{
  "title": "Midpoint Formula (Coordinate Geometry) - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "What is the midpoint of the line segment joining (2,3) and (4,7)?", "options": {"A": "(3,5)", "B": "(5,3)", "C": "(6,10)", "D": "(1,2)"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "The midpoint of the line segment joining (0,0) and (8,6) is:", "options": {"A": "(4,3)", "B": "(3,4)", "C": "(8,6)", "D": "(6,8)"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "If the midpoint of (2,3) and (x,5) is (4,4), then x = ?", "options": {"A": "6", "B": "5", "C": "4", "D": "3"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "The midpoint of (-1,2) and (3,-4) is:", "options": {"A": "(1,-1)", "B": "(-1,1)", "C": "(2,-2)", "D": "(-2,2)"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "If M(2,3) is the midpoint of A(1,2) and B(x,y), then B is:", "options": {"A": "(3,4)", "B": "(4,3)", "C": "(2,3)", "D": "(1,1)"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "The midpoint of (a,b) and (-a,-b) is:", "options": {"A": "(0,0)", "B": "(a,b)", "C": "(-a,-b)", "D": "(2a,2b)"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "If P(2,3) and Q(4,k) have midpoint (3,5), then k = ?", "options": {"A": "7", "B": "6", "C": "5", "D": "4"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "The midpoint of (5,-2) and (-3,8) is:", "options": {"A": "(1,3)", "B": "(3,1)", "C": "(2,6)", "D": "(6,2)"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "If A(1,2), B(3,4) and C(x,y) are collinear with B as midpoint of AC, then C is:", "options": {"A": "(5,6)", "B": "(6,5)", "C": "(4,5)", "D": "(5,4)"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "The midpoint formula for points (x₁,y₁) and (x₂,y₂) is:", "options": {"A": "((x₁+x₂)/2, (y₁+y₂)/2)", "B": "((x₁-x₂)/2, (y₁-y₂)/2)", "C": "(x₁+x₂, y₁+y₂)", "D": "((x₁+x₂)/3, (y₁+y₂)/3)"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "The midpoint of (0,4) and (6,0) is:", "options": {"A": "(3,2)", "B": "(2,3)", "C": "(4,6)", "D": "(6,4)"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "If midpoint of (k,3) and (5,k) is (4,4), then k = ?", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "The midpoint of (-2,-3) and (4,5) is:", "options": {"A": "(1,1)", "B": "(-1,-1)", "C": "(2,2)", "D": "(-2,-2)"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "If A(2,5) and B(4,9), then midpoint M divides AB in ratio:", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "The midpoint of (3.5, 2.5) and (4.5, 3.5) is:", "options": {"A": "(4,3)", "B": "(3,4)", "C": "(5,2)", "D": "(2,5)"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "If M(0,0) is midpoint of A(-2,3) and B(a,b), then B is:", "options": {"A": "(2,-3)", "B": "(-2,3)", "C": "(2,3)", "D": "(-2,-3)"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "The midpoint of (100,200) and (300,400) is:", "options": {"A": "(200,300)", "B": "(300,200)", "C": "(400,600)", "D": "(600,400)"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "If A(1,1), B(2,2) and C(3,3) are three points, the midpoint of AC is:", "options": {"A": "(2,2)", "B": "(1,1)", "C": "(3,3)", "D": "(4,4)"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "The midpoint of (x₁,y₁) and (x₂,y₂) divides the segment in ratio:", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "m:n"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "If P(3,4) and Q(7,10), then coordinates of midpoint are:", "options": {"A": "(5,7)", "B": "(7,5)", "C": "(10,14)", "D": "(14,10)"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Find the midpoint of the line segment joining the points (3,-4) and (-5,8).", "options": {"A": "(-1,2)", "B": "(1,-2)", "C": "(-2,1)", "D": "(2,-1)"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "If (4,5) is the midpoint of (2,3) and (6,k), find k.", "options": {"A": "7", "B": "8", "C": "9", "D": "10"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "The midpoint of segment AB is (3,4). If A is (1,2), find