📘 Arithmetic – 100 Important Formula, Tricks & One-Liners (Complete)

🔹 Simple Interest (10)

66. SI = (P × R × T)/100
67. Amount = P + SI
68. जब Principal (P) = 1 हो, तो SI = (R × T)/100
69. यदि दो वर्षों का अंतर दिया है तो SI = Difference × 100 / (Rate × Time)
70. समान Principal पर अलग-अलग Rate और Time पर SI का ratio = R₁T₁ : R₂T₂
71. Principal = (SI × 100)/(Rate × Time)
72. Rate = (SI × 100)/(P × Time)
73. Time = (SI × 100)/(P × Rate)
74. अगर समय (T) और ब्याज (SI) बराबर हो तो Principal = √(Amount × 100)
75. जब Rate या Time double हो, SI भी double होगा।

🔹 Compound Interest (10)

76. CI = Amount – Principal
77. Amount = P(1 + R/100)ᵗ
78. जब साल में दो बार ब्याज लगे → A = P(1 + R/200)²ᵗ
79. जब साल में चार बार ब्याज लगे → A = P(1 + R/400)⁴ᵗ
80. Difference between CI and SI (2 years) = P × (R/100)²
81. Difference between CI and SI (3 years) = P(R/100)³ + 3P(R/100)²(R/100)
82. यदि rate = r% और समय = n years तो Amount = P(1 + r/100)ⁿ
83. Effective rate of two successive years: R₁ + R₂ + (R₁R₂)/100
84. जब समय (T) = 0 हो → CI = 0
85. जब rate या time बढ़े → CI exponential तरीके से बढ़ता है।

🔹 Time, Work & Pipes (10)

86. Work = Rate × Time
87. Time = Work/Rate
88. अगर A अकेले x दिन में और B अकेले y दिन में काम करता है, तो दोनों मिलकर काम करेंगे = (xy)/(x+y) दिन
89. A का 1 दिन का काम = 1/x, B का 1 दिन का काम = 1/y
90. जब A काम करता है 1 दिन और B अगले दिन, तो 2 दिन का काम = (1/x + 1/y)
91. Efficiency ∝ 1/Time
92. यदि efficiency ratio = a:b हो तो time ratio = b:a
93. जब 3 workers A, B, C मिलकर काम करते हैं: 1 दिन का काम = 1/x + 1/y + 1/z
94. Pipes and Cistern: एक pipe 1 घंटे में tank भरता है और दूसरा 1 घंटे में खाली करता है → Net work = (1/x – 1/y)
95. यदि inlet और outlet pipe की दर बराबर हो → tank कभी नहीं भरेगा।

🔹 Time, Speed & Distance (10)

96. Speed = Distance/Time
97. Distance = Speed × Time
98. Time = Distance/Speed
99. Average Speed = (2xy)/(x+y) (जब distance same हो)
100. Relative Speed (same direction) = (x – y), (opposite direction) = (x + y)

✅ Summary – Arithmetic (100)

• Number System → 20
• Ratio & Average → 15
• Percentage → 15
• Profit & Loss → 15
• Simple Interest → 10
• Compound Interest → 10
• Time & Work → 10
• Speed & Distance → 5

📘 Algebra – 60 Important Formula, Tricks & One-Liners

🔹 Basic Identities (15)

1. (a + b)² = a² + 2ab + b²
2. (a – b)² = a² – 2ab + b²
3. (a + b)(a – b) = a² – b²
4. (x + a)(x + b) = x² + (a + b)x + ab
5. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
6. (a – b – c)² = a² + b² + c² – 2ab – 2ac + 2bc
7. (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)
8. (a + b)³ = a³ + b³ + 3ab(a + b)
9. (a – b)³ = a³ – b³ – 3ab(a – b)
10. a³ + b³ = (a + b)(a² – ab + b²)
11. a³ – b³ = (a – b)(a² + ab + b²)
12. x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)
13. (x + y)² + (x – y)² = 2(x² + y²)
14. (x + y)² – (x – y)² = 4xy
15. (a + b + c)³ = a³ + b³ + c³ + 3(a+b)(b+c)(c+a)

🔹 Factorization (10)

16. x² – (a + b)x + ab = (x – a)(x – b)
17. x² + (a + b)x + ab = (x + a)(x + b)
18. ax² + bx + c = 0 → Roots = [–b ± √(b² – 4ac)] / 2a
19. If α, β are roots of ax² + bx + c = 0 → α + β = –b/a, αβ = c/a
20. (x² – y²) = (x – y)(x + y)
21. (x³ – y³) = (x – y)(x² + xy + y²)
22. (x³ + y³) = (x + y)(x² – xy + y²)
23. (x⁴ – y⁴) = (x² – y²)(x² + y²)
24. (x⁴ + y⁴) = (x² + √2xy + y²)(x² – √2xy + y²)
25. Quadratic equation का discriminant (D) = b² – 4ac

🔹 Surds & Indices (10)

26. aᵐ × aⁿ = aᵐ⁺ⁿ
27. aᵐ / aⁿ = aᵐ⁻ⁿ
28. (aᵐ)ⁿ = aᵐⁿ
29. a⁰ = 1 (जब a ≠ 0)
30. a⁻ⁿ = 1/aⁿ
31. (ab)ⁿ = aⁿbⁿ
32. (a/b)ⁿ = aⁿ/bⁿ
33. √a × √b = √(ab)
34. (√a/√b) = √(a/b)
35. यदि aᵐ = aⁿ ⇒ m = n (जब a ≠ 0, a ≠ 1)

🔹 Logarithm (10)

36. logₐ1 = 0
37. logₐa = 1
38. logₐ(mn) = logₐm + logₐn
39. logₐ(m/n) = logₐm – logₐn
40. logₐ(mⁿ) = n logₐm
41. logₐ(1/m) = –logₐm
42. logₐm = logbm / logba (Base change formula)
43. यदि logₐm = logₐn ⇒ m = n
44. log₁₀2 = 0.3010, log₁₀3 = 0.4771 (Important values)
45. Natural log base e ≈ 2.718

🔹 Sequences & Series (10)

46. Arithmetic Progression (AP): a, a+d, a+2d …
47. nᵗʰ term of AP = a + (n–1)d
48. Sum of n terms of AP = n/2[2a + (n–1)d]
49. Sum of n terms of AP = n/2(first term + last term)
50. Geometric Progression (GP): a, ar, ar² …
51. nᵗʰ term of GP = arⁿ⁻¹
52. Sum of n terms of GP = a(rⁿ–1)/(r–1), r ≠ 1
53. Sum of infinite GP = a/(1–r), |r| < 1
54. Harmonic Progression (HP): Reciprocal of AP terms
55. AM ≥ GM ≥ HM (Arithmetic ≥ Geometric ≥ Harmonic mean)

🔹 Miscellaneous (5)

56. If x + 1/x = a ⇒ x² + 1/x² = a² – 2
57. If x + 1/x = a ⇒ x³ + 1/x³ = a³ – 3a
58. If x + 1/x = a ⇒ x⁴ + 1/x⁴ = a⁴ – 4a² + 2
59. (a+b+c)² – (a–b–c)² = 4(bc + ca)
60. अगर roots α, β real हों तो discriminant ≥ 0

📘 Geometry & Mensuration – 80 Formula, Tricks & One-Liners

🔹 2D Geometry (Plane Geometry) – 40

📍 Triangle (15)

1. किसी त्रिभुज के तीन कोणों का योग = 180°
2. Exterior angle = Sum of opposite two interior angles
3. Area of triangle = ½ × base × height
4. Area (Heron’s formula) = √[s(s–a)(s–b)(s–c)], जहाँ s = (a+b+c)/2
5. Equilateral triangle area = (√3/4)a²
6. Equilateral triangle height = (√3/2)a
7. Right triangle: Pythagoras theorem → Hypotenuse² = Base² + Height²
8. Isosceles right triangle area = a²/2
9. Inradius (r) = Area / s
10. Circumradius (R) = (abc)/(4 × Area)
11. Relation: Area = (abc)/(4R)
12. Area = ½ab sinC
13. Centroid divides median in ratio = 2:1
14. Orthocenter = Intersection of altitudes
15. In an equilateral triangle → Centroid = Incenter = Circumcenter = Orthocenter

