[ { "objective_code": "MATH.S1.INT.BASIC", "stem_en": "Calculate: (-8) + 5", "stem_zh": "計算：(-8) + 5", "option_a_en": "-13", "option_b_en": "-3", "option_c_en": "3", "option_d_en": "13", "option_a_zh": "-13", "option_b_zh": "-3", "option_c_zh": "3", "option_d_zh": "13", "answer": "B", "explanation_en": "When adding a positive to a negative, subtract the smaller from the larger and keep the sign of the larger: 8 - 5 = 3, and since 8 > 5, the result is negative: -3.", "explanation_zh": "負數加正數：用大數減小數，結果取絕對值較大的符號。8 - 5 = 3，因為8 > 5，結果為負：-3。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.INT.BASIC", "stem_en": "Calculate: (-6) × (-4)", "stem_zh": "計算：(-6) × (-4)", "option_a_en": "-24", "option_b_en": "-10", "option_c_en": "10", "option_d_en": "24", "option_a_zh": "-24", "option_b_zh": "-10", "option_c_zh": "10", "option_d_zh": "24", "answer": "D", "explanation_en": "Negative × negative = positive. (-6) × (-4) = +(6 × 4) = 24.", "explanation_zh": "負數乘以負數結果為正數。(-6) × (-4) = +(6 × 4) = 24。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.INT.BASIC", "stem_en": "Which of the following is correct?", "stem_zh": "以下哪項是正確的？", "option_a_en": "(-3)² = -9", "option_b_en": "(-3)² = 9", "option_c_en": "-3² = 9", "option_d_en": "(-3)³ = 27", "option_a_zh": "(-3)² = -9", "option_b_zh": "(-3)² = 9", "option_c_zh": "-3² = 9", "option_d_zh": "(-3)³ = 27", "answer": "B", "explanation_en": "(-3)² = (-3) × (-3) = 9. Note: -3² = -(3²) = -9, so C is wrong. (-3)³ = -27, so D is wrong.", "explanation_zh": "(-3)² = (-3) × (-3) = 9。注意：-3² = -(3²) = -9，所以C錯。(-3)³ = -27，所以D錯。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.INT.BASIC", "stem_en": "Simplify: 3 - (-5) + (-2)", "stem_zh": "化簡：3 - (-5) + (-2)", "option_a_en": "-4", "option_b_en": "0", "option_c_en": "6", "option_d_en": "10", "option_a_zh": "-4", "option_b_zh": "0", "option_c_zh": "6", "option_d_zh": "10", "answer": "C", "explanation_en": "3 - (-5) + (-2) = 3 + 5 - 2 = 6. Subtracting a negative is the same as adding a positive.", "explanation_zh": "3 - (-5) + (-2) = 3 + 5 - 2 = 6。減去一個負數等於加上一個正數。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.FRACTION.BASIC", "stem_en": "Which fraction is equivalent to 4/6?", "stem_zh": "哪個分數與 4/6 相等？", "option_a_en": "2/3", "option_b_en": "3/4", "option_c_en": "8/10", "option_d_en": "6/8", "option_a_zh": "2/3", "option_b_zh": "3/4", "option_c_zh": "8/10", "option_d_zh": "6/8", "answer": "A", "explanation_en": "4/6 simplifies to 2/3 by dividing both numerator and denominator by 2.", "explanation_zh": "4/6 的分子分母都除以2，得到 2/3，所以兩者相等。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.FRACTION.BASIC", "stem_en": "What is 3/4 as a decimal?", "stem_zh": "3/4 等於多少小數？", "option_a_en": "0.25", "option_b_en": "0.34", "option_c_en": "0.75", "option_d_en": "1.33", "option_a_zh": "0.25", "option_b_zh": "0.34", "option_c_zh": "0.75", "option_d_zh": "1.33", "answer": "C", "explanation_en": "3 ÷ 4 = 0.75. You can also think of it as 75/100 = 0.75.", "explanation_zh": "3 ÷ 4 = 0.75。也可以想成 3/4 = 75/100 = 0.75。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.FRACTION.ADD", "stem_en": "Calculate: 3/8 + 2/8", "stem_zh": "計算：3/8 + 2/8", "option_a_en": "5/16", "option_b_en": "5/8", "option_c_en": "6/16", "option_d_en": "1", "option_a_zh": "5/16", "option_b_zh": "5/8", "option_c_zh": "6/16", "option_d_zh": "1", "answer": "B", "explanation_en": "Same denominator fractions: add the numerators, keep the denominator. 3 + 2 = 5, so the answer is 5/8.", "explanation_zh": "同分母分數相加：只加分子，分母不變。3 + 2 = 5，答案是 5/8。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.FRACTION.ADD", "stem_en": "Calculate: 1/3 + 1/4", "stem_zh": "計算：1/3 + 1/4", "option_a_en": "2/7", "option_b_en": "1/6", "option_c_en": "7/12", "option_d_en": "5/12", "option_a_zh": "2/7", "option_b_zh": "1/6", "option_c_zh": "7/12", "option_d_zh": "5/12", "answer": "C", "explanation_en": "Find a common denominator: LCD of 3 and 4 is 12. 1/3 = 4/12, 1/4 = 3/12. Sum = 7/12.", "explanation_zh": "找公分母：3和4的最小公倍數是12。1/3 = 4/12，1/4 = 3/12。相加 = 7/12。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.FRACTION.ADD", "stem_en": "Calculate: 5/6 - 1/4", "stem_zh": "計算：5/6 - 1/4", "option_a_en": "4/2", "option_b_en": "7/12", "option_c_en": "1/3", "option_d_en": "2/3", "option_a_zh": "4/2", "option_b_zh": "7/12", "option_c_zh": "1/3", "option_d_zh": "2/3", "answer": "B", "explanation_en": "LCD of 6 and 4 is 12. 5/6 = 10/12, 1/4 = 3/12. 10/12 - 3/12 = 7/12.", "explanation_zh": "6和4的最小公倍數是12。5/6 = 10/12，1/4 = 3/12。10/12 - 3/12 = 7/12。