B.", "options": {"A": "(5,6)", "B": "(6,5)", "C": "(4,5)", "D": "(5,4)"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Find the value of k if midpoint of (k,2k) and (3k,4k) is (10,15).", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "If A(2,3), B(4,k) and C(6,7) are collinear and B is midpoint of AC, find k.", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "The midpoint of (a+b, a-b) and (a-b, a+b) is:", "options": {"A": "(a,b)", "B": "(b,a)", "C": "(2a,2b)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Find the midpoint of the points (2/3, 3/4) and (4/3, 5/4).", "options": {"A": "(1,1)", "B": "(2,2)", "C": "(1/2,1/2)", "D": "(3/2,3/2)"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "If (p,q) is midpoint of (1,2) and (5,6), then (p,q) = ?", "options": {"A": "(3,4)", "B": "(4,3)", "C": "(2,3)", "D": "(3,2)"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "The midpoint of the line joining (sinθ, cosθ) and (cosθ, sinθ) is:", "options": {"A": "((sinθ+cosθ)/2, (sinθ+cosθ)/2)", "B": "(sinθ+cosθ, sinθ+cosθ)", "C": "(1,1)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Find the coordinates of point B if A(-2,3) and midpoint M(1,4).", "options": {"A": "(4,5)", "B": "(5,4)", "C": "(3,6)", "D": "(6,3)"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "The midpoint of vertices of triangle A(1,2), B(3,4), C(5,6) with midpoints of sides are:", "options": {"A": "(2,3), (3,4), (4,5)", "B": "(2,3), (4,5), (3,5)", "C": "(3,4), (4,5), (5,6)", "D": "(1,2), (3,4), (5,6)"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If M(2,3) divides AB in ratio 1:1 where A(0,1), then B = ?", "options": {"A": "(4,5)", "B": "(5,4)", "C": "(3,4)", "D": "(4,3)"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "The midpoint of (√2, √3) and (√8, √12) is:", "options": {"A": "((√2+2√2)/2, (√3+2√3)/2)", "B": "(√2, √3)", "C": "(2√2, 2√3)", "D": "(√10, √15)"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Find k if midpoint of (k+1, k+2) and (k+3, k+4) is (5,6).", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "The midpoint of line joining origin to point (a,b) is:", "options": {"A": "(a/2, b/2)", "B": "(a,b)", "C": "(2a,2b)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If A(1,3), B(2,5) and C(3,7) are collinear, midpoint of AC is:", "options": {"A": "(2,5)", "B": "(1,3)", "C": "(3,7)", "D": "(4,9)"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "The midpoint of (0.25, 0.75) and (0.75, 0.25) is:", "options": {"A": "(0.5,0.5)", "B": "(0.25,0.25)", "C": "(0.75,0.75)", "D": "(1,1)"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Find the midpoint of points where x-axis and y-axis intersect the line 2x+3y=6.", "options": {"A": "(1.5,1)", "B": "(1,1.5)", "C": "(2,3)", "D": "(3,2)"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "If midpoint of (2a,4) and (-2,2b) is (1,3), then (a,b) = ?", "options": {"A": "(2,1)", "B": "(1,2)", "C": "(3,4)", "D": "(4,3)"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "The midpoint of line segment joining centers of circles x²+y²=4 and x²+y²=16 is:", "options": {"A": "(0,0)", "B": "(2,2)", "C": "(4,4)", "D": "(8,8)"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "Find the midpoint of the line segment joining the points (a cosθ, b sinθ) and (a cosφ, b sinφ).", "options": {"A": "(a(cosθ+cosφ)/2, b(sinθ+sinφ)/2)", "B": "(a cosθ cosφ, b sinθ sinφ)", "C": "(a(cosθ-cosφ)/2, b(sinθ-sinφ)/2)", "D": "(a, b)"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "If (3,4) is midpoint of (5,7) and (a,b), and (5,6) is midpoint of (3,4) and (c,d), find (a,b,c,d).", "options": {"A": "(1,1,7,8)", "B": "(1,1,8,7)", "C": "(2,2,7,8)", "D": "(2,2,8,7)"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "The midpoint of the line joining the points (at², 2at) and (a/t², -2a/t) is:", "options": {"A": "(a(t²+1/t²)/2, a(t-1/t))", "B": "(a(t²+1/t²), a(t-1/t))", "C": "(a(t²-1/t²)/2, a(t+1/t))", "D": "(a, a)"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "Find the coordinates of the point which divides the line joining (1,-2) and (4,7) externally in ratio 2:1 and verify it's same as midpoint formula for special case.", "options": {"A": "(7,16)", "B": "Midpoint is special case of section formula when m:n=1:1", "C": "(5,12)", "D": "Both A and B"}, "correct_answer": "D"},
