{
"title": "Complementary Angles - Grade 10 CBSE",
"total_questions": 100,
"questions": [
{"id": 1, "difficulty": "easy", "question": "If θ = 30°, then its complementary angle is:", "options": {"A": "45°", "B": "90°", "C": "60°", "D": "150°"}, "correct_answer": "C"},
{"id": 2, "difficulty": "easy", "question": "Two angles are complementary if their sum is:", "options": {"A": "270°", "B": "90°", "C": "180°", "D": "360°"}, "correct_answer": "B"},
{"id": 3, "difficulty": "easy", "question": "Complement of 45° is:", "options": {"A": "90°", "B": "135°", "C": "45°", "D": "60°"}, "correct_answer": "C"},
{"id": 4, "difficulty": "easy", "question": "sin(90° - θ) = ?", "options": {"A": "tanθ", "B": "cosecθ", "C": "cosθ", "D": "sinθ"}, "correct_answer": "C"},
{"id": 5, "difficulty": "easy", "question": "cos(90° - A) = ?", "options": {"A": "secA", "B": "tanA", "C": "cotA", "D": "sinA"}, "correct_answer": "D"},
{"id": 6, "difficulty": "easy", "question": "tan(90° - θ) = ?", "options": {"A": "cotθ", "B": "sinθ", "C": "tanθ", "D": "cosθ"}, "correct_answer": "A"},
{"id": 7, "difficulty": "easy", "question": "If α + β = 90°, then α and β are called:", "options": {"A": "Supplementary", "B": "Complementary", "C": "Adjacent", "D": "Vertical"}, "correct_answer": "B"},
{"id": 8, "difficulty": "easy", "question": "Complement of 15° is:", "options": {"A": "165°", "B": "105°", "C": "85°", "D": "75°"}, "correct_answer": "D"},
{"id": 9, "difficulty": "easy", "question": "sec(90° - θ) = ?", "options": {"A": "secθ", "B": "sinθ", "C": "tanθ", "D": "cosecθ"}, "correct_answer": "D"},
{"id": 10, "difficulty": "easy", "question": "cosec(90° - A) = ?", "options": {"A": "cotA", "B": "cosA", "C": "secA", "D": "tanA"}, "correct_answer": "C"},
{"id": 11, "difficulty": "easy", "question": "cot(90° - θ) = ?", "options": {"A": "cosθ", "B": "cosecθ", "C": "sinθ", "D": "tanθ"}, "correct_answer": "D"},
{"id": 12, "difficulty": "easy", "question": "If sin 25° = cos θ, then θ = ?", "options": {"A": "25°", "B": "155°", "C": "115°", "D": "65°"}, "correct_answer": "D"},
{"id": 13, "difficulty": "easy", "question": "If tan 35° = cot α, then α = ?", "options": {"A": "35°", "B": "145°", "C": "125°", "D": "55°"}, "correct_answer": "D"},
{"id": 14, "difficulty": "easy", "question": "sin 60° = cos ?", "options": {"A": "60°", "B": "120°", "C": "30°", "D": "150°"}, "correct_answer": "C"},
{"id": 15, "difficulty": "easy", "question": "cos 80° = sin ?", "options": {"A": "100°", "B": "170°", "C": "10°", "D": "80°"}, "correct_answer": "C"},
{"id": 16, "difficulty": "easy", "question": "tan 20° = cot ?", "options": {"A": "160°", "B": "70°", "C": "110°", "D": "20°"}, "correct_answer": "B"},
{"id": 17, "difficulty": "easy", "question": "sec 40° = cosec ?", "options": {"A": "40°", "B": "140°", "C": "130°", "D": "50°"}, "correct_answer": "D"},
{"id": 18, "difficulty": "easy", "question": "cosec 70° = sec ?", "options": {"A": "70°", "B": "110°", "C": "160°", "D": "20°"}, "correct_answer": "D"},