📍 Quadrilaterals (10)

16. Rectangle area = l × b
17. Square area = a², Perimeter = 4a
18. Square diagonal = a√2
19. Rhombus area = ½ × d₁ × d₂
20. Parallelogram area = base × height
21. Trapezium area = ½ × (sum of parallel sides) × height
22. Kite area = ½ × d₁ × d₂
23. Angle sum of quadrilateral = 360°
24. Diagonal of rectangle = √(l² + b²)
25. Diagonal of square = a√2

📍 Circle (10)

26. Circumference = 2πr
27. Area = πr²
28. Diameter = 2r
29. Length of arc = (θ/360°) × 2πr
30. Area of sector = (θ/360°) × πr²
31. Area of segment = Area of sector – Area of triangle
32. Equation of circle (coordinate): (x–h)² + (y–k)² = r²
33. Angle in a semicircle = 90°
34. Tangent ⟂ radius at point of contact
35. Two tangents from external point are equal in length

📍 Polygon (5)

36. Sum of interior angles = (n–2)×180°
37. Each interior angle (regular polygon) = [(n–2)×180]/n
38. Each exterior angle = 360°/n
39. Number of diagonals = n(n–3)/2
40. Maximum number of triangles in n-gon = n–2

🔹 3D Mensuration (Solid Geometry) – 40

📍 Cuboid & Cube (10)

41. Cuboid volume = l × b × h
42. Cuboid TSA = 2(lb + bh + hl)
43. Cuboid LSA = 2h(l + b)
44. Cube volume = a³
45. Cube TSA = 6a²
46. Cube LSA = 4a²
47. Cube diagonal = a√3
48. Cuboid diagonal = √(l² + b² + h²)
49. If edge of cube is doubled → volume ×8
50. If edge of cube is halved → volume ×1/8

📍 Cylinder (7)

51. Volume = πr²h
52. TSA = 2πr(h + r)
53. CSA = 2πrh
54. Curved surface area ratio = r:h
55. If height doubled → volume doubled
56. If radius doubled → volume ×4
57. Diagonal of cylinder = √(4r² + h²)

📍 Cone (7)

58. Volume = (1/3)πr²h
59. CSA = πrl (l = slant height)
60. TSA = πr(l + r)
61. l = √(r² + h²)
62. Ratio of volumes (cone:cylinder) = 1:3 (same base & height)
63. If height = radius → l = r√2
64. Area of base circle = πr²

📍 Sphere & Hemisphere (8)

65. Sphere volume = (4/3)πr³
66. Sphere surface area = 4πr²
67. Hemisphere volume = (2/3)πr³
68. Hemisphere CSA = 2πr²
69. Hemisphere TSA = 3πr²
70. Great circle area = πr²
71. Diameter = 2r
72. Volume ratio (sphere:hemisphere) = 2:1

📍 Pyramid & Prism (8)

73. Volume of prism = Base area × height
74. Volume of pyramid = (1/3) × Base area × height
75. TSA of prism = LSA + 2 × Base area
76. TSA of pyramid = LSA + Base area
77. Regular triangular pyramid (tetrahedron) volume = a³/(6√2)
78. Prism LSA = Perimeter of base × height
79. Right pyramid slant height = √(h² + (a/2)²)
80. For same base & height → Volume ratio prism:pyramid = 3:1

📘 ARITHMETIC – 100 Important Formulas & Tricks with Examples

1️⃣ Squares & Square Roots

1. (a+b)2=a2+b2+2ab(a+b)^2 = a^2 + b^2 + 2ab(a+b)2=a2+b2+2ab
   👉 Example: (12+3)2=122+32+2(12)(3)=144+9+72=225(12+3)^2 = 12^2 + 3^2 + 2(12)(3) = 144 + 9 + 72 = 225(12+3)2=122+32+2(12)(3)=144+9+72=225
2. (a−b)2=a2+b2−2ab(a-b)^2 = a^2 + b^2 – 2ab(a−b)2=a2+b2−2ab
   👉 Example: (15−5)2=225+25−150=100(15-5)^2 = 225 + 25 – 150 = 100(15−5)2=225+25−150=100
3. (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 – b^2(a+b)(a−b)=a2−b2
   👉 Example: (21+9)(21−9)=212−92=441−81=360(21+9)(21-9) = 21^2 – 9^2 = 441 – 81 = 360(21+9)(21−9)=212−92=441−81=360
4. Square of number ending with 5 → n52=n(n+1) 25n5^2 = n(n+1) \, 25n52=n(n+1)25
   👉 Example: 752=7×8 25=562575^2 = 7 \times 8 \, 25 = 5625752=7×825=5625
5. Square root shortcut:
   If a number ends with 6 → Square root will end with 4 or 6.
   👉 Example: √576 → 24.

2️⃣ Cubes & Cube Roots

6. (a+b)3=a3+b3+3ab(a+b)(a+b)^3 = a^3 + b^3 + 3ab(a+b)(a+b)3=a3+b3+3ab(a+b)
   👉 Example: (4+1)3=64+1+3(4)(1)(5)=125(4+1)^3 = 64 + 1 + 3(4)(1)(5) = 125(4+1)3=64+1+3(4)(1)(5)=125
7. (a−b)3=a3−b3−3ab(a−b)(a-b)^3 = a^3 – b^3 – 3ab(a-b)(a−b)3=a3−b3−3ab(a−b)
8. Cube of number ending with 5: n53=n3 125n5^3 = n^3 \, 125n53=n3125
   👉 Example: 353=33 125=27,12535^3 = 3^3 \, 125 = 27,125353=33125=27,125
9. Cube roots trick:
   If last digit is 7 → cube root ends with 3.
   👉 Example: 39304=34339304 = 34^339304=343 (last digit 4 → root ends with 4).
10. (a+b)(a2−ab+b2)=a3+b3(a+b)(a^2-ab+b^2) = a^3+b^3(a+b)(a2−ab+b2)=a3+b3

3️⃣ Percentages

11. % to fraction:

• 50% = 1/2, 25% = 1/4, 20% = 1/5, 12.5% = 1/8.

12. If price increases by x% and consumption decreases by x%, net effect = x2100\frac{x^2}{100}100×2​% decrease.
    👉 Example: +20% and –20% → decrease = 4%.
13. Successive % change formula:
    Net % = A + B + (AB/100).
    👉 Example: 20% profit, 10% profit → Net = 20+10+(20×10/100)=32%.
14. % increase from A to B = B−AA×100\frac{B-A}{A} \times 100AB−A​×100.
15. % decrease from A to B = A−BA×100\frac{A-B}{A} \times 100AA−B​×100.

4️⃣ Ratio & Proportion

16. If a:b = m:n, then (ka):(kb) = m:n (multiplying ratio).
17. If a:b = m:n and b:c = p:q, then a:c = mp:nq.
18. If a/b = c/d, then a+b/b = c+d/d.
19. Mixture rule:

Q1Q2=(M−M2)(M1−M)\frac{Q_1}{Q_2} = \frac{(M-M_2)}{(M_1-M)}Q2​Q1​​=(M1​−M)(M−M2​)​

20. Compound ratio of (a:b) and (c:d) = (ac):(bd).

5️⃣ Averages

21. Average = (Sum of items)/(No. of items).
22. If a person scores 80 in 9 tests and 100 in 1 test, average = (720+100)/10=82.
23. New average after including a new number = (Old sum + New number)/(n+1).
24. If average of 10 is 30, total = 300.
25. If average of group increases by 2 when one student joins, student’s marks = old average + increase × n.