    {"id": 45, "difficulty": "hard", "question": "The midpoints of sides of triangle are (2,3), (4,5), (6,7). Find vertices.", "options": {"A": "(0,1), (4,9), (8,5)", "B": "(1,1), (5,5), (9,9)", "C": "(2,2), (6,6), (10,10)", "D": "(3,3), (7,7), (11,11)"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "If (a,b) is midpoint of line segment joining (x₁,y₁) and (x₂,y₂), prove that x₁+x₂=2a and y₁+y₂=2b.", "options": {"A": "This is definition of midpoint", "B": "Follows from section formula", "C": "Both A and B", "D": "None of these"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "hard", "question": "The line joining (1,2) and (3,4) is divided by midpoint M. If M divides line in ratio:", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "Find midpoint of line joining (a secθ, b tanθ) and (a secφ, b tanφ).", "options": {"A": "(a(secθ+secφ)/2, b(tanθ+tanφ)/2)", "B": "(a secθ secφ, b tanθ tanφ)", "C": "(a(secθ-secφ)/2, b(tanθ-tanφ)/2)", "D": "(a, b)"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "If vertices of parallelogram are (1,2), (3,4), (5,6), (7,8), midpoints of diagonals are:", "options": {"A": "(4,5) and (4,5)", "B": "(3,4) and (5,6)", "C": "(2,3) and (6,7)", "D": "(1,2) and (7,8)"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "The midpoint of focal chord of parabola y²=4ax joining (at₁², 2at₁) and (at₂², 2at₂) with t₁t₂=-1 is:", "options": {"A": "(a(t₁²+t₂²)/2, a(t₁+t₂))", "B": "(a, 2a)", "C": "(2a, 4a)", "D": "(a(t₁+t₂), 2a)"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Find midpoint of (3,-5) and (-7,9).", "options": {"A": "(-2,2)", "B": "(2,-2)", "C": "(-5,7)", "D": "(5,-7)"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "If (3,k) is midpoint of (1,4) and (5,6), find k.", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "The midpoint of segment PQ is (4,5). If P is (2,3), find Q.", "options": {"A": "(6,7)", "B": "(7,6)", "C": "(5,6)", "D": "(6,5)"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "Find value of k if midpoint of (2k,3k) and (4k,5k) is (9,12).", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "If A(1,2), B(3,k) and C(5,8) are collinear with B as midpoint, find k.", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "The midpoint of (x+y, x-y) and (x-y, x+y) is:", "options": {"A": "(x,y)", "B": "(y,x)", "C": "(2x,2y)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "Find midpoint of (3/4, 5/6) and (5/4, 7/6).", "options": {"A": "(1,1)", "B": "(2,2)", "C": "(1/2,1/2)", "D": "(3/2,3/2)"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "If (p,q) is midpoint of (2,4) and (6,8), then (p,q)=?", "options": {"A": "(4,6)", "B": "(6,4)", "C": "(3,5)", "D": "(5,3)"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "The midpoint of line joining (cosθ, sinθ) and (sinθ, cosθ) is:", "options": {"A": "((cosθ+sinθ)/2, (sinθ+cosθ)/2)", "B": "(cosθ+sinθ, sinθ+cosθ)", "C": "(1,1)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "Find coordinates of B if A(3,5) and midpoint M(6,8).", "options": {"A": "(9,11)", "B": "(11,9)", "C": "(8,10)", "D": "(10,8)"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "hard", "question": "Find midpoint of line joining (a coshθ, b sinhθ) and (a coshφ, b sinhφ).", "options": {"A": "(a(coshθ+coshφ)/2, b(sinhθ+sinhφ)/2)", "B": "(a coshθ coshφ, b sinhθ sinhφ)", "C": "(a(coshθ-coshφ)/2, b(sinhθ-sinhφ)/2)", "D": "(a, b)"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "hard", "question": "If (4,5) is midpoint of (6,7) and (a,b), and (6,7) is midpoint of (4,5) and (c,d), find (a,b,c,d).", "options": {"A": "(2,3,8,9)", "B": "(2,3,9,8)", "C": "(3,4,8,9)", "D": "(3,4,9,8)"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "hard", "question": "The midpoint of line joining (am², 2am) and (a/n², -2a/n) is:", "options": {"A": "(a(m²+1/n²)/2, a(m-1/n))", "B": "(a(m²+1/n²), a(m-1/n))", "C": "(a(m²-1/n²)/2, a(m+1/n))", "D": "(a, a)"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "hard", "question": "Find coordinates of point dividing (2,3) and (5,8) externally in ratio 3:2 and compare with midpoint.", "options": {"A": "(11,18)", "B": "Midpoint is when m=n", "C": "(9,14)", "D": "Both A and B"}, "correct_answer": "B"},