{"id": 19, "difficulty": "easy", "question": "cot 10° = tan ?", "options": {"A": "10°", "B": "100°", "C": "80°", "D": "170°"}, "correct_answer": "C"},
{"id": 20, "difficulty": "easy", "question": "If A + B = 90°, then sin A = ?", "options": {"A": "sin B", "B": "tan B", "C": "cosec B", "D": "cos B"}, "correct_answer": "D"},
{"id": 21, "difficulty": "medium", "question": "If sin 3θ = cos (θ - 10°), where 3θ and (θ-10°) are acute, find θ.", "options": {"A": "30°", "B": "25°", "C": "35°", "D": "20°"}, "correct_answer": "B"},
{"id": 22, "difficulty": "medium", "question": "If tan 2θ = cot (θ + 18°), where 2θ and (θ+18°) are acute, find θ.", "options": {"A": "36°", "B": "24°", "C": "54°", "D": "18°"}, "correct_answer": "B"},
{"id": 23, "difficulty": "medium", "question": "Evaluate: sin 25° cos 65° + cos 25° sin 65°", "options": {"A": "0", "B": "1", "C": "2", "D": "√2"}, "correct_answer": "B"},
{"id": 24, "difficulty": "medium", "question": "Simplify: tan 48° tan 23° tan 42° tan 67°", "options": {"A": "0", "B": "1", "C": "√3", "D": "2"}, "correct_answer": "B"},
{"id": 25, "difficulty": "medium", "question": "If sin A = cos B, where A and B are acute, then A + B = ?", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
{"id": 26, "difficulty": "medium", "question": "Evaluate: cos 38° cos 52° - sin 38° sin 52°", "options": {"A": "1", "B": "0", "C": "0.5", "D": "2"}, "correct_answer": "B"},
{"id": 27, "difficulty": "medium", "question": "If sec 4A = cosec (A - 20°), where 4A is acute, find A.", "options": {"A": "20°", "B": "22°", "C": "25°", "D": "30°"}, "correct_answer": "B"},
{"id": 28, "difficulty": "medium", "question": "Simplify: sin 70°/cos 20° + cos 55°/sin 35°", "options": {"A": "1", "B": "2", "C": "√2", "D": "0.5"}, "correct_answer": "B"},
{"id": 29, "difficulty": "medium", "question": "If tan 2θ = cot 3θ, where both are acute, find θ.", "options": {"A": "15°", "B": "18°", "C": "20°", "D": "22°"}, "correct_answer": "B"},
{"id": 30, "difficulty": "medium", "question": "Evaluate: sec 35°/cosec 55° + cot 78°/tan 12°", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "C"},
{"id": 31, "difficulty": "medium", "question": "If cos (40° + x) = sin 30°, find x.", "options": {"A": "10°", "B": "20°", "C": "30°", "D": "40°"}, "correct_answer": "B"},
{"id": 32, "difficulty": "medium", "question": "Simplify: sin (90°-θ) cos (90°-θ) / (tan θ)", "options": {"A": "sin θ", "B": "cos θ", "C": "cot θ", "D": "1"}, "correct_answer": "C"},
{"id": 33, "difficulty": "medium", "question": "If sin (θ+34°) = cos (θ-16°), find θ.", "options": {"A": "30°", "B": "36°", "C": "40°", "D": "45°"}, "correct_answer": "B"},
{"id": 34, "difficulty": "medium", "question": "Evaluate: tan 15° tan 25° tan 45° tan 65° tan 75°", "options": {"A": "0", "B": "1", "C": "√3", "D": "2"}, "correct_answer": "B"},
{"id": 35, "difficulty": "medium", "question": "If cosec (θ+20°) = sec (3θ-40°), find θ.", "options": {"A": "25°", "B": "27.5°", "C": "30°", "D": "35°"}, "correct_answer": "B"},