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.FRACTION.MULT", "stem_en": "Calculate: 2/3 × 3/4", "stem_zh": "計算：2/3 × 3/4", "option_a_en": "5/7", "option_b_en": "6/7", "option_c_en": "1/2", "option_d_en": "2/4", "option_a_zh": "5/7", "option_b_zh": "6/7", "option_c_zh": "1/2", "option_d_zh": "2/4", "answer": "C", "explanation_en": "Multiply numerators and denominators: (2×3)/(3×4) = 6/12 = 1/2.", "explanation_zh": "分子相乘、分母相乘：(2×3)/(3×4) = 6/12 = 1/2。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.FRACTION.MULT", "stem_en": "Calculate: 3/5 ÷ 2/5", "stem_zh": "計算：3/5 ÷ 2/5", "option_a_en": "6/25", "option_b_en": "1/5", "option_c_en": "3/2", "option_d_en": "5/3", "option_a_zh": "6/25", "option_b_zh": "1/5", "option_c_zh": "3/2", "option_d_zh": "5/3", "answer": "C", "explanation_en": "Division by a fraction = multiply by its reciprocal. 3/5 ÷ 2/5 = 3/5 × 5/2 = 15/10 = 3/2.", "explanation_zh": "除以一個分數等於乘以它的倒數。3/5 ÷ 2/5 = 3/5 × 5/2 = 15/10 = 3/2。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.DECIMAL.OPS", "stem_en": "Calculate: 1.2 × 0.3", "stem_zh": "計算：1.2 × 0.3", "option_a_en": "0.036", "option_b_en": "0.36", "option_c_en": "3.6", "option_d_en": "36", "option_a_zh": "0.036", "option_b_zh": "0.36", "option_c_zh": "3.6", "option_d_zh": "36", "answer": "B", "explanation_en": "12 × 3 = 36. Count 2 decimal places total (1 from 1.2, 1 from 0.3), giving 0.36.", "explanation_zh": "12 × 3 = 36。共有2位小數（1.2有1位，0.3有1位），所以答案是0.36。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.DECIMAL.OPS", "stem_en": "Calculate: 4.8 ÷ 0.4", "stem_zh": "計算：4.8 ÷ 0.4", "option_a_en": "0.12", "option_b_en": "1.2", "option_c_en": "12", "option_d_en": "120", "option_a_zh": "0.12", "option_b_zh": "1.2", "option_c_zh": "12", "option_d_zh": "120", "answer": "C", "explanation_en": "Multiply both by 10 to eliminate decimals: 48 ÷ 4 = 12.", "explanation_zh": "分子分母同乘10去掉小數：48 ÷ 4 = 12。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.PERCENT.BASIC", "stem_en": "What is 15% of 200?", "stem_zh": "200的15%是多少？", "option_a_en": "15", "option_b_en": "30", "option_c_en": "115", "option_d_en": "300", "option_a_zh": "15", "option_b_zh": "30", "option_c_zh": "115", "option_d_zh": "300", "answer": "B", "explanation_en": "15% of 200 = (15/100) × 200 = 0.15 × 200 = 30.", "explanation_zh": "200的15% = (15/100) × 200 = 0.15 × 200 = 30。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.PERCENT.BASIC", "stem_en": "A shirt costs $80. After a 25% discount, what is the sale price?", "stem_zh": "一件恤衫原價$80，折扣25%後售價是多少？", "option_a_en": "$20", "option_b_en": "$55", "option_c_en": "$60", "option_d_en": "$100", "option_a_zh": "$20", "option_b_zh": "$55", "option_c_zh": "$60", "option_d_zh": "$100", "answer": "C", "explanation_en": "Discount = 25% of 80 = 20. Sale price = 80 - 20 = $60.", "explanation_zh": "折扣額 = 80 × 25% = 20。售價 = 80 - 20 = $60。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.PERCENT.BASIC", "stem_en": "Express 0.065 as a percentage.", "stem_zh": "將 0.065 化成百分數。", "option_a_en": "0.065%", "option_b_en": "0.65%", "option_c_en": "6.5%", "option_d_en": "65%", "option_a_zh": "0.065%", "option_b_zh": "0.65%", "option_c_zh": "6.5%", "option_d_zh": "65%", "answer": "C", "explanation_en": "Multiply by 100: 0.065 × 100 = 6.5%.", "explanation_zh": "乘以100：0.065 × 100 = 6.5%。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.RATIO.BASIC", "stem_en": "Simplify the ratio 12 : 18.", "stem_zh": "化簡比 12 : 18。", "option_a_en": "1 : 2", "option_b_en": "2 : 3", "option_c_en": "3 : 2", "option_d_en": "6 : 9", "option_a_zh": "1 : 2", "option_b_zh": "2 : 3", "option_c_zh": "3 : 2", "option_d_zh": "6 : 9", "answer": "B", "explanation_en": "HCF of 12 and 18 is 6. 12÷6 : 18÷6 = 2 : 3.", "explanation_zh": "12和18的最大公因數是6。12÷6 : 18÷6 = 2 : 3。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.RATIO.BASIC", "stem_en": "Alice and Bob share $120 in the ratio 3:5. How much does Bob receive?", "stem_zh": "Alice和Bob按3:5分$120，Bob得多少？", "option_a_en": "$45", "option_b_en": "$60", "option_c_en": "$75", "option_d_en": "$80", "option_a_zh": "$45", "option_b_zh": "$60", "option_c_zh": "$75", "option_d_zh": "$80", "answer": "C", "explanation_en": "Total parts = 3 + 5 = 8. Bob's share = (5/8) × 120 = $75.", "explanation_zh": "份數共 = 3 + 5 = 8。Bob的份額 = (5/8) × 120 = $75。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.ALGEBRA.INTRO", "stem_en": "If x = 4, what is the value of 3x - 5?", "stem_zh": "若 x = 4，3x - 5 的值是多少？", "option_a_en": "7", "option_b_en": "8", "option_c_en": "12", "option_d_en": "17", "option_a_zh": "7", "option_b_zh": "8", "option_c_zh": "12", "option_d_zh": "17", "answer": "A", "explanation_en": "Substitute x = 4: 3(4) - 5 = 12 - 5 = 7.", "explanation_zh": "代入 x = 4：3(4) - 5 = 12 - 5 = 7。