{
  "version": 1,
  "note": "Jeopardy boards — pure classroom projection mode for no-device schools. 5 categories × 5 point tiers (100/200/300/400/500). Difficulty increases with points. Teacher clicks a cell, question shows full-screen, students answer aloud / on whiteboards.",
  "tiers": [100, 200, 300, 400, 500],
  "courses": {
    "slai": {
      "label": "SL AI",
      "categories": [
        { "title": "Number & Algebra", "cells": [
          { "q": "Round 4,529 to 3 significant figures.", "a": "4530" },
          { "q": "Marcus invests £2000 at 3% p.a. simple interest for 4 years. Total interest?", "a": "£240" },
          { "q": "Solve $\\log_{10}(x) = 2$.", "a": "x = 100" },
          { "q": "A car depreciates by 8% per year from €14,000. Value after 3 years (to nearest €).", "a": "€10,913" },
          { "q": "Percentage error: measured 24.7 m, true 24.0 m. Give the percentage error to 3 s.f.", "a": "2.92%" }
        ]},
        { "title": "Functions", "cells": [
          { "q": "$f(x) = 2x + 5$. Find $f(3)$.", "a": "11" },
          { "q": "Find the y-intercept of $y = 3(0.8)^x$.", "a": "3" },
          { "q": "$f(x) = x^2 - 4x + 3$. Roots?", "a": "x = 1, x = 3" },
          { "q": "The temperature $T$ satisfies $T = 20 + 15\\cos(0.5t)$. Amplitude?", "a": "15" },
          { "q": "$f(x) = a b^x$ with $f(0) = 5$ and $f(1) = 20$. Find $a$ and $b$.", "a": "a = 5, b = 4" }
        ]},
        { "title": "Geometry & Trig", "cells": [
          { "q": "Area of a rectangle 8 cm × 5 cm.", "a": "40 cm²" },
          { "q": "In a right triangle, opposite = 4, hypotenuse = 5. Find $\\sin\\theta$.", "a": "0.8 or 4/5" },
          { "q": "Volume of a cylinder r = 3 cm, h = 10 cm (3 s.f., use π = 3.142).", "a": "283 cm³" },
          { "q": "Triangle sides 7, 9, 12. Largest angle (3 s.f.).", "a": "87.3°" },
          { "q": "A pyramid has square base 6 m, slant height 8 m. Surface area (excl. base)?", "a": "96 m²" }
        ]},
        { "title": "Statistics", "cells": [
          { "q": "Median of 3, 8, 4, 10, 6.", "a": "6" },
          { "q": "Interpret r = -0.94 (choose: strong-negative / weak / strong-positive).", "a": "Strong negative correlation" },
          { "q": "Data: 2, 5, 5, 7, 11. Population standard deviation to 3 s.f.", "a": "3.03" },
          { "q": "Regression line $y = 2.4x - 1.5$. Predict y when x = 15.", "a": "34.5" },
          { "q": "χ² = 8.9, critical value 7.815. Reject H₀? Yes or No?", "a": "Yes — reject H₀" }
        ]},
        { "title": "Calculus", "cells": [
          { "q": "Differentiate $f(x) = 3x^2$.", "a": "f'(x) = 6x" },
          { "q": "Gradient of $y = x^3 - 2x$ at $x = 2$.", "a": "10" },
          { "q": "$\\int 4x \\, dx$.", "a": "2x² + C" },
          { "q": "Area under $y = x^2$ from 0 to 3.", "a": "9" },
          { "q": "For $f(x) = x^3 - 3x$, find x-coordinates of stationary points.", "a": "x = 1 and x = -1" }
        ]}
      ]
    },
    "slaa": {
      "label": "SL AA",
      "categories": [
        { "title": "Sequences", "cells": [
          { "q": "Arithmetic: u₁ = 5, d = 3. Find u₁₀.", "a": "32" },
          { "q": "Geometric: u₁ = 2, r = 3. Find u₅.", "a": "162" },
          { "q": "Sum of first 20 terms of 4, 7, 10, 13, ...", "a": "650" },