{"id": 36, "difficulty": "medium", "question": "Simplify: [sin 35°/cos 55°]² + [cos 55°/sin 35°]²", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
{"id": 37, "difficulty": "medium", "question": "If cot (2θ+10°) = tan (3θ-20°), find θ.", "options": {"A": "18°", "B": "20°", "C": "22°", "D": "25°"}, "correct_answer": "B"},
{"id": 38, "difficulty": "medium", "question": "Evaluate: sin 18°/cos 72° + √3 (tan 10° tan 30° tan 80°)", "options": {"A": "1", "B": "2", "C": "√3", "D": "1/√3"}, "correct_answer": "A"},
{"id": 39, "difficulty": "medium", "question": "If cos (2θ-30°) = sin (θ+24°), find θ.", "options": {"A": "28°", "B": "32°", "C": "36°", "D": "42°"}, "correct_answer": "B"},
{"id": 40, "difficulty": "medium", "question": "Simplify: (cos 70°/sin 20°) + (cos 55°.cosec 35°)/(tan 5°.tan 45°.tan 85°)", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
{"id": 41, "difficulty": "medium", "question": "If tan (3θ-15°) = cot (θ+25°), find θ.", "options": {"A": "15°", "B": "20°", "C": "25°", "D": "30°"}, "correct_answer": "B"},
{"id": 42, "difficulty": "medium", "question": "Evaluate: sin 39° - cos 51°", "options": {"A": "0", "B": "1", "C": "0.5", "D": "-0.5"}, "correct_answer": "A"},
{"id": 43, "difficulty": "medium", "question": "If sec (5θ) = cosec (θ+18°), where 5θ is acute, find θ.", "options": {"A": "9°", "B": "12°", "C": "15°", "D": "18°"}, "correct_answer": "B"},
{"id": 44, "difficulty": "medium", "question": "Simplify: tan 1° tan 2° tan 3° ... tan 89°", "options": {"A": "0", "B": "1", "C": "2", "D": "Undefined"}, "correct_answer": "B"},
{"id": 45, "difficulty": "medium", "question": "If sin (θ+45°) = cos (θ-15°), find tan θ.", "options": {"A": "1", "B": "√3", "C": "1/√3", "D": "√2"}, "correct_answer": "C"},
{"id": 46, "difficulty": "medium", "question": "Evaluate: cos² 20° + cos² 70°", "options": {"A": "0", "B": "1", "C": "2", "D": "0.5"}, "correct_answer": "B"},
{"id": 47, "difficulty": "medium", "question": "If cot (2α) = tan (α+30°), find α.", "options": {"A": "15°", "B": "20°", "C": "25°", "D": "30°"}, "correct_answer": "B"},
{"id": 48, "difficulty": "medium", "question": "Simplify: sin (90°-A) tan (90°-A) sec A / (cos A cot A cosec (90°-A))", "options": {"A": "0", "B": "1", "C": "tan A", "D": "cot A"}, "correct_answer": "B"},
{"id": 49, "difficulty": "medium", "question": "If sin (3x+10°) = cos (2x-20°), find x.", "options": {"A": "18°", "B": "20°", "C": "22°", "D": "24°"}, "correct_answer": "B"},
{"id": 50, "difficulty": "medium", "question": "Evaluate: tan 48° tan 42° + sin 22° cosec 22° - cos 68° sec 68°", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "B"},
{"id": 51, "difficulty": "medium", "question": "If cosec (4θ) = sec (θ+15°), where 4θ is acute, find θ.", "options": {"A": "10°", "B": "15°", "C": "20°", "D": "25°"}, "correct_answer": "B"},
{"id": 52, "difficulty": "medium", "question": "Simplify: [cos (90°-θ) sin (90°-θ)] / [tan (90°-θ)]", "options": {"A": "sin θ", "B": "cos θ", "C": "sin² θ", "D": "cos² θ"}, "correct_answer": "D"},