    {"id": 65, "difficulty": "hard", "question": "The midpoints of sides of triangle are (1,2), (3,4), (5,6). Find vertices.", "options": {"A": "(-1,0), (3,8), (7,4)", "B": "(0,0), (4,4), (8,8)", "C": "(1,1), (5,5), (9,9)", "D": "(2,2), (6,6), (10,10)"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "hard", "question": "If (c,d) is midpoint of (x₁,y₁) and (x₂,y₂), then x₁+x₂=? and y₁+y₂=?", "options": {"A": "2c, 2d", "B": "c/2, d/2", "C": "c, d", "D": "2c, d"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "hard", "question": "The line joining (2,3) and (4,5) has midpoint M dividing in ratio:", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "hard", "question": "Find midpoint of line joining (a cosecθ, b cotθ) and (a cosecφ, b cotφ).", "options": {"A": "(a(cosecθ+cosecφ)/2, b(cotθ+cotφ)/2)", "B": "(a cosecθ cosecφ, b cotθ cotφ)", "C": "(a(cosecθ-cosecφ)/2, b(cotθ-cotφ)/2)", "D": "(a, b)"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "hard", "question": "If vertices of quadrilateral are (2,3), (4,5), (6,7), (8,9), midpoints of diagonals are:", "options": {"A": "(5,6) and (5,6)", "B": "(4,5) and (6,7)", "C": "(3,4) and (7,8)", "D": "(2,3) and (8,9)"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "hard", "question": "The midpoint of chord of ellipse x²/a²+y²/b²=1 joining (a cosθ, b sinθ) and (a cosφ, b sinφ) is:", "options": {"A": "(a(cosθ+cosφ)/2, b(sinθ+sinφ)/2)", "B": "(a cosθ cosφ, b sinθ sinφ)", "C": "(a, b)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "medium", "question": "Find midpoint of (5,-8) and (-3,12).", "options": {"A": "(1,2)", "B": "(2,1)", "C": "(4,10)", "D": "(10,4)"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "If (4,k) is midpoint of (2,5) and (6,7), find k.", "options": {"A": "6", "B": "7", "C": "8", "D": "9"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "The midpoint of segment RS is (5,6). If R is (3,4), find S.", "options": {"A": "(7,8)", "B": "(8,7)", "C": "(6,7)", "D": "(7,6)"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "Find value of k if midpoint of (3k,4k) and (5k,6k) is (12,15).", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "If A(2,4), B(4,k) and C(6,10) are collinear with B as midpoint, find k.", "options": {"A": "7", "B": "8", "C": "9", "D": "10"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "The midpoint of (p+q, p-q) and (p-q, p+q) is:", "options": {"A": "(p,q)", "B": "(q,p)", "C": "(2p,2q)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Find midpoint of (5/6, 7/8) and (7/6, 9/8).", "options": {"A": "(1,1)", "B": "(2,2)", "C": "(1/2,1/2)", "D": "(3/2,3/2)"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "If (p,q) is midpoint of (3,6) and (7,10), then (p,q)=?", "options": {"A": "(5,8)", "B": "(8,5)", "C": "(4,7)", "D": "(7,4)"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "The midpoint of line joining (tanθ, cotθ) and (cotθ, tanθ) is:", "options": {"A": "((tanθ+cotθ)/2, (cotθ+tanθ)/2)", "B": "(tanθ+cotθ, cotθ+tanθ)", "C": "(1,1)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "Find coordinates of B if A(4,6) and midpoint M(7,9).", "options": {"A": "(10,12)", "B": "(12,10)", "C": "(9,11)", "D": "(11,9)"}, "correct_answer": "A"},
