{
  "course_questions": [
    {
      "id": "a2-u01-q01",
      "unit": 1,
      "skill": "function evaluation",
      "answer": "3",
      "explanation": "Substitute 2: 2(2)²−3(2)+1=3."
    },
    {
      "id": "a2-u01-q02",
      "unit": 1,
      "skill": "function evaluation",
      "answer": "28",
      "explanation": "Substitute -3: 2(-3)²−3(-3)+1=28."
    },
    {
      "id": "a2-u01-q03",
      "unit": 1,
      "skill": "function evaluation",
      "answer": "21",
      "explanation": "Substitute 4: 2(4)²−3(4)+1=21."
    },
    {
      "id": "a2-u01-q04",
      "unit": 1,
      "skill": "domain",
      "answer": "all real x except 5",
      "explanation": "The denominator cannot equal zero, so x≠5."
    },
    {
      "id": "a2-u01-q05",
      "unit": 1,
      "skill": "transformations",
      "answer": "right 4 and up 2",
      "explanation": "Inside subtraction shifts right; outside addition shifts up."
    },
    {
      "id": "a2-u01-q06",
      "unit": 1,
      "skill": "composition",
      "answer": "2x²+1",
      "explanation": "f(g(x))=2(x²)+1."
    },
    {
      "id": "a2-u01-q07",
      "unit": 1,
      "skill": "inverse",
      "answer": "(x+6)/3",
      "explanation": "Swap x and y and solve for y."
    },
    {
      "id": "a2-u01-q08",
      "unit": 1,
      "skill": "inverse",
      "answer": "it is not one-to-one on all real numbers",
      "explanation": "Without restriction, most positive outputs have two inputs."
    },
    {
      "id": "a2-u01-q09",
      "unit": 1,
      "skill": "absolute value",
      "answer": "(2,0)",
      "explanation": "The absolute value parent is shifted right 2."
    },
    {
      "id": "a2-u01-q10",
      "unit": 1,
      "skill": "piecewise",
      "answer": "0",
      "explanation": "At x=0 use the x≥0 rule."
    },
    {
      "id": "a2-u01-q11",
      "unit": 1,
      "skill": "range",
      "answer": "y≥−3",
      "explanation": "A square is nonnegative, so the minimum output is −3."
    },
    {
      "id": "a2-u02-q01",
      "unit": 2,
      "skill": "degree",
      "answer": "6",
      "explanation": "The largest exponent is 6."
    },
    {
      "id": "a2-u02-q02",
      "unit": 2,
      "skill": "polynomial operations",
      "answer": "5x²+4x−3",
      "explanation": "Combine like powers."
    },
    {
      "id": "a2-u02-q03",
      "unit": 2,
      "skill": "polynomial operations",
      "answer": "x²+2x−15",
      "explanation": "Distribute: x²+5x−3x−15."
    },
    {
      "id": "a2-u02-q04",
      "unit": 2,
      "skill": "complex powers",
      "answer": "−i",
      "explanation": "27 mod 4=3, so i²⁷=i³=−i."
    },
    {
      "id": "a2-u02-q05",
      "unit": 2,
      "skill": "complex operations",
      "answer": "7−4i",
      "explanation": "Add real and imaginary parts separately."
    },
    {
      "id": "a2-u02-q06",
      "unit": 2,
      "skill": "complex operations",
      "answer": "2i",
      "explanation": "1+2i+i²=2i."
    },
    {
      "id": "a2-u02-q07",
      "unit": 2,
      "skill": "complex roots",
      "answer": "x=±4i",
      "explanation": "x²=−16 gives x=±4i."
    },
    {
      "id": "a2-u02-q08",
      "unit": 2,
      "skill": "conjugate",
      "answer": "3+5i",
      "explanation": "Change the sign of the imaginary part."
    },
    {
      "id": "a2-u02-q09",
      "unit": 2,
      "skill": "modulus",
      "answer": "10",
      "explanation": "√(6²+(−8)²)=√100=10."
    },
    {
      "id": "a2-u02-q10",
      "unit": 2,
      "skill": "conjugate zeros",
      "answer": "2−3i",
      "explanation": "Nonreal zeros of real-coefficient polynomials occur in conjugate pairs."
    },
    {
      "id": "a2-u02-q11",
      "unit": 2,
      "skill": "end behavior",
      "answer": "both ends down",
      "explanation": "Even degree with negative leading coefficient sends both ends to −∞."
    },
    {
      "id": "a2-u02-q12",
      "unit": 2,
      "skill": "FTA",
      "answer": "7",
      "explanation": "The Fundamental Theorem of Algebra gives 7 complex zeros counting multiplicity."
