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

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

2. 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.

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

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

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

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

5. 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.

6. 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).

7. 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.

8. 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.

9. 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).

10. 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.

11. 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.

12. 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).

13. 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.

14. 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 $a=b$).

15. 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.

16. 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.

17. 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).

18. 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.

19. 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 $^{n}C_{0} = {}^{n}C_{n} = 1$.

20. 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 $0 \leq P(E) \leq 1$.

21. 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 $P(A \cup B) = P(A) + P(B)$.

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

23. 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 $n$.

24. 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.

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