{"id": 53, "difficulty": "medium", "question": "If tan (2θ-10°) = cot (3θ+20°), find θ.", "options": {"A": "12°", "B": "14°", "C": "16°", "D": "18°"}, "correct_answer": "C"},
{"id": 54, "difficulty": "medium", "question": "Evaluate: sin² 5° + sin² 10° + sin² 15° + ... + sin² 85°", "options": {"A": "8", "B": "8.5", "C": "9", "D": "9.5"}, "correct_answer": "D"},
{"id": 55, "difficulty": "medium", "question": "If cos (θ+20°) = sin (θ+10°), find θ.", "options": {"A": "25°", "B": "30°", "C": "35°", "D": "40°"}, "correct_answer": "B"},
{"id": 56, "difficulty": "medium", "question": "Simplify: (sin 72° + cos 18°)(sin 72° - cos 18°)", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
{"id": 57, "difficulty": "medium", "question": "If sec (3α) = cosec (2α+30°), find α.", "options": {"A": "10°", "B": "12°", "C": "15°", "D": "18°"}, "correct_answer": "B"},
{"id": 58, "difficulty": "medium", "question": "Evaluate: cos 1° cos 2° cos 3° ... cos 90°", "options": {"A": "0", "B": "1", "C": "0.5", "D": "Undefined"}, "correct_answer": "A"},
{"id": 59, "difficulty": "medium", "question": "If sin (45°+θ) = cos (θ-15°), find tan θ.", "options": {"A": "1", "B": "√3", "C": "1/√3", "D": "0"}, "correct_answer": "C"},
{"id": 60, "difficulty": "medium", "question": "Simplify: tan 5° tan 15° tan 25° ... tan 85°", "options": {"A": "0", "B": "1", "C": "√3", "D": "Undefined"}, "correct_answer": "B"},
{"id": 61, "difficulty": "medium", "question": "If cot (θ+15°) = tan (θ+45°), find θ.", "options": {"A": "10°", "B": "15°", "C": "20°", "D": "25°"}, "correct_answer": "B"},
{"id": 62, "difficulty": "medium", "question": "Evaluate: sin 35° cos 55° + cos 35° sin 55° - 1", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "B"},
{"id": 63, "difficulty": "medium", "question": "If sin (2A) = cos (3A-30°), find A.", "options": {"A": "20°", "B": "24°", "C": "30°", "D": "36°"}, "correct_answer": "B"},
{"id": 64, "difficulty": "medium", "question": "Simplify: [sin (90°-θ) cos θ + cos (90°-θ) sin θ] / tan θ", "options": {"A": "sin θ", "B": "cos θ", "C": "1", "D": "cot θ"}, "correct_answer": "D"},
{"id": 65, "difficulty": "medium", "question": "If tan (45°+θ) = cot (30°+θ), find θ.", "options": {"A": "5°", "B": "7.5°", "C": "10°", "D": "15°"}, "correct_answer": "B"},
{"id": 66, "difficulty": "medium", "question": "Evaluate: (cos 70°/sin 20°)² + (sin 59°/cos 31°)²", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
{"id": 67, "difficulty": "medium", "question": "If cosec (θ+10°) = sec (2θ-20°), find θ.", "options": {"A": "30°", "B": "33.33°", "C": "40°", "D": "50°"}, "correct_answer": "B"},
{"id": 68, "difficulty": "medium", "question": "Simplify: sin 25° cos 65° + sin 65° cos 25°", "options": {"A": "0", "B": "0.5", "C": "1", "D": "√3/2"}, "correct_answer": "C"},
{"id": 69, "difficulty": "medium", "question": "If cos (3θ-10°) = sin (θ+30°), find θ.", "options": {"A": "15°", "B": "17.5°", "C": "20°", "D": "25°"}, "correct_answer": "B"},
{"id": 70, "difficulty": "medium", "question": "Evaluate: tan 20° tan 40° tan 50° tan 70°", "options": {"A": "0", "B": "1", "C": "√3", "D": "1/√3"}, "correct_answer": "B"},
