full-course.json
json Container
READ_ONLY_ACCESS[ { "id": "maths1-w01-000", "front": "Power Set cardinality of set A with $n$ elements", "back": "$2^n$" }, { "id": "maths1-w01-001", "front": "Cartesian Product cardinality $|A \\times B|$", "back": "$|A| \\times |B|$" }, { "id": "maths1-w01-002", "front": "De Morgan's Law for $(A \\cup B)'$", "back": "$A' \\cap B'$" }, { "id": "maths1-w01-003", "front": "De Morgan's Law for $(A \\cap B)'$", "back": "$A' \\cup B'$" }, { "id": "maths1-w01-004", "front": "Equivalence Relation properties", "back": "Reflexive, Symmetric, and Transitive" }, { "id": "maths1-w01-005", "front": "Partial Order properties", "back": "Reflexive, Antisymmetric, and Transitive" }, { "id": "maths1-w01-006", "front": "Antisymmetric condition", "back": "If $(a, b) \\in R$ and $(b, a) \\in R$, then $a = b$" }, { "id": "maths1-w01-007", "front": "Injective (one-to-one) condition", "back": "$f(x_1) = f(x_2) \\implies x_1 = x_2$" }, { "id": "maths1-w01-008", "front": "Surjective (onto) condition", "back": "Range of $f$ = Codomain of $f$" }, { "id": "maths1-w01-009", "front": "Bijective function", "back": "Both Injective and Surjective (invertible)" }, { "id": "maths1-w02-000", "front": "Distance between $(x_1, y_1)$ and $(x_2, y_2)$", "back": "$\\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$" }, { "id": "maths1-w02-001", "front": "Slope of line through $(x_1, y_1)$ and $(x_2, y_2)$", "back": "$m = \\frac{y_2 - y_1}{x_2 - x_1}$" }, { "id": "maths1-w02-002", "front": "Condition for parallel lines with slopes $m_1, m_2$", "back": "$m_1 = m_2$" }, { "id": "maths1-w02-003", "front": "Condition for perpendicular lines with slopes $m_1, m_2$", "back": "$m_1 m_2 = -1 \\implies m_2 = -\\frac{1}{m_1}$" }, { "id": "maths1-w02-004", "front": "Slope of general line $ax + by + c = 0$", "back": "$m = -\\frac{a}{b}$" }, { "id": "maths1-w02-005", "front": "Perpendicular distance from $(x_0, y_0)$ to $ax + by + c = 0$", "back": "$\\frac{|ax_0 + by_0 + c|}{\\sqrt{a^2 + b^2}}$" }, { "id": "maths1-w02-006", "front": "Midpoint of segment between $(x_1, y_1)$ and $(x_2, y_2)$", "back": "$(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2})$" }, { "id": "maths1-w03-000", "front": "Vertex $x$-coordinate for $y = ax^2 + bx + c$", "back": "$x = -\\frac{b}{2a}$" }, { "id": "maths1-w03-001", "front": "Discriminant formula for $ax^2 + bx + c = 0$", "back": "$D = b^2 - 4ac$" }, { "id": "maths1-w03-002", "front": "Roots nature when discriminant $D > 0$", "back": "Two distinct real roots" }, { "id": "maths1-w03-003", "front": "Roots nature when discriminant $D = 0$", "back": "One real repeated root (tangent to axis)" }, { "id": "maths1-w03-004", "front": "Roots nature when discriminant $D < 0$", "back": "Two complex conjugate roots (no x-intercepts)" }, { "id": "maths1-w03-005", "front": "Vieta's formula for sum of roots", "back": "$\\alpha + \\beta = -\\frac{b}{a}$" }, { "id": "maths1-w03-006", "front": "Vieta's formula for product of roots", "back": "$\\alpha \\beta = \\frac{c}{a}$" }, { "id": "maths1-w03-007", "front": "Parabola opening direction if $a > 0$", "back": "Upward (has a global minimum at vertex)" }, { "id": "maths1-w03-008", "front": "Parabola opening direction if $a < 0$", "back": "Downward (has a global maximum at vertex)" }, { "id": "maths1-w04-000", "front": "Remainder Theorem for $P(x)$ divided by $x - c$", "back": "Remainder is $P(c)$" }, { "id": "maths1-w04-001", "front": "Factor Theorem condition", "back": "$x - c$ is a factor of $P(x)$ if and only if $P(c) = 0$" }, { "id": "maths1-w04-002", "front": "Behavior of graph at root with even multiplicity", "back": "Touches x-axis and turns around (tangent)" }, { "id": "maths1-w04-003", "front": "Behavior of graph at root with odd multiplicity", "back": "Crosses the x-axis" }, { "id": "maths1-w04-004", "front": "Maximum number of turning points for degree $n$ polynomial", "back": "$n - 1$" }, { "id": "maths1-w04-005", "front": "End