{"title": "Quadratic Equations (Algebra) - Grade 10 ICSE", "total_questions": 60, "questions": [{"id": 1, "difficulty": "easy", "question": "What is the standard form of a quadratic equation? 📝", "options": {"A": "ax² + bx + c = 0", "B": "ax + b = 0", "C": "y = mx + c", "D": "x² + y² = r²"}, "correct_answer": "A"}, {"id": 2, "difficulty": "easy", "question": "Identify the quadratic equation: 🧐", "options": {"A": "3x + 5 = 0", "B": "2x² - 5x + 3 = 0", "C": "x³ + 2x + 1 = 0", "D": "4/x + 2 = 0"}, "correct_answer": "B"}, {"id": 3, "difficulty": "easy", "question": "What is the degree of a quadratic equation? 🔢", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "C"}, {"id": 4, "difficulty": "easy", "question": "If x = 2 is a root of x² - kx + 6 = 0, what is k? ⚖️", "options": {"A": "3", "B": "5", "C": "4", "D": "7"}, "correct_answer": "B"}, {"id": 5, "difficulty": "easy", "question": "The sum of the roots of x² - 8x + 15 = 0 is? ➕", "options": {"A": "15", "B": "-8", "C": "8", "D": "-15"}, "correct_answer": "C"}, {"id": 6, "difficulty": "easy", "question": "The product of the roots of 2x² + 5x - 3 = 0 is? ✖️", "options": {"A": "5/2", "B": "-3/2", "C": "3/2", "D": "-5/2"}, "correct_answer": "B"}, {"id": 7, "difficulty": "easy", "question": "Which is a quadratic equation? 🤔", "options": {"A": "√x + 2 = 5", "B": "x² + 3x = 10", "C": "x⁴ + 2x² = 0", "D": "|x| - 3 = 0"}, "correct_answer": "B"}, {"id": 8, "difficulty": "easy", "question": "The constant term in x² - 7x + 12 = 0 is? 🔤", "options": {"A": "1", "B": "-7", "C": "12", "D": "x²"}, "correct_answer": "C"}, {"id": 9, "difficulty": "easy", "question": "What is the coefficient of x² in 5 - 3x² + x = 0? 🔍", "options": {"A": "5", "B": "1", "C": "-3", "D": "0"}, "correct_answer": "C"}, {"id": 10, "difficulty": "easy", "question": "If one root of x² + px + 8 = 0 is 2, find p. 🎯", "options": {"A": "-6", "B": "6", "C": "4", "D": "-4"}, "correct_answer": "A"}, {"id": 11, "difficulty": "easy", "question": "The roots of (x - 3)(x + 2) = 0 are? 🌱", "options": {"A": "3, 2", "B": "-3, 2", "C": "3, -2", "D": "-3, -2"}, "correct_answer": "C"}, {"id": 12, "difficulty": "easy", "question": "Which value of x satisfies x² = 9? ✅", "options": {"A": "3 only", "B": "-3 only", "C": "0", "D": "3 and -3"}, "correct_answer": "D"}, {"id": 13, "difficulty": "easy", "question": "If x² = 25, then x = ? 🔄", "options": {"A": "5", "B": "-5", "C": "±5", "D": "10"}, "correct_answer": "C"}, {"id": 14, "difficulty": "easy", "question": "The quadratic equation whose roots are 1 and -1 is? 🎪", "options": {"A": "x² + 1 = 0", "B": "x² - 1 = 0", "C": "x² + x + 1 = 0", "D": "x² - x - 1 = 0"}, "correct_answer": "B"}, {"id": 15, "difficulty": "easy", "question": "A quadratic equation always has how many roots? 🎭", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "C"}, {"id": 16, "difficulty": "easy", "question": "What is the nature of roots if discriminant D > 0? 🌈", "options": {"A": "Real & Equal", "B": "Real & Distinct", "C": "Imaginary", "D": "No roots"}, "correct_answer": "B"}, {"id": 17, "difficulty": "easy", "question": "For x² + 2x + 1 = 0, the discriminant is? 