PAF Aeronautical Engineering Mathematics Syllabus

For $ax^{2}+bx+c=0$ with roots $\alpha$ and $\beta$, state the sum and product of the roots.

Sum: $\alpha + \beta = -\dfrac{b}{a}$. Product: $\alpha\beta = \dfrac{c}{a}$.

How do you form a quadratic equation given its roots $\alpha$ and $\beta$?

$x^{2} - (\alpha + \beta)x + \alpha\beta = 0$, i.e. $x^{2} - (\text{sum})x + (\text{product}) = 0$.

If $\alpha, \beta$ are roots of $ax^{2}+bx+c=0$, express $\alpha^{2}+\beta^{2}$ in terms of the coefficients.

$\alpha^{2}+\beta^{2} = (\alpha+\beta)^{2} - 2\alpha\beta = \dfrac{b^{2}}{a^{2}} - \dfrac{2c}{a} = \dfrac{b^{2}-2ac}{a^{2}}$.

Define a matrix and state what is meant by its order (dimension).

A matrix is a rectangular array of numbers arranged in rows and columns. Its order is $m \times n$, where $m$ is the number of rows and $n$ the number of columns.

Distinguish a diagonal matrix, a scalar matrix and an identity matrix.

A diagonal matrix has all off-diagonal entries zero. A scalar matrix is a diagonal matrix with all diagonal entries equal. An identity matrix $I$ is a scalar matrix with diagonal entries all equal to 1.

Define the transpose of a matrix and state what makes a matrix symmetric vs. skew-symmetric.

The transpose $A^{T}$ swaps rows and columns. $A$ is symmetric if $A^{T}=A$; it is skew-symmetric if $A^{T}=-A$ (which forces zero diagonal).

State the rule and the dimension condition for matrix multiplication $AB$.

$AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$; an $m\times n$ times $n\times p$ matrix yields an $m\times p$ matrix, with $(AB)_{ij} = \sum_{k} a_{ik}b_{kj}$.

Is matrix multiplication commutative? State two related identities for transposes.

No, in general $AB \neq BA$. Transpose identities: $(AB)^{T} = B^{T}A^{T}$ and $(A+B)^{T} = A^{T}+B^{T}$.

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

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

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

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

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 $\det(A) = ad - bc \neq 0$.

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. $\det(A) \neq 0$.

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.

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 $D \neq 0$.

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.

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).

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.

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.

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 $Ax+B$.

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.

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.

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.

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: $\text{AM} = \dfrac{x_1 + x_2 + \cdots + x_n}{n}$.

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.

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 $\dfrac{a(r^{n}-1)}{r-1}$).

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).

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 $a = b$.

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).

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

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

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 $P(E') = 1 - P(E)$.

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 $n \in \mathbb{N}$.

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 $T_{r+1} = {}^{n}C_{r}\, a^{n-r} b^{r}$.