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.ALGEBRA.INTRO", "stem_en": "Simplify: 5a + 3b - 2a + b", "stem_zh": "化簡：5a + 3b - 2a + b", "option_a_en": "3a + 4b", "option_b_en": "3a + 2b", "option_c_en": "7a + 4b", "option_d_en": "3a - 4b", "option_a_zh": "3a + 4b", "option_b_zh": "3a + 2b", "option_c_zh": "7a + 4b", "option_d_zh": "3a - 4b", "answer": "A", "explanation_en": "Collect like terms: (5a - 2a) + (3b + b) = 3a + 4b.", "explanation_zh": "合併同類項：(5a - 2a) + (3b + b) = 3a + 4b。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.ALGEBRA.EQUATION1", "stem_en": "Solve: 3x + 5 = 17", "stem_zh": "解方程：3x + 5 = 17", "option_a_en": "x = 2", "option_b_en": "x = 4", "option_c_en": "x = 6", "option_d_en": "x = 7", "option_a_zh": "x = 2", "option_b_zh": "x = 4", "option_c_zh": "x = 6", "option_d_zh": "x = 7", "answer": "B", "explanation_en": "3x + 5 = 17 → 3x = 12 → x = 4. Subtract 5 from both sides, then divide by 3.", "explanation_zh": "3x + 5 = 17 → 3x = 12 → x = 4。兩邊減5，再除以3。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.ALGEBRA.EQUATION1", "stem_en": "Solve: 2(x - 3) = 8", "stem_zh": "解方程：2(x - 3) = 8", "option_a_en": "x = 1", "option_b_en": "x = 4", "option_c_en": "x = 7", "option_d_en": "x = 11", "option_a_zh": "x = 1", "option_b_zh": "x = 4", "option_c_zh": "x = 7", "option_d_zh": "x = 11", "answer": "C", "explanation_en": "Expand: 2x - 6 = 8 → 2x = 14 → x = 7. Or divide both sides by 2 first: x - 3 = 4 → x = 7.", "explanation_zh": "展開：2x - 6 = 8 → 2x = 14 → x = 7。或先除以2：x - 3 = 4 → x = 7。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.ALGEBRA.EQUATION1", "stem_en": "Solve: 5x - 3 = 2x + 9", "stem_zh": "解方程：5x - 3 = 2x + 9", "option_a_en": "x = 2", "option_b_en": "x = 3", "option_c_en": "x = 4", "option_d_en": "x = 6", "option_a_zh": "x = 2", "option_b_zh": "x = 3", "option_c_zh": "x = 4", "option_d_zh": "x = 6", "answer": "C", "explanation_en": "Move x terms to left: 5x - 2x = 9 + 3 → 3x = 12 → x = 4.", "explanation_zh": "移項：5x - 2x = 9 + 3 → 3x = 12 → x = 4。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S2.ALGEBRA.POLY", "stem_en": "Expand: (x + 3)(x + 2)", "stem_zh": "展開：(x + 3)(x + 2)", "option_a_en": "x² + 5x + 5", "option_b_en": "x² + 5x + 6", "option_c_en": "x² + 6x + 6", "option_d_en": "x² + 6", "option_a_zh": "x² + 5x + 5", "option_b_zh": "x² + 5x + 6", "option_c_zh": "x² + 6x + 6", "option_d_zh": "x² + 6", "answer": "B", "explanation_en": "FOIL: x·x + x·2 + 3·x + 3·2 = x² + 2x + 3x + 6 = x² + 5x + 6.", "explanation_zh": "展開：x·x + x·2 + 3·x + 3·2 = x² + 2x + 3x + 6 = x² + 5x + 6。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.ALGEBRA.POLY", "stem_en": "Factorise: x² + 7x + 12", "stem_zh": "因式分解：x² + 7x + 12", "option_a_en": "(x+3)(x+4)", "option_b_en": "(x+2)(x+6)", "option_c_en": "(x+1)(x+12)", "option_d_en": "(x+6)(x+2)", "option_a_zh": "(x+3)(x+4)", "option_b_zh": "(x+2)(x+6)", "option_c_zh": "(x+1)(x+12)", "option_d_zh": "(x+6)(x+2)", "answer": "A", "explanation_en": "Find two numbers that multiply to 12 and add to 7: 3 and 4. So (x+3)(x+4).", "explanation_zh": "找兩個數，乘積為12，和為7：3和4。所以 (x+3)(x+4)。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.ALGEBRA.POLY", "stem_en": "Expand: (2x - 1)²", "stem_zh": "展開：(2x - 1)²", "option_a_en": "4x² + 1", "option_b_en": "4x² - 4x + 1", "option_c_en": "4x² - 2x + 1", "option_d_en": "2x² - 4x + 1", "option_a_zh": "4x² + 1", "option_b_zh": "4x² - 4x + 1", "option_c_zh": "4x² - 2x + 1", "option_d_zh": "2x² - 4x + 1", "answer": "B", "explanation_en": "(a-b)² = a² - 2ab + b². Here a=2x, b=1: (2x)² - 2(2x)(1) + 1² = 4x² - 4x + 1.", "explanation_zh": "用公式 (a-b)² = a² - 2ab + b²。這裡 a=2x, b=1：(2x)² - 2(2x)(1) + 1² = 4x² - 4x + 1。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.ALGEBRA.EQUATION2", "stem_en": "Solve the simultaneous equations: x + y = 5 and x - y = 1.", "stem_zh": "解聯立方程：x + y = 5 及 x - y = 1。", "option_a_en": "x=2, y=3", "option_b_en": "x=3, y=2", "option_c_en": "x=4, y=1", "option_d_en": "x=1, y=4", "option_a_zh": "x=2, y=3", "option_b_zh": "x=3, y=2", "option_c_zh": "x=4, y=1", "option_d_zh": "x=1, y=4", "answer": "B", "explanation_en": "Add the two equations: 2x = 6, so x = 3. Substitute: 3 + y = 5, so y = 2.", "explanation_zh": "兩式相加：2x = 6，所以 x = 3。代入：3 + y = 5，所以 y = 2。