    },
    {
      "id": "a2-u03-q01",
      "unit": 3,
      "skill": "remainder theorem",
      "answer": "9",
      "explanation": "P(2)=8−4+5=9."
    },
    {
      "id": "a2-u03-q02",
      "unit": 3,
      "skill": "factor theorem",
      "answer": "x+1",
      "explanation": "P(c)=0 means x−c is a factor; c=−1."
    },
    {
      "id": "a2-u03-q03",
      "unit": 3,
      "skill": "rational root theorem",
      "answer": "3/2",
      "explanation": "Candidates are factors of 6 divided by factors of 2."
    },
    {
      "id": "a2-u03-q04",
      "unit": 3,
      "skill": "division",
      "answer": "x+3",
      "explanation": "x²−9=(x−3)(x+3)."
    },
    {
      "id": "a2-u03-q05",
      "unit": 3,
      "skill": "multiplicity",
      "answer": "3",
      "explanation": "The exponent on the factor x−2 is 3."
    },
    {
      "id": "a2-u03-q06",
      "unit": 3,
      "skill": "graph behavior",
      "answer": "touches and turns",
      "explanation": "Even multiplicity preserves sign across the zero."
    },
    {
      "id": "a2-u03-q07",
      "unit": 3,
      "skill": "biquadratic",
      "answer": "x=±1,±2",
      "explanation": "Let u=x²: (u−1)(u−4)=0."
    },
    {
      "id": "a2-u03-q08",
      "unit": 3,
      "skill": "division",
      "answer": "x²+2x+4",
      "explanation": "Use the difference of cubes."
    },
    {
      "id": "a2-u03-q09",
      "unit": 3,
      "skill": "zeros factors",
      "answer": "(x+1)(x−2)(x−4)",
      "explanation": "A zero r corresponds to factor x−r."
    },
    {
      "id": "a2-u03-q10",
      "unit": 3,
      "skill": "modelling",
      "answer": "excellent fit inside data but unstable extreme behavior",
      "explanation": "High-degree polynomials can fit known points yet behave poorly outside them."
    },
    {
      "id": "a2-u03-q11",
      "unit": 3,
      "skill": "synthetic division",
      "answer": "−4",
      "explanation": "x+4=x−(−4)."
    },
    {
      "id": "a2-u03-q12",
      "unit": 3,
      "skill": "multiplicity",
      "answer": "cross with flattening",
      "explanation": "Odd multiplicity crosses; multiplicity above 1 flattens."
    },
    {
      "id": "a2-u04-q01",
      "unit": 4,
      "skill": "rational simplify",
      "answer": "(x+3)/x, x≠0,3",
      "explanation": "Factor and cancel x−3 while preserving original restrictions."
    },
    {
      "id": "a2-u04-q02",
      "unit": 4,
      "skill": "restrictions",
      "answer": "x=−2 and x=2",
      "explanation": "x²−4=(x−2)(x+2)."
    },
    {
      "id": "a2-u04-q03",
      "unit": 4,
      "skill": "rational addition",
      "answer": "(2x+1)/[x(x+1)]",
      "explanation": "Use the LCD x(x+1)."
    },
    {
      "id": "a2-u04-q04",
      "unit": 4,
      "skill": "holes",
      "answer": "a hole",
      "explanation": "Cancellation removes the blow-up but preserves an excluded input."
    },
    {
      "id": "a2-u04-q05",
      "unit": 4,
      "skill": "asymptote",
      "answer": "x=5",
      "explanation": "The denominator is zero at x=5."
    },
    {
      "id": "a2-u04-q06",
      "unit": 4,
      "skill": "asymptote",
      "answer": "y=2",
      "explanation": "The vertical shift sets the horizontal asymptote."
    },
    {
      "id": "a2-u04-q07",
      "unit": 4,
      "skill": "variation",
      "answer": "24",
      "explanation": "k=xy=24."
    },
    {
      "id": "a2-u04-q08",
      "unit": 4,
      "skill": "work rate",
      "answer": "2 h",
      "explanation": "1/T=1/6+1/3=1/2."
    },
    {
      "id": "a2-u04-q09",
      "unit": 4,
      "skill": "extraneous",
      "answer": "check candidates in the original equation",
      "explanation": "Clearing denominators may create invalid candidates."
    },
    {
      "id": "a2-u04-q10",
      "unit": 4,
      "skill": "graph behavior",
      "answer": "yes",
      "explanation": "Horizontal asymptotes describe end behavior, not forbidden values."
