GATE Biomedical 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 Biomedical Engineering, not a summary of it.

1. 1

   Linear Algebra

   3 topics

   • Matrix algebra
   • Systems of linear equations
   • Eigenvalues and Eigenvectors
2. 2

   Calculus

   9 topics

   • Mean value theorems
   • Theorems of integral calculus
   • Partial derivatives
   • Maxima and minima
   • Multiple integrals
   • Fourier series
   • Vector identities
   • Line, surface and volume integrals
   • Stokes, Gauss and Green's theorems
3. 3

   Differential Equations

   6 topics

   • First order linear and nonlinear differential equations
   • Higher order linear differential equations with constant coefficients
   • Method of separation of variables
   • Cauchy's and Euler's equations
   • Initial and boundary value problems
   • Solution of partial differential equations
4. 4

   Analysis of Complex Variables

   4 topics

   • Analytic functions
   • Cauchy's integral theorem and integral formula
   • Taylor's and Laurent's series
   • Residue theorem
5. 5

   Probability and Statistics

   9 topics

   • Sampling theorems
   • Conditional probability
   • Mean, median, mode and standard deviation
   • Random variables
   • Discrete and continuous distributions: normal, Poisson and binomial distributions
   • Tests of Significance
   • Statistical power analysis
   • Sample size estimation
   • Linear Regression and correlation analysis
6. 6

   Numerical Methods

   4 topics

   • Matrix inversion
   • Numerical solutions of nonlinear algebraic equations
   • Iterative methods for solving differential equations
   • Numerical integration

Engineering Mathematics flashcards for GATE Biomedical Engineering

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

1. What does the rank of a matrix represent, and how is it found?

   The rank is the maximum number of linearly independent rows (or columns) of a matrix. It is found by reducing the matrix to row echelon form and counting the number of non-zero rows.

2. State the condition for a square matrix $A$ to be invertible (non-singular).

   $A$ is invertible if and only if $\det(A) \neq 0$. Equivalently, its rank equals its order $n$, and it has no zero eigenvalue.

3. Define a symmetric and a skew-symmetric matrix.

   A matrix is symmetric if $A^{T} = A$ (so $a_{ij} = a_{ji}$). It is skew-symmetric if $A^{T} = -A$ (so $a_{ij} = -a_{ji}$, and all diagonal entries are $0$).

4. What is the formula for the inverse of a non-singular matrix $A$ using its adjoint?

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

5. State the trace and determinant relationships with eigenvalues for an $n \times n$ matrix.

   The trace equals the sum of eigenvalues, $\operatorname{tr}(A) = \sum_{i=1}^{n} \lambda_{i}$, and the determinant equals their product, $\det(A) = \prod_{i=1}^{n} \lambda_{i}$.

6. How is a system of linear equations $Ax = b$ classified as consistent or inconsistent?

   It is consistent (has a solution) if $\operatorname{rank}(A) = \operatorname{rank}([A \mid b])$. It is inconsistent if $\operatorname{rank}(A) < \operatorname{rank}([A \mid b])$.

7. For a consistent system $Ax = b$ with $n$ unknowns, when is the solution unique versus infinitely many?

   If $\operatorname{rank}(A) = \operatorname{rank}([A \mid b]) = n$, the solution is unique. If the common rank $r < n$, there are infinitely many solutions with $n - r$ free parameters.

8. State Cramer's rule for solving $Ax = b$.

   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 $b$.

9. When does the homogeneous system $Ax = 0$ have a non-trivial solution?

   It has non-trivial solutions if and only if $\det(A) = 0$ (i.e. $\operatorname{rank}(A) < n$). If $\det(A) \neq 0$, only the trivial solution $x = 0$ exists.

10. Define eigenvalue and eigenvector of a square matrix $A$.

    A scalar $\lambda$ is an eigenvalue and non-zero vector $x$ an eigenvector if $Ax = \lambda x$. They are found from the characteristic equation $\det(A - \lambda I) = 0$.

11. State the Cayley-Hamilton theorem.

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

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

    The eigenvalues are simply the entries on the main diagonal.

13. How do eigenvalues of $A^{-1}$, $A^{k}$, and $A^{T}$ relate to those of $A$?

    If $\lambda$ is an eigenvalue of $A$: $A^{-1}$ has eigenvalue $\frac{1}{\lambda}$, $A^{k}$ has eigenvalue $\lambda^{k}$, and $A^{T}$ has the same eigenvalues $\lambda$ as $A$.

14. 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.

15. State Rolle's theorem.

    If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $c \in (a,b)$ such that $f'(c) = 0$.

16. State the Lagrange (Mean Value) theorem.

    If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ with $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

17. State Cauchy's mean value theorem.

    If $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x) \neq 0$, then there exists $c \in (a,b)$ with $$\frac{f'(c)}{g'(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}.$$

18. State the Fundamental Theorem of Calculus (both parts).

    Part 1: If $F(x) = \int_{a}^{x} f(t)\,dt$, then $F'(x) = f(x)$. Part 2: If $F$ is an antiderivative of $f$, then $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.

19. State the Mean Value Theorem for integrals.

    If $f$ is continuous on $[a,b]$, there exists $c \in [a,b]$ such that $$\int_{a}^{b} f(x)\,dx = f(c)\,(b - a).$$

20. What is the geometric meaning of the partial derivative $\frac{\partial f}{\partial x}$?

    It is the rate of change (slope) of $f(x,y)$ in the $x$-direction, holding $y$ constant; the slope of the tangent to the surface in a plane parallel to the $xz$-plane.

21. State the chain rule for $z = f(x,y)$ where $x = x(t)$ and $y = y(t)$.

    $$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}.$$

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