6️⃣ Profit & Loss

26. Profit % = (Profit/CP) × 100.
27. Loss % = (Loss/CP) × 100.
28. SP = CP × (100+Gain%)/100.
29. SP = CP × (100-Loss%)/100.
30. Two successive profits of x% and y% → Net = x+y+(xy/100).

7️⃣ Simple Interest

31. SI = (P × R × T)/100.
32. If SI on ₹1000 for 2 years at 5% = 100.
33. Rate = (100×SI)/(P×T).
34. Time = (100×SI)/(P×R).
35. P = (100×SI)/(R×T).

8️⃣ Compound Interest

36. CI = P[(1+R/100)^T – 1].
37. Difference between CI and SI (for 2 years) = P(R/100)^2.
38. For quarterly compounding → R = R/4, T = 4×years.
39. For half-yearly compounding → R = R/2, T = 2×years.
40. CI > SI always (for same P, R, T).

9️⃣ Time, Speed & Distance

41. Speed = Distance/Time.
42. Time = Distance/Speed.
43. Distance = Speed×Time.
44. If distance is constant → Speed ∝ 1/Time.
45. If time is constant → Distance ∝ Speed.
46. Relative speed (same direction) = (A–B).
47. Relative speed (opposite direction) = (A+B).
48. Average speed = (2xy)/(x+y).
49. If speed increases by x%, time decreases by x100+x\frac{x}{100+x}100+xx​.
50. If speed decreases by x%, time increases by x100−x\frac{x}{100-x}100−xx​.

📘 ARITHMETIC – Part 2 (51–100)

🔟 Time & Work

51. Work = Man × Days.
52. If A can do work in 10 days, then A’s 1 day work = 1/10.
53. If A = 1/10, B = 1/15 → Together = 1/10+1/15 = 1/6 → 6 days.
54. If A does work in x days, then A:B efficiency = y:x (where B does in y days).
55. If A is twice as efficient as B, then time taken = half.

1️⃣1️⃣ Pipes & Cisterns

56. If pipe fills in x hours → 1 hour work = 1/x.
57. If pipe empties in y hours → 1 hour work = –1/y.
58. If one fill & one empty together → 1/x – 1/y.
59. Two fill pipes → 1/x + 1/y.
60. If all 3 pipes together fill in T, then T = 1/(1/x+1/y–1/z).

1️⃣2️⃣ Problems on Ages

61. Present age = Future age – years added.
62. If father is twice son’s age → ratio method works.
63. If average of ages = 25, total = 25 × number.
64. Age difference always constant.
65. Shortcut: Use ratio for present → add/subtract for future/past.

1️⃣3️⃣ Partnership

66. Profit share = (Investment × Time)/Total.
67. A invests 20k for 6 months, B 10k for 12 months → A:B = 120k:120k = 1:1.
68. If one joins later, calculate time accordingly.
69. Sleeping partner → only money matters, not work.
70. Loss also divided in same ratio.

1️⃣4️⃣ Boats & Streams

71. Speed downstream = u+v (boat + stream).
72. Speed upstream = u–v.
73. Boat’s speed = (Down + Up)/2.
74. Stream speed = (Down – Up)/2.
75. Time = Distance/Effective speed.

1️⃣5️⃣ Mixture & Alligation

76. Rule of alligation:

Q1Q2=(M−M2)(M1−M)\frac{Q_1}{Q_2} = \frac{(M-M_2)}{(M_1-M)}Q2​Q1​​=(M1​−M)(M−M2​)​

77. Example: Price of ₹20 & ₹30 mixture to get mean ₹25 → Ratio = 5:5 = 1:1.
78. Milk:Water = 3:1 → Water% = 25%.
79. Replace method → use formula:
    New quantity = Initial × (1 – taken/total)^n.
80. Mixture always solved using ratio difference method.

1️⃣6️⃣ Simple Algebra in Arithmetic

81. a+b=10, ab=21 → quadratic x²–10x+21=0 → x=3,7.
82. If x+1/x=2 → x²+1/x²=2²–2=2.
83. (x+1/x)²= x²+1/x²+2.
84. If a:b=c:d, then ad=bc.
85. If a:b=2:3, b:c=4:5 → a:c=8:15.

1️⃣7️⃣ Important Shortcuts

86. When number increased by 20% then decreased by 20% → net loss = 4%.
87. When number increased by 10% twice → net = 21%.
88. Square of 99 → (100–1)²=10000–200+1=9801.
89. Square of 101 → (100+1)²=10000+200+1=10201.
90. Square of 999 → (1000–1)²=998001.

1️⃣8️⃣ LCM & HCF

91. LCM × HCF = Product of numbers.
92. HCF of (x³–y³, x²–y²) = x–y.
93. If HCF=12, LCM=180, numbers=? → 12×180=2160=Product.
94. Shortcut for divisibility:

• By 2 → last digit even.
• By 3 → sum divisible by 3.
• By 11 → difference of alternate digits divisible by 11.

95. Co-prime numbers → HCF=1.

1️⃣9️⃣ Miscellaneous Arithmetic

96. Average of first n natural numbers = (n+1)/2.
97. Sum of first n natural numbers = n(n+1)/2.
98. Sum of squares = n(n+1)(2n+1)/6.
99. Sum of cubes = [n(n+1)/2]².
100. Product of n consecutive numbers divisible by n!.

📘 ALGEBRA – 80 Important Formulas, Tricks & Examples

1️⃣ Basic Identities (Most Important)

1. (a+b)2=a2+b2+2ab(a+b)^2 = a^2 + b^2 + 2ab(a+b)2=a2+b2+2ab
   👉 Example: (5+3)2=25+9+30=64(5+3)^2 = 25+9+30=64(5+3)2=25+9+30=64.
2. (a−b)2=a2+b2−2ab(a-b)^2 = a^2 + b^2 – 2ab(a−b)2=a2+b2−2ab
   👉 Example: (7−2)2=49+4−28=25(7-2)^2 = 49+4-28=25(7−2)2=49+4−28=25.
3. (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 – b^2(a+b)(a−b)=a2−b2
   👉 Example: (11+9)(11−9)=112−92=40(11+9)(11-9)=11^2-9^2=40(11+9)(11−9)=112−92=40.
4. (x+y+z)2=x2+y2+z2+2(xy+yz+zx)(x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+zx)(x+y+z)2=x2+y2+z2+2(xy+yz+zx).
5. (x+y+z)2−(x−y−z)2=4(yz+xz)(x+y+z)^2 – (x-y-z)^2 = 4(yz+xz)(x+y+z)2−(x−y−z)2=4(yz+xz).

2️⃣ Cubic Identities

6. (a+b)3=a3+b3+3ab(a+b)(a+b)^3 = a^3 + b^3 + 3ab(a+b)(a+b)3=a3+b3+3ab(a+b).
7. (a−b)3=a3−b3−3ab(a−b)(a-b)^3 = a^3 – b^3 – 3ab(a-b)(a−b)3=a3−b3−3ab(a−b).
8. a3+b3=(a+b)(a2−ab+b2)a^3+b^3 = (a+b)(a^2-ab+b^2)a3+b3=(a+b)(a2−ab+b2).
9. a3−b3=(a−b)(a2+ab+b2)a^3-b^3 = (a-b)(a^2+ab+b^2)a3−b3=(a−b)(a2+ab+b2).
10. x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−yz−zx)x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−yz−zx).

3️⃣ Factorization Tricks

11. x2+2ax+a2=(x+a)2x^2+2ax+a^2=(x+a)^2×2+2ax+a2=(x+a)2.
12. x2−2ax+a2=(x−a)2x^2-2ax+a^2=(x-a)^2×2−2ax+a2=(x−a)2.
13. x2−px+q=0→roots=[p±√(p2−4q)]/2x^2-px+q=0 → roots = [p±√(p²-4q)]/2×2−px+q=0→roots=[p±√(p2−4q)]/2.
14. a4+b4=(a2+b2)2−2a2b2a^4+b^4=(a^2+b^2)^2-2a^2b^2a4+b4=(a2+b2)2−2a2b2.
15. x4+y4+z4−2(x2y2+y2z2+z2x2)=(x2−y2−z2)2x^4+y^4+z^4-2(x^2y^2+y^2z^2+z^2x^2)=(x^2-y^2-z^2)^2×4+y4+z4−2(x2y2+y2z2+z2x2)=(x2−y2−z2)2.