behavior of polynomial is determined by", "back": "The leading term $a_n x^n$" }, { "id": "maths1-w05-000", "front": "Vertical Line Test determines", "back": "Whether a curve in the plane represents a valid function" }, { "id": "maths1-w05-001", "front": "Horizontal Line Test determines", "back": "Whether a function is injective (and therefore invertible)" }, { "id": "maths1-w05-002", "front": "Composite function $(f \\circ g)(x)$ definition", "back": "$f(g(x))$" }, { "id": "maths1-w05-003", "front": "Graph of $f^{-1}(x)$ relative to $f(x)$", "back": "Reflected across the line $y = x$" }, { "id": "maths1-w05-004", "front": "Condition for existence of inverse function $f^{-1}$", "back": "The function must be bijective (one-to-one and onto)" }, { "id": "maths1-w05-005", "front": "Exponential function $f(x) = a^x$ domain and range ($a > 0, a \\neq 1$)", "back": "Domain: $\\mathbb{R}$, Range: $(0, \\infty)$" }, { "id": "maths1-w06-000", "front": "Definition of $\\log_b(x) = y$", "back": "$b^y = x$ ($x > 0, b > 0, b \\neq 1$)" }, { "id": "maths1-w06-001", "front": "Product rule for logarithms: $\\log_b(xy)$", "back": "$\\log_b(x) + \\log_b(y)$" }, { "id": "maths1-w06-002", "front": "Quotient rule for logarithms: $\\log_b(x/y)$", "back": "$\\log_b(x) - \\log_b(y)$" }, { "id": "maths1-w06-003", "front": "Power rule for logarithms: $\\log_b(x^k)$", "back": "$k \\log_b(x)$" }, { "id": "maths1-w06-004", "front": "Change of base formula for $\\log_b(a)$", "back": "$\\frac{\\ln(a)}{\\ln(b)} = \\frac{\\log_{10}(a)}{\\log_{10}(b)}$" }, { "id": "maths1-w06-005", "front": "Natural logarithm $\\ln(x)$ base", "back": "Base $e \\approx 2.71828$" }, { "id": "maths1-w06-006", "front": "Dangerous trap when solving log equations", "back": "Must check for extraneous roots (log argument must be $> 0$)" }, { "id": "maths1-w07-000", "front": "Limit of sequence $\\lim_{n\\to\\infty} \\frac{1}{n}$", "back": "$0$" }, { "id": "maths1-w07-001", "front": "Limit at infinity of rational function where deg(numerator) < deg(denominator)", "back": "$0$" }, { "id": "maths1-w07-002", "front": "Limit at infinity of rational function where deg(numerator) = deg(denominator)", "back": "Ratio of leading coefficients $\\frac{a_n}{b_n}$" }, { "id": "maths1-w07-003", "front": "Three conditions for continuity of $f(x)$ at $x = c$", "back": "1. $f(c)$ is defined; 2. $\\lim_{x\\to c} f(x)$ exists; 3. $\\lim_{x\\to c} f(x) = f(c)$" }, { "id": "maths1-w07-004", "front": "Two-sided limit $\\lim_{x\\to c} f(x) = L$ exists iff", "back": "$\\lim_{x\\to c^-} f(x) = \\lim_{x\\to c^+} f(x) = L$" }, { "id": "maths1-w08-000", "front": "Definition of derivative $f'(x)$ from first principles", "back": "$\\lim_{h\\to 0} \\frac{f(x+h) - f(x)}{h}$" }, { "id": "maths1-w08-001", "front": "Relationship between differentiability and continuity", "back": "Differentiable $\\implies$ Continuous (converse is FALSE)" }, { "id": "maths1-w08-002", "front": "Power rule for differentiation $\\frac{d}{dx}[x^n]$", "back": "$n x^{n-1}$" }, { "id": "maths1-w08-003", "front": "Product rule for differentiation $\\frac{d}{dx}[f(x)g(x)]$", "back": "$f'(x)g(x) + f(x)g'(x)$" }, { "id": "maths1-w08-004", "front": "Quotient rule for differentiation $\\frac{d}{dx}\\left[\\frac{f(x)}{g(x)}\\right]$", "back": "$\\frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$" }, { "id": "maths1-w08-005", "front": "Chain rule for differentiation $\\frac{d}{dx}[f(g(x))]$", "back": "$f'(g(x)) \\cdot g'(x)$" }, { "id": "maths1-w08-006", "front": "Critical point of $f(x)$ definition", "back": "Point in domain where $f'(x) = 0$ or $f'(x)$ is undefined" }, { "id": "maths1-w08-007", "front": "L'Hôpital's rule condition", "back": "Applicable only to indeterminate forms $\\frac{0}{0}$ or $\\frac{\\pm\\infty}{\\pm\\infty}$" }, { "id": "maths1-w09-000", "front": "Fundamental Theorem of Calculus (Part 1)", "back": "$\\frac{d}{dx} \\left[\\int_a^x