GATE Environmental Engineering Mathematics Foundation Syllabus

Mathematics Foundation syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics Foundation in GATE Environmental Engineering, not a summary of it.

1. 1

   Linear Algebra

   3 topics

   • Determinants and Matrices
   • Systems of Linear Equations
   • Eigenvalues and Eigenvectors
2. 2

   Calculus

   10 topics

   • Functions
   • Limit
   • Continuity
   • Differentiability
   • Local Maxima and Minima
   • Taylor Series
   • Tests for Convergence
   • Definite and Indefinite Integrals
   • Application of Definite Integral to Obtain Area and Volume
   • Partial and Total Derivatives
3. 3

   Differential Equations

   4 topics

   • Linear and Non-linear First Order Ordinary Differential Equations (ODE)
   • Higher Order Linear ODEs with Constant Coefficients
   • Cauchy's and Euler's Equations
   • Laplace Transform and its Application in Solving Linear ODEs
4. 4

   Probability and Statistics

   12 topics

   • Descriptive Statistics
   • Measurement of Central Tendency
   • Dispersion
   • Skewness and Kurtosis
   • Probability Concepts
   • Conditional Probability
   • Bayes Theorem
   • Risk and Reliability
   • Probability Distributions
   • Correlation
   • Single and Multiple Regression Models
   • Hypothesis Testing (t-test, F-test, chi-square test)

Mathematics Foundation flashcards for GATE Environmental Engineering

19 of 52 cards from the Mathematics Foundation 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 condition on the determinant for a square matrix $A$ to be invertible (non-singular).

   $A$ is invertible if and only if $\det(A) \neq 0$. If $\det(A) = 0$, the matrix is singular and has no inverse.

3. How are the determinant and trace of a matrix related to its eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$?

   $\det(A) = \prod_{i=1}^{n} \lambda_i$ (product of eigenvalues) and $\operatorname{tr}(A) = \sum_{i=1}^{n} \lambda_i$ (sum of eigenvalues).

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

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

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

   For $\det(A) \neq 0$, each unknown is $x_i = \dfrac{\det(A_i)}{\det(A)}$, 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 $[A|b]$, classify the solutions of a linear system $A\vec{x}=\vec{b}$.

   If $\operatorname{rank}(A) \neq \operatorname{rank}([A|b])$: no solution (inconsistent). If $\operatorname{rank}(A) = \operatorname{rank}([A|b]) = n$ (number of unknowns): unique solution. If equal but $< n$: infinitely many solutions.

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

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

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

   A scalar $\lambda$ is an eigenvalue and non-zero vector $\vec{x}$ an eigenvector if $A\vec{x} = \lambda \vec{x}$, meaning $A$ scales $\vec{x}$ without changing its direction.

9. What is the characteristic equation used to find the eigenvalues of a matrix $A$?

   $\det(A - \lambda I) = 0$, where $I$ is the identity matrix.

10. State the Cayley–Hamilton theorem.

    Every square matrix satisfies its own characteristic equation. If the characteristic polynomial is $p(\lambda)$, then $p(A) = 0$.

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

    The eigenvalues are exactly the entries on the main diagonal.

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

    All eigenvalues of a real symmetric matrix are real, and eigenvectors for distinct eigenvalues are orthogonal.

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

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

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

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

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 $g'(x) \neq 0$ near $a$, then $\lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)}$.

16. 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., $|x|$ at $x=0$).

17. Give the limit definition of the derivative $f'(x)$.

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

18. State the first-derivative test conditions for a local maximum and local minimum of $f$ at a critical point $x=c$.

    At $c$ where $f'(c)=0$: if $f'$ changes from $+$ to $-$, $c$ is a local maximum; if $f'$ changes from $-$ to $+$, $c$ is a local minimum.

19. State the second-derivative test for classifying a critical point $x=c$ where $f'(c)=0$.

    If $f''(c) > 0$, $c$ is a local minimum; if $f''(c) < 0$, $c$ is a local maximum; if $f''(c) = 0$, the test is inconclusive.

See more Mathematics Foundation flashcards →

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