📊", "options": {"A": "0", "B": "4", "C": "8", "D": "-4"}, "correct_answer": "A"}, {"id": 18, "difficulty": "easy", "question": "If D = 0, the roots are? ⚡", "options": {"A": "Distinct", "B": "Imaginary", "C": "Equal", "D": "None"}, "correct_answer": "C"}, {"id": 19, "difficulty": "easy", "question": "The quadratic formula is? 🧮", "options": {"A": "-b ± √(b² - 4ac) / 2a", "B": "b ± √(b² - 4ac) / 2a", "C": "-b ± √(b² + 4ac) / 2a", "D": "b ± √(b² + 4ac) / 2a"}, "correct_answer": "A"}, {"id": 20, "difficulty": "easy", "question": "Which of these is NOT quadratic? ❌", "options": {"A": "x(x+1)=0", "B": "x² = 4", "C": "x + 1/x = 2", "D": "3x² - 2x = 0"}, "correct_answer": "C"}, {"id": 21, "difficulty": "medium", "question": "Find the roots of 2x² - 7x + 3 = 0 by factorization. 🔨", "options": {"A": "3, 1/2", "B": "3, 2", "C": "1, 3/2", "D": "-3, -1/2"}, "correct_answer": "A"}, {"id": 22, "difficulty": "medium", "question": "Solve: x² - 5x + 6 = 0 🌟", "options": {"A": "2, 3", "B": "-2, -3", "C": "1, 6", "D": "-1, -6"}, "correct_answer": "A"}, {"id": 23, "difficulty": "medium", "question": "If α and β are roots of x² - 6x + 8 = 0, find α² + β². 🧠", "options": {"A": "20", "B": "36", "C": "28", "D": "12"}, "correct_answer": "A"}, {"id": 24, "difficulty": "medium", "question": "Form a quadratic equation whose roots are 2 and -3. 🏗️", "options": {"A": "x² + x - 6 = 0", "B": "x² - x - 6 = 0", "C": "x² + x + 6 = 0", "D": "x² - x + 6 = 0"}, "correct_answer": "A"}, {"id": 25, "difficulty": "medium", "question": "Solve using formula: x² + 4x - 5 = 0 📐", "options": {"A": "1, -5", "B": "-1, 5", "C": "2, -3", "D": "-2, 3"}, "correct_answer": "A"}, {"id": 26, "difficulty": "medium", "question": "Find k if x² + kx + 9 = 0 has equal roots. ⚓", "options": {"A": "±6", "B": "±3", "C": "±9", "D": "±12"}, "correct_answer": "A"}, {"id": 27, "difficulty": "medium", "question": "Solve: √2x² + 7x + 5√2 = 0 🔵", "options": {"A": "-√2, -5/√2", "B": "√2, 5/√2", "C": "2, 5", "D": "-2, -5"}, "correct_answer": "A"}, {"id": 28, "difficulty": "medium", "question": "Find the value of p for which 4x² - px + 9 = 0 has real & equal roots. 🎭", "options": {"A": "±12", "B": "±6", "C": "±9", "D": "±8"}, "correct_answer": "A"}, {"id": 29, "difficulty": "medium", "question": "If one root is double the other in 2x² - 9x + k = 0, find k. 🎪", "options": {"A": "9", "B": "6", "C": "8", "D": "4"}, "correct_answer": "A"}, {"id": 30, "difficulty": "medium", "question": "Solve: 1/(x+1) + 2/(x+2) = 4/(x+4) 🤯", "options": {"A": "0, -5/3", "B": "0, 5/3", "C": "1, 2", "D": "No solution"}, "correct_answer": "A"}, {"id": 31, "difficulty": "medium", "question": "Find the discriminant of 3x² - 2x + 1/3 = 0. 🎨", "options": {"A": "0", "B": "1", "C": "2", "D": "4"}, "correct_answer": "A"}, {"id": 32, "difficulty": "medium", "question": "If α,β are roots of 3x² - 6x + 4 = 0, find α/β + β/α. 