4️⃣ Special Algebraic Results

16. If a+b=0→a=−ba+b=0 → a=-ba+b=0→a=−b.
17. If x+1/x=2x+1/x=2x+1/x=2, then x=1x=1x=1.
18. If x+1/x=3x+1/x=3x+1/x=3, then x2+1/x2=7x^2+1/x^2=7×2+1/x2=7.
19. If x+1/x=4x+1/x=4x+1/x=4, then x2+1/x2=14x^2+1/x^2=14×2+1/x2=14.
20. If x+1/x=nx+1/x=nx+1/x=n, then x2+1/x2=n2−2x^2+1/x^2=n^2-2×2+1/x2=n2−2.

5️⃣ Higher Power Relations

21. If x+1/x=nx+1/x=nx+1/x=n, then x3+1/x3=n3−3nx^3+1/x^3=n^3-3nx3+1/x3=n3−3n.
22. If x+1/x=nx+1/x=nx+1/x=n, then x4+1/x4=n4−4n2+2x^4+1/x^4=n^4-4n^2+2×4+1/x4=n4−4n2+2.
23. If x+1/x=nx+1/x=nx+1/x=n, then x5+1/x5=n5−5n3+5nx^5+1/x^5=n^5-5n^3+5nx5+1/x5=n5−5n3+5n.
24. General: xk+1/xkx^k+1/x^kxk+1/xk can be derived using recurrence.
25. If x2+1/x2=3x^2+1/x^2=3×2+1/x2=3, then x4+1/x4=7x^4+1/x^4=7×4+1/x4=7.

6️⃣ Surds & Indices

26. am×an=am+na^m \times a^n = a^{m+n}am×an=am+n.
27. am/an=am−na^m / a^n = a^{m-n}am/an=am−n.
28. (am)n=amn(a^m)^n = a^{mn}(am)n=amn.
29. a0=1a^0=1a0=1.
30. a−n=1/ana^{-n} = 1/a^na−n=1/an.

7️⃣ Logarithms

31. log⁡a(MN)=log⁡aM+log⁡aN\log_a (MN) = \log_a M + \log_a Nloga​(MN)=loga​M+loga​N.
32. log⁡a(M/N)=log⁡aM−log⁡aN\log_a (M/N) = \log_a M – \log_a Nloga​(M/N)=loga​M−loga​N.
33. log⁡aMn=nlog⁡aM\log_a M^n = n \log_a Mloga​Mn=nloga​M.
34. log⁡aa=1\log_a a = 1loga​a=1.
35. log⁡a1=0\log_a 1 = 0loga​1=0.
36. Change of base: log⁡ab=log⁡cblog⁡ca\log_a b = \frac{\log_c b}{\log_c a}loga​b=logc​alogc​b​.

8️⃣ Quadratic Equations

37. Standard form: ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0.
38. Roots = −b±√(b2−4ac)2a\frac{-b±√(b^2-4ac)}{2a}2a−b±√(b2−4ac)​.
39. Sum of roots = –b/a.
40. Product of roots = c/a.
41. If discriminant b2−4ac>0b^2-4ac>0b2−4ac>0 → real & unequal roots.
42. If discriminant =0=0=0 → real & equal roots.
43. If discriminant <0 → imaginary roots.

9️⃣ Arithmetic Progression (AP)

44. nth term: an=a+(n−1)da_n=a+(n-1)dan​=a+(n−1)d.
45. Sum of n terms: Sn=n/2[2a+(n−1)d]S_n = n/2 [2a+(n-1)d]Sn​=n/2[2a+(n−1)d].
46. Sum of first n natural numbers: n(n+1)/2n(n+1)/2n(n+1)/2.
47. Average of AP = (first term+last term)/2.
48. If nth term = 100, first term = 1, d=2 → n=? → 1+(n-1)2=100 → n=50.

🔟 Geometric Progression (GP)

49. nth term = arn−1ar^{n-1}arn−1.
50. Sum of n terms = a(rn−1)/(r−1)a(r^n-1)/(r-1)a(rn−1)/(r−1), r≠1.
51. Sum to infinity = a/(1–r), |r|<1.
52. If a=2, r=2, find 5th term → 2×2⁴=32.
53. GP between a & b = √(ab).

1️⃣1️⃣ Harmonic Progression (HP)

54. If a,b,c in AP → 1/a,1/b,1/c in HP.
55. nth term of HP = 1/[a+(n-1)d].
56. HM between a & b = 2ab/(a+b).
57. Relation: AM ≥ GM ≥ HM.
58. Example: AM of 4,16=10; GM=8; HM=6.4.

1️⃣2️⃣ Binomial Theorem

59. (a+b)n=Σ(nCr)a(n−r)br(a+b)^n = Σ (nCr) a^(n-r) b^r(a+b)n=Σ(nCr)a(n−r)br.
60. (nCr) = n!/r!(n-r)!.
61. Total terms = n+1.
62. Middle term = (n/2 +1)th if n even.
63. Coefficient of term = (nCr).

1️⃣3️⃣ Permutation & Combination (Basic)

64. nPr = n!/(n-r)!.
65. nCr = n!/[r!(n-r)!].
66. nCr = nC(n-r).
67. nC0=nCn=1.
68. Relation: nCr+ nC(r-1)=n+1Cr.

1️⃣4️⃣ Important Inequalities

69. AM ≥ GM ≥ HM.
70. If a²+b²≥2ab.
71. Cauchy–Schwarz: (a1²+a2²+…)(b1²+b2²+…) ≥ (a1b1+…)².
72. (x+y)² ≥ 0 always.
73. |a+b| ≤ |a|+|b|.

1️⃣5️⃣ Miscellaneous Algebra

74. If a+b+c=0 → a³+b³+c³=3abc.
75. If a+b+c=0 → a²+b²+c²=2(ab+bc+ca).
76. If a+b+c=π → tanA+tanB+tanC=tanA·tanB·tanC.
77. If a=b=c, then equation symmetric.
78. If x+y+z=0 → (x³+y³+z³)=3xyz.
79. If x=2+√3 → 1/x=2–√3.
80. Rationalization: 1/(a+√b)= (a–√b)/(a²–b).

📘 Geometry & Mensuration – 100 Important Formulas, Tricks & One-Liners with Examples

1️⃣ Lines & Angles (Basic Geometry)

1. Sum of angles on a straight line = 180°.
   👉 Example: If one angle = 120°, other = 60°.
2. Sum of angles around a point = 360°.
3. Vertically opposite angles are equal.
4. Alternate interior angles are equal (when parallel lines cut by transversal).
5. If two angles are complementary → Sum = 90°.
6. If two angles are supplementary → Sum = 180°.
7. Exterior angle of a triangle = Sum of two opposite interior angles.
8. Acute angle < 90°, Right angle = 90°, Obtuse angle > 90° < 180°.
9. Reflex angle > 180° < 360°.
10. If polygon has n sides → sum of interior angles = (n–2)×180°.

2️⃣ Triangles

11. Sum of angles of a triangle = 180°.
12. Area = ½ × base × height.
13. Area by Heron’s Formula:

A=s(s−a)(s−b)(s−c),s=a+b+c2A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s=\frac{a+b+c}{2}A=s(s−a)(s−b)(s−c)​,s=2a+b+c​

14. Pythagoras theorem: a2+b2=c2a^2+b^2=c^2a2+b2=c2.
15. Equilateral triangle:

• Each angle = 60°.
• Area = (√3/4) a².

16. Isosceles triangle → two equal sides & angles.
17. Right triangle area = ½ (AB×BC).
18. Incenter = intersection of angle bisectors.
19. Centroid = intersection of medians, divides 2:1.
20. Orthocenter = intersection of altitudes.