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.ALGEBRA.EQUATION2", "stem_en": "Solve: 2x + y = 7 and x + 2y = 8.", "stem_zh": "解聯立方程：2x + y = 7 及 x + 2y = 8。", "option_a_en": "x=1, y=5", "option_b_en": "x=2, y=3", "option_c_en": "x=3, y=2", "option_d_en": "x=2, y=2", "option_a_zh": "x=1, y=5", "option_b_zh": "x=2, y=3", "option_c_zh": "x=3, y=2", "option_d_zh": "x=2, y=2", "answer": "B", "explanation_en": "From eq1: y = 7 - 2x. Sub into eq2: x + 2(7-2x) = 8 → x + 14 - 4x = 8 → -3x = -6 → x = 2, y = 3.", "explanation_zh": "由式1：y = 7 - 2x。代入式2：x + 2(7-2x) = 8 → -3x = -6 → x = 2，y = 3。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.INEQUAL.BASIC", "stem_en": "Solve: 2x - 3 > 5", "stem_zh": "解不等式：2x - 3 > 5", "option_a_en": "x > 1", "option_b_en": "x > 4", "option_c_en": "x < 4", "option_d_en": "x ≥ 4", "option_a_zh": "x > 1", "option_b_zh": "x > 4", "option_c_zh": "x < 4", "option_d_zh": "x ≥ 4", "answer": "B", "explanation_en": "2x - 3 > 5 → 2x > 8 → x > 4. Add 3 to both sides, then divide by 2 (positive, no sign flip).", "explanation_zh": "2x - 3 > 5 → 2x > 8 → x > 4。兩邊加3，再除以2（正數，不需反號）。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.INEQUAL.BASIC", "stem_en": "Solve: -3x ≤ 12", "stem_zh": "解不等式：-3x ≤ 12", "option_a_en": "x ≤ -4", "option_b_en": "x ≥ -4", "option_c_en": "x ≤ 4", "option_d_en": "x ≥ 4", "option_a_zh": "x ≤ -4", "option_b_zh": "x ≥ -4", "option_c_zh": "x ≤ 4", "option_d_zh": "x ≥ 4", "answer": "B", "explanation_en": "Divide by -3 and FLIP the inequality: x ≥ -4. When dividing by a negative, the inequality sign reverses.", "explanation_zh": "除以-3，不等號方向要反轉：x ≥ -4。除以負數時不等號方向要改變。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S1.MEASURE.PERIMETER", "stem_en": "A rectangle has length 8 cm and width 5 cm. What is its area?", "stem_zh": "一個長方形長8 cm，闊5 cm，面積是多少？", "option_a_en": "13 cm²", "option_b_en": "26 cm²", "option_c_en": "40 cm²", "option_d_en": "80 cm²", "option_a_zh": "13 cm²", "option_b_zh": "26 cm²", "option_c_zh": "40 cm²", "option_d_zh": "80 cm²", "answer": "C", "explanation_en": "Area of rectangle = length × width = 8 × 5 = 40 cm².", "explanation_zh": "長方形面積 = 長 × 闊 = 8 × 5 = 40 cm²。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.MEASURE.PERIMETER", "stem_en": "A triangle has base 10 cm and height 6 cm. What is its area?", "stem_zh": "一個三角形底邊10 cm，高6 cm，面積是多少？", "option_a_en": "16 cm²", "option_b_en": "30 cm²", "option_c_en": "60 cm²", "option_d_en": "120 cm²", "option_a_zh": "16 cm²", "option_b_zh": "30 cm²", "option_c_zh": "60 cm²", "option_d_zh": "120 cm²", "answer": "B", "explanation_en": "Area of triangle = ½ × base × height = ½ × 10 × 6 = 30 cm².", "explanation_zh": "三角形面積 = ½ × 底 × 高 = ½ × 10 × 6 = 30 cm²。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.MEASURE.CIRCLE", "stem_en": "A circle has radius 7 cm. What is its circumference? (Use π ≈ 22/7)", "stem_zh": "一個圓半徑7 cm，圓周是多少？（用 π ≈ 22/7）", "option_a_en": "14 cm", "option_b_en": "22 cm", "option_c_en": "44 cm", "option_d_en": "154 cm", "option_a_zh": "14 cm", "option_b_zh": "22 cm", "option_c_zh": "44 cm", "option_d_zh": "154 cm", "answer": "C", "explanation_en": "Circumference = 2πr = 2 × (22/7) × 7 = 44 cm.", "explanation_zh": "圓周 = 2πr = 2 × (22/7) × 7 = 44 cm。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.GEOM.ANGLES", "stem_en": "Two parallel lines are cut by a transversal. If one co-interior angle is 110°, what is the other?", "stem_zh": "兩平行線被截線切割，若一個同側內角為110°，另一個是多少？", "option_a_en": "70°", "option_b_en": "80°", "option_c_en": "110°", "option_d_en": "180°", "option_a_zh": "70°", "option_b_zh": "80°", "option_c_zh": "110°", "option_d_zh": "180°", "answer": "A", "explanation_en": "Co-interior angles (same-side interior angles) between parallel lines are supplementary: they add to 180°. So 180° - 110° = 70°.", "explanation_zh": "平行線的同側內角（共軛角）相加等於180°。所以 180° - 110° = 70°。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.GEOM.TRIANGLE", "stem_en": "In a triangle, two angles are 55° and 75°. What is the third angle?", "stem_zh": "一個三角形兩個角分別是55°和75°，第三個角是多少？", "option_a_en": "40°", "option_b_en": "50°", "option_c_en": "60°", "option_d_en": "130°", "option_a_zh": "40°", "option_b_zh": "50°", "option_c_zh": "60°", "option_d_zh": "130°", "answer": "B", "explanation_en": "Angle sum of a triangle = 180°. Third angle = 180° - 55° - 75° = 50°.", "explanation_zh": "三角形內角和 = 180°。第三角 = 180° - 55° - 75° = 50°。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S2.GEOM.QUADRILATERAL", "stem_en": "What is the sum of the interior angles of a quadrilateral?", "stem_zh": "四邊形的內角和是多少？", "option_a_en": "180°", "option_b_en": "270°", "option_c_en": "360°", "option_d_en": "540°", "option_a_zh": "180°", "option_b_zh": "270°", "option_c_zh": "360°", "option_d_zh": "540°", "answer": "C", "explanation_en": "Any quadrilateral can be split into 2 triangles. Sum = 2 × 180° = 360°.", "explanation_zh": "任何四邊形可分成2個三角形。