    },
    {
      "id": "a2-u04-q11",
      "unit": 4,
      "skill": "variation",
      "answer": "y=kx",
      "explanation": "Direct variation is proportional."
    },
    {
      "id": "a2-u04-q12",
      "unit": 4,
      "skill": "LCD",
      "answer": "x(x−2)",
      "explanation": "Use each distinct factor to its highest power."
    },
    {
      "id": "a2-u05-q01",
      "unit": 5,
      "skill": "rational exponents",
      "answer": "9",
      "explanation": "(∛27)²=9."
    },
    {
      "id": "a2-u05-q02",
      "unit": 5,
      "skill": "radicals",
      "answer": "6√2",
      "explanation": "72=36·2."
    },
    {
      "id": "a2-u05-q03",
      "unit": 5,
      "skill": "absolute value",
      "answer": "|x|",
      "explanation": "The principal square root is nonnegative."
    },
    {
      "id": "a2-u05-q04",
      "unit": 5,
      "skill": "radical operations",
      "answer": "3√2",
      "explanation": "3·2√2−3√2=3√2."
    },
    {
      "id": "a2-u05-q05",
      "unit": 5,
      "skill": "radical equations",
      "answer": "x=4",
      "explanation": "Squaring gives candidates 4 and −1; only 4 checks."
    },
    {
      "id": "a2-u05-q06",
      "unit": 5,
      "skill": "domain",
      "answer": "x≥3",
      "explanation": "The radicand must be nonnegative."
    },
    {
      "id": "a2-u05-q07",
      "unit": 5,
      "skill": "rationalize",
      "answer": "√2/2",
      "explanation": "Multiply numerator and denominator by √2."
    },
    {
      "id": "a2-u05-q08",
      "unit": 5,
      "skill": "transformations",
      "answer": "(3,5)",
      "explanation": "The parent endpoint shifts right 3 and up 5."
    },
    {
      "id": "a2-u05-q09",
      "unit": 5,
      "skill": "cube root",
      "answer": "x=14",
      "explanation": "Cube: 2x−1=27."
    },
    {
      "id": "a2-u05-q10",
      "unit": 5,
      "skill": "distance",
      "answer": "10",
      "explanation": "√(6²+8²)=10."
    },
    {
      "id": "a2-u05-q11",
      "unit": 5,
      "skill": "extraneous",
      "answer": "squaring both sides",
      "explanation": "Squaring is not reversible without checking signs."
    },
    {
      "id": "a2-u05-q12",
      "unit": 5,
      "skill": "exponents radicals",
      "answer": "√(x³)",
      "explanation": "Denominator 2 gives square root; numerator 3 gives power."
    },
    {
      "id": "a2-u06-q01",
      "unit": 6,
      "skill": "growth rate",
      "answer": "6%",
      "explanation": "The factor 1.06 equals 1+0.06."
    },
    {
      "id": "a2-u06-q02",
      "unit": 6,
      "skill": "decay rate",
      "answer": "18%",
      "explanation": "1−0.82=0.18."
    },
    {
      "id": "a2-u06-q03",
      "unit": 6,
      "skill": "exponential equation",
      "answer": "x=3",
      "explanation": "16=2⁴, so x+1=4."
    },
    {
      "id": "a2-u06-q04",
      "unit": 6,
      "skill": "common base",
      "answer": "x=3/2",
      "explanation": "3^(2x)=3³."
    },
    {
      "id": "a2-u06-q05",
      "unit": 6,
      "skill": "continuous growth",
      "answer": "A=Pe^(rt)",
      "explanation": "The base e models continuous compounding."
    },
    {
      "id": "a2-u06-q06",
      "unit": 6,
      "skill": "model type",
      "answer": "exponential",
      "explanation": "Exponential outputs multiply by a constant factor."
    },
    {
      "id": "a2-u06-q07",
      "unit": 6,
      "skill": "compound interest",
      "answer": "compounding periods per year",
      "explanation": "n specifies compounding frequency."
    },
    {
      "id": "a2-u06-q08",
      "unit": 6,
      "skill": "growth factor",
      "answer": "2",
      "explanation": "Each step multiplies by 2."
    },
    {
      "id": "a2-u06-q09",
      "unit": 6,
      "skill": "model comparison",
      "answer": "constant first differences",
      "explanation": "Linear change has constant first differences."
    },
    {
      "id": "a2-u06-q10",
      "unit": 6,
      "skill": "model comparison",
      "answer": "constant second differences",
      "explanation": "Quadratic tables have constant second differences."