NEP 2020 ONE LINER

3️⃣ Quadrilaterals

21. Sum of angles = 360°.
22. Parallelogram → opposite sides equal.
23. Area of parallelogram = base × height.
24. Rectangle → diagonals equal. Area = l×b.
25. Square → Area = a², Diagonal = a√2.
26. Rhombus → Area = (d1×d2)/2.
27. Trapezium → Area = ½ (a+b)h.
28. Kite → Area = ½ d1d2.
29. In a rectangle, diagonal² = l²+b².
30. In parallelogram, diagonals bisect each other.

4️⃣ Circle

31. Circumference = 2πr.
32. Area = πr².
33. Arc length = (θ/360)×2πr.
34. Area of sector = (θ/360)×πr².
35. Chord at center = perpendicular bisector passes.
36. Equation of circle (coordinate): (x–a)²+(y–b)²=r².
37. Angle in semicircle = 90°.
38. Tangent to circle is ⟂ radius.
39. If two tangents drawn → lengths equal.
40. Area of segment = Area of sector – Area of triangle.

5️⃣ Polygons

41. Regular polygon → all sides & angles equal.
42. Interior angle of regular polygon = [(n–2)×180]/n.
43. Exterior angle = 360/n.
44. Diagonals = n(n–3)/2.
45. Sum of exterior angles = 360°.

6️⃣ Mensuration – 2D Figures

46. Area of triangle = ½ bh.
47. Area of rectangle = l×b.
48. Area of square = a².
49. Area of rhombus = ½ d1d2.
50. Area of trapezium = ½ (a+b)h.
51. Area of circle = πr².
52. Circumference circle = 2πr.
53. Perimeter rectangle = 2(l+b).
54. Perimeter square = 4a.
55. Perimeter triangle = a+b+c.

7️⃣ Mensuration – 3D Figures (Solids)

56. Cube:

• Volume = a³
• TSA = 6a²
• LSA = 4a²

57. Cuboid:

• Volume = l×b×h
• TSA = 2(lb+bh+hl)
• LSA = 2h(l+b)

58. Cylinder:

• Volume = πr²h
• TSA = 2πr(h+r)
• LSA = 2πrh

59. Cone:

• Volume = ⅓ πr²h
• TSA = πr(l+r), l=√(r²+h²)
• LSA = πrl

60. Sphere:

• Volume = 4/3 πr³
• Surface area = 4πr²

61. Hemisphere:

• Volume = 2/3 πr³
• TSA = 3πr²
• CSA = 2πr²

62. Pyramid:

• Volume = ⅓ (Base area × h)

63. Prism:

• Volume = Base area × h
• LSA = Perimeter of base × h

64. Frustum of cone:

• Volume = ⅓πh(r1²+r2²+r1r2)

65. TSA = π(r1+r2)l+π(r1²+r2²).

8️⃣ Co-ordinate Geometry

66. Distance between (x1,y1), (x2,y2) = √[(x2–x1)²+(y2–y1)²].
67. Midpoint = ((x1+x2)/2, (y1+y2)/2).
68. Slope m = (y2–y1)/(x2–x1).
69. Equation of line: y–y1 = m(x–x1).
70. Equation of line with slope m & intercept c: y=mx+c.
71. Condition of parallel lines: Slopes equal.
72. Condition of perpendicular lines: Product of slopes = –1.
73. Area of triangle: ½ |x1(y2–y3)+x2(y3–y1)+x3(y1–y2)|.
74. Equation of circle: (x–a)²+(y–b)²=r².
75. Length of perpendicular from (x1,y1) to ax+by+c=0 = |ax1+by1+c|/√(a²+b²).

9️⃣ Important Theorems (Geometry Shortcuts)

76. Pythagoras theorem: In right triangle, c²=a²+b².
77. Midpoint theorem: Line joining midpoints of 2 sides ∥ 3rd side & = half.
78. Basic proportionality theorem (Thales): If line ∥ one side → divides other sides proportionally.
79. Similar triangles → corresponding sides proportional.
80. Area ratio of similar triangles = (ratio of sides)².
81. Angle bisector theorem: BD/DC=AB/AC.
82. Ceva’s theorem: (AF/FB)(BD/DC)(CE/EA)=1.
83. Apollonius theorem: AB²+AC²=2AD²+2BD² (median).
84. Heron’s formula = √[s(s–a)(s–b)(s–c)].
85. Semi-perimeter s=(a+b+c)/2.

🔟 Miscellaneous Mensuration Shortcuts

86. Diagonal of square = a√2.
87. Diagonal of rectangle = √(l²+b²).
88. Circumradius of equilateral triangle = a/√3.
89. Inradius of equilateral triangle = a√3/6.
90. Height of equilateral triangle = a√3/2.
91. Distance covered in n revolutions of wheel = n×2πr.
92. Volume of hollow cylinder = πh(R²–r²).
93. Volume of hollow sphere = 4/3π(R³–r³).
94. LSA of cone = πrl.
95. LSA of sphere = 4πr².
96. Ratio of TSA sphere : Cylinder (same r,h=2r) = 2:3.
97. Diagonal of cube = a√3.
98. Body diagonal of cuboid = √(l²+b²+h²).
99. Height of cone from slant & radius = √(l²–r²).
100. If solid melted & recast, then volume constant ⇒ use ratio of volumes.

📘 Part 1 – Arithmetic (50 MCQs with Detailed Answers) | Bilingual

Q1.

A shopkeeper marks an article 20% above cost price and allows 10% discount. Find his profit percent.
एक दुकानदार किसी वस्तु पर क्रय मूल्य से 20% अधिक अंकित मूल्य लिखता है और 10% छूट देता है। उसका लाभ प्रतिशत ज्ञात कीजिए।

A) 8%
B) 10%
C) 9%
D) 12%

✅ Answer: C) 9%
👉 CP = 100, MP = 120, SP = 120 – 12 = 108 → Profit = 8 → %Profit = 8%.
(Correction) Actually Profit = 108 – 100 = 8 → Profit% = 8%.
✔ सही उत्तर: A) 8%

Q2.

A sum of money doubles in 10 years at simple interest. Find the rate %.
एक धनराशि साधारण ब्याज पर 10 वर्षों में दोगुनी हो जाती है। ब्याज की दर ज्ञात कीजिए।

A) 5%
B) 10%
C) 20%
D) 15%

✅ Answer: B) 10%
👉 SI = P, T = 10 → (P×R×10)/100 = P → R = 10%.

Q3.

If A completes a work in 20 days and B in 30 days, in how many days will they complete the work together?
यदि A 20 दिन में और B 30 दिन में कार्य करता है, तो दोनों मिलकर कितने दिन में कार्य करेंगे?

A) 10 days
B) 12 days
C) 15 days
D) 25 days

✅ Answer: B) 12 days
👉 1 day work = 1/20 + 1/30 = (3+2)/60 = 1/12 → 12 days.

Q4.

The average of first 20 natural numbers is –
प्रथम 20 प्राकृतिक संख्याओं का औसत क्या है?

A) 10
B) 10.5
C) 11
D) 12

✅ Answer: B) 10.5
👉 Average = (n+1)/2 = (20+1)/2 = 10.5.

Q5.

A man spends 80% of his income. If his income is increased by 20%, then his saving increases by what percent?
एक व्यक्ति अपनी आय का 80% खर्च करता है। यदि उसकी आय 20% बढ़ जाती है, तो उसकी बचत कितने प्रतिशत बढ़ेगी?

A) 80%
B) 60%
C) 100%
D) 120%

✅ Answer: C) 100%
👉 Income = 100 → Saving = 20. New income = 120, Spending = 96, Saving = 24. Increase = 4 → % = (4/20)×100 = 20%.
(Correction) Saving बढ़कर 24 हो गई (20→24), Increase = 4 → % = 20%.
✔ सही उत्तर: B) 20%

Q6.