內角和 = 2 × 180° = 360°。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.GEOM.3D", "stem_en": "A cuboid has length 5 cm, width 4 cm, and height 3 cm. What is its volume?", "stem_zh": "一個長方體長5 cm，闊4 cm，高3 cm，體積是多少？", "option_a_en": "12 cm³", "option_b_en": "47 cm³", "option_c_en": "60 cm³", "option_d_en": "94 cm³", "option_a_zh": "12 cm³", "option_b_zh": "47 cm³", "option_c_zh": "60 cm³", "option_d_zh": "94 cm³", "answer": "C", "explanation_en": "Volume of cuboid = l × w × h = 5 × 4 × 3 = 60 cm³.", "explanation_zh": "長方體體積 = 長 × 闊 × 高 = 5 × 4 × 3 = 60 cm³。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.COORD.BASIC", "stem_en": "What is the midpoint of the line segment joining A(2, 4) and B(8, 10)?", "stem_zh": "連結 A(2, 4) 和 B(8, 10) 的線段中點是什麼？", "option_a_en": "(3, 5)", "option_b_en": "(5, 7)", "option_c_en": "(6, 7)", "option_d_en": "(10, 14)", "option_a_zh": "(3, 5)", "option_b_zh": "(5, 7)", "option_c_zh": "(6, 7)", "option_d_zh": "(10, 14)", "answer": "B", "explanation_en": "Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+8)/2, (4+10)/2) = (5, 7).", "explanation_zh": "中點 = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+8)/2, (4+10)/2) = (5, 7)。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.COORD.BASIC", "stem_en": "What is the gradient of the line passing through (1, 2) and (3, 8)?", "stem_zh": "經過 (1, 2) 和 (3, 8) 的直線斜率是多少？", "option_a_en": "2", "option_b_en": "3", "option_c_en": "4", "option_d_en": "6", "option_a_zh": "2", "option_b_zh": "3", "option_c_zh": "4", "option_d_zh": "6", "answer": "B", "explanation_en": "Gradient = (y₂-y₁)/(x₂-x₁) = (8-2)/(3-1) = 6/2 = 3.", "explanation_zh": "斜率 = (y₂-y₁)/(x₂-x₁) = (8-2)/(3-1) = 6/2 = 3。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S1.DATA.BASIC", "stem_en": "In a bar chart, the bar for Monday has height 15 and Tuesday has height 20. What fraction of the two-day total does Monday represent?", "stem_zh": "在柱形圖中，星期一柱高15，星期二柱高20。星期一佔兩天總數的幾分之幾？", "option_a_en": "1/4", "option_b_en": "3/7", "option_c_en": "3/4", "option_d_en": "15/35", "option_a_zh": "1/4", "option_b_zh": "3/7", "option_c_zh": "3/4", "option_d_zh": "15/35", "answer": "B", "explanation_en": "Monday's fraction = 15/(15+20) = 15/35 = 3/7.", "explanation_zh": "星期一佔 = 15/(15+20) = 15/35 = 3/7。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.DATA.AVERAGE", "stem_en": "The scores of 5 students are: 70, 80, 65, 90, 75. What is the mean?", "stem_zh": "5位同學的分數為：70, 80, 65, 90, 75。平均分是多少？", "option_a_en": "72", "option_b_en": "75", "option_c_en": "76", "option_d_en": "80", "option_a_zh": "72", "option_b_zh": "75", "option_c_zh": "76", "option_d_zh": "80", "answer": "C", "explanation_en": "Mean = (70 + 80 + 65 + 90 + 75) / 5 = 380 / 5 = 76.", "explanation_zh": "平均數 = (70 + 80 + 65 + 90 + 75) / 5 = 380 / 5 = 76。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S1.DATA.AVERAGE", "stem_en": "The median of 3, 7, 9, 12, 15 is:", "stem_zh": "3, 7, 9, 12, 15 的中位數是：", "option_a_en": "7", "option_b_en": "9", "option_c_en": "12", "option_d_en": "9.2", "option_a_zh": "7", "option_b_zh": "9", "option_c_zh": "12", "option_d_zh": "9.2", "answer": "B", "explanation_en": "The median is the middle value when data is ordered. With 5 values, the 3rd is the median: 9.", "explanation_zh": "中位數是排列後中間的值。5個數據，第3個是中位數：9。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S1" }, { "objective_code": "MATH.S2.PROB.BASIC", "stem_en": "A bag has 3 red and 5 blue balls. One is drawn at random. What is the probability of drawing a red ball?", "stem_zh": "一個袋子有3個紅球和5個藍球，隨機取一個，取到紅球的概率是多少？", "option_a_en": "1/3", "option_b_en": "3/5", "option_c_en": "3/8", "option_d_en": "5/8", "option_a_zh": "1/3", "option_b_zh": "3/5", "option_c_zh": "3/8", "option_d_zh": "5/8", "answer": "C", "explanation_en": "P(red) = number of red / total = 3 / (3+5) = 3/8.", "explanation_zh": "P(紅) = 紅球數 / 總球數 = 3 / (3+5) = 3/8。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S2.PROB.BASIC", "stem_en": "A fair die is rolled. What is the probability of getting a number greater than 4?", "stem_zh": "擲一個公正骰子，得到大於4的數的概率是多少？", "option_a_en": "1/6", "option_b_en": "1/3", "option_c_en": "1/2", "option_d_en": "2/3", "option_a_zh": "1/6", "option_b_zh": "1/3", "option_c_zh": "1/2", "option_d_zh": "2/3", "answer": "B", "explanation_en": "Numbers greater than 4 on a die: {5, 6} — that is 2 out of 6. P = 2/6 = 1/3.", "explanation_zh": "骰子中大於4的數：{5, 6}，共2個。