GATE CS & IT Engineering Engineering Mathematics Flashcards

24 sample cards from the Engineering Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. How many ways can you choose $r$ items from $n$ types with repetition allowed (combinations with repetition)?

   $\binom{n+r-1}{r}$.

2. State the Pigeonhole Principle in its generalized form.

   If $n$ objects are placed into $k$ boxes, then at least one box contains at least $\left\lceil \frac{n}{k} \right\rceil$ objects.

3. State the Binomial Theorem.

   $(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}$.

4. Solve the homogeneous linear recurrence $a_n = 5a_{n-1} - 6a_{n-2}$ using its characteristic equation.

   Characteristic equation: $r^{2} - 5r + 6 = 0$, roots $r=2,3$. General solution: $a_n = A\cdot 2^{n} + B\cdot 3^{n}$.

5. For a linear homogeneous recurrence with a repeated characteristic root $r$ of multiplicity $2$, what form does the solution take?

   $a_n = (A + Bn)\, r^{n}$.

6. What is the ordinary generating function for the sequence $a_n = 1$ for all $n \geq 0$?

   $\sum_{n=0}^{\infty} x^{n} = \frac{1}{1-x}$, valid for $|x|<1$.

7. Give the generating function $\sum_{n=0}^{\infty} \binom{n+k}{k} x^{n}$.

   $\frac{1}{(1-x)^{k+1}}$.

8. How do you find the closed form of a sequence from its generating function?

   Express the generating function as a rational function, decompose into partial fractions, then expand each term as a known power series; the coefficient of $x^{n}$ gives $a_n$.

9. What does it mean for a square matrix to be singular, in terms of its determinant?

   A matrix $A$ is singular (non-invertible) if and only if $\det(A) = 0$.

10. State the formula for the inverse of a non-singular matrix $A$ using its adjugate.

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

11. How does the determinant of a product behave, and what is $\det(A^{-1})$?

    $\det(AB) = \det(A)\det(B)$, and $\det(A^{-1}) = \frac{1}{\det(A)}$.

12. For an $n\times n$ matrix, how do $\det(kA)$ and $\det(A^{T})$ relate to $\det(A)$?

    $\det(kA) = k^{n}\det(A)$ and $\det(A^{T}) = \det(A)$.

13. According to the Rouché-Capelli theorem, when is a system $A\vec{x}=\vec{b}$ consistent, and when does it have a unique solution?

    It is consistent iff $\operatorname{rank}(A) = \operatorname{rank}([A\,|\,\vec{b}])$. The solution is unique iff this common rank equals the number of unknowns $n$; otherwise there are infinitely many solutions.

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

    A non-trivial solution exists iff $\det(A)=0$, i.e. iff $\operatorname{rank}(A)$ is less than the number of unknowns.

15. Define eigenvalues and eigenvectors via the equation $A\vec{x} = \lambda \vec{x}$, and give the characteristic equation.

    A non-zero vector $\vec{x}$ with $A\vec{x}=\lambda\vec{x}$ is an eigenvector with eigenvalue $\lambda$. Eigenvalues solve the characteristic equation $\det(A - \lambda I) = 0$.

16. How do the sum and product of all eigenvalues of a matrix $A$ relate to its trace and determinant?

    $\sum \lambda_i = \operatorname{trace}(A)$ and $\prod \lambda_i = \det(A)$.

17. What are the eigenvalues of a triangular matrix, and what is the eigenvalue property of a real symmetric matrix?

    The eigenvalues of a triangular (or diagonal) matrix are its diagonal entries. A real symmetric matrix has all real eigenvalues and mutually orthogonal eigenvectors.

18. State the Cayley-Hamilton theorem.

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

19. What is LU decomposition, and what is its main computational use?

    It factors a square matrix as $A = LU$, where $L$ is lower triangular and $U$ is upper triangular. It is used to solve $A\vec{x}=\vec{b}$ efficiently via forward substitution ($L\vec{y}=\vec{b}$) then back substitution ($U\vec{x}=\vec{y}$), especially for multiple right-hand sides.

20. State 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 $|f(x)-L| < \epsilon$.

21. State L'Hôpital's rule for evaluating an indeterminate limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

    If $\lim_{x\to a}\frac{f(x)}{g(x)}$ is of form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then it equals $\lim_{x\to a}\frac{f'(x)}{g'(x)}$, provided the latter limit exists.

22. State the condition for a function $f$ to be continuous at a point $x=a$.

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

23. What is the relationship between differentiability and continuity of a function at a point?

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

24. State the first-derivative test and second-derivative test for a local extremum at a critical point $x=c$ where $f'(c)=0$.

    First-derivative test: if $f'$ changes from $+$ to $-$ at $c$, it is a local maximum; from $-$ to $+$, a local minimum. Second-derivative test: $f''(c)<0 \Rightarrow$ local maximum, $f''(c)>0 \Rightarrow$ local minimum; $f''(c)=0$ is inconclusive.

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Engineering Mathematics flashcards FAQ