Additional Mathematics for Senior High Schools Year 2 - Review Flashcards

Comprehensive vocabulary flashcards covering sets, binomial expansions, sequences, polynomials, circles, vectors, matrices, statistics, logarithms, trigonometry, and calculus.

De Morgan's Laws

Rules used to relate intersections and unions through complements; e.g., $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

Binomial Theorem

A formula for expanding expressions of the form $(a + b)^n$, given by $\sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$.

Binomial Coefficient ($nCr$)

The number of ways to choose $r$ elements from $n$ elements, calculated as $\frac{n!}{(n-r)!r!}$.

Sequence

A set of numbers arranged in a specific, given order where each number is called a term.

Arithmetic Progression (AP)

A sequence with a common difference $d$. The sum of the first $n$ terms is $S_n = \frac{n}{2}[2a + (n-1)d]$.

Geometric Progression (GP)

A sequence with a common ratio $r$. The sum of the first $n$ terms is $S_n = \frac{a(1-r^n)}{1-r}$ for $r \neq 1$.

Convergent Series

A series whose sum approaches a finite value as the number of terms increases indefinitely; occurs in a GP if |r| < 1.

Sum to Infinity ($S_\infty$)

The limiting value of the sum of a convergent geometric series, given by $S_\infty = \frac{a}{1-r}$.

Rational Zero Theorem

A theorem used to list potential rational zeros of a polynomial by dividing the factors of the constant term by factors of the leading coefficient.

Multiplicity

The number of times a particular root appears in the factorization of a polynomial, affecting the graph's behavior at the $x$-axis.

Standard Equation of a Circle

The representation $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Dot Product (Scalar Product)

An operation on two vectors returning a scalar: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta)$ or $a_1b_1 + a_2b_2$.

Scalar Projection

The magnitude of the vector component of $\mathbf{a}$ in the direction of $\mathbf{b}$, calculated as $\text{Proj}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$.

Determinant (2x2 Matrix)

For a matrix $A = \begin{pmatrix} a &amp; b \ c &amp; d \end{pmatrix}$, the value calculated as $ad - bc$.

Singular Matrix

A square matrix that has a determinant of zero and therefore no inverse.

Bivariate Data

Data involving exactly two variables, often analyzed to determine cause-and-effect relationships.

Spearman’s Rank Correlation Coefficient ($r_s$)

A statistical measure of the monotonic relationship between ranked variables: $r_s = 1 - \frac{6\sum d^2}{n(n^2 - 1)}$.

Exponential Growth/Decay Formula

Modeled by $A = P(1 \pm r)^t$, used for finance, population studies, and radioactive decay.

Pythagorean Identities

Fundamental trigonometric relationships: $\sin^2(\beta) + \cos^2(\beta) = 1$, $1 + \tan^2(\beta) = \sec^2(\beta)$, and $1 + \cot^2(\beta) = \csc^2(\beta)$.

Power Rule (Differentiation)

If $y = ax^n$, then $\frac{dy}{dx} = nax^{n-1}$.

Product Rule

The derivative rule for $y = uv$ given by $\frac{dy}{dx} = v\frac{du}{dx} + u\frac{dv}{dx}$.

Stationary Point

A point on a curve where the gradient $\frac{dy}{dx}$ is equal to zero, potentially a maximum, minimum, or point of inflection.

Second Derivative Test

Using $\frac{d^2y}{dx^2}$ to classify stationary points: if > 0 it is a minimum; if < 0 it is a maximum.

Fundamental Theorem of Calculus

The principle that relates differentiation and integration: $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$, where $F$ is the antiderivative.

Mutually Exclusive Events

Events that cannot occur at the same time; for these events, $P(A \cup B) = P(A) + P(B)$.

Permutations ($nPr$)

Arrangements of items where the specific order is important: $nPr = \frac{n!}{(n-r)!}$.

Combinations ($nCr$)

Selections of items from a larger group where the order of selection does not matter.