{"title": "Solving by Formula (Algebra) - Grade 10 ICSE", "total_questions": 60, "questions": [{"id": 1, "difficulty": "easy", "question": "Solve using formula: x² - 5x + 6 = 0 📝", "options": {"A": "x = 2, 3", "B": "x = -2, -3", "C": "x = 1, 6", "D": "x = -1, -6"}, "correct_answer": "A"}, {"id": 2, "difficulty": "easy", "question": "The quadratic formula is? 🧮", "options": {"A": "x = [-b ± √(b²-4ac)]/(2a)", "B": "x = [b ± √(b²+4ac)]/(2a)", "C": "x = [-b ± √(b²+4ac)]/(2a)", "D": "x = [b ± √(b²-4ac)]/(2a)"}, "correct_answer": "A"}, {"id": 3, "difficulty": "easy", "question": "Solve: 2x² - 7x + 3 = 0 using formula 🔢", "options": {"A": "x = 3, 1/2", "B": "x = -3, -1/2", "C": "x = 2, 3/2", "D": "x = 1, 3/2"}, "correct_answer": "A"}, {"id": 4, "difficulty": "easy", "question": "Discriminant of ax²+bx+c=0 is? 🔍", "options": {"A": "b²-4ac", "B": "b²+4ac", "C": "4ac-b²", "D": "√(b²-4ac)"}, "correct_answer": "A"}, {"id": 5, "difficulty": "easy", "question": "Solve: x² + 4x - 5 = 0 by formula ➕", "options": {"A": "x = 1, -5", "B": "x = -1, 5", "C": "x = 2, -3", "D": "x = -2, 3"}, "correct_answer": "A"}, {"id": 6, "difficulty": "easy", "question": "For x² - 8x + 15 = 0, discriminant D = ? ⚖️", "options": {"A": "4", "B": "64", "C": "2", "D": "0"}, "correct_answer": "A"}, {"id": 7, "difficulty": "easy", "question": "Solve: 3x² - 5x - 2 = 0 using formula 🎯", "options": {"A": "x = 2, -1/3", "B": "x = -2, 1/3", "C": "x = 1, -2/3", "D": "x = -1, 2/3"}, "correct_answer": "A"}, {"id": 8, "difficulty": "easy", "question": "If D>0, roots are? 🌱", "options": {"A": "Real & distinct", "B": "Real & equal", "C": "Imaginary", "D": "None"}, "correct_answer": "A"}, {"id": 9, "difficulty": "easy", "question": "Solve: 4x² - 9 = 0 by formula 🔄", "options": {"A": "x = ±3/2", "B": "x = ±2/3", "C": "x = ±9/4", "D": "x = ±4/9"}, "correct_answer": "A"}, {"id": 10, "difficulty": "easy", "question": "For 2x²+5x-3=0, D = ? 📊", "options": {"A": "49", "B": "25", "C": "1", "D": "0"}, "correct_answer": "A"}, {"id": 11, "difficulty": "easy", "question": "Solve: x² - 10x + 25 = 0 by formula 🎭", "options": {"A": "x = 5 (double)", "B": "x = -5", "C": "x = 5, -5", "D": "x = 10, 2.5"}, "correct_answer": "A"}, {"id": 12, "difficulty": "easy", "question": "If D=0, roots are? ⚡", "options": {"A": "Real & equal", "B": "Real & distinct", "C": "Imaginary", "D": "No roots"}, "correct_answer": "A"}, {"id": 13, "difficulty": "easy", "question": "Solve: x² - 3x = 0 by formula 🌟", "options": {"A": "x = 0, 3", "B": "x = 0, -3", "C": "x = 3 only", "D": "x = 1, 3"}, "correct_answer": "A"}, {"id": 14, "difficulty": "easy", "question": "For x²+6x+9=0, D = ? 🎨", "options": {"A": "0", "B": "36", "C": "12", "D": "-12"}, "correct_answer": "A"}, {"id": 15, "difficulty": "easy", "question": "Solve: 5x² - 20 = 0 using formula ⚓", "options": {"A": "x = ±2", "B": "x = ±4", "C": "x = ±√5", "D": "x = ±20"}, "correct_answer": "A"}, {"id": 16, "difficulty": "easy", "question": "If D<0, roots are? 