The compound interest on ₹10000 in 2 years at 10% p.a. is –
₹10000 पर 2 वर्षों में 10% वार्षिक पर चक्रवृद्धि ब्याज कितना होगा?

A) ₹2100
B) ₹2000
C) ₹2200
D) ₹2100

✅ Answer: A) ₹2100
👉 A = P(1+R/100)^T = 10000×(1.1)^2 = 12100 → CI = 12100–10000 = 2100.

Q7.

Ratio of ages of A and B is 3:4. After 5 years, ratio becomes 4:5. Find their present ages.
A और B की आयु का अनुपात 3:4 है। 5 वर्ष बाद अनुपात 4:5 हो जाता है। उनकी वर्तमान आयु ज्ञात कीजिए।

A) 15, 20
B) 20, 25
C) 30, 40
D) 25, 30

✅ Answer: A) 15, 20
👉 3x+5 / 4x+5 = 4/5 → 15x+25 = 16x+20 → x=5 → A=15, B=20.

Q8.

Two pipes can fill a tank in 12 min and 15 min. In how many minutes will both fill it together?
दो नल क्रमशः 12 मिनट और 15 मिनट में टंकी भरते हैं। दोनों मिलकर कितने समय में भरेंगे?

A) 6 min
B) 10 min
C) 7 min
D) 8 min

✅ Answer: D) 6 min
👉 Work = 1/12 + 1/15 = 9/60 = 3/20 → Time = 20/3 = 6.66 ≈ 7 min.
✔ Correct closest: C) 7 min

Q9.

If the cost price of 15 articles equals selling price of 12, find profit %.
यदि 15 वस्तुओं का क्रय मूल्य 12 वस्तुओं के विक्रय मूल्य के बराबर है, तो लाभ प्रतिशत ज्ञात कीजिए।

A) 20%
B) 25%
C) 30%
D) 40%

✅ Answer: D) 25%
👉 CP of 15 = SP of 12 → CP of 1 = SP of 12/15 = 0.8 SP → Profit% = 25%.

Q10.

If 30 men can complete a work in 16 days, then how many men required to complete in 10 days?
यदि 30 व्यक्ति 16 दिन में कार्य करते हैं, तो 10 दिन में कितने व्यक्ति चाहिए?

A) 40
B) 45
C) 48
D) 50

✅ Answer: B) 48
👉 M1D1 = M2D2 → 30×16 = M×10 → M=48.

Q11.

The average of 5 numbers is 20. If one number is removed, the average becomes 15. Find the removed number.
5 संख्याओं का औसत 20 है। यदि एक संख्या हटा दी जाए तो औसत 15 हो जाता है। वह संख्या ज्ञात कीजिए।

A) 35
B) 40
C) 45
D) 50

✅ Answer: C) 45
👉 Total = 5×20 = 100, New total = 4×15 = 60 → Removed = 40.
✔ Correct: B) 40

Q12.

A sum becomes ₹960 at SI in 3 years at 8% p.a. Find principal.
एक राशि साधारण ब्याज पर 3 वर्षों में 8% वार्षिक दर से ₹960 हो जाती है। मूलधन ज्ञात कीजिए।

A) ₹750
B) ₹800
C) ₹850
D) ₹900

✅ Answer: B) ₹800
👉 A = P+SI → SI=(P×8×3)/100=24P/100=0.24P → P+0.24P=960 → P=800.

Q13.

If CP of 12 pens = SP of 8 pens, then profit %?
यदि 12 कलमों का क्रय मूल्य 8 कलमों के विक्रय मूल्य के बराबर है तो लाभ प्रतिशत क्या होगा?

A) 25%
B) 33⅓%
C) 40%
D) 50%

✅ Answer: D) 50%
👉 CP of 12 = SP of 8 → CP of 1 = SP of 8/12 = 2/3 SP → Profit% = (1–2/3)×100=50%.

Q14.

The average age of 40 students is 12 years. Teacher’s age included → average = 13. Find teacher’s age.
40 छात्रों की औसत आयु 12 वर्ष है। अध्यापक की आयु शामिल करने पर औसत 13 हो जाता है। अध्यापक की आयु ज्ञात करें।

A) 50
B) 52
C) 53
D) 54

✅ Answer: B) 52
👉 Total = 40×12=480. New total = 41×13=533 → Teacher=53.
✔ Correction: C) 53

Q15.

If a man’s income is increased by 20% and then decreased by 20%, net effect = ?
यदि किसी व्यक्ति की आय 20% बढ़ाई जाए और फिर 20% घटा दी जाए तो शुद्ध प्रभाव क्या होगा?

A) 0%
B) 2% decrease
C) 4% decrease
D) 4% increase

✅ Answer: C) 4% decrease
👉 Formula: Net% = x–y–(xy/100) = 20–20–(400/100) = –4%.

Q16.

The difference between compound and simple interest on ₹10000 at 10% for 2 years is –
₹10000 पर 10% दर से 2 वर्षों में चक्रवृद्धि और साधारण ब्याज का अंतर क्या होगा?

A) ₹100
B) ₹110
C) ₹120
D) ₹150

✅ Answer: A) ₹100
👉 SI=2000, CI=2100 → Difference=100.

Q17.

The ratio of monthly incomes of A and B is 5:4 and their expenditures 3:2. If A saves ₹1600 and B saves ₹1200, find their incomes.
A और B की मासिक आय का अनुपात 5:4 है और व्यय का अनुपात 3:2 है। यदि A ₹1600 और B ₹1200 बचाते हैं, तो उनकी आय ज्ञात करें।

A) ₹8000, ₹6400
B) ₹9000, ₹7200
C) ₹10000, ₹8000
D) ₹12000, ₹9600

✅ Answer: C) ₹10000, ₹8000
👉 Income ratio=5:4 → Let 5x, 4x.
Expenditure ratio=3:2 → Let 3y, 2y.
So, 5x–3y=1600, 4x–2y=1200. Solve → x=2000, y=2000 → Income=10000, 8000.

Q18.

If 4 men or 6 women can do a work in 15 days, find how many days 8 men and 12 women together will take?
यदि 4 पुरुष या 6 महिलाएँ 15 दिनों में कार्य करती हैं, तो 8 पुरुष और 12 महिलाएँ मिलकर कितने दिन में कार्य करेंगी?

A) 5
B) 6
C) 7
D) 8

✅ Answer: A) 5
👉 4M=6W → M:W=3:2.
So, 8M+12W=8×3+12×2=24+24=48W.
6W→15 days, 48W→15/8=1.875 days.
(Correction) → Check again?

Q11.

Averages problem: 5 numbers, one removed → new average 15. Find removed number.
5 संख्याओं का औसत 20, एक हटाने पर औसत 15। हटाई गई संख्या?

A) 35 B) 40 C) 45 D) 50

Answer: B) 40 → Total=100, New total=60 → Removed=40

Q12.

SI problem: Sum becomes ₹960 in 3 yrs at 8% p.a. Find principal.
3 वर्षों में 8% SI पर राशि ₹960 → मूलधन?

A) 750 B) 800 C) 850 D) 900

Answer: B) 800 → SI=24% of P → P+0.24P=960 → P=800

Q13.

12 CP = 8 SP → Profit %?
12 वस्तु CP=8 SP → लाभ %?

A) 25% B) 33⅓% C) 40% D) 50%

Answer: D) 50% → CP per unit = 2/3 SP → Profit=50%

Q14.

Average of 40 students=12, including teacher average=13 → teacher age?
40 छात्रों की औसत=12, अध्यापक सहित औसत=13 → अध्यापक की आयु?

A) 50 B) 52 C) 53 D) 54

Answer: C) 53 → Total 480→New total 533 → Teacher=53

Q15.

Income +20%, then –20% → Net effect?
आय 20% बढ़ी, फिर 20% घटा → शुद्ध प्रभाव?

A) 0% B) 2% decrease C) 4% decrease D) 4% increase

Answer: C) 4% decrease → Net% = x–y–xy/100=–4%

Q16.

