{
  "title": "Definition_and_terms(Arithmetic_Progressions) - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Which of these is an Arithmetic Progression?", "options": {"A": "2,4,8,16,...", "B": "3,6,9,12,...", "C": "1,4,9,16,...", "D": "2,5,10,17,..."}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "The sequence 5,8,11,14,... has common difference:", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "In AP 7,10,13,16,... the first term is:", "options": {"A": "7", "B": "10", "C": "13", "D": "16"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Which term describes the constant difference between consecutive terms in AP?", "options": {"A": "First term", "B": "Common difference", "C": "nth term", "D": "Sum"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "In sequence -3,0,3,6,..., common difference is:", "options": {"A": "-3", "B": "0", "C": "3", "D": "6"}, "correct_answer": "C"},
    {"id": 6, "difficulty": "easy", "question": "AP where first term is 2 and common difference is 5: 4th term?", "options": {"A": "12", "B": "15", "C": "17", "D": "19"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "The numbers 10,7,4,1,... form AP with common difference:", "options": {"A": "3", "B": "-3", "C": "7", "D": "-7"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "In AP, if a=4, d=3, then 3rd term is:", "options": {"A": "7", "B": "10", "C": "13", "D": "16"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Which is NOT an AP?", "options": {"A": "1,3,5,7,...", "B": "2,5,8,11,...", "C": "3,6,12,24,...", "D": "10,7,4,1,..."}, "correct_answer": "C"},
    {"id": 10, "difficulty": "easy", "question": "In AP 20,16,12,8,..., common difference is:", "options": {"A": "4", "B": "-4", "C": "8", "D": "-8"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "First term of AP: 100,95,90,85,... is:", "options": {"A": "85", "B": "90", "C": "95", "D": "100"}, "correct_answer": "D"},
    {"id": 12, "difficulty": "easy", "question": "Common difference in AP: ½, 1, 1½, 2,... is:", "options": {"A": "½", "B": "1", "C": "1½", "D": "2"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Next term in AP: 0.2, 0.5, 0.8, 1.1,... is:", "options": {"A": "1.3", "B": "1.4", "C": "1.5", "D": "1.6"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "AP with negative common difference:", "options": {"A": "1,4,7,10,...", "B": "15,12,9,6,...", "C": "2,4,8,16,...", "D": "5,10,15,20,..."}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "In AP: x, x+3, x+6, x+9,..., common difference:", "options": {"A": "x", "B": "3", "C": "6", "D": "9"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "If a=10, d=-2, 2nd term is:", "options": {"A": "8", "B": "6", "C": "4", "D": "2"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Sequence where each term increases by same amount:", "options": {"A": "Geometric Progression", "B": "Arithmetic Progression", "C": "Harmonic Progression", "D": "Fibonacci Sequence"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "In AP 3.5, 4.0, 4.5, 5.0,..., common difference:", "options": {"A": "0.2", "B": "0.3", "C": "0.4", "D": "0.5"}, "correct_answer": "D"},
    {"id": 19, "difficulty": "easy", "question": "AP starting from 50 with d=5: 2nd term?", "options": {"A": "55", "B": "60", "C": "65", "D": "70"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Which represents common difference 'd' in AP?", "options": {"A": "a₂ - a₁", "B": "a₁ - a₂", "C": "a₁ + a₂", "D": "a₁ × a₂"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "For what value of k will k+2, 4k-6, 3k-2 be three consecutive terms of AP?", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "If 2x, x+10, 3x+2 are in AP, find x.", "options": {"A": "6", "B": "8", "C": "10", "D": "12"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Which term of AP: 3,8,13,18,... is 78?", "options": {"A": "14", "B": "15", "C": "16", "D": "17"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "If 4th term of AP is 14 and 9th term is 29, find first term.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "D"},
    {"id": 25, "difficulty": "medium", "question": "How many terms are in AP: 7,13,19,...,205?", "options": {"A": "32", "B": "33", "C": "34", "D": "35"}, "correct_answer": "C"},
