GATE Agricultural 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 Agricultural Engineering, not a summary of it.

1. 1

   Linear Algebra

   5 topics

   • Matrices and Determinants
   • Linear and Orthogonal Transformations
   • Caley Hamilton Theorem
   • Eigen Values and Eigen Vectors
   • Solutions of Linear Equations
2. 2

   Calculus

   9 topics

   • Limit, Continuity and Differentiability
   • Partial Derivatives
   • Homogeneous Function - Euler’s Theorem on Homogeneous Functions
   • Total Differentiation
   • Maxima and Minima of Function with Several Independent Variables
   • Sequences and Series - Infinite Series, Tests for Convergence
   • Fourier Series
   • Taylor Series
   • MacLaurin Series
3. 3

   Vector Calculus

   8 topics

   • Vector Differentiation
   • Scalar and Vector Point Functions
   • Vector Differential Operators - del, Gradient
   • Divergence and Curl
   • Physical Interpretations-Line, Surface and Volume Integrals
   • Stokes' Theorem
   • Gauss' Theorem
   • Green’s Theorem
4. 4

   Differential Equations

   5 topics

   • Linear and Non-linear First Order Ordinary Differential Equations (ODE)
   • Homogeneous Differential Equations
   • Higher Order Linear ODEs with Constant Coefficients
   • Laplace Transforms and Their Inverse
   • Partial Differential Equations - Laplace, Heat and Wave Equations
5. 5

   Probability and Statistics

   6 topics

   • Mean, Median, Mode and Standard Deviation
   • Random Variables
   • Poisson Distribution
   • Normal Distribution
   • Binomial Distribution
   • Correlation and Regression Analysis
6. 6

   Numerical Methods

   3 topics

   • Solutions of Linear and Non-linear Algebraic Equations
   • Numerical Integration - Trapezoidal and Simpson’s Rule
   • Numerical Solutions of ODEs

Engineering Mathematics flashcards for GATE Agricultural Engineering

21 of 50 cards from the Engineering Mathematics deck — real questions with worked answers.

1. What is the definition of 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 a square matrix $A$ for it to be invertible (non-singular).

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

3. What is the formula for the inverse of a non-singular matrix $A$ in terms of its adjoint?

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

4. Define the rank of a matrix.

   The rank of a matrix is the order of its largest non-zero minor, equivalently the number of linearly independent rows (or columns).

5. What is an orthogonal matrix, and what is its key property?

   A real square matrix $A$ is orthogonal if $A^{T}A=AA^{T}=I$, i.e. $A^{-1}=A^{T}$. Its determinant is $\det(A)=\pm 1$ and it preserves lengths and angles.

6. What characterizes an orthogonal (linear) transformation geometrically?

   An orthogonal transformation preserves the inner product (hence lengths and angles); $\vec{y}=A\vec{x}$ with $A$ orthogonal so that $\|A\vec{x}\|=\|\vec{x}\|$.

7. State the Cayley–Hamilton theorem.

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

8. How can the Cayley–Hamilton theorem be used to find $A^{-1}$ for a $2\times2$ matrix with characteristic equation $\lambda^{2}-c_{1}\lambda+c_{0}=0$?

   From $A^{2}-c_{1}A+c_{0}I=0$, multiply by $A^{-1}$: $A-c_{1}I+c_{0}A^{-1}=0$, so $A^{-1}=\dfrac{1}{c_{0}}\,(c_{1}I-A)$ (valid when $c_{0}=\det A\neq 0$).

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

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

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

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

11. For an $n\times n$ matrix, how do the sum and product of eigenvalues relate to the matrix?

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

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

    They are simply the entries on the main diagonal.

13. State Cramer's rule for solving a system $A\vec{x}=\vec{b}$ with non-singular $A$.

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

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

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

15. 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. $\operatorname{rank}(A)<n$, the number of unknowns).

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

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

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

    $f$ is continuous at $a$ if $\lim_{x\to a} f(x)=f(a)$ (the limit exists, $f(a)$ is defined, and they are equal).

18. State the relationship between differentiability and continuity.

    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 $x=0$).

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

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

20. State the equality (Clairaut/Schwarz) condition for mixed second-order partial derivatives.

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

21. Define a homogeneous function of degree $n$.

    $f(x,y)$ is homogeneous of degree $n$ if $f(tx,ty)=t^{n} f(x,y)$ for all $t>0$.

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