🌀", "options": {"A": "Imaginary", "B": "Real", "C": "Equal", "D": "Zero"}, "correct_answer": "A"}, {"id": 17, "difficulty": "easy", "question": "Solve: x² + 7x + 12 = 0 by formula 🧩", "options": {"A": "x = -3, -4", "B": "x = 3, 4", "C": "x = 2, 6", "D": "x = -2, -6"}, "correct_answer": "A"}, {"id": 18, "difficulty": "easy", "question": "For x²-49=0, D = ? 📐", "options": {"A": "196", "B": "0", "C": "49", "D": "7"}, "correct_answer": "A"}, {"id": 19, "difficulty": "easy", "question": "Solve: 2x² - 8 = 0 using formula 🔨", "options": {"A": "x = ±2", "B": "x = ±4", "C": "x = ±√8", "D": "x = ±√2"}, "correct_answer": "A"}, {"id": 20, "difficulty": "easy", "question": "Nature of roots of x²+2x+3=0 is? 🌈", "options": {"A": "Imaginary", "B": "Real & equal", "C": "Real & distinct", "D": "None"}, "correct_answer": "A"}, {"id": 21, "difficulty": "medium", "question": "Solve using formula: √2x² + 7x + 5√2 = 0 🔵", "options": {"A": "x = -√2, -5/√2", "B": "x = √2, 5/√2", "C": "x = 2, 5", "D": "x = -2, -5"}, "correct_answer": "A"}, {"id": 22, "difficulty": "medium", "question": "Find discriminant of 3x² - 2x + 1/3 = 0 🎪", "options": {"A": "0", "B": "1", "C": "4", "D": "8"}, "correct_answer": "A"}, {"id": 23, "difficulty": "medium", "question": "Solve: 4√3x² + 5x - 2√3 = 0 by formula 🌐", "options": {"A": "x = √3/4, -2/√3", "B": "x = -√3/4, 2/√3", "C": "x = 2√3, -√3/2", "D": "x = -2√3, √3/2"}, "correct_answer": "A"}, {"id": 24, "difficulty": "medium", "question": "If roots of kx² - 5x + k = 0 are equal, find k using D=0. 🔑", "options": {"A": "k = ±5/2", "B": "k = 5", "C": "k = 2/5", "D": "k = 0"}, "correct_answer": "A"}, {"id": 25, "difficulty": "medium", "question": "Solve: 2x² + 5√3x + 6 = 0 by formula 🧠", "options": {"A": "x = -2√3, -√3/2", "B": "x = 2√3, √3/2", "C": "x = -2, -√3", "D": "x = 2, √3"}, "correct_answer": "A"}, {"id": 26, "difficulty": "medium", "question": "For what p does 4x² - px + 9 = 0 have equal roots? 🎲", "options": {"A": "p = ±12", "B": "p = ±6", "C": "p = ±9", "D": "p = ±8"}, "correct_answer": "A"}, {"id": 27, "difficulty": "medium", "question": "Solve: x² - (√3+√2)x + √6 = 0 by formula 🧿", "options": {"A": "x = √3, √2", "B": "x = -√3, -√2", "C": "x = √3, -√2", "D": "x = -√3, √2"}, "correct_answer": "A"}, {"id": 28, "difficulty": "medium", "question": "Find k if x² + kx + 9 = 0 has equal roots. ⚙️", "options": {"A": "k = ±6", "B": "k = ±3", "C": "k = ±9", "D": "k = ±12"}, "correct_answer": "A"}, {"id": 29, "difficulty": "medium", "question": "Solve: 2√2x² + 9x + 5√2 = 0 using formula 🪐", "options": {"A": "x = -√2, -5/(2√2)", "B": "x = √2, 5/(2√2)", "C": "x = -2√2, -5/√2", "D": "x = 2√2, 5/√2"}, "correct_answer": "A"}, {"id": 30, "difficulty": "medium", "question": "For (m-12)x² + 2(m-12)x + 2 = 0 to have equal roots, m = ? 🎯", "options": {"A": "14", "B": "12", "C": "10", "D": "8"}, "correct_answer": "A"}, {"id": 31, "difficulty": "medium", "question": "Solve: (x²+1)² - x² = 0 using formula? (Hint: substitute) 🏆", "options": {"A": "No real roots", "B": "x = ±1", "C": "x = 0", "D": "x = ±i"}, "correct_answer": "A"}, {"id": 32, "difficulty": "medium", "question": "If roots of px² + qx + q = 0 are in ratio α/β+β/α+1=0, find p:q. 