    },
    {
      "id": "a2-u06-q11",
      "unit": 6,
      "skill": "residuals",
      "answer": "to detect patterns the model misses",
      "explanation": "Residual patterns reveal systematic lack of fit."
    },
    {
      "id": "a2-u06-q12",
      "unit": 6,
      "skill": "model limits",
      "answer": "extrapolation",
      "explanation": "It lies far beyond observed data."
    },
    {
      "id": "a2-u07-q01",
      "unit": 7,
      "skill": "log definition",
      "answer": "5",
      "explanation": "2⁵=32."
    },
    {
      "id": "a2-u07-q02",
      "unit": 7,
      "skill": "conversion",
      "answer": "3^−2=1/9",
      "explanation": "Logarithmic form states the exponent."
    },
    {
      "id": "a2-u07-q03",
      "unit": 7,
      "skill": "common log",
      "answer": "3",
      "explanation": "10³=1000."
    },
    {
      "id": "a2-u07-q04",
      "unit": 7,
      "skill": "natural log",
      "answer": "−2",
      "explanation": "ln and e^x are inverses."
    },
    {
      "id": "a2-u07-q05",
      "unit": 7,
      "skill": "log properties",
      "answer": "log_b M + log_b N",
      "explanation": "Products become sums."
    },
    {
      "id": "a2-u07-q06",
      "unit": 7,
      "skill": "log properties",
      "answer": "log(a+b)=log a+log b",
      "explanation": "There is no sum property."
    },
    {
      "id": "a2-u07-q07",
      "unit": 7,
      "skill": "log equation",
      "answer": "x=9",
      "explanation": "x−1=8."
    },
    {
      "id": "a2-u07-q08",
      "unit": 7,
      "skill": "domain",
      "answer": "x>4",
      "explanation": "The argument must be positive."
    },
    {
      "id": "a2-u07-q09",
      "unit": 7,
      "skill": "change base",
      "answer": "ln10/ln2",
      "explanation": "Use log_k(x)/log_k(b)."
    },
    {
      "id": "a2-u07-q10",
      "unit": 7,
      "skill": "solve exponential",
      "answer": "t=ln7/0.4",
      "explanation": "Take ln of both sides and divide by 0.4."
    },
    {
      "id": "a2-u07-q11",
      "unit": 7,
      "skill": "log scale",
      "answer": "1000",
      "explanation": "10³=1000."
    },
    {
      "id": "a2-u07-q12",
      "unit": 7,
      "skill": "domain check",
      "answer": "some may make a log argument nonpositive",
      "explanation": "Algebraic manipulation can produce domain-invalid candidates."
    },
    {
      "id": "a2-u08-q01",
      "unit": 8,
      "skill": "nonlinear systems",
      "answer": "−1 and 3",
      "explanation": "Set x²=2x+3 and factor."
    },
    {
      "id": "a2-u08-q02",
      "unit": 8,
      "skill": "sum product",
      "answer": "2 and 3",
      "explanation": "They are roots of t²−5t+6=0."
    },
    {
      "id": "a2-u08-q03",
      "unit": 8,
      "skill": "geometry systems",
      "answer": "no",
      "explanation": "A line intersects a circle in at most two points."
    },
    {
      "id": "a2-u08-q04",
      "unit": 8,
      "skill": "matrix dimensions",
      "answer": "3×2",
      "explanation": "Rows are listed first."
    },
    {
      "id": "a2-u08-q05",
      "unit": 8,
      "skill": "matrix multiplication",
      "answer": "columns of A equal rows of B",
      "explanation": "Inner dimensions must match."
    },
    {
      "id": "a2-u08-q06",
      "unit": 8,
      "skill": "matrix multiplication",
      "answer": "no",
      "explanation": "AB and BA may differ or one may be undefined."
    },
    {
      "id": "a2-u08-q07",
      "unit": 8,
      "skill": "determinant",
      "answer": "ad−bc",
      "explanation": "Use diagonal products."
    },
    {
      "id": "a2-u08-q08",
      "unit": 8,
      "skill": "inverse",
      "answer": "determinant is nonzero",
      "explanation": "Zero determinant means singular."
    },
    {
      "id": "a2-u08-q09",
      "unit": 8,
      "skill": "row reduction",
      "answer": "no solution",
      "explanation": "It represents 0=5."
    },
    {
      "id": "a2-u08-q10",
      "unit": 8,
      "skill": "systems",
      "answer": "intersections",
      "explanation": "A solution satisfies every equation."