🌀", "options": {"A": "1", "B": "2", "C": "0", "D": "3"}, "correct_answer": "A"}, {"id": 33, "difficulty": "medium", "question": "Solve: x² - (√3 + √2)x + √6 = 0 🌐", "options": {"A": "√3, √2", "B": "-√3, -√2", "C": "√3, -√2", "D": "-√3, √2"}, "correct_answer": "A"}, {"id": 34, "difficulty": "medium", "question": "For what value of m does (m-12)x² + 2(m-12)x + 2 = 0 have equal roots? ⚙️", "options": {"A": "14", "B": "12", "C": "10", "D": "8"}, "correct_answer": "A"}, {"id": 35, "difficulty": "medium", "question": "Find the roots of 2x² + 5√3x + 6 = 0. 🧩", "options": {"A": "-2√3, -√3/2", "B": "2√3, √3/2", "C": "-2, -√3", "D": "2, √3"}, "correct_answer": "A"}, {"id": 36, "difficulty": "medium", "question": "Solve: (x²+1)² - x² = 0 has how many real roots? 🎯", "options": {"A": "0", "B": "2", "C": "4", "D": "1"}, "correct_answer": "A"}, {"id": 37, "difficulty": "medium", "question": "If roots of kx² - 5x + k = 0 are equal, find k. 🔑", "options": {"A": "±5/2", "B": "5", "C": "2/5", "D": "0"}, "correct_answer": "A"}, {"id": 38, "difficulty": "medium", "question": "Solve: 2(x² - 6) = 3(x - 4). 🧨", "options": {"A": "3, -4", "B": "-3, 4", "C": "0, 3", "D": "0, -4"}, "correct_answer": "A"}, {"id": 39, "difficulty": "medium", "question": "Find the sum of reciprocals of roots of 2x² - 4x + 5 = 0. 🍬", "options": {"A": "4/5", "B": "5/4", "C": "2/5", "D": "5/2"}, "correct_answer": "A"}, {"id": 40, "difficulty": "medium", "question": "If α+β=5 and αβ=6, find the quadratic equation. 🌻", "options": {"A": "x² - 5x + 6 = 0", "B": "x² + 5x + 6 = 0", "C": "x² - 5x - 6 = 0", "D": "x² + 5x - 6 = 0"}, "correct_answer": "A"}, {"id": 41, "difficulty": "hard", "question": "If the equation x² - ax + b = 0 has roots α,β and α² + β² = 29, α³ + β³ = 133, find a and b. 🚀", "options": {"A": "a=7, b=10", "B": "a=5, b=6", "C": "a=10, b=7", "D": "a=6, b=5"}, "correct_answer": "A"}, {"id": 42, "difficulty": "hard", "question": "Solve for x: √(x+1) + √(x-1) = 1. 🧿", "options": {"A": "No real solution", "B": "x=1", "C": "x=0", "D": "x=5/4"}, "correct_answer": "A"}, {"id": 43, "difficulty": "hard", "question": "If the roots of px² + qx + q = 0 are α and β such that α/β + β/α + 1 = 0, find p:q. 🎲", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:2"}, "correct_answer": "A"}, {"id": 44, "difficulty": "hard", "question": "Find the condition that roots of ax² + 2bx + c = 0 are in the ratio m:n. 📏", "options": {"A": "b²mn = ac(m+n)²", "B": "b² = ac", "C": "b² = mn ac", "D": "b(m+n) = a c mn"}, "correct_answer": "A"}, {"id": 45, "difficulty": "hard", "question": "If α,β are roots of x² - px + q = 0, find the value of α⁴ + β⁴. 🏆", "options": {"A": "p⁴ - 4p²q + 2q²", "B": "p⁴ + 4p²q + 2q²", "C": "p⁴ - 2p²q + q²", "D": "p⁴ + 2p²q + q²"}, "correct_answer": "A"}, {"id": 46, "difficulty": "hard", "question": "Solve: |x² - 5x + 6| = x - 2. 🎭", "options": {"A": "x = 2, 4", "B": "x = 2 only", "C": "x = 4 only", "D": "x = 1, 3"}, "correct_answer": "A"}, {"id": 47, "difficulty": "hard", "question": "If every root of x² + ax + b = 0 is decreased by 1, it becomes x² + px + q = 0. Find a,b,p,q relation. 