P = 2/6 = 1/3。", "difficulty_estimate": 1, "question_type": "MCQ", "grade": "S2" }, { "objective_code": "MATH.S3.NUMBER.RATIONAL", "stem_en": "Which of the following is an irrational number?", "stem_zh": "以下哪個是無理數？", "option_a_en": "√4", "option_b_en": "√9", "option_c_en": "√7", "option_d_en": "22/7", "option_a_zh": "√4", "option_b_zh": "√9", "option_c_zh": "√7", "option_d_zh": "22/7", "answer": "C", "explanation_en": "√4 = 2 and √9 = 3 are rational. 22/7 is a rational fraction. √7 cannot be expressed as a ratio of integers, so it is irrational.", "explanation_zh": "√4 = 2 和 √9 = 3 是有理數。22/7 是有理分數。√7 不能表示為整數之比，因此是無理數。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.NUMBER.RATIONAL", "stem_en": "Simplify: √50", "stem_zh": "化簡：√50", "option_a_en": "5", "option_b_en": "5√2", "option_c_en": "10√5", "option_d_en": "25√2", "option_a_zh": "5", "option_b_zh": "5√2", "option_c_zh": "10√5", "option_d_zh": "25√2", "answer": "B", "explanation_en": "√50 = √(25 × 2) = √25 × √2 = 5√2.", "explanation_zh": "√50 = √(25 × 2) = √25 × √2 = 5√2。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.ALGEBRA.QUADRATIC", "stem_en": "Solve: x² - 5x + 6 = 0", "stem_zh": "解方程：x² - 5x + 6 = 0", "option_a_en": "x = 1 or x = 6", "option_b_en": "x = 2 or x = 3", "option_c_en": "x = -2 or x = -3", "option_d_en": "x = 2 or x = -3", "option_a_zh": "x = 1 或 x = 6", "option_b_zh": "x = 2 或 x = 3", "option_c_zh": "x = -2 或 x = -3", "option_d_zh": "x = 2 或 x = -3", "answer": "B", "explanation_en": "Factorise: x² - 5x + 6 = (x - 2)(x - 3) = 0. So x = 2 or x = 3.", "explanation_zh": "因式分解：x² - 5x + 6 = (x - 2)(x - 3) = 0。所以 x = 2 或 x = 3。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.ALGEBRA.QUADRATIC", "stem_en": "The discriminant of ax² + bx + c = 0 is b² - 4ac. If b² - 4ac < 0, the equation has:", "stem_zh": "方程 ax² + bx + c = 0 的判別式為 b² - 4ac。若 b² - 4ac < 0，方程有：", "option_a_en": "Two equal real roots", "option_b_en": "Two distinct real roots", "option_c_en": "No real roots", "option_d_en": "One real root", "option_a_zh": "兩個相等實根", "option_b_zh": "兩個不相等實根", "option_c_zh": "無實根", "option_d_zh": "一個實根", "answer": "C", "explanation_en": "When discriminant < 0, the square root of a negative number is not real, so the equation has no real roots.", "explanation_zh": "當判別式 < 0 時，無法對負數取實數平方根，故方程無實根。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.ALGEBRA.QUADRATIC", "stem_en": "Use the quadratic formula to find the solutions of x² - 4x - 5 = 0.", "stem_zh": "用求根公式求 x² - 4x - 5 = 0 的解。", "option_a_en": "x = 5 or x = -1", "option_b_en": "x = -5 or x = 1", "option_c_en": "x = 5 or x = 1", "option_d_en": "x = -5 or x = -1", "option_a_zh": "x = 5 或 x = -1", "option_b_zh": "x = -5 或 x = 1", "option_c_zh": "x = 5 或 x = 1", "option_d_zh": "x = -5 或 x = -1", "answer": "A", "explanation_en": "a=1, b=-4, c=-5. Discriminant = 16 + 20 = 36. x = (4 ± 6)/2. So x = 5 or x = -1.", "explanation_zh": "a=1, b=-4, c=-5。判別式 = 16 + 20 = 36。x = (4 ± 6)/2。所以 x = 5 或 x = -1。", "difficulty_estimate": 4, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.ALGEBRA.FUNCTIONS", "stem_en": "The graph of y = x² - 4x + 3 has its vertex at:", "stem_zh": "y = x² - 4x + 3 的頂點在：", "option_a_en": "(2, -1)", "option_b_en": "(-2, -1)", "option_c_en": "(2, 1)", "option_d_en": "(-2, 1)", "option_a_zh": "(2, -1)", "option_b_zh": "(-2, -1)", "option_c_zh": "(2, 1)", "option_d_zh": "(-2, 1)", "answer": "A", "explanation_en": "Complete the square: y = (x-2)² - 4 + 3 = (x-2)² - 1. Vertex is at (2, -1).", "explanation_zh": "配方：y = (x-2)² - 4 + 3 = (x-2)² - 1。頂點為 (2, -1)。", "difficulty_estimate": 4, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.ALGEBRA.FUNCTIONS", "stem_en": "The axis of symmetry of y = -2x² + 8x - 5 is:", "stem_zh": "y = -2x² + 8x - 5 的對稱軸是：", "option_a_en": "x = -2", "option_b_en": "x = 2", "option_c_en": "x = 4", "option_d_en": "x = -4", "option_a_zh": "x = -2", "option_b_zh": "x = 2", "option_c_zh": "x = 4", "option_d_zh": "x = -4", "answer": "B", "explanation_en": "The axis of symmetry is x = -b/(2a) = -8/(2×(-2)) = -8/(-4) = 2.", "explanation_zh": "對稱軸為 x = -b/(2a) = -8/(2×(-2)) = -8/(-4) = 2。", "difficulty_estimate": 4, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.ALGEBRA.FRACTIONS", "stem_en": "Simplify: (x² - 9) / (x + 3)", "stem_zh": "化簡：(x² - 9) / (x + 3)", "option_a_en": "x - 3", "option_b_en": "x + 3", "option_c_en": "x - 9", "option_d_en": "x²", "option_a_zh": "x - 3", "option_b_zh": "x + 3", "option_c_zh": "x - 9", "option_d_zh": "x²", "answer": "A", "explanation_en": "x² - 9 = (x+3)(x-3). Dividing by (x+3) gives (x-3), where x ≠ -3.", "explanation_zh": "x² - 9 = (x+3)(x-3)。