PAF Aeronautical Engineering Mathematics Flashcards

24 sample cards from the Mathematics deck

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

1. Give the formula for the determinant of a $2\times 2$ matrix.

   For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad - bc$.

2. State the determinant multiplication property and the rule for $\det(A^{T})$.

   $\det(AB) = \det(A)\det(B)$, and $\det(A^{T}) = \det(A)$.

3. Give the formula for the inverse of a $2\times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

   $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $\det(A) = ad - bc \neq 0$.

4. What general formula gives the inverse of a square matrix, and when does an inverse exist?

   $A^{-1} = \dfrac{1}{\det(A)}\,\operatorname{adj}(A)$, where $\operatorname{adj}(A)$ is the adjugate. The inverse exists iff $A$ is non-singular, i.e. $\det(A) \neq 0$.

5. Define the rank of a matrix.

   The rank of a matrix is the maximum number of linearly independent rows (equivalently, columns); it equals the order of the largest non-zero minor, or the number of non-zero rows in row-echelon form.

6. State Cramer's rule for the system $a_1x + b_1y = c_1$, $a_2x + b_2y = c_2$.

   $x = \dfrac{D_x}{D}$, $y = \dfrac{D_y}{D}$, where $D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$, $D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}$, $D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}$, with $D \neq 0$.

7. How do you solve the matrix equation $AX = B$ using the inverse method?

   If $A$ is non-singular, $X = A^{-1}B$, where $X$ is the column vector of unknowns and $B$ the column vector of constants.

8. For a square linear system, what does the determinant of the coefficient matrix tell you about the solution?

   If $\det(A) \neq 0$, there is a unique solution. If $\det(A) = 0$, the system has either no solution (inconsistent) or infinitely many solutions (dependent).

9. What is the partial fraction decomposition form for a non-repeated linear factor, e.g. $\frac{1}{(x-a)(x-b)}$?

   $\dfrac{1}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}$, where $A, B$ are constants found by equating numerators.

10. What is the partial fraction form for a repeated linear factor $(x-a)^{2}$ in the denominator?

    $\dfrac{A}{x-a} + \dfrac{B}{(x-a)^{2}}$ — one term for each power up to the multiplicity.

11. What is the partial fraction form for an irreducible quadratic factor $(x^{2}+px+q)$?

    $\dfrac{Ax + B}{x^{2}+px+q}$ — the numerator over an irreducible quadratic is linear, of the form $Ax+B$.

12. Before applying partial fractions, what condition must the rational function satisfy, and what if it fails?

    The fraction must be proper (degree of numerator < degree of denominator). If improper, first perform polynomial long division, then decompose the remainder fraction.

13. Define an arithmetic progression (AP) and give its general $n$th term.

    An AP has a constant common difference $d$ between consecutive terms. The $n$th term is $a_n = a + (n-1)d$, where $a$ is the first term.

14. State the formula for the sum of the first $n$ terms of an arithmetic progression.

    $S_n = \dfrac{n}{2}\big[2a + (n-1)d\big] = \dfrac{n}{2}(a + l)$, where $a$ is the first term and $l$ the last term.

15. What is the arithmetic mean of two numbers $a$ and $b$, and what is it for $n$ values?

    For two numbers: $\text{AM} = \dfrac{a+b}{2}$. For $n$ values: $\text{AM} = \dfrac{x_1 + x_2 + \cdots + x_n}{n}$.

16. Define a geometric progression (GP) and give its general $n$th term.

    A GP has a constant common ratio $r$ between consecutive terms. The $n$th term is $a_n = a r^{n-1}$, where $a$ is the first term.

17. State the formula for the sum of the first $n$ terms of a geometric progression.

    $S_n = \dfrac{a(1 - r^{n})}{1 - r}$ for $r \neq 1$ (equivalently $\dfrac{a(r^{n}-1)}{r-1}$).

18. State the formula for the sum of an infinite geometric series and its convergence condition.

    $S_\infty = \dfrac{a}{1 - r}$, valid only when $|r| < 1$ (otherwise the series diverges).

19. What is the geometric mean of two positive numbers $a$ and $b$, and how does it compare to the arithmetic mean?

    $\text{GM} = \sqrt{ab}$. By the AM–GM inequality, $\dfrac{a+b}{2} \geq \sqrt{ab}$, with equality iff $a = b$.

20. State the formulas for the number of permutations $^{n}P_{r}$ and combinations $^{n}C_{r}$.

    $^{n}P_{r} = \dfrac{n!}{(n-r)!}$ (order matters); $^{n}C_{r} = \dfrac{n!}{r!\,(n-r)!}$ (order does not matter).

21. Distinguish permutations from combinations, and give the relationship between them.

    Permutations count ordered arrangements; combinations count unordered selections. They are related by $^{n}P_{r} = r! \cdot {}^{n}C_{r}$.

22. State the classical definition of probability of an event $E$, and the formula for the complement.

    $P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$, with $0 \leq P(E) \leq 1$ and $P(E') = 1 - P(E)$.

23. State the three steps of the principle of mathematical induction.

    (1) Base case: verify the statement holds for $n = 1$. (2) Inductive hypothesis: assume it holds for $n = k$. (3) Inductive step: prove it then holds for $n = k+1$; conclude it holds for all $n \in \mathbb{N}$.

24. State the binomial theorem for $(a+b)^{n}$ and give the general term.

    $(a+b)^{n} = \sum_{r=0}^{n} {}^{n}C_{r}\, a^{n-r} b^{r}$. The general (\,$(r+1)$th\,) term is $T_{r+1} = {}^{n}C_{r}\, a^{n-r} b^{r}$.

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