          { "q": "S∞ of $8, 4, 2, 1, ...$ ", "a": "16" },
          { "q": "Coefficient of $x^3$ in $(2 + x)^5$.", "a": "40" }
        ]},
        { "title": "Trig", "cells": [
          { "q": "$\\sin(30°) = ?$", "a": "1/2" },
          { "q": "$\\cos(60°) = ?$", "a": "1/2" },
          { "q": "Solve $\\sin(x) = 0.5$ for $x \\in [0°, 360°]$.", "a": "x = 30°, 150°" },
          { "q": "Simplify $\\sin^2\\theta + \\cos^2\\theta$.", "a": "1" },
          { "q": "Solve $2\\cos^2 x - 1 = 0$ for $x \\in [0, 2\\pi]$.", "a": "π/4, 3π/4, 5π/4, 7π/4" }
        ]},
        { "title": "Functions", "cells": [
          { "q": "$f(x) = 2x + 1$, $g(x) = x^2$. $(f \\circ g)(2) = ?$", "a": "9" },
          { "q": "$f(x) = 3x - 6$. Inverse $f^{-1}(x) = ?$", "a": "(x+6)/3" },
          { "q": "Discriminant of $x^2 + 4x + 5 = 0$?", "a": "-4" },
          { "q": "Vertex of $y = -2(x-3)^2 + 5$.", "a": "(3, 5)" },
          { "q": "Solve $\\log_2(x) + \\log_2(x-2) = 3$.", "a": "x = 4" }
        ]},
        { "title": "Probability", "cells": [
          { "q": "P(rolling a prime on a fair die).", "a": "1/2" },
          { "q": "P(A) = 0.3, P(B) = 0.4, independent. P(A ∩ B) = ?", "a": "0.12" },
          { "q": "5 red, 3 blue balls. Draw 2 without replacement. P(both red)?", "a": "5/14" },
          { "q": "X ~ B(6, 0.5). P(X = 3) as a fraction with denominator 64.", "a": "20/64 (or 5/16)" },
          { "q": "X ~ N(50, 4). Find P(X > 55) to 3 s.f.", "a": "0.00621" }
        ]},
        { "title": "Calculus", "cells": [
          { "q": "Differentiate $f(x) = 5x^2$.", "a": "f'(x) = 10x" },
          { "q": "$\\frac{d}{dx}[\\sin(x)]$?", "a": "cos(x)" },
          { "q": "$\\int_1^3 2x \\, dx$.", "a": "8" },
          { "q": "Product rule for $y = x \\sin(x)$: $\\frac{dy}{dx} = ?$", "a": "sin(x) + x cos(x)" },
          { "q": "$\\int \\frac{1}{x} dx$.", "a": "ln|x| + C" }
        ]}
      ]
    },
    "hlai": {
      "label": "HL AI",
      "categories": [
        { "title": "Matrices", "cells": [
          { "q": "det([[2,1],[3,4]]) = ?", "a": "5" },
          { "q": "Matrix product [[1,2],[3,4]] × [[5],[6]] = ?", "a": "[[17],[39]]" },
          { "q": "Inverse of [[2,0],[0,3]].", "a": "[[0.5,0],[0,1/3]]" },
          { "q": "Eigenvalues of [[3,1],[0,2]].", "a": "λ = 3 and λ = 2" },
          { "q": "In a Markov chain with matrix M = [[0.7,0.4],[0.3,0.6]], find the steady state.", "a": "[4/7, 3/7]" }
        ]},
        { "title": "Complex", "cells": [
          { "q": "$|3 + 4i|$?", "a": "5" },
          { "q": "$(2 + i)(1 - i)$?", "a": "3 - i" },
          { "q": "arg(1 + i) in radians?", "a": "π/4" },
          { "q": "$z = 4 \\text{cis}(\\pi/3)$. Cartesian form?", "a": "2 + 2√3 i" },
          { "q": "$z^2 = -9$. Solutions?", "a": "z = ±3i" }
        ]},
        { "title": "Graphs", "cells": [
          { "q": "A graph has 5 vertices, each of degree 4. Total edges?", "a": "10" },
          { "q": "Complete graph K₆ — edges?", "a": "15" },
          { "q": "Kruskal on 4 edges of weights 2, 3, 5, 7 (forming a spanning tree of 4 nodes) — total weight?", "a": "17" },
          { "q": "Adjacency matrix squared gives what count?", "a": "Number of walks of length 2 between vertices" },