🚀", "options": {"A": "p:q = 1:2", "B": "p:q = 2:1", "C": "p:q = 1:1", "D": "p:q = 3:2"}, "correct_answer": "A"}, {"id": 33, "difficulty": "medium", "question": "Solve: 3x² + 2√6x + 2 = 0 by formula 🌠", "options": {"A": "x = -√6/3 (double)", "B": "x = √6/3", "C": "x = ±√6/3", "D": "x = -√6/3, √6/3"}, "correct_answer": "A"}, {"id": 34, "difficulty": "medium", "question": "Find k so that (k+1)x² - 2(k-1)x + 1 = 0 has equal roots. 🎖️", "options": {"A": "k = 0, 3", "B": "k = 1, 2", "C": "k = 0 only", "D": "k = 3 only"}, "correct_answer": "A"}, {"id": 35, "difficulty": "medium", "question": "Solve: √3x² + 10x + 8√3 = 0 by formula 🏔️", "options": {"A": "x = -4/√3, -2√3", "B": "x = 4/√3, 2√3", "C": "x = -2√3, -√3/2", "D": "x = 2√3, √3/2"}, "correct_answer": "A"}, {"id": 36, "difficulty": "medium", "question": "If equation x² - 2px + 8p-15 = 0 has equal roots, find p. 🌉", "options": {"A": "p = 3, 5", "B": "p = 2, 6", "C": "p = 4, 5", "D": "p = 1, 7"}, "correct_answer": "A"}, {"id": 37, "difficulty": "medium", "question": "Solve: 4x² - 4a²x + (a⁴-b⁴) = 0 by formula 🛸", "options": {"A": "x = (a²±b²)/2", "B": "x = a²±b²", "C": "x = (a±b)/2", "D": "x = a²/2 ± b²"}, "correct_answer": "A"}, {"id": 38, "difficulty": "medium", "question": "For what k does (k-5)x² - 2kx + (k+4)=0 have equal roots? 🎇", "options": {"A": "k = 1, 10", "B": "k = 2, 8", "C": "k = 3, 6", "D": "k = 4, 5"}, "correct_answer": "A"}, {"id": 39, "difficulty": "medium", "question": "Solve: abx² + (b²-ac)x - bc = 0 by formula 🌌", "options": {"A": "x = -b/a, c/b", "B": "x = b/a, -c/b", "C": "x = -b/a, -c/b", "D": "x = b/a, c/b"}, "correct_answer": "A"}, {"id": 40, "difficulty": "medium", "question": "If roots of x² - (k+6)x + 2(2k-1)=0 have sum = half of product, find k. 🎡", "options": {"A": "k = 7", "B": "k = 5", "C": "k = 3", "D": "k = 1"}, "correct_answer": "A"}, {"id": 41, "difficulty": "hard", "question": "Solve: 2^(2x+2) - 33×2^x + 8 = 0 using substitution and formula. ⚡", "options": {"A": "x = 3, -2", "B": "x = 2, -3", "C": "x = 0, 1", "D": "x = 1, 2"}, "correct_answer": "A"}, {"id": 42, "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": 43, "difficulty": "hard", "question": "Solve: √(x+1) + √(2x+3) = √(12x+16) by squaring and formula. 🚁", "options": {"A": "x = 1", "B": "x = 2", "C": "x = 3", "D": "x = 4"}, "correct_answer": "A"}, {"id": 44, "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 in terms of p,q. 🪐", "options": {"A": "r = (2/9)(2p+q)(2q+p)", "B": "r = pq", "C": "r = p+q", "D": "r = 0"}, "correct_answer": "A"}, {"id": 45, "difficulty": "hard", "question": "Solve: (x²+3x)² - 8(x²+3x) - 20 = 0 using substitution and formula. 🎯", "options": {"A": "x = -5, -2, 1, 2", "B": "x = -1, -2, 5", "C": "x = 0, -3", "D": "x = 2, 3"}, "correct_answer": "A"}, {"id": 46, "difficulty": "hard", "question": "If every root of x²+ax+b=0 is decreased by 1 to get x²+px+q=0, find 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": 47, "difficulty": "hard", "question": "Solve: |x² - 5x + 6| = x-2 by considering cases and formula. 