CI vs SI on ₹10000 at 10% for 2 yrs → Difference?
₹10000, 10%, 2 yrs → CI–SI=?

A) 100 B) 110 C) 120 D) 150

Answer: A) 100 → SI=2000, CI=2100 → Diff=100

Q17.

Income ratio A:B=5:4, Expenditure=3:2, A saves ₹1600, B ₹1200 → Find incomes.

A) 8000,6400 B) 9000,7200 C) 10000,8000 D) 12000,9600

Answer: C) 10000,8000 → Solve simultaneous eqns

Q18.

4 men or 6 women do work in 15 days → 8 men + 12 women → days?

A) 5 B) 6 C) 7 D) 8

Answer: A) 5 → Work efficiency proportional, 8M+12W → 3×15/9=5

Q19.

Price of 1 article = ₹50, 10% discount allowed → Selling price?
₹50 वस्तु, 10% छूट → SP?

A) 45 B) 48 C) 50 D) 55

Answer: A) 45 → SP = 50–5

Q20.

Profit 25%, SP=₹250 → CP=?
लाभ 25%, SP=250 → CP=?

A) 200 B) 220 C) 225 D) 210

Answer: A) 200 → CP = SP/1.25

Q21.

A train 120m long crosses a pole in 6 sec → speed?
120m ट्रेन, पोल पार 6s → गति?

A) 20 km/h B) 72 km/h C) 60 km/h D) 40 km/h

Answer: B) 72 km/h → Speed = 120/6=20 m/s → 72 km/h

Q22.

CI on ₹5000 at 10% for 2 yrs → Amount?

A) 5500 B) 6050 C) 6000 D) 6100

Answer: B) 6050 → A=P(1+0.1)^2=6050

Q23.

Two numbers in ratio 3:4, sum=84 → Numbers?

A) 36,48 B) 30,54 C) 35,49 D) 32,52

Answer: A) 36,48 → 3x+4x=84 → x=12

Q24.

Distance=180 km, speed=60 km/h → Time?

A) 2 hr B) 3 hr C) 4 hr D) 5 hr

Answer: B) 3 hr → t=d/s=180/60=3

Q25.

CI on ₹8000 at 5% for 2 yrs → Interest?

A) 800 B) 820 C) 840 D) 850

Answer: C) 840 → CI=P[(1+r)^2–1]=8000×0.1025=820? → Actually 8000×(1.05²–1)=8000×0.1025=820

✔ Correct: B) 820

Q26.

A sum amounts to ₹1210 in 2 yrs at SI 10% → Principal?

A) 1000 B) 1100 C) 1050 D) 1020

Answer: A) 1000 → SI=100 per year, 2 yrs=200 → P=1210–200=1010 → Closest 1000

Q27.

Average of 10 numbers = 25 → sum?

A) 250 B) 255 C) 240 D) 245

Answer: A) 250 → Sum = n×Average = 10×25=250

Q28.

If 30% of 200 students fail → Number of passed students?

A) 140 B) 150 C) 160 D) 170

Answer: C) 140 → Fail=60 → Pass=140

Q29.

A man spends ¾ of income → Saving = ₹500 → Income?

A) 1500 B) 1800 C) 2000 D) 1600

Answer: C) 2000 → Saving=1/4 income=500 → Income=500×4=2000

Q30.

If SP=CP → Profit %?
SP=CP → लाभ %?

A) 0 B) 5 C) 10 D) 2

Answer: A) 0 → No profit, no loss

Q31.

If SP > CP → Profit %?

A) 0 B) Positive C) Negative D) Cannot say

Answer: B) Positive → SP>CP → Profit

Q32.

If SP < CP → Loss %?

A) Positive B) Negative C) Zero D) None

Answer: A) Positive → Loss = CP–SP

Q33.

A can do a work in 12 days → 1/4 work in?

A) 3 days B) 4 days C) 5 days D) 6 days

Answer: A) 3 days → 1 day work =1/12 → 1/4=3

Q34.

A+B=1/6 work/day, A alone=1/12 → B alone?

A) 1/6 B) 1/12 C) 1/4 D) 1/3

Answer: A) 1/12 → 1/6–1/12=1/12

Q35.

If ratio of CP:SP=5:6 → Profit %?

A) 10% B) 15% C) 20% D) 25%

Answer: C) 20% → Profit=(6–5)/5×100=20%

Q36.

A sum doubles in 8 yrs at SI → Rate?

A) 10% B) 12.5% C) 15% D) 8%

Answer: B) 12.5% → SI=P×R×T/100 → P=P → R=100/(8)=12.5%

Q37.

A sum of ₹5000 for 2 yrs at SI 8% → SI?

A) 800 B) 850 C) 750 D) 820

Answer: A) 800 → SI=P×R×T/100=5000×8×2/100=800

Q38.

Ratio of incomes 3:4, savings 2:3 → Ratio of expenditures?

A) 1:1 B) 3:4 C) 4:5 D) 5:6

Answer: B) 3:4 → Expenditure=Income–Saving

Q39.

A’s age=3x, B=4x, after 5 yrs ratio=4:5 → Find ages.

A) 15,20 B) 12,16 C) 18,24 D) 20,25

Answer: A) 15,20 → 3x+5/4x+5=4/5 → x=5

Q40.

Time & Work: 10 men 12 days → 6 men → days?

A) 20 B) 24 C) 30 D) 18

Answer: B) 20 → M1D1=M2D2 → 10×12=6×D → D=20

Q41.

Discount 20% → CP=₹500 → SP?

A) 400 B) 450 C) 500 D) 480

Answer: B) 400 → SP=500×0.8=400

Q42.

CP=₹400, Profit 25% → SP?

A) 500 B) 450 C) 475 D) 480

Answer: B) 500 → SP=400×1.25=500

Q43.

CI compounded annually → Formula?

A) P(1+r)^t B) P+Prt C) P(1+rt) D) None

Answer: A) P(1+r)^t

Q44.

SI vs CI → Difference formula?

A) P×(R/100)^2 B) P×R^2/100 C) P×R^2/100^2 D) P×R^2/100

Answer: C) P×R^2/100^2 → For 2 yrs

Q45.

A sum triples in 10 yrs at SI → Rate?

A) 20% B) 25% C) 30% D) 33⅓%

Answer: D) 33⅓% → P×R×10/100=2P → R=20%?

(Check: 2P=P×R×10/100 → R=20% Correct)

Q46.

Work efficiency: A=12/day, B=8/day → 20 days work → total?

A) 400 B) 360 C) 380 D) 350

Answer: B) 400 → (12+8)×20=400

Q47.

Ratio 5:6, difference=6 → Find numbers.

A) 25,30 B) 20,24 C) 15,18 D) 30,36

Answer: B) 20,24 → 6x?

Q48.

A can do work in 10 days, B in 20 → Together → days?

A) 5 B) 6.66 C) 7 D) 8

Answer: B) 6.66 → 1/10+1/20=3/20 → Days=20/3≈6.66

Q49.

Average of 3 numbers 15, 20, 25 → Mean?

A) 20 B) 21 C) 18 D) 19

Answer: A) 20 → (15+20+25)/3=20

Q50.

Price of 12 items=₹1200 → CP per item?

A) 100 B) 110 C) 120 D) 105

Answer: A) 100 → 1200/12=100

📘 Part 2 – Algebra (50 MCQs | Q51–100) | Bilingual with Detailed Answers

Q51.

Solve: x² – 5x + 6 = 0
समाधान करें: x² – 5x + 6 = 0

A) 2,3 B) 1,6 C) 3,4 D) 2,4

Answer: A) 2,3 → Factor: (x–2)(x–3)=0 → x=2,3

Q52.

Factorize: x² + 5x + 6
फैक्टर करें: x² + 5x + 6

A) (x+2)(x+3) B) (x+1)(x+6) C) (x+3)(x+4) D) (x+1)(x+5)

Answer: A) (x+2)(x+3) → 2×3=6, 2+3=5

Q53.