{"id": 71, "difficulty": "medium", "question": "If sin (60°-θ) = cos (30°+θ), find tan θ.", "options": {"A": "0", "B": "1", "C": "√3", "D": "1/√3"}, "correct_answer": "B"},
{"id": 72, "difficulty": "medium", "question": "Simplify: (sin 49°/cos 41°)² + (cos 41°/sin 49°)²", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
{"id": 73, "difficulty": "medium", "question": "If cot (2θ) = tan (4θ-30°), find θ.", "options": {"A": "15°", "B": "20°", "C": "25°", "D": "30°"}, "correct_answer": "B"},
{"id": 74, "difficulty": "medium", "question": "Evaluate: cos 57° sin 33° + sin 57° cos 33°", "options": {"A": "0", "B": "0.5", "C": "1", "D": "√2"}, "correct_answer": "C"},
{"id": 75, "difficulty": "medium", "question": "If sec (2α+10°) = cosec (3α-20°), find α.", "options": {"A": "18°", "B": "20°", "C": "22°", "D": "25°"}, "correct_answer": "B"},
{"id": 76, "difficulty": "medium", "question": "Simplify: tan 10° tan 80° + tan 20° tan 70° + tan 30° tan 60°", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
{"id": 77, "difficulty": "medium", "question": "If sin (θ+30°) = cos (θ-20°), find θ.", "options": {"A": "35°", "B": "40°", "C": "45°", "D": "50°"}, "correct_answer": "B"},
{"id": 78, "difficulty": "medium", "question": "Evaluate: sin² 25° + sin² 65°", "options": {"A": "0", "B": "0.5", "C": "1", "D": "2"}, "correct_answer": "C"},
{"id": 79, "difficulty": "medium", "question": "If tan (3θ-15°) = cot (θ+45°), find θ.", "options": {"A": "12°", "B": "15°", "C": "18°", "D": "20°"}, "correct_answer": "B"},
{"id": 80, "difficulty": "medium", "question": "Simplify: (cos 70°/sin 20°) + (cos 55°.cosec 35°)/2", "options": {"A": "1", "B": "1.5", "C": "2", "D": "2.5"}, "correct_answer": "C"},
{"id": 81, "difficulty": "hard", "question": "If sin (θ+20°) = cos (3θ-40°) and both angles are acute, find θ.", "options": {"A": "25°", "B": "27.5°", "C": "30°", "D": "32.5°"}, "correct_answer": "B"},
{"id": 82, "difficulty": "hard", "question": "Evaluate: tan 1° tan 2° tan 3° ... tan 89° without calculator.", "options": {"A": "0", "B": "1", "C": "√2", "D": "Undefined"}, "correct_answer": "B"},
{"id": 83, "difficulty": "hard", "question": "If sin θ = cos (2θ+30°), find all possible values of θ between 0° and 90°.", "options": {"A": "20° only", "B": "30° only", "C": "20° and 60°", "D": "20° and 70°"}, "correct_answer": "A"},
{"id": 84, "difficulty": "hard", "question": "Simplify: [sin (70°-θ) cos (20°+θ) + cos (70°-θ) sin (20°+θ)] / cos θ", "options": {"A": "tan θ", "B": "cot θ", "C": "1", "D": "sin θ"}, "correct_answer": "C"},
{"id": 85, "difficulty": "hard", "question": "If tan (2θ+15°) = cot (3θ-15°), find the value of cosec (5θ).", "options": {"A": "√2", "B": "2", "C": "2/√3", "D": "√2/2"}, "correct_answer": "A"},
{"id": 86, "difficulty": "hard", "question": "Prove that: cos A/(1-sin A) + cos A/(1+sin A) = 2 sec A", "options": {"A": "Identity is true", "B": "Identity is false", "C": "True only for A=45°", "D": "Can't be determined"}, "correct_answer": "A"},