    },
    {
      "id": "a2-u08-q11",
      "unit": 8,
      "skill": "matrix equation",
      "answer": "A⁻¹B",
      "explanation": "Multiply on the left by A⁻¹."
    },
    {
      "id": "a2-u08-q12",
      "unit": 8,
      "skill": "verification",
      "answer": "extraneous solutions",
      "explanation": "Squaring can create invalid candidates."
    },
    {
      "id": "a2-u09-q01",
      "unit": 9,
      "skill": "arithmetic sequence",
      "answer": "83",
      "explanation": "7+19·4=83."
    },
    {
      "id": "a2-u09-q02",
      "unit": 9,
      "skill": "geometric sequence",
      "answer": "1/3",
      "explanation": "Divide a term by its predecessor."
    },
    {
      "id": "a2-u09-q03",
      "unit": 9,
      "skill": "sequence formula",
      "answer": "aₙ=a₁+(n−1)d",
      "explanation": "Arithmetic terms add d each step."
    },
    {
      "id": "a2-u09-q04",
      "unit": 9,
      "skill": "sequence formula",
      "answer": "aₙ=a₁r^(n−1)",
      "explanation": "Geometric terms multiply by r."
    },
    {
      "id": "a2-u09-q05",
      "unit": 9,
      "skill": "arithmetic series",
      "answer": "210",
      "explanation": "There are 10 terms; 10(3+39)/2=210."
    },
    {
      "id": "a2-u09-q06",
      "unit": 9,
      "skill": "geometric series",
      "answer": "80",
      "explanation": "Finite geometric sum equals 80."
    },
    {
      "id": "a2-u09-q07",
      "unit": 9,
      "skill": "convergence",
      "answer": "|r|<1",
      "explanation": "Terms must shrink to zero."
    },
    {
      "id": "a2-u09-q08",
      "unit": 9,
      "skill": "infinite series",
      "answer": "24",
      "explanation": "12/(1−1/2)=24."
    },
    {
      "id": "a2-u09-q09",
      "unit": 9,
      "skill": "sigma",
      "answer": "24",
      "explanation": "3+5+7+9=24."
    },
    {
      "id": "a2-u09-q10",
      "unit": 9,
      "skill": "binomial theorem",
      "answer": "6",
      "explanation": "Pascal row is 1,4,6,4,1."
    },
    {
      "id": "a2-u09-q11",
      "unit": 9,
      "skill": "difference table",
      "answer": "quadratic",
      "explanation": "Degree 2 patterns have constant second differences."
    },
    {
      "id": "a2-u09-q12",
      "unit": 9,
      "skill": "recursive",
      "answer": "aₙ=aₙ₋₁+d",
      "explanation": "Recursion uses the previous term."
    },
    {
      "id": "a2-u10-q01",
      "unit": 10,
      "skill": "radians",
      "answer": "5π/6",
      "explanation": "Multiply by π/180."
    },
    {
      "id": "a2-u10-q02",
      "unit": 10,
      "skill": "degrees",
      "answer": "60°",
      "explanation": "Multiply by 180/π."
    },
    {
      "id": "a2-u10-q03",
      "unit": 10,
      "skill": "unit circle",
      "answer": "(cosθ,sinθ)",
      "explanation": "Cosine is horizontal, sine vertical."
    },
    {
      "id": "a2-u10-q04",
      "unit": 10,
      "skill": "exact trig",
      "answer": "−1/2",
      "explanation": "Reference π/6 in quadrant III."
    },
    {
      "id": "a2-u10-q05",
      "unit": 10,
      "skill": "exact trig",
      "answer": "1",
      "explanation": "Both sine and cosine are negative, so tangent is positive."
    },
    {
      "id": "a2-u10-q06",
      "unit": 10,
      "skill": "period",
      "answer": "2π",
      "explanation": "Sine repeats every 2π."
    },
    {
      "id": "a2-u10-q07",
      "unit": 10,
      "skill": "amplitude",
      "answer": "3",
      "explanation": "Amplitude is |A|."
    },
    {
      "id": "a2-u10-q08",
      "unit": 10,
      "skill": "period",
      "answer": "π",
      "explanation": "2π/2=π."
    },
    {
      "id": "a2-u10-q09",
      "unit": 10,
      "skill": "tangent",
      "answer": "x=π/2+kπ",
      "explanation": "Cosine is zero there."
    },
    {
      "id": "a2-u10-q10",
      "unit": 10,
      "skill": "inverse trig",
      "answer": "π/6",
      "explanation": "Principal arcsine lies in [−π/2,π/2]."
