state a property of its diagonal entries.

   A square matrix is skew-symmetric if $A^{T} = -A$, so $a_{ij} = -a_{ji}$. Hence all diagonal entries satisfy $a_{ii} = 0$.

3. What is an orthogonal matrix, and what is its inverse?

   A square matrix $A$ is orthogonal if $A^{T}A = AA^{T} = I$. Therefore $A^{-1} = A^{T}$, and $\det(A) = \pm 1$.

4. State the condition for a square matrix to be invertible (nonsingular).

   $A$ is invertible if and only if $\det(A) \neq 0$, equivalently its rank equals its order and its rows/columns are linearly independent.

5. What is the rank of a matrix?

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

6. Give the formula for the inverse of a nonsingular matrix in terms of its adjugate.

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

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

   $A\vec{x} = \lambda\vec{x}$ with $\vec{x} \neq \vec{0}$. Eigenvalues are the roots of the characteristic equation $\det(A - \lambda I) = 0$.

8. What does the sum of the eigenvalues of a matrix equal?

   The sum of the eigenvalues equals the trace of the matrix: $\sum_i \lambda_i = \operatorname{tr}(A) = \sum_i a_{ii}$.

9. What does the product of the eigenvalues of a matrix equal?

   The product of the eigenvalues equals the determinant: $\prod_i \lambda_i = \det(A)$.

10. State the Cayley–Hamilton theorem.

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

11. What are the eigenvalues of a triangular (or diagonal) matrix?

    They are exactly the diagonal entries of the matrix.

12. What can be said about the eigenvalues of a real symmetric matrix?

    All its eigenvalues are real, and eigenvectors corresponding to distinct eigenvalues are orthogonal.

13. State the Rouché–Capelli (consistency) condition for the linear system $A\vec{x} = \vec{b}$.

    The system is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}([A \mid b])$. If this common rank equals the number of unknowns there is a unique solution; if it is less, there are infinitely many.

14. When does a homogeneous system $A\vec{x} = \vec{0}$ have a nontrivial solution?

    When $\det(A) = 0$ (for a square system), i.e. when the coefficient matrix is singular / rank is less than the number of unknowns.

15. State Cramer's rule for the solution of $A\vec{x} = \vec{b}$.

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

16. Define the limit of a function: what does $\lim_{x \to a} f(x) = L$ mean?

    For every $\varepsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \varepsilon$. The limit exists only if the left- and right-hand limits are equal.

17. State the condition for a function $f$ to be continuous at a point $x = a$.

    $f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$, which requires $f(a)$ to exist, the limit to exist, and the two to be equal.

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. State the definition of the derivative of $f$ at $x$ as a limit.

    $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ provided this limit exists.

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

    $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$ provided the right-hand limit exists and the form is indeterminate.

21. Write the Taylor series expansion of $f(x)$ about $x = a$.

    $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^{n} = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^{2} + \cdots$$

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