          { "q": "A Hamiltonian cycle on K₄ has how many distinct cycles (up to rotation/reflection)?", "a": "3" }
        ]},
        { "title": "Statistics", "cells": [
          { "q": "Sample mean 50, SD 5, n = 100. Standard error of the mean?", "a": "0.5" },
          { "q": "χ² test — degrees of freedom for 4×3 contingency table.", "a": "6" },
          { "q": "z-value for a 95% two-tailed CI.", "a": "1.96" },
          { "q": "Poisson(λ=2). Give P(X = 0) to 3 s.f.", "a": "0.135" },
          { "q": "Two-sample t-test p = 0.02, α = 0.05. Conclusion?", "a": "Reject H₀ (significant)" }
        ]},
        { "title": "Calculus", "cells": [
          { "q": "$\\frac{d}{dx}[\\ln x]$?", "a": "1/x" },
          { "q": "$\\int_0^1 3x^2 dx$.", "a": "1" },
          { "q": "Volume of revolution of $y = x$ around x-axis, from 0 to 3.", "a": "9π" },
          { "q": "Euler's method $y' = y$, $y(0) = 1$, $h = 0.1$. Find $y(0.1)$.", "a": "1.1" },
          { "q": "Coupled DEs: $x' = 2y$, $y' = -2x$. Type of trajectory?", "a": "Circular orbits" }
        ]}
      ]
    },
    "hlaa": {
      "label": "HL AA",
      "categories": [
        { "title": "Complex", "cells": [
          { "q": "$|3 + 4i|$?", "a": "5" },
          { "q": "$(1 + i)^2$?", "a": "2i" },
          { "q": "arg(-1) in radians?", "a": "π" },
          { "q": "By De Moivre: $(\\text{cis}(\\pi/6))^6$?", "a": "cis(π) = -1" },
          { "q": "Non-real 3rd roots of unity?", "a": "cis(2π/3) and cis(4π/3)" }
        ]},
        { "title": "Vectors", "cells": [
          { "q": "(1,2,2)·(2,0,1)", "a": "4" },
          { "q": "|(3, 4, 0)|", "a": "5" },
          { "q": "(1,0,0) × (0,1,0) = ?", "a": "(0, 0, 1)" },
          { "q": "Angle between (1,1,0) and (1,0,1) (to 3 s.f. degrees).", "a": "60°" },
          { "q": "Line through (1,0,2) parallel to (2,1,-1). Vector equation?", "a": "r = (1,0,2) + t(2,1,-1)" }
        ]},
        { "title": "Calculus", "cells": [
          { "q": "$\\frac{d}{dx}[\\ln x]$?", "a": "1/x" },
          { "q": "$\\int \\sec^2 x \\, dx$", "a": "tan(x) + C" },
          { "q": "$\\int x e^x dx$ (by parts).", "a": "xe^x - e^x + C" },
          { "q": "$\\int \\frac{1}{1+x^2} dx$?", "a": "arctan(x) + C" },
          { "q": "Solve DE $\\frac{dy}{dx} = \\frac{y}{x}$, $y(1)=1$.", "a": "y = x" }
        ]},
        { "title": "Series & Proof", "cells": [
          { "q": "Sum $1 + 2 + ... + 100$.", "a": "5050" },
          { "q": "Coefficient of $x^3$ in $(1 + x)^7$.", "a": "35" },
          { "q": "Maclaurin: first 3 terms of $e^x$.", "a": "1 + x + x²/2" },
          { "q": "By induction, first case for $n = 1$ of $1^2 + 2^2 + ... + n^2 = \\frac{n(n+1)(2n+1)}{6}$?", "a": "LHS = 1, RHS = 1·2·3/6 = 1 ✓" },
          { "q": "Prove $\\sqrt{2}$ is irrational — technique?", "a": "Proof by contradiction" }
        ]},
        { "title": "Trig", "cells": [
          { "q": "$\\sec(0) = ?$", "a": "1" },
          { "q": "$\\tan(\\pi/4) = ?$", "a": "1" },
          { "q": "$\\sin(2\\theta) = ?$ (identity)", "a": "2 sin(θ)cos(θ)" },
          { "q": "$\\cos(A+B) = ?$", "a": "cos A cos B − sin A sin B" },
          { "q": "Solve $\\cos(2x) = 0$ for $x \\in [0, 2\\pi]$.", "a": "π/4, 3π/4, 5π/4, 7π/4" }
        ]}
      ]
    }
  }
}