    },
    {
      "id": "a2-u10-q11",
      "unit": 10,
      "skill": "vectors",
      "answer": "10",
      "explanation": "√(36+64)=10."
    },
    {
      "id": "a2-u10-q12",
      "unit": 10,
      "skill": "arc length",
      "answer": "6π",
      "explanation": "s=rθ=8·3π/4."
    },
    {
      "id": "a2-u11-q01",
      "unit": 11,
      "skill": "identity",
      "answer": "sin²x+cos²x=1",
      "explanation": "It follows from the unit circle."
    },
    {
      "id": "a2-u11-q02",
      "unit": 11,
      "skill": "identity",
      "answer": "sinx",
      "explanation": "tanx=sinx/cosx."
    },
    {
      "id": "a2-u11-q03",
      "unit": 11,
      "skill": "sum identity",
      "answer": "(√6+√2)/4",
      "explanation": "Use sin(45°+30°)."
    },
    {
      "id": "a2-u11-q04",
      "unit": 11,
      "skill": "double angle",
      "answer": "24/25",
      "explanation": "2sinx cosx=24/25."
    },
    {
      "id": "a2-u11-q05",
      "unit": 11,
      "skill": "trig equation",
      "answer": "π/6 and 5π/6",
      "explanation": "Sine is 1/2 in quadrants I and II."
    },
    {
      "id": "a2-u11-q06",
      "unit": 11,
      "skill": "general solution",
      "answer": "x=−π/4+kπ",
      "explanation": "Tangent period is π."
    },
    {
      "id": "a2-u11-q07",
      "unit": 11,
      "skill": "law of sines",
      "answer": "ASA, AAS, or SSA",
      "explanation": "Law of Sines requires a known opposite pair."
    },
    {
      "id": "a2-u11-q08",
      "unit": 11,
      "skill": "law of cosines",
      "answer": "SAS or SSS",
      "explanation": "Law of Cosines fits SAS and SSS."
    },
    {
      "id": "a2-u11-q09",
      "unit": 11,
      "skill": "ambiguous case",
      "answer": "SSA",
      "explanation": "SSA may yield 0, 1, or 2 triangles."
    },
    {
      "id": "a2-u11-q10",
      "unit": 11,
      "skill": "law of cosines",
      "answer": "7",
      "explanation": "c²=25+64−80(1/2)=49."
    },
    {
      "id": "a2-u11-q11",
      "unit": 11,
      "skill": "precision",
      "answer": "to reduce accumulated error",
      "explanation": "Intermediate rounding can distort final results."
    },
    {
      "id": "a2-u11-q12",
      "unit": 11,
      "skill": "bearings",
      "answer": "north toward east",
      "explanation": "Compass bearings start at north or south."
    },
    {
      "id": "a2-u12-q01",
      "unit": 12,
      "skill": "combinations",
      "answer": "210",
      "explanation": "10C4=210."
    },
    {
      "id": "a2-u12-q02",
      "unit": 12,
      "skill": "permutations",
      "answer": "840",
      "explanation": "7P4=7·6·5·4=840."
    },
    {
      "id": "a2-u12-q03",
      "unit": 12,
      "skill": "complement",
      "answer": "0.73",
      "explanation": "Complement is 1−0.27."
    },
    {
      "id": "a2-u12-q04",
      "unit": 12,
      "skill": "union",
      "answer": "0.8",
      "explanation": "0.6+0.5−0.3=0.8."
    },
    {
      "id": "a2-u12-q05",
      "unit": 12,
      "skill": "conditional probability",
      "answer": "P(A∩B)/P(B)",
      "explanation": "Restrict the sample space to B."
    },
    {
      "id": "a2-u12-q06",
      "unit": 12,
      "skill": "binomial",
      "answer": "constant success probability",
      "explanation": "Binomial trials have fixed n and constant p."
    },
    {
      "id": "a2-u12-q07",
      "unit": 12,
      "skill": "binomial mean",
      "answer": "28",
      "explanation": "Mean=np=28."
    },
    {
      "id": "a2-u12-q08",
      "unit": 12,
      "skill": "z score",
      "answer": "(x−μ)/σ",
      "explanation": "Standardize distance from the mean."
    },
    {
      "id": "a2-u12-q09",
      "unit": 12,
      "skill": "z interpretation",
      "answer": "1.5 standard deviations below the mean",
      "explanation": "The sign gives direction from the mean."
    },
    {
      "id": "a2-u12-q10",
      "unit": 12,
      "skill": "sampling bias",
      "answer": "self-selection bias",
      "explanation": "People choose whether to participate."
