ETEA Engineering Test Mathematics Syllabus

Define the transpose of a matrix and give the transpose of a product.

The transpose $A^{t}$ swaps rows and columns: $(A^{t})_{ij} = a_{ji}$. For products $(AB)^{t} = B^{t} A^{t}$ (order reverses).

Distinguish a symmetric matrix from a skew-symmetric matrix.

A symmetric matrix satisfies $A^{t} = A$ (so $a_{ij}=a_{ji}$). A skew-symmetric matrix satisfies $A^{t} = -A$ (so $a_{ij}=-a_{ji}$ and all diagonal entries are $0$).

How do you compute the determinant of a $2\times2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

$$\det = ad - bc.$$

List three key properties of determinants used to simplify their evaluation.

1) Swapping two rows (or columns) changes the sign. 2) If a row/column is multiplied by $k$, the determinant is multiplied by $k$. 3) A determinant with two identical (or proportional) rows/columns equals $0$. Also $\det(AB)=\det(A)\det(B)$ and $\det(A^{t})=\det(A)$.

What does it mean for a matrix to be singular versus non-singular?

A square matrix is singular if $\det(A) = 0$ (it has no inverse), and non-singular (invertible) if $\det(A) \neq 0$.

Give the formula for the inverse of a $2\times2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

$$A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, \quad ad - bc \neq 0.$$

What is the general formula for the inverse of a square matrix using the adjoint?

$$A^{-1} = \frac{1}{\det(A)}\,\operatorname{adj}(A),$$ where $\operatorname{adj}(A)$ is the transpose of the matrix of cofactors, valid when $\det(A) \neq 0$.

State Cramer's rule for the system $a_1 x + b_1 y = c_1$, $a_2 x + b_2 y = c_2$.

With $D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \neq 0$: $$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D},$$ where $D_x$ and $D_y$ replace the $x$- and $y$-coefficient column respectively with the constants column.

State the quadratic formula for $ax^{2} + bx + c = 0$.

$$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}, \quad a \neq 0.$$

What is the discriminant of a quadratic, and how does its sign classify the roots?

The discriminant is $\Delta = b^{2} - 4ac$. If $\Delta > 0$: two distinct real roots; if $\Delta = 0$: one repeated (equal) real root; if $\Delta < 0$: two complex conjugate roots. A perfect-square $\Delta>0$ gives rational roots.

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

$$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}.$$

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

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

What form of partial fraction corresponds to a non-repeated linear factor $(ax+b)$ in the denominator?

A single term $\dfrac{A}{ax+b}$, where $A$ is a constant to be determined.

How do you decompose a fraction with a repeated linear factor $(ax+b)^{2}$ and with an irreducible quadratic factor $(x^{2}+c)$?

Repeated linear: $\dfrac{A}{ax+b} + \dfrac{B}{(ax+b)^{2}}$. Irreducible quadratic: $\dfrac{Ax + B}{x^{2}+c}$ (linear numerator over the quadratic).

What must be true before applying partial fractions, and how is an improper fraction handled?

The fraction must be proper (numerator degree < denominator degree). If improper, first perform polynomial long division so the result is a polynomial plus a proper remainder fraction, then decompose the proper part.

Give the formulas for the $n$th term and the sum of the first $n$ terms of an arithmetic progression.

$$a_n = a + (n-1)d, \qquad S_n = \frac{n}{2}\big[2a + (n-1)d\big] = \frac{n}{2}(a + a_n),$$ where $a$ is the first term and $d$ the common difference.

What is the arithmetic mean of two numbers $a$ and $b$, and the condition for three terms to be in A.P.?

The arithmetic mean is $A = \frac{a+b}{2}$. Three terms $a, b, c$ are in A.P. iff $2b = a + c$ (each term minus the previous is constant).

Give the $n$th term and the sum of the first $n$ terms of a geometric progression.

$$a_n = a r^{\,n-1}, \qquad S_n = \frac{a(1 - r^{n})}{1 - r}, \; r \neq 1,$$ where $a$ is the first term and $r$ the common ratio.

What is the sum to infinity of a geometric series, and when does it converge?

$$S_\infty = \frac{a}{1 - r}, \quad \text{valid only when } |r| < 1.$$ If $|r| \geq 1$ the series diverges.

Define the geometric mean of two positive numbers and the condition for three terms to be in G.P.

The geometric mean is $G = \sqrt{ab}$. Three terms $a,b,c$ are in G.P. iff $b^{2} = ac$ (the ratio of successive terms is constant).

What is a harmonic progression, and how is its $n$th term found?

A harmonic progression (H.P.) is a sequence whose reciprocals form an arithmetic progression. To find the $n$th term, take reciprocals to get the A.P., apply $a + (n-1)d$, then take the reciprocal of the result.

Give the harmonic mean of two numbers and the relationship among A, G, H.

The harmonic mean is $H = \dfrac{2ab}{a+b}$. For two positive numbers, $G^{2} = A \cdot H$ and the inequality $A \geq G \geq H$ holds (equality when $a=b$).

What is the formula for the number of permutations of $n$ distinct objects taken $r$ at a time?

$$^{n}P_{r} = \frac{n!}{(n-r)!}.$$ Order matters in permutations.

How many distinct arrangements are there of $n$ objects where some are alike (e.g. $p$ alike of one kind, $q$ of another)?

$$\frac{n!}{p!\,q!\,\cdots}.$$ Divide $n!$ by the factorials of the counts of each repeated type.

How many circular permutations are there of $n$ distinct objects?

$(n-1)!$ arrangements, because rotations are considered identical (fixing one object removes the rotational symmetry).

State the formula for combinations of $n$ objects taken $r$ at a time and how it relates to permutations.

$$^{n}C_{r} = \frac{n!}{r!\,(n-r)!} = \frac{^{n}P_{r}}{r!}.$$ In combinations order does not matter, so each selection counts once rather than $r!$ times.

State two key identities for combinations.

Symmetry: $^{n}C_{r} = {}^{n}C_{n-r}$. Pascal's rule: $^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r}$. Also $^{n}C_{0} = {}^{n}C_{n} = 1$.

Define the classical (theoretical) probability of an event $E$.

$$P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}} = \frac{n(E)}{n(S)},$$ where $0 \leq P(E) \leq 1$.

State the addition rule of probability for two events, and how it simplifies for mutually exclusive events.

$$P(A \cup B) = P(A) + P(B) - P(A \cap B).$$ If $A$ and $B$ are mutually exclusive then $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

State the multiplication rule of probability for independent events and the complement rule.

For independent events $P(A \cap B) = P(A)\,P(B)$. The complement rule is $P(E') = 1 - P(E)$.

State the two steps of the Principle of Mathematical Induction.

To prove $P(n)$ for all $n \geq 1$: (1) Base step — show $P(1)$ is true. (2) Inductive step — assume $P(k)$ is true (the hypothesis) and use it to prove $P(k+1)$. Then $P(n)$ holds for all natural numbers $n$.

Use induction-derived results: what are the closed forms for $\sum_{k=1}^{n} k$ and $\sum_{k=1}^{n} k^{2}$?

$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \qquad \sum_{k=1}^{n} k^{2} = \frac{n(n+1)(2n+1)}{6}.$$ Both are classic statements provable by mathematical induction.