GATE Biotechnology 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 Biotechnology, not a summary of it.

1. 1

   Linear Algebra

   3 topics

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

   Calculus

   3 topics

   • Limits, Continuity and Differentiability
   • Partial Derivatives, Maxima and Minima
   • Sequences and Series
     • Test for Convergence
3. 3

   Differential Equations

   4 topics

   • Linear and Nonlinear First Order ODEs
   • Higher Order ODEs with Constant Coefficients
   • Cauchy’s and Euler’s Equations
   • Laplace Transforms
4. 4

   Probability and Statistics

   4 topics

   • Mean, Median, Mode and Standard Deviation
   • Random Variables
   • Poisson, Normal and Binomial Distributions
   • Correlation and Regression Analysis
5. 5

   Numerical Methods

   3 topics

   • Solution of Linear and Nonlinear Algebraic Equations
   • Integration by Trapezoidal and Simpson’s Rule
   • Single Step Method for Differential Equations

Engineering Mathematics flashcards for GATE Biotechnology

21 of 61 cards from the Engineering Mathematics deck — real questions with worked answers.

1. What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

   $\det = ad - bc$.

2. For a square matrix $A$, when does the inverse $A^{-1}$ exist?

   $A^{-1}$ exists if and only if $A$ is non-singular, i.e. $\det(A) \neq 0$.

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

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

4. How does $\det(AB)$ relate to $\det(A)$ and $\det(B)$ for square matrices of the same order?

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

5. What is the relationship between $\det(A^{T})$ and $\det(A)$?

   $\det(A^{T}) = \det(A)$.

6. What does the rank of a matrix represent?

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

7. State Cramer's Rule for solving $A\vec{x} = \vec{b}$ (with $\det(A) \neq 0$).

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

8. Using the rank criterion, when is the system $A\vec{x} = \vec{b}$ consistent?

   It is consistent if and only if $\text{rank}(A) = \text{rank}([A\,|\,\vec{b}])$ (the augmented matrix).

9. For a system of $n$ equations $A\vec{x}=\vec{b}$, give the conditions for a unique solution versus infinitely many solutions.

   Unique solution: $\text{rank}(A) = \text{rank}([A|\vec{b}]) = n$. Infinitely many: $\text{rank}(A) = \text{rank}([A|\vec{b}]) < n$. No solution: $\text{rank}(A) \neq \text{rank}([A|\vec{b}])$.

10. When does a homogeneous system $A\vec{x} = \vec{0}$ have non-trivial solutions?

    When $\det(A) = 0$, i.e. $\text{rank}(A) < n$ (the number of unknowns).

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

    A non-zero vector $\vec{x}$ with $A\vec{x} = \lambda \vec{x}$ is an eigenvector, and the scalar $\lambda$ is the corresponding eigenvalue.

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

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

13. How are the trace and determinant of $A$ related to its eigenvalues $\lambda_{1},\dots,\lambda_{n}$?

    $\text{trace}(A) = \sum_{i} \lambda_{i}$ and $\det(A) = \prod_{i} \lambda_{i}$.

14. State the Cayley-Hamilton theorem.

    Every square matrix satisfies its own characteristic equation: substituting $A$ into $\det(A - \lambda I) = 0$ yields the zero matrix.

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

    They are all real, and eigenvectors corresponding to distinct eigenvalues are orthogonal.

16. Give the formal $\epsilon$-$\delta$ definition of $\lim_{x \to a} f(x) = L$.

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

17. State the three conditions for $f$ to be continuous at $x = a$.

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

18. What is the relationship between differentiability and continuity of a function?

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

19. State the definition of the derivative $f'(x)$ as a limit.

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

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

    If $\lim \frac{f(x)}{g(x)}$ is of form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim \dfrac{f(x)}{g(x)} = \lim \dfrac{f'(x)}{g'(x)}$, provided the latter limit exists.

21. For $z = f(x,y)$, what is the total differential $dz$?

    $dz = \dfrac{\partial f}{\partial x}\,dx + \dfrac{\partial f}{\partial y}\,dy$.

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