🔗", "options": {"A": "p = a - 2, q = b - a + 1", "B": "p = a + 2, q = b + a + 1", "C": "p = 2 - a, q = 1 - b", "D": "p = a, q = b"}, "correct_answer": "A"}, {"id": 48, "difficulty": "hard", "question": "Find all real x satisfying: x² + 3|x| + 2 = 0. 🤖", "options": {"A": "No real solution", "B": "x = -1, -2", "C": "x = 1, 2", "D": "x = ±1, ±2"}, "correct_answer": "A"}, {"id": 49, "difficulty": "hard", "question": "If α,β are roots of x² - 2x + 3 = 0, find the equation whose roots are α³ - 3α² + 5α - 2 and β³ - 3β² + 5β - 2. 🧬", "options": {"A": "x² - 3x + 2 = 0", "B": "x² + 3x + 2 = 0", "C": "x² - 2x + 3 = 0", "D": "x² + 2x + 3 = 0"}, "correct_answer": "A"}, {"id": 50, "difficulty": "hard", "question": "For what real a do the equations x² + ax + 1 = 0 and x² + x + a = 0 have a common root? 🌌", "options": {"A": "a = -2", "B": "a = 1", "C": "a = 2", "D": "a = 0"}, "correct_answer": "A"}, {"id": 51, "difficulty": "hard", "question": "Solve: 2^(2x+2) - 33 * 2^(x) + 8 = 0. ⚡", "options": {"A": "x = 3, -2", "B": "x = 2, -3", "C": "x = 0, 1", "D": "x = 1, 2"}, "correct_answer": "A"}, {"id": 52, "difficulty": "hard", "question": "If the quadratic (sinθ)x² + (2cosθ)x + (cosθ+sinθ)/2 = 0 has equal roots, find θ. 🎡", "options": {"A": "θ = nπ ± π/4", "B": "θ = nπ", "C": "θ = nπ/2", "D": "θ = nπ/3"}, "correct_answer": "A"}, {"id": 53, "difficulty": "hard", "question": "Find k so that x² - (k+6)x + 2(2k-1) = 0 has sum of roots = half of product of roots. 🪐", "options": {"A": "k = 7", "B": "k = 5", "C": "k = 3", "D": "k = 1"}, "correct_answer": "A"}, {"id": 54, "difficulty": "hard", "question": "If α,β are roots of x² - px + r = 0 and α/2, 2β are roots of x² - qx + r = 0, find r. 🌠", "options": {"A": "r = (2/9)(2p+q)(2q+p)", "B": "r = pq", "C": "r = p+q", "D": "r = 0"}, "correct_answer": "A"}, {"id": 55, "difficulty": "hard", "question": "Solve: (x² + 3x)² - 8(x²+3x) - 20 = 0. 🚁", "options": {"A": "x = -5, -2, 1, 2", "B": "x = -1, -2, 5", "C": "x = 0, -3", "D": "x = 2, 3"}, "correct_answer": "A"}, {"id": 56, "difficulty": "hard", "question": "Find the cubic equation whose roots are squares of the roots of x³ - 3x² + 2x - 1 = 0. 🎇", "options": {"A": "y³ - 5y² + 6y - 1 = 0", "B": "y³ + 5y² + 6y + 1 = 0", "C": "y³ - 6y² + 5y - 1 = 0", "D": "y³ - y² + y - 1 = 0"}, "correct_answer": "A"}, {"id": 57, "difficulty": "hard", "question": "If α,β,γ,δ are roots of x⁴ - 8x³ + 24x² - 32x + 16 = 0, find ∑α². 🎖️", "options": {"A": "32", "B": "24", "C": "16", "D": "8"}, "correct_answer": "A"}, {"id": 58, "difficulty": "hard", "question": "For real x, max value of (x² + 4x + 9)/(x² + 4x + 5) is? 🏔️", "options": {"A": "3", "B": "2", "C": "5", "D": "9"}, "correct_answer": "A"}, {"id": 59, "difficulty": "hard", "question": "If a,b,c are real and a+b+c=0, prove that the roots of ax²+bx+c=0 are real. But as MCQ: Condition for real roots? 🌉", "options": {"A": "Always real if a≠0", "B": "Imaginary", "C": "Equal", "D": "No roots"}, "correct_answer": "A"}, {"id": 60, "difficulty": "hard", "question": "Find all x satisfying √(x+3-4√(x-1)) + √(x+8-6√(x-1)) = 1. 🛸", "options": {"A": "x ∈ [5,10]", "B": "x=5 only", "C": "x=10 only", "D": "No solution"}, "correct_answer": "A"}]}