    {"id": 26, "difficulty": "medium", "question": "If a=5 and a₅=25, find common difference d.", "options": {"A": "4", "B": "5", "C": "6", "D": "7"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "The 10th term of AP: 2,7,12,... is:", "options": {"A": "42", "B": "45", "C": "47", "D": "50"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "If a₃=15 and a₁₀=43 in AP, find a₁₅.", "options": {"A": "65", "B": "67", "C": "69", "D": "71"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "For AP with a=8, d=3, which term is 83?", "options": {"A": "24", "B": "25", "C": "26", "D": "27"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "If 3,a,b are in AP, then b-a equals:", "options": {"A": "a-3", "B": "3-a", "C": "3", "D": "a"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "If 2x+1, 4x, 13-x are in AP, find x.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "AP with 5th term 20 and 10th term 35. Find 15th term.", "options": {"A": "45", "B": "50", "C": "55", "D": "60"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "If aₙ = 4n+1, is this AP? If yes, find a and d.", "options": {"A": "Yes, a=5,d=4", "B": "Yes, a=4,d=5", "C": "No, not AP", "D": "Yes, a=1,d=4"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "If a=3, d=-2, which term is -15?", "options": {"A": "9", "B": "10", "C": "11", "D": "12"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "For AP: 21,18,15,..., -81. Number of terms?", "options": {"A": "33", "B": "34", "C": "35", "D": "36"}, "correct_answer": "C"},
    {"id": 36, "difficulty": "medium", "question": "If a₂=13 and a₅=22 in AP, find a₇.", "options": {"A": "28", "B": "29", "C": "30", "D": "31"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "For AP with d=5 and a₁₀=47, find a₁.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "If a₃+a₈=40 and a₅+a₁₀=60, find a₁.", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "AP where 3rd term is 4 times 1st term and 2nd term is 9. Find AP.", "options": {"A": "3,9,15,...", "B": "6,9,12,...", "C": "5,9,13,...", "D": "4,9,14,..."}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "If 5 times 5th term equals 8 times 8th term, find 13th term.", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "For AP: √2, √8, √18, √32,... find 10th term.", "options": {"A": "√200", "B": "10√2", "C": "√162", "D": "9√2"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "If aₙ = 3n-2, find a₅ - a₃.", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "D"},
    {"id": 43, "difficulty": "medium", "question": "AP with a₆=19 and a₁₇=41. Find a₁₁.", "options": {"A": "27", "B": "29", "C": "31", "D": "33"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "medium", "question": "If a=2p+1, d=2p, find a₅.", "options": {"A": "10p+1", "B": "8p+1", "C": "6p+1", "D": "4p+1"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "For AP: 1/3, 5/3, 9/3, 13/3,... find a₈.", "options": {"A": "25/3", "B": "27/3", "C": "29/3", "D": "31/3"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "medium", "question": "If a₈=26 and a₁₃=46, find d.", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "Three numbers in AP sum to 15 and product to 80. Find numbers.", "options": {"A": "2,5,8", "B": "3,5,7", "C": "4,5,6", "D": "1,5,9"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "medium", "question": "If (x+1), 3x, (4x+2) in AP, find x.", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "AP where a₅-a₂=12. Find d.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "C"},
    {"id": 50, "difficulty": "medium", "question": "If a=7, last term l=84, d=7, find n.", "options": {"A": "10", "B": "11", "C": "12", "D": "13"}, "correct_answer": "C"},
    {"id": 51, "difficulty": "medium", "question": "For AP with a₃=7 and a₇=15, find a₁₀.", "options": {"A": "20", "B": "21", "C": "22", "D": "23"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "medium", "question": "If aₙ = 2n+3, verify if AP. Find a and d.", "options": {"A": "Yes, a=5,d=2", "B": "Yes, a=3,d=2", "C": "No", "D": "Yes, a=2,d=3"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "AP where 4th term is 0. Find relation between a and d.", "options": {"A": "a=3d", "B": "a=-3d", "C": "a=4d", "D": "a=-4d"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "medium", "question": "If a₅=30 and a₁₂=65, find a₂₀.", "options": {"A": "105", "B": "110", "C": "115", "D": "120"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "For AP: 0.6, 1.7, 2.8,... find a₆.", "options": {"A": "4.0", "B": "4.9", "C": "5.0", "D": "5.1"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "If 3, x, y, 18 in AP, find x and y.", "options": {"A": "8,13", "B": "7,12", "C": "6,11", "D": "5,10"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "AP with a=1, d=4. Which term equals 97?", "options": {"A": "23", "B": "24", "C": "25", "D": "26"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "medium", "question": "If aₙ = 5-2n, find a₄.", "options": {"A": "-3", "B": "-5", "C": "-7", "D": "-9"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "For AP where a₈=2a₄ and a₁₂=20, find a.