    },
    {
      "id": "a2-u12-q11",
      "unit": 12,
      "skill": "causation",
      "answer": "no",
      "explanation": "Confounding and design matter."
    },
    {
      "id": "a2-u12-q12",
      "unit": 12,
      "skill": "margin of error",
      "answer": "sampling uncertainty under stated assumptions",
      "explanation": "It does not automatically include systematic bias."
    },
    {
      "id": "a2-u13-q01",
      "unit": 13,
      "skill": "model question",
      "answer": "a specific decision, variables, and measurable outcome",
      "explanation": "The question must be answerable and decision-relevant."
    },
    {
      "id": "a2-u13-q02",
      "unit": 13,
      "skill": "model selection",
      "answer": "exponential",
      "explanation": "Repeated multiplication is exponential."
    },
    {
      "id": "a2-u13-q03",
      "unit": 13,
      "skill": "model selection",
      "answer": "sinusoidal",
      "explanation": "Sinusoidal functions model cycles."
    },
    {
      "id": "a2-u13-q04",
      "unit": 13,
      "skill": "reproducibility",
      "answer": "to make the analysis reproducible",
      "explanation": "Another person should be able to repeat the work."
    },
    {
      "id": "a2-u13-q05",
      "unit": 13,
      "skill": "sensitivity",
      "answer": "tests how conclusions change when assumptions or parameters vary",
      "explanation": "It reveals fragility or robustness."
    },
    {
      "id": "a2-u13-q06",
      "unit": 13,
      "skill": "limitations",
      "answer": "limited generalizability",
      "explanation": "Population and context may differ."
    },
    {
      "id": "a2-u13-q07",
      "unit": 13,
      "skill": "model defense",
      "answer": "structure, residuals, interpretability, and domain fit",
      "explanation": "Model choice needs multiple forms of evidence."
    },
    {
      "id": "a2-u13-q08",
      "unit": 13,
      "skill": "sensitivity",
      "answer": "sensitive",
      "explanation": "The conclusion is not robust."
    },
    {
      "id": "a2-u13-q09",
      "unit": 13,
      "skill": "communication",
      "answer": "result, uncertainty, limitations, and conditions",
      "explanation": "Transparent communication distinguishes evidence from recommendation."
    },
    {
      "id": "a2-u13-q10",
      "unit": 13,
      "skill": "defense",
      "answer": "explain choices, answer critique, and revise when needed",
      "explanation": "Defense tests reasoning and communication."
    },
    {
      "id": "a2-u13-q11",
      "unit": 13,
      "skill": "model comparison",
      "answer": "to test whether the conclusion depends on model choice",
      "explanation": "Competing models reveal structural uncertainty."
    },
    {
      "id": "a2-u13-q12",
      "unit": 13,
      "skill": "revision",
      "answer": "revise the model or conclusion transparently",
      "explanation": "Revision is part of mathematical inquiry."
    },
    {
      "id": "a2-u01-q12",
      "unit": 1,
      "skill": "range",
      "answer": "y≥−2",
      "explanation": "The square root is nonnegative, then the graph shifts down 2."
    }
  ],
  "diagnostic": [
    {
      "id": "a2-d-01",
      "skill": "expressions",
      "answer": "8x",
      "explanation": "Combine like terms."
    },
    {
      "id": "a2-d-02",
      "skill": "evaluation",
      "answer": "5",
      "explanation": "2(4)−3=5."
    },
    {
      "id": "a2-d-03",
      "skill": "linear equations",
      "answer": "x=7",
      "explanation": "Add 7 and divide by 4."
    },
    {
      "id": "a2-d-04",
      "skill": "linear equations",
      "answer": "x=7",
      "explanation": "Divide by 3, then add 2."
    },
    {
      "id": "a2-d-05",
      "skill": "linear equations",
      "answer": "x=4",
      "explanation": "12=3x."
    },
    {
      "id": "a2-d-06",
      "skill": "inequalities",
      "answer": "x>−4",
      "explanation": "Dividing by a negative reverses the sign."
    },
    {
      "id": "a2-d-07",
      "skill": "slope",
      "answer": "2",
      "explanation": "(11−3)/(5−1)=2."
    },
    {
      "id": "a2-d-08",
      "skill": "linear functions",
      "answer": "y=3x−2",
      "explanation": "Use y=mx+b."
    },
    {
      "id": "a2-d-09",
      "skill": "systems",
      "answer": "(4,3)",
      "explanation": "Add equations: 2x=8."
    },
    {
      "id": "a2-d-10",
      "skill": "factoring",
      "answer": "(x−3)(x+3)",
      "explanation": "Difference of squares."