除以 (x+3) 得 (x-3)，其中 x ≠ -3。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.PERCENT.ADVANCED", "stem_en": "HK$10,000 is invested at 5% compound interest per annum for 2 years. What is the amount at the end?", "stem_zh": "港幣10,000元以年利率5%複利投資2年，最終金額是多少？", "option_a_en": "HK$11,000", "option_b_en": "HK$11,025", "option_c_en": "HK$10,250", "option_d_en": "HK$11,250", "option_a_zh": "港幣11,000元", "option_b_zh": "港幣11,025元", "option_c_zh": "港幣10,250元", "option_d_zh": "港幣11,250元", "answer": "B", "explanation_en": "A = P(1+r)ⁿ = 10000 × (1.05)² = 10000 × 1.1025 = 11025.", "explanation_zh": "A = P(1+r)ⁿ = 10000 × (1.05)² = 10000 × 1.1025 = 11025。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.GEOM.PYTHAGORAS", "stem_en": "In a right-angled triangle, the two legs are 6 cm and 8 cm. What is the hypotenuse?", "stem_zh": "直角三角形兩條直角邊分別為6厘米和8厘米，斜邊是多少？", "option_a_en": "10 cm", "option_b_en": "12 cm", "option_c_en": "14 cm", "option_d_en": "√100 cm", "option_a_zh": "10厘米", "option_b_zh": "12厘米", "option_c_zh": "14厘米", "option_d_zh": "√100厘米", "answer": "A", "explanation_en": "By Pythagoras: c² = 6² + 8² = 36 + 64 = 100. So c = 10 cm. (Note: √100 cm = 10 cm, so A and D are equivalent but A is the simplified form.)", "explanation_zh": "由畢氏定理：c² = 6² + 8² = 36 + 64 = 100。所以 c = 10厘米。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.GEOM.PYTHAGORAS", "stem_en": "A ladder 13 m long leans against a vertical wall. The foot of the ladder is 5 m from the base of the wall. How high does the ladder reach?", "stem_zh": "一架13米長的梯子靠在垂直牆上，梯腳距牆腳5米，梯子能到達多高？", "option_a_en": "8 m", "option_b_en": "10 m", "option_c_en": "12 m", "option_d_en": "18 m", "option_a_zh": "8米", "option_b_zh": "10米", "option_c_zh": "12米", "option_d_zh": "18米", "answer": "C", "explanation_en": "h² = 13² - 5² = 169 - 25 = 144. h = √144 = 12 m.", "explanation_zh": "h² = 13² - 5² = 169 - 25 = 144。h = √144 = 12米。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.TRIG.BASIC", "stem_en": "In a right-angled triangle, the side opposite to angle θ is 3 and the hypotenuse is 5. What is sin θ?", "stem_zh": "在直角三角形中，角θ的對邊是3，斜邊是5，sin θ 是多少？", "option_a_en": "3/4", "option_b_en": "4/5", "option_c_en": "3/5", "option_d_en": "4/3", "option_a_zh": "3/4", "option_b_zh": "4/5", "option_c_zh": "3/5", "option_d_zh": "4/3", "answer": "C", "explanation_en": "sin θ = opposite / hypotenuse = 3/5.", "explanation_zh": "sin θ = 對邊 / 斜邊 = 3/5。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.TRIG.BASIC", "stem_en": "What is the exact value of tan 45°?", "stem_zh": "tan 45° 的準確值是多少？", "option_a_en": "1/2", "option_b_en": "√3/2", "option_c_en": "√3", "option_d_en": "1", "option_a_zh": "1/2", "option_b_zh": "√3/2", "option_c_zh": "√3", "option_d_zh": "1", "answer": "D", "explanation_en": "In a 45°-45°-90° triangle with legs 1 and 1, tan 45° = opposite/adjacent = 1/1 = 1.", "explanation_zh": "在45°-45°-90°三角形中，兩直角邊為1，tan 45° = 對邊/鄰邊 = 1/1 = 1。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.TRIG.BASIC", "stem_en": "In triangle ABC with right angle at C, angle A = 30° and the hypotenuse AB = 10 cm. Find BC.", "stem_zh": "三角形ABC中，直角在C，角A = 30°，斜邊AB = 10厘米。求BC。", "option_a_en": "5 cm", "option_b_en": "5√3 cm", "option_c_en": "10√3 cm", "option_d_en": "√3 cm", "option_a_zh": "5厘米", "option_b_zh": "5√3厘米", "option_c_zh": "10√3厘米", "option_d_zh": "√3厘米", "answer": "A", "explanation_en": "BC is opposite to angle A. sin 30° = BC/AB = BC/10. BC = 10 × sin 30° = 10 × 0.5 = 5 cm.", "explanation_zh": "BC 是角A的對邊。sin 30° = BC/AB = BC/10。BC = 10 × sin 30° = 10 × 0.5 = 5厘米。", "difficulty_estimate": 4, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.TRIG.APPLICATIONS", "stem_en": "A person stands 50 m from the base of a building. The angle of elevation to the top is 60°. How tall is the building? (tan 60° = √3 ≈ 1.732)", "stem_zh": "一人站在距大廈50米處，仰角為60°，大廈有多高？（tan 60° = √3 ≈ 1.732）", "option_a_en": "50 m", "option_b_en": "50√3 m", "option_c_en": "100 m", "option_d_en": "25√3 m", "option_a_zh": "50米", "option_b_zh": "50√3米", "option_c_zh": "100米", "option_d_zh": "25√3米", "answer": "B", "explanation_en": "tan 60° = height / 50. Height = 50 × tan 60° = 50√3 m ≈ 86.6 m.", "explanation_zh": "tan 60° = 高度 / 50。