GATE Textile 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 Textile 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

   10 topics

   • Limit, continuity and differentiability
   • Successive differentiation
   • Partial differentiation
   • Maxima and minima
   • Errors and approximations
   • Definite and improper integrals
   • Sequences and series
   • Test for convergence
   • Power series
   • Taylor series
3. 3

   Differential Equations

   6 topics

   • First order linear and non-linear differential equations
   • Higher order linear differential equations with constant coefficients
   • Euler-Cauchy equation
   • Partial differential equations
   • Wave and heat equations
   • Laplace's equation
4. 4

   Probability and Statistics

   9 topics

   • Random variables
   • Poisson, binomial and normal distributions
   • Mean, mode, median, standard deviation
   • Confidence interval
   • Test of hypothesis
   • Correlation analysis
   • Regression analysis
   • Analysis of variance
   • Control charts
5. 5

   Numerical Methods

   3 topics

   • Numerical solutions of linear and non-linear algebraic equations
   • Numerical integration by trapezoidal and Simpson's rules
   • Single-step and multi-step numerical methods for differential equations

ENGINEERING MATHEMATICS flashcards for GATE Textile Engineering

21 of 55 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 the key property relating the determinant of a matrix product $AB$ to the determinants of $A$ and $B$.

   $\det(AB) = \det(A)\,\det(B)$.

3. What is the condition on $\det(A)$ for a square matrix $A$ to be invertible (non-singular)?

   $A$ is invertible if and only if $\det(A) \neq 0$.

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

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

5. What does it mean for a matrix to be symmetric versus skew-symmetric?

   Symmetric: $A^{T} = A$. Skew-symmetric: $A^{T} = -A$ (so its diagonal entries are all $0$).

6. How does the determinant change when two rows of a matrix are interchanged?

   The determinant changes sign (is multiplied by $-1$).

7. What is the rank of a matrix?

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

8. State Cramer's rule for solving the system $A\vec{x} = \vec{b}$ with $\det(A) \neq 0$.

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

9. Using rank, state the consistency condition for the linear system $A\vec{x} = \vec{b}$.

   The system is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}([A \mid \vec{b}])$.

10. For a consistent system $A\vec{x}=\vec{b}$ in $n$ unknowns, when is the solution unique versus infinitely many?

    Unique if $\operatorname{rank}(A) = n$; infinitely many ($n - r$ free parameters) if $\operatorname{rank}(A) = r < n$.

11. What condition guarantees a homogeneous system $A\vec{x} = \vec{0}$ has a non-trivial solution?

    A non-trivial solution exists if and only if $\det(A) = 0$ (i.e. $\operatorname{rank}(A) < n$).

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

    A scalar $\lambda$ and non-zero vector $\vec{x}$ satisfying $A\vec{x} = \lambda \vec{x}$ are an eigenvalue and corresponding eigenvector of $A$.

13. What is the characteristic equation used to find eigenvalues of $A$?

    $\det(A - \lambda I) = 0$.

14. How are the trace and determinant of $A$ related to its eigenvalues?

    $\operatorname{trace}(A) = \sum \lambda_{i}$ (sum of eigenvalues) and $\det(A) = \prod \lambda_{i}$ (product of eigenvalues).

15. State the Cayley-Hamilton theorem.

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

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

    They are all real, and eigenvectors for distinct eigenvalues are mutually orthogonal.

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

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

18. What three conditions must hold for $f$ to be continuous at $x = a$?

    $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

19. What is the relationship between differentiability and continuity at a point?

    Differentiability at a point implies continuity there, but continuity does not imply differentiability (e.g. $f(x) = |x|$ at $x = 0$).

20. Define the derivative $f'(x)$ as a limit.

    $f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$.

21. State Leibniz's rule for the $n$-th derivative of a product $(uv)$.

    $(uv)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} u^{(n-k)} v^{(k)}$.

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