    },
    {
      "id": "a2-d-11",
      "skill": "factoring",
      "answer": "(x+2)(x+3)",
      "explanation": "2·3=6 and 2+3=5."
    },
    {
      "id": "a2-d-12",
      "skill": "quadratics",
      "answer": "x=2 or 3",
      "explanation": "Factor (x−2)(x−3)."
    },
    {
      "id": "a2-d-13",
      "skill": "discriminant",
      "answer": "−12",
      "explanation": "b²−4ac=16−28."
    },
    {
      "id": "a2-d-14",
      "skill": "quadratics",
      "answer": "(2,5)",
      "explanation": "Vertex form gives (h,k)."
    },
    {
      "id": "a2-d-15",
      "skill": "exponents",
      "answer": "x⁸",
      "explanation": "Add exponents."
    },
    {
      "id": "a2-d-16",
      "skill": "exponents",
      "answer": "x⁵",
      "explanation": "Subtract exponents."
    },
    {
      "id": "a2-d-17",
      "skill": "exponents",
      "answer": "x⁶",
      "explanation": "Multiply exponents."
    },
    {
      "id": "a2-d-18",
      "skill": "exponents",
      "answer": "1/8",
      "explanation": "Negative exponent means reciprocal."
    },
    {
      "id": "a2-d-19",
      "skill": "scientific notation",
      "answer": "4.2×10⁻⁴",
      "explanation": "Move decimal four places right."
    },
    {
      "id": "a2-d-20",
      "skill": "radicals",
      "answer": "5√2",
      "explanation": "50=25·2."
    },
    {
      "id": "a2-d-21",
      "skill": "radicals",
      "answer": "x=49",
      "explanation": "Square both sides."
    },
    {
      "id": "a2-d-22",
      "skill": "domain",
      "answer": "x≠−2",
      "explanation": "Denominator cannot be zero."
    },
    {
      "id": "a2-d-23",
      "skill": "domain",
      "answer": "x≥4",
      "explanation": "Radicand must be nonnegative."
    },
    {
      "id": "a2-d-24",
      "skill": "functions",
      "answer": "11",
      "explanation": "2(5)+1=11."
    },
    {
      "id": "a2-d-25",
      "skill": "composition",
      "answer": "11",
      "explanation": "g(4)=8, then f(8)=11."
    },
    {
      "id": "a2-d-26",
      "skill": "functions",
      "answer": "each input has exactly one output",
      "explanation": "Function definition."
    },
    {
      "id": "a2-d-27",
      "skill": "percent",
      "answer": "20",
      "explanation": "0.25·80=20."
    },
    {
      "id": "a2-d-28",
      "skill": "percent change",
      "answer": "20%",
      "explanation": "Increase 10 divided by original 50."
    },
    {
      "id": "a2-d-29",
      "skill": "statistics",
      "answer": "summarize and predict a trend",
      "explanation": "It models association."
    },
    {
      "id": "a2-d-30",
      "skill": "statistics",
      "answer": "5",
      "explanation": "Sum 20 divided by 4."
    },
    {
      "id": "a2-d-31",
      "skill": "statistics",
      "answer": "3",
      "explanation": "Middle ordered value."
    },
    {
      "id": "a2-d-32",
      "skill": "probability",
      "answer": "1/2",
      "explanation": "Three of six outcomes."
    },
    {
      "id": "a2-d-33",
      "skill": "absolute value",
      "answer": "7",
      "explanation": "Distance is absolute difference."
    },
    {
      "id": "a2-d-34",
      "skill": "absolute value",
      "answer": "x=±5",
      "explanation": "Both numbers are distance 5 from zero."
    },
    {
      "id": "a2-d-35",
      "skill": "number fluency",
      "answer": "0.75",
      "explanation": "3÷4=0.75."
    },
    {
      "id": "a2-d-36",
      "skill": "proportions",
      "answer": "x=10",
      "explanation": "Cross multiply: 3x=30."
    },
    {
      "id": "a2-d-37",
      "skill": "linear functions",
      "answer": "7",
      "explanation": "Set x=0."
    },
    {
      "id": "a2-d-38",
      "skill": "functions",
      "answer": "an x-intercept",
      "explanation": "At a zero, output is 0."
    },
    {
      "id": "a2-d-39",
      "skill": "exponential",
      "answer": "3",
      "explanation": "At x=0, y=3."
    },
    {
      "id": "a2-d-40",
      "skill": "sequences",
      "answer": "2,6,18,54",
      "explanation": "Each term is multiplied by 3."
    }
  ]
}