高度 = 50 × tan 60° = 50√3 米 ≈ 86.6 米。", "difficulty_estimate": 4, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.GEOM.CIRCLES", "stem_en": "AB is a diameter of a circle, and C is a point on the circle. What is angle ACB?", "stem_zh": "AB是圓的直徑，C是圓上的一點，角ACB是多少？", "option_a_en": "45°", "option_b_en": "60°", "option_c_en": "90°", "option_d_en": "180°", "option_a_zh": "45°", "option_b_zh": "60°", "option_c_zh": "90°", "option_d_zh": "180°", "answer": "C", "explanation_en": "Angle in a semicircle theorem: the angle subtended by a diameter at any point on the circle is 90°.", "explanation_zh": "半圓上的圓周角定理：直徑所對的圓周角等於90°。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.GEOM.CIRCLES", "stem_en": "The angle subtended by arc PQ at the centre O is 80°. What is the angle subtended by the same arc at a point R on the major arc?", "stem_zh": "弧PQ對圓心O的圓心角是80°，同一弧對大弧上的點R的圓周角是多少？", "option_a_en": "160°", "option_b_en": "80°", "option_c_en": "40°", "option_d_en": "20°", "option_a_zh": "160°", "option_b_zh": "80°", "option_c_zh": "40°", "option_d_zh": "20°", "answer": "C", "explanation_en": "The angle at the centre is twice the angle at the circumference on the same arc. So angle at R = 80°/2 = 40°.", "explanation_zh": "圓心角等於同弧上圓周角的兩倍。所以角R = 80°/2 = 40°。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.COORD.LINES", "stem_en": "What is the slope of the line passing through (1, 2) and (3, 8)?", "stem_zh": "過點(1, 2)和(3, 8)的直線斜率是多少？", "option_a_en": "2", "option_b_en": "3", "option_c_en": "4", "option_d_en": "6", "option_a_zh": "2", "option_b_zh": "3", "option_c_zh": "4", "option_d_zh": "6", "answer": "B", "explanation_en": "Slope = (y₂ - y₁)/(x₂ - x₁) = (8 - 2)/(3 - 1) = 6/2 = 3.", "explanation_zh": "斜率 = (y₂ - y₁)/(x₂ - x₁) = (8 - 2)/(3 - 1) = 6/2 = 3。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.COORD.LINES", "stem_en": "A line has slope 2 and passes through (0, -3). Its equation is:", "stem_zh": "一條直線斜率為2，過點(0, -3)，方程是：", "option_a_en": "y = 2x + 3", "option_b_en": "y = 2x - 3", "option_c_en": "y = -2x + 3", "option_d_en": "y = 3x - 2", "option_a_zh": "y = 2x + 3", "option_b_zh": "y = 2x - 3", "option_c_zh": "y = -2x + 3", "option_d_zh": "y = 3x - 2", "answer": "B", "explanation_en": "Using y = mx + c with m = 2 and c = -3 (the y-intercept): y = 2x - 3.", "explanation_zh": "用 y = mx + c，m = 2，c = -3（y軸截距）：y = 2x - 3。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.DATA.DISPERSION", "stem_en": "The data set is: 4, 7, 2, 9, 3. What is the range?", "stem_zh": "數據集為：4, 7, 2, 9, 3。全距是多少？", "option_a_en": "5", "option_b_en": "6", "option_c_en": "7", "option_d_en": "9", "option_a_zh": "5", "option_b_zh": "6", "option_c_zh": "7", "option_d_zh": "9", "answer": "C", "explanation_en": "Range = max - min = 9 - 2 = 7.", "explanation_zh": "全距 = 最大值 - 最小值 = 9 - 2 = 7。", "difficulty_estimate": 2, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.DATA.DISPERSION", "stem_en": "The lower quartile (Q1) of 2, 4, 5, 7, 8, 10 is:", "stem_zh": "數據 2, 4, 5, 7, 8, 10 的下四分位數 (Q1) 是：", "option_a_en": "4", "option_b_en": "4.5", "option_c_en": "5", "option_d_en": "6", "option_a_zh": "4", "option_b_zh": "4.5", "option_c_zh": "5", "option_d_zh": "6", "answer": "B", "explanation_en": "The data is already ordered. For 6 values, Q1 is the median of the lower half {2, 4, 5} = 4. Wait — Q1 = (4+5)/2 = 4.5 using the interpolation method for even-count data.", "explanation_zh": "數據已排序。對於6個數值，Q1 是下半部分 {2, 4, 5} 的中間值，採內插法：Q1 = (4+5)/2 = 4.5。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.PROB.ADVANCED", "stem_en": "Events A and B are mutually exclusive. P(A) = 0.3, P(B) = 0.4. What is P(A or B)?", "stem_zh": "事件A和B是互斥的。P(A) = 0.3，P(B) = 0.4。P(A或B)是多少？", "option_a_en": "0.12", "option_b_en": "0.58", "option_c_en": "0.70", "option_d_en": "1.0", "option_a_zh": "0.12", "option_b_zh": "0.58", "option_c_zh": "0.70", "option_d_zh": "1.0", "answer": "C", "explanation_en": "For mutually exclusive events: P(A or B) = P(A) + P(B) = 0.3 + 0.4 = 0.7.", "explanation_zh": "互斥事件：P(A或B) = P(A) + P(B) = 0.3 + 0.4 = 0.7。", "difficulty_estimate": 3, "question_type": "MCQ", "grade": "S3" }, { "objective_code": "MATH.S3.PROB.ADVANCED", "stem_en": "A bag has 3 red and 2 blue balls. Two balls are drawn without replacement. What is the probability both are red?", "stem_zh": "袋中有3個紅球和2個藍球，不放回地取出兩個，兩個都是紅球的概率是多少？", "option_a_en": "9/25", "option_b_en": "6/20", "option_c_en": "3/10", "option_d_en": "1/2", "option_a_zh": "9/25", "option_b_zh": "6/20", "option_c_zh": "3/10", "option_d_zh": "1/2", "answer": "C", "explanation_en": "P(1st red) = 3/5. P(2nd red | 1st red) = 2/4 = 1/2. P(both red) = 3/5 × 1/2 = 3/10.", "explanation_zh": "P(第1個紅) = 3/5。P(第2個紅|第1個紅) = 2/4 = 1/2。P(都是紅) = 3/5 × 1/2 = 3/10。", "difficulty_estimate": 4, "question_type": "MCQ", "grade": "S3" } ]