🎭", "options": {"A": "x = 2, 4", "B": "x = 2 only", "C": "x = 4 only", "D": "x = 1, 3"}, "correct_answer": "A"}, {"id": 48, "difficulty": "hard", "question": "If a,b,c are real and a+b+c=0, using discriminant show roots of ax²+bx+c=0 are real. MCQ: Condition? 🌈", "options": {"A": "Always real if a≠0", "B": "Imaginary", "C": "Equal", "D": "No roots"}, "correct_answer": "A"}, {"id": 49, "difficulty": "hard", "question": "Solve: √(x+3-4√(x-1)) + √(x+8-6√(x-1)) = 1 by substitution and formula. 🤯", "options": {"A": "x ∈ [5,10]", "B": "x = 5 only", "C": "x = 10 only", "D": "No solution"}, "correct_answer": "A"}, {"id": 50, "difficulty": "hard", "question": "If α,β are roots of x² - 2x + 3 = 0, find equation with roots α³-3α²+5α-2 and β³-3β²+5β-2 using formula relations. 🧬", "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": 51, "difficulty": "hard", "question": "Solve: 4^(x+1) - 6^x - 2×9^(x) = 0 using substitution y=2^x/3^x and formula. 🎲", "options": {"A": "x = 0", "B": "x = 1", "C": "x = 2", "D": "x = log2(3)"}, "correct_answer": "A"}, {"id": 52, "difficulty": "hard", "question": "If the equations x²+ax+1=0 and x²+x+a=0 have a common root, find a. 🧿", "options": {"A": "a = -2", "B": "a = 1", "C": "a = 2", "D": "a = 0"}, "correct_answer": "A"}, {"id": 53, "difficulty": "hard", "question": "Solve: (x²+1)/x + (x)/(x²+1) = 5/2 using substitution and formula. ⚙️", "options": {"A": "x = 2±√3, (1±√3)/2", "B": "x = 1, 2", "C": "x = 0, 1", "D": "x = ±1"}, "correct_answer": "A"}, {"id": 54, "difficulty": "hard", "question": "If roots of ax²+2bx+c=0 are in ratio m:n, find condition. 📏", "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": 55, "difficulty": "hard", "question": "Solve: √(1+√(1+√x)) = 2 by repeated squaring and formula. 🚀", "options": {"A": "x = 255", "B": "x = 256", "C": "x = 16", "D": "x = 225"}, "correct_answer": "A"}, {"id": 56, "difficulty": "hard", "question": "If α,β are roots of x² - px + q = 0, find α⁴ + β⁴ in terms of p,q using formula relations. 🏆", "options": {"A": "p⁴ - 4p²q + 2q²", "B": "p⁴ + 4p²q + 2q²", "C": "p⁴ - 2p²q + q²", "D": "p⁴ + 2p²q + q²"}, "correct_answer": "A"}, {"id": 57, "difficulty": "hard", "question": "Solve: (2x²+3x+2)/(2x²+3x+1) = (3x²+4x+3)/(3x²+4x+2) using substitution and formula. 🎖️", "options": {"A": "x = -1, -2/3", "B": "x = 0, -1", "C": "x = 1, 2/3", "D": "x = -1/2, -1"}, "correct_answer": "A"}, {"id": 58, "difficulty": "hard", "question": "If α,β are roots of ax²+bx+c=0, find equation whose roots are α²+β² and α²β² using formula. 🏔️", "options": {"A": "a²y² - (b²-2ac)a y + c² = 0", "B": "a²y² - b² y + c² = 0", "C": "y² - (b²-2ac)/a² y + c²/a² = 0", "D": "y² - (b²/a²) y + c²/a² = 0"}, "correct_answer": "A"}, {"id": 59, "difficulty": "hard", "question": "Solve: (x+1)(x+2)(x+3)(x+4) = 24 by grouping and formula. 🌉", "options": {"A": "x = 0, -5", "B": "x = 1, -6", "C": "x = 2, -7", "D": "x = 0, -6"}, "correct_answer": "A"}, {"id": 60, "difficulty": "hard", "question": "If the cubic x³-3x²+2x-1=0 has roots α,β,γ, find the cubic whose roots are α²,β²,γ² using symmetric sums and formula. 🛸", "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"}]}