High School Mathematics Review Notes

Comprehensive practice flashcards covering Romanian high school curriculum topics including Progressions, Logarithms, Functions, Trigonometry, Analytic Geometry, Matrix Algebra, and Calculus.

Arithmetic Progression General Term

$$a_n = a_1 + (n-1) \times r$$

Arithmetic Progression Sum of First $n$ Terms

$S_n = \frac{(a_1 + a_n) \times n}{2}$ or $S_n = \frac{(2a_1 + (n-1) \times r) \times n}{2}$

Geometric Progression Condition for Three Consecutive Numbers

$$b_n^2 = b_{n-1} \times b_{n+1}$$

Logarithm Existence Conditions

For $\log_a(x)$, the conditions are $a > 0$, $a \neq 1$, and $x > 0$.

Change of Base Formula (Logarithms)

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

Complex Number Algebraic Form

$z = a + bi$, where $a$ is the real part ($Re(z)$) and $bi$ is the imaginary part ($Im(z) = b$).

Modulus of a Complex Number ($|z|$)

$$|z| = \sqrt{a^2 + b^2}$$

Imaginary Unit Powers

$i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$

Floor Function ($[x]$)

The largest integer less than or equal to $x$, satisfying $[x] \leq x < [x] + 1$.

Injective Function (Definition)

A function where $f(x_1) = f(x_2) \implies x_1 = x_2$.

Bijective Function

A function that is both injective and surjective.

Coordinates of the Quadratic Vertex ($V$)

$$V\left(\frac{-b}{2a}, \frac{-\Delta}{4a}\right)$$

Vieté's Relations (2nd Degree)

$x_1 + x_2 = \frac{-b}{a}$ and $x_1 \times x_2 = \frac{c}{a}$

Arrangements Formula ($A_n^k$)

$$A_n^k = \frac{n!}{(n-k)!}$$

Combinations Formula ($C_n^k$)

$$C_n^k = \frac{n!}{k! \times (n-k)!}$$

Newton's Binomial Expansion General Term ($T_{k+1}$)

$$T_{k+1} = C_n^k \times a^{n-k} \times b^k$$

Compound Interest Formula ($S_n$)

$$S_n = S \times (1 + \frac{r}{100})^n$$

Slope of a Line through Two Points

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Area of a Triangle (Analytic Geometry)

$Area = \frac{1}{2} \times |\Delta|$, where $\Delta$ is the determinant of the vertices' coordinates.

Fundamental Identity of Trigonometry

$$\sin^2(x) + \cos^2(x) = 1$$

Law of Cosines

$$a^2 = b^2 + c^2 - 2bc \times \cos(A)$$

Trace of a Square Matrix ($Tr(A)$)

The sum of the elements on the principal diagonal.

Hamilton-Cayley Relation (2nd Order)

$$A^2 - Tr(A) \times A + \det(A) \times I_2 = O_2$$

Group (Definition)

A set with a composition law that is closed, associative, has an identity element, and every element is symmetric/invertible.

Leibniz-Newton Formula

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Volume of a Solid of Revolution

$$V = \pi \times \int_a^b f^2(x)\,dx$$

Derivative of $\ln(x)$ ($x > 0$)

$$\frac{1}{x}$$

L'Hospital's Rule

$\lim_{x \to x_0} \frac{f(x)}{g(x)} = \lim_{x \to x_0} \frac{f'(x)}{g'(x)}$ for cases $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Theorem of Rolle (Hypotheses)

$f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$. There exists $c \in (a,b)$ such that $f'(c) = 0$.