{"id": 87, "difficulty": "hard", "question": "If sin (θ+φ) = 1/2 and cos (θ-φ) = 1/2, find θ and φ (acute angles).", "options": {"A": "θ=45°, φ=15°", "B": "θ=30°, φ=30°", "C": "θ=45°, φ=0°", "D": "θ=60°, φ=30°"}, "correct_answer": "A"},
{"id": 88, "difficulty": "hard", "question": "Evaluate without calculator: sin² 5° + sin² 10° + ... + sin² 85° + sin² 90°", "options": {"A": "9", "B": "9.5", "C": "10", "D": "10.5"}, "correct_answer": "B"},
{"id": 89, "difficulty": "hard", "question": "If tan (θ+15°) = √3, find the value of θ and hence evaluate: sin θ + cos θ", "options": {"A": "1", "B": "√3", "C": "(√3+1)/2", "D": "(√3-1)/2"}, "correct_answer": "C"},
{"id": 90, "difficulty": "hard", "question": "Simplify: (sin θ sec θ)² + (cos θ cosec θ)² - (tan θ + cot θ)²", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "B"},
{"id": 91, "difficulty": "hard", "question": "If sec 4A = cosec (A-20°), where 4A is an acute angle, find A and also find tan 3A.", "options": {"A": "22°, 1", "B": "22°, √3", "C": "22°, 1/√3", "D": "22°, 2"}, "correct_answer": "A"},
{"id": 92, "difficulty": "hard", "question": "Prove that: (1+tan² A)/(1+cot² A) = [(1-tan A)/(1-cot A)]²", "options": {"A": "Identity is true", "B": "Identity is false", "C": "True for A=45° only", "D": "Can't be proved"}, "correct_answer": "A"},
{"id": 93, "difficulty": "hard", "question": "If sin (A-B) = 1/2 and cos (A+B) = 1/2, find A and B where A>B and both acute.", "options": {"A": "A=45°, B=15°", "B": "A=60°, B=30°", "C": "A=45°, B=30°", "D": "A=30°, B=15°"}, "correct_answer": "A"},
{"id": 94, "difficulty": "hard", "question": "Evaluate: tan 20° tan 40° tan 60° tan 80°", "options": {"A": "1", "B": "√3", "C": "3", "D": "3√3"}, "correct_answer": "C"},
{"id": 95, "difficulty": "hard", "question": "If cot (θ+20°) = tan (θ+10°), find θ and evaluate sin 2θ.", "options": {"A": "30°, √3/2", "B": "30°, 1/2", "C": "30°, 1", "D": "30°, 0"}, "correct_answer": "B"},
{"id": 96, "difficulty": "hard", "question": "Prove that: √[(1+sin A)/(1-sin A)] = sec A + tan A", "options": {"A": "Identity is true", "B": "Identity is false", "C": "True for A=0° only", "D": "Can't be determined"}, "correct_answer": "A"},
{"id": 97, "difficulty": "hard", "question": "If cos (θ+30°) = sin (2θ-20°), find θ and evaluate tan (θ+15°).", "options": {"A": "26.67°, 1/√3", "B": "26.67°, √3", "C": "26.67°, 1", "D": "26.67°, 2"}, "correct_answer": "C"},
{"id": 98, "difficulty": "hard", "question": "Simplify: [cos⁴ A - sin⁴ A] / [cos² A - sin² A]", "options": {"A": "1", "B": "cos² A", "C": "sin² A", "D": "cos 2A"}, "correct_answer": "A"},
{"id": 99, "difficulty": "hard", "question": "If tan 2θ = cot (θ+24°), find θ and evaluate sec 3θ.", "options": {"A": "22°, √2", "B": "22°, 2", "C": "22°, 2/√3", "D": "22°, √2/2"}, "correct_answer": "A"},
{"id": 100, "difficulty": "hard", "question": "Prove that: sin θ/(1+cos θ) + (1+cos θ)/sin θ = 2 cosec θ", "options": {"A": "Identity is true", "B": "Identity is false", "C": "True for θ=45° only", "D": "Can't be proved"}, "correct_answer": "A"}
]
}