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "D"},
    {"id": 60, "difficulty": "medium", "question": "If (a-b), (a), (a+b) in AP, find common difference.", "options": {"A": "a", "B": "b", "C": "2b", "D": "2a"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "AP: log2, log4, log8, log16,... Is this AP? If yes, find d.", "options": {"A": "Yes, d=log2", "B": "Yes, d=log4", "C": "No", "D": "Yes, d=1"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "medium", "question": "If 2a, 3a, 6-1 in AP, find a.", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "AP with a₃+a₇=30 and a₂+a₈=34. Find a.", "options": {"A": "8", "B": "9", "C": "10", "D": "11"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "If a=10, d=-0.5, find a₆.", "options": {"A": "7.5", "B": "7.0", "C": "6.5", "D": "6.0"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "For AP: 100, 96, 92,... find first negative term.", "options": {"A": "24th", "B": "25th", "C": "26th", "D": "27th"}, "correct_answer": "D"},
    {"id": 66, "difficulty": "medium", "question": "If 4/5, a, 2 in AP, find a.", "options": {"A": "7/5", "B": "6/5", "C": "1", "D": "4/5"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "medium", "question": "AP with a₅=22 and a₁₅=62. Find a₁₀.", "options": {"A": "40", "B": "42", "C": "44", "D": "46"}, "correct_answer": "B"},
    {"id": 68, "difficulty": "medium", "question": "If aₙ = 3+4n, find a₅-a₂.", "options": {"A": "8", "B": "10", "C": "12", "D": "14"}, "correct_answer": "C"},
    {"id": 69, "difficulty": "medium", "question": "For AP: 1/2, 3/2, 5/2,... find a₉.", "options": {"A": "15/2", "B": "17/2", "C": "19/2", "D": "21/2"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "medium", "question": "If a₇-a₃=24, find d.", "options": {"A": "4", "B": "5", "C": "6", "D": "7"}, "correct_answer": "C"},
    {"id": 71, "difficulty": "medium", "question": "AP where a=π, d=π. Find a₄.", "options": {"A": "2π", "B": "3π", "C": "4π", "D": "5π"}, "correct_answer": "C"},
    {"id": 72, "difficulty": "medium", "question": "If 2, a, b, 14 in AP, find a and b.", "options": {"A": "5,8", "B": "6,10", "C": "4,8", "D": "3,8"}, "correct_answer": "B"},
    {"id": 73, "difficulty": "medium", "question": "For AP with a₆=19 and a₁₁=34, find a₁.", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "If aₙ = n²+1, is this AP?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "medium", "question": "AP: √3, √12, √27,... find a₅.", "options": {"A": "√48", "B": "√75", "C": "5√3", "D": "6√3"}, "correct_answer": "C"},
    {"id": 76, "difficulty": "medium", "question": "If a₈=2a₄ and a₃=10, find a.", "options": {"A": "4", "B": "5", "C": "6", "D": "7"}, "correct_answer": "C"},
    {"id": 77, "difficulty": "medium", "question": "For AP: 0.1, 0.3, 0.5,... find a₇.", "options": {"A": "1.1", "B": "1.3", "C": "1.5", "D": "1.7"}, "correct_answer": "B"},
    {"id": 78, "difficulty": "medium", "question": "If a+b, a, a-b in AP, find d.", "options": {"A": "a", "B": "b", "C": "-b", "D": "-a"}, "correct_answer": "C"},
    {"id": 79, "difficulty": "medium", "question": "AP with a=3.5, d=0, find a₅.", "options": {"A": "0", "B": "3.5", "C": "7", "D": "10.5"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "medium", "question": "If a₃+a₆=29 and a₂+a₅=26, find a.", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "C"},