    {"id": 81, "difficulty": "hard", "question": "Find midpoint of line joining (a e^θ, b e^{-θ}) and (a e^φ, b e^{-φ}).", "options": {"A": "(a(e^θ+e^φ)/2, b(e^{-θ}+e^{-φ})/2)", "B": "(a e^{θ+φ}, b e^{-(θ+φ)})", "C": "(a(e^θ-e^φ)/2, b(e^{-θ}-e^{-φ})/2)", "D": "(a, b)"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "If (5,6) is midpoint of (7,8) and (a,b), and (7,8) is midpoint of (5,6) and (c,d), find (a,b,c,d).", "options": {"A": "(3,4,9,10)", "B": "(3,4,10,9)", "C": "(4,5,9,10)", "D": "(4,5,10,9)"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "The midpoint of line joining (ap², 2ap) and (aq², 2aq) is:", "options": {"A": "(a(p²+q²)/2, a(p+q))", "B": "(a(p²+q²), a(p+q))", "C": "(a(p²-q²)/2, a(p-q))", "D": "(a, a)"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Find coordinates of point dividing (3,5) and (7,11) externally in ratio 4:3 and compare with midpoint.", "options": {"A": "(19,29)", "B": "Midpoint is when m=n=1", "C": "(16,24)", "D": "Both A and B"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "The midpoints of sides of triangle are (2,4), (4,6), (6,8). Find vertices.", "options": {"A": "(0,2), (4,10), (8,6)", "B": "(1,3), (5,7), (9,11)", "C": "(2,4), (6,8), (10,12)", "D": "(3,5), (7,9), (11,13)"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "If (e,f) is midpoint of (x₁,y₁) and (x₂,y₂), then x₁+x₂=? and y₁+y₂=?", "options": {"A": "2e, 2f", "B": "e/2, f/2", "C": "e, f", "D": "2e, f"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "The line joining (3,4) and (5,6) has midpoint M dividing in ratio:", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Find midpoint of line joining (a secθ, b tanθ) and (a secφ, b tanφ).", "options": {"A": "(a(secθ+secφ)/2, b(tanθ+tanφ)/2)", "B": "(a secθ secφ, b tanθ tanφ)", "C": "(a(secθ-secφ)/2, b(tanθ-tanφ)/2)", "D": "(a, b)"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "If vertices of parallelogram are (3,4), (5,6), (7,8), (9,10), midpoints of diagonals are:", "options": {"A": "(6,7) and (6,7)", "B": "(5,6) and (7,8)", "C": "(4,5) and (8,9)", "D": "(3,4) and (9,10)"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "The midpoint of chord of hyperbola x²/a²-y²/b²=1 joining (a secθ, b tanθ) and (a secφ, b tanφ) is:", "options": {"A": "(a(secθ+secφ)/2, b(tanθ+tanφ)/2)", "B": "(a secθ secφ, b tanθ tanφ)", "C": "(a, b)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "medium", "question": "Find midpoint of (7,-10) and (-5,14).", "options": {"A": "(1,2)", "B": "(2,1)", "C": "(6,12)", "D": "(12,6)"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "medium", "question": "If (5,k) is midpoint of (3,6) and (7,8), find k.", "options": {"A": "7", "B": "8", "C": "9", "D": "10"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "medium", "question": "The midpoint of segment TU is (6,7). If T is (4,5), find U.", "options": {"A": "(8,9)", "B": "(9,8)", "C": "(7,8)", "D": "(8,7)"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "medium", "question": "Find value of k if midpoint of (4k,5k) and (6k,7k) is (15,18).", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "medium", "question": "If A(3,6), B(5,k) and C(7,12) are collinear with B as midpoint, find k.", "options": {"A": "9", "B": "10", "C": "11", "D": "12"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "medium", "question": "The midpoint of (m+n, m-n) and (m-n, m+n) is:", "options": {"A": "(m,n)", "B": "(n,m)", "C": "(2m,2n)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "medium", "question": "Find midpoint of (7/8, 9/10) and (9/8, 11/10).", "options": {"A": "(1,1)", "B": "(2,2)", "C": "(1/2,1/2)", "D": "(3/2,3/2)"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "medium", "question": "If (p,q) is midpoint of (4,8) and (8,12), then (p,q)=?", "options": {"A": "(6,10)", "B": "(10,6)", "C": "(5,9)", "D": "(9,5)"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "medium", "question": "The midpoint of line joining (sinθ, cosθ) and (cosθ, sinθ) lies on line:", "options": {"A": "y=x", "B": "y=-x", "C": "x+y=1", "D": "x-y=1"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "medium", "question": "Find coordinates of B if A(5,7) and midpoint M(8,10).", "options": {"A": "(11,13)", "B": "(13,11)", "C": "(10,12)", "D": "(12,10)"}, "correct_answer": "A"}
  ]
}