f(t) dt\\right] = f(x)$" }, { "id": "maths1-w09-001", "front": "Fundamental Theorem of Calculus (Part 2)", "back": "$\\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$" }, { "id": "maths1-w09-002", "front": "Power rule for integration $\\int x^n dx$ ($n \\neq -1$)", "back": "$\\frac{x^{n+1}}{n+1} + C$" }, { "id": "maths1-w09-003", "front": "Indefinite integral $\\int \\frac{1}{x} dx$", "back": "$\\ln|x| + C$" }, { "id": "maths1-w09-004", "front": "Indefinite integral $\\int e^{kx} dx$", "back": "$\\frac{1}{k} e^{kx} + C$" }, { "id": "maths1-w09-005", "front": "Geometric meaning of $\\int_a^b f(x) dx$", "back": "Net signed area between curve $y = f(x)$ and x-axis" }, { "id": "maths1-w09-006", "front": "u-substitution is the integral counterpart of", "back": "The Chain Rule in differentiation" }, { "id": "maths1-w10-000", "front": "Handshaking Lemma for undirected graph $G = (V, E)$", "back": "$\\sum_{v \\in V} \\deg(v) = 2|E|$" }, { "id": "maths1-w10-001", "front": "Maximum number of edges in simple undirected graph with $n$ vertices", "back": "$\\binom{n}{2} = \\frac{n(n-1)}{2}$" }, { "id": "maths1-w10-002", "front": "Maximum number of directed edges in simple digraph with $n$ vertices", "back": "$n(n-1)$" }, { "id": "maths1-w10-003", "front": "Breadth-First Search (BFS) primary data structure", "back": "Queue (FIFO) — finds shortest paths in unweighted graphs" }, { "id": "maths1-w10-004", "front": "Depth-First Search (DFS) primary data structure", "back": "Stack (LIFO) or recursion" }, { "id": "maths1-w10-005", "front": "Number of edges in a tree with $n$ vertices", "back": "$n - 1$" }, { "id": "maths1-w10-006", "front": "DAG stands for", "back": "Directed Acyclic Graph" }, { "id": "maths1-w10-007", "front": "A directed graph has a topological sort if and only if", "back": "It contains no directed cycles (it is a DAG)" }, { "id": "maths1-w11-000", "front": "Relaxation step for edge $(u, v)$ with weight $w(u, v)$", "back": "If $dist[u] + w(u, v) < dist[v]$: $dist[v] = dist[u] + w(u, v)$" }, { "id": "maths1-w11-001", "front": "Dijkstra's algorithm requirement", "back": "All edge weights must be nonnegative ($w(u, v) \\ge 0$)" }, { "id": "maths1-w11-002", "front": "Bellman–Ford algorithm number of relaxation rounds", "back": "$|V| - 1$ rounds over all edges" }, { "id": "maths1-w11-003", "front": "How Bellman–Ford detects negative-weight cycles", "back": "A negative cycle exists if an edge can still be relaxed in the $|V|$-th round" }, { "id": "maths1-w11-004", "front": "Adjacency matrix power $(A^k)_{ij}$ interpretation", "back": "Number of walks of length exactly $k$ from vertex $i$ to vertex $j$" }, { "id": "maths1-w11-005", "front": "Transitive closure definition", "back": "Reachability matrix indicating if a path of any length exists from $i$ to $j$" }, { "id": "maths1-w11-006", "front": "Algorithm for single-source shortest path in DAG with arbitrary weights", "back": "Topological sort + single relaxation pass in $O(|V| + |E|)$" }, { "id": "maths1-w12-000", "front": "Equivalence Relation vs Partial Order distinction", "back": "Equivalence is Symmetric; Partial Order is Antisymmetric" }, { "id": "maths1-w12-001", "front": "Vertex of parabola vs Critical point of derivative", "back": "Vertex $x = -b/(2a)$ is where $f'(x) = 2ax + b = 0$" }, { "id": "maths1-w12-002", "front": "Shortest path algorithm decision: Unweighted graph", "back": "Breadth-First Search (BFS) in $O(|V| + |E|)$" }, { "id": "maths1-w12-003", "front": "Shortest path algorithm decision: Nonnegative weights", "back": "Dijkstra's Algorithm in $O((|V| + |E|)\\log |V|)$" }, { "id": "maths1-w12-004", "front": "Shortest path algorithm decision: General weights (negative edges allowed)", "back": "Bellman–Ford Algorithm in $O(|V| \\cdot |E|)$" }, { "id": "maths1-w12-005", "front": "Fundamental theorem linking derivatives and integrals", "back": "Fundamental Theorem of Calculus (FTC): accumulation vs rate of change" } ]