    {"id": 81, "difficulty": "hard", "question": "If pth term of AP is q and qth term is p, find (p+q)th term.", "options": {"A": "0", "B": "1", "C": "p+q", "D": "p-q"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "If m times mth term equals n times nth term of AP with m≠n, find (m+n)th term.", "options": {"A": "0", "B": "a", "C": "m+n", "D": "mn"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "If a₁, a₂,..., aₙ are in AP with common difference d, find value of ∑(aₖ - aₖ₋₁) from k=2 to n.", "options": {"A": "(n-1)d", "B": "nd", "C": "a+(n-1)d", "D": "S_n"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "If S_n = 3n²+2n, find a_n (nth term).", "options": {"A": "6n-1", "B": "6n+1", "C": "6n-2", "D": "6n+2"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "If ratio of sums of m and n terms of AP is m²:n², find ratio of mth and nth terms.", "options": {"A": "(2m-1):(2n-1)", "B": "(2m+1):(2n+1)", "C": "m:n", "D": "m²:n²"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "If a,b,c are in AP, prove that a²(b+c), b²(c+a), c²(a+b) are in AP. For a=1,b=2,c=3, verify.", "options": {"A": "5,16,33 in AP", "B": "5,14,33 in AP", "C": "5,16,35 in AP", "D": "5,14,35 in AP"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "If 1/(b+c), 1/(c+a), 1/(a+b) are in AP, prove that a²,b²,c² are in AP. For a=1,b=2,c=3, check.", "options": {"A": "1,4,9 in AP", "B": "1,4,9 not in AP", "C": "1,2,3 in AP", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 88, "difficulty": "hard", "question": "If a,b,c are in AP, prove that (a-c)² = 4(b²-ac). For a=2,b=4,c=6, verify.", "options": {"A": "16=16", "B": "16=20", "C": "20=20", "D": "20=16"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "If pth, qth, rth terms of AP are a,b,c respectively, prove that a(q-r)+b(r-p)+c(p-q)=0. For AP 2,5,8,... with p=1,q=2,r=3, verify.", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "If S_m = S_n for m≠n in AP, prove that S_(m+n)=0. For AP 3,5,7,... find m,n such that S_m=S_n.", "options": {"A": "m=3,n=5", "B": "m=4,n=6", "C": "m=5,n=7", "D": "m=6,n=8"}, "correct_answer": "B"},
    {"id": 91, "difficulty": "hard", "question": "If a,b,c are in AP, prove that (a+2b-c)(2b+c-a)(c+a-b)=4abc. For a=1,b=2,c=3, verify.", "options": {"A": "LHS=24,RHS=24", "B": "LHS=20,RHS=24", "C": "LHS=24,RHS=20", "D": "LHS=20,RHS=20"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Find condition that roots of equation (b-c)x²+(c-a)x+(a-b)=0 are equal, given a,b,c in AP.", "options": {"A": "a=b=c", "B": "a=2b-c", "C": "2b=a+c", "D": "b²=ac"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "If a,b,c are in AP and a²,b²,c² are in AP, prove that a=b=c or a,b,c are in GP. For a=1,b=2,c=3, check.", "options": {"A": "Satisfies both", "B": "Satisfies only first", "C": "Satisfies only second", "D": "Satisfies neither"}, "correct_answer": "D"},
    {"id": 94, "difficulty": "hard", "question": "If 1/a, 1/b, 1/c are in AP, prove that (b+c)/a, (c+a)/b, (a+b)/c are in AP. For a=1,b=2,c=3, verify.", "options": {"A": "5,2,4/3 in AP", "B": "5,2,5/3 in AP", "C": "5,2,2 in AP", "D": "5,2,1 in AP"}, "correct_answer": "B"},
    {"id": 95, "difficulty": "hard", "question": "If a,b,c are in AP, prove that a(1/b+1/c), b(1/c+1/a), c(1/a+1/b) are in AP. For a=1,b=2,c=3, check.", "options": {"A": "4/3,5/3,2 in AP", "B": "4/3,5/3,8/3 in AP", "C": "4/3,2,8/3 in AP", "D": "5/3,2,8/3 in AP"}, "correct_answer": "B"},
    {"id": 96, "difficulty": "hard", "question": "If x,y,z are in AP, prove that (x+2y-z)(2y+z-x)(z+x-y)=4xyz. For x=1,y=2,z=3, verify.", "options": {"A": "LHS=24,RHS=24", "B": "LHS=20,RHS=24", "C": "LHS=24,RHS=20", "D": "LHS=20,RHS=20"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "If a,b,c are in AP, prove that (a-c)²/(b²-ac)=4. For a=2,b=4,c=6, verify.", "options": {"A": "16/4=4", "B": "16/8=2", "C": "16/12=4/3", "D": "16/16=1"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "If p,q,r are in AP, prove that p²(q+r), q²(r+p), r²(p+q) are in AP. For p=1,q=2,r=3, verify.", "options": {"A": "5,16,33 in AP", "B": "5,14,33 in AP", "C": "5,16,35 in AP", "D": "5,14,35 in AP"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "If a,b,c are in AP, prove that 1/√b+√c, 1/√c+√a, 1/√a+√b are in AP. For a=1,b=4,c=7, check.", "options": {"A": "1/3,1/4,1/5 in AP", "B": "1/3,1/4,1/5 not in AP", "C": "1/2,1/3,1/4 in AP", "D": "1/2,1/3,1/4 not in AP"}, "correct_answer": "B"},
    {"id": 100, "difficulty": "hard", "question": "If S_n = an²+bn, prove that AP exists. Find a and d in terms of a and b.", "options": {"A": "a=a+b, d=2a", "B": "a=b, d=2a", "C": "a=a, d=2a", "D": "a=b-a, d=2a"}, "correct_answer": "A"}
  ]
}