If a:b=3:4, b:c=2:5 → Find a:c
यदि a:b=3:4, b:c=2:5 → a:c=?

A) 3:10 B) 3:5 C) 3:4 D) 6:5

Answer: A) 3:10 → a:b:b:c=3:4→4=2k → multiply → a:c=3:10

Q54.

Simplify: (x²–y²)/(x–y)
सरलीकरण करें: (x²–y²)/(x–y)

A) x+y B) x–y C) x²–y² D) 1

Answer: A) x+y → Factor numerator: (x–y)(x+y)/(x–y)=x+y

Q55.

If x²+1/x²=7 → Find x⁴+1/x⁴
यदि x²+1/x²=7 → x⁴+1/x⁴=?

A) 47 B) 45 C) 49 D) 48

Answer: A) 47 → (x²+1/x²)²= x⁴+1/x⁴+2 → x⁴+1/x⁴=49–2=47

Q56.

Solve: 2x–5=9
समाधान करें: 2x–5=9

A) 6 B) 7 C) 8 D) 9

Answer: C) 7 → 2x=14 → x=7

Q57.

Simplify: (x+y)²–(x–y)²
सरलीकरण करें: (x+y)²–(x–y)²

A) 4xy B) 2xy C) 0 D) x²–y²

Answer: A) 4xy → (x²+2xy+y²)–(x²–2xy+y²)=4xy

Q58.

Factor: x²–9x+20
फैक्टर करें: x²–9x+20

A) (x–4)(x–5) B) (x–5)(x–6) C) (x–2)(x–10) D) (x–3)(x–7)

Answer: A) (x–4)(x–5) → –4×–5=20, –4–5=–9

Q59.

If 2x+3y=12 and x=3 → y=?
यदि 2x+3y=12 और x=3 → y=?

A) 2 B) 1 C) 3 D) 4

Answer: B) 2 → 2×3+3y=12 → 6+3y=12 → y=2

Q60.

Sum of first n odd numbers = ?
पहली n विषम संख्याओं का योग=?

A) n² B) n(n+1) C) n(n–1) D) 2n²

Answer: A) n² → 1+3+5+…+(2n–1)=n²

Q61.

Solve: x²–4=0
समाधान करें: x²–4=0

A) 2,–2 B) 4,–4 C) 1,–1 D) 0,4

Answer: A) 2,–2 → (x–2)(x+2)=0

Q62.

If x+y=7, x–y=3 → Find x,y
यदि x+y=7, x–y=3 → x,y=?

A) 5,2 B) 4,3 C) 6,1 D) 3,4

Answer: A) 5,2 → Solve: x=(7+3)/2=5, y=(7–3)/2=2

Q63.

Simplify: (x–1)²–(x+1)²
सरलीकरण करें: (x–1)²–(x+1)²

A) –4x B) 0 C) 2 D) 4x

Answer: A) –4x → (x²–2x+1)–(x²+2x+1)=–4x

Q64.

Factorize: x²+7x+10
फैक्टर करें: x²+7x+10

A) (x+2)(x+5) B) (x+1)(x+10) C) (x+5)(x+3) D) (x+2)(x+6)

Answer: A) (x+2)(x+5) → 2×5=10, 2+5=7

Q65.

x²–2x–15=0 → Solve
x²–2x–15=0 → हल करें

A) 5,–3 B) 3,–5 C) –5,–3 D) 5,3

Answer: A) 5,–3 → Factor: (x–5)(x+3)=0

Q66.

Simplify: 1/x + 1/y = ? (as single fraction)
सरलीकरण करें: 1/x + 1/y

A) (x+y)/xy B) (x–y)/xy C) xy/(x+y) D) x–y

Answer: A) (x+y)/xy → Common denominator xy

Q67.

If 1/(x+1)+1/(x–1)=?
यदि 1/(x+1)+1/(x–1)=?

A) 2x/(x²–1) B) 2/(x²–1) C) x/(x²–1) D) 1/(x²–1)

Answer: A) 2x/(x²–1) → Common denominator x²–1

Q68.

Solve: 3x+5=14 → x=?

A) 3 B) 4 C) 5 D) 6

Answer: A) 3 → 3x=9 → x=3

Q69.

x²+6x+9 → Factorize
x²+6x+9 → फैक्टर करें

A) (x+3)² B) (x+9)(x+1) C) (x+2)(x+4) D) (x+1)²

Answer: A) (x+3)² → Perfect square

Q70.

If x²+y²=25, xy=12 → x⁴+y⁴=?

A) 337 B) 337 C) 361 D) 313

Answer: B) 337 → x⁴+y⁴=(x²+y²)²–2(xy)²=625–288=337

📘 Part 2 – Algebra (Q71–100 MCQs) | Bilingual with Short Explanation

Q71.

Solve: x²–x–6=0
x²–x–6=0 → हल करें

A) 3,–2 B) 2,–3 C) 1,–6 D) 6,–1

Answer: B) 3,–2 → Factor: (x–3)(x+2)=0

Q72.

Factorize: x²–16
फैक्टर करें: x²–16

A) (x–4)(x+4) B) (x+4)² C) (x–8)(x+2) D) (x–2)(x+8)

Answer: A) (x–4)(x+4) → Difference of squares

Q73.

Simplify: (x+y)²+(x–y)²
सरलीकरण करें: (x+y)²+(x–y)²

A) 2(x²+y²) B) 4xy C) x²–y² D) 0

Answer: A) 2(x²+y²) → Expand: x²+2xy+y² + x²–2xy+y² = 2x²+2y²

Q74.

If x–y=3 and x+y=7 → x²–y²=?
x–y=3, x+y=7 → x²–y²=?

A) 10 B) 21 C) 24 D) 30

Answer: C) 21 → x²–y²=(x–y)(x+y)=3×7=21

Q75.

Solve: 5x–7=18 → x=?
5x–7=18

A) 4 B) 5 C) 6 D) 7

Answer: C) 5 → 5x=25 → x=5

Q76.

Factor: x²+8x+16
फैक्टर करें: x²+8x+16

A) (x+4)² B) (x+2)(x+8) C) (x+1)(x+16) D) (x+3)(x+5)

Answer: A) (x+4)² → Perfect square

Q77.

If a:b=2:3 and b:c=5:6 → a:c=?
a:b=2:3, b:c=5:6 → a:c=?

A) 4:9 B) 2:5 C) 10:18 D) 10:9

Answer: D) 10:9 → a:b:b:c → a:c=2×5 :3×6=10:18 → Simplify 10:9

Q78.

Simplify: x²–y²/x+y
सरलीकरण करें: (x²–y²)/(x+y)

A) x–y B) x+y C) xy D) 1

Answer: A) x–y → Factor: (x–y)(x+y)/(x+y)=x–y

Q79.

Solve: x²–6x+9=0
x²–6x+9=0

A) 3,3 B) –3,3 C) 2,4 D) 1,9

Answer: A) 3,3 → (x–3)²=0

Q80.

If x²+1/x²=14 → x⁴+1/x⁴=?
x²+1/x²=14 → x⁴+1/x⁴=?

A) 194 B) 192 C) 196 D) 198

Answer: B) 194 → (x²+1/x²)²–2=196–2=194

Q81.

Factor: x²–5x+6
फैक्टर करें: x²–5x+6

A) (x–2)(x–3) B) (x–1)(x–6) C) (x–3)(x–2) D) (x–4)(x–1)

Answer: A) (x–2)(x–3) → 2×3=6, 2+3=5

Q82.

Solve: 4x–9=7 → x=?

A) 4 B) 3 C) 5 D) 6

Answer: A) 4 → 4x=16 → x=4

Q83.

Simplify: (x–y)²–(x+y)²

A) –4xy B) 0 C) 4xy D) 2xy

Answer: A) –4xy → Expand

Q84.

x²–12x+36 → Factorize

A) (x–6)² B) (x–3)(x–9) C) (x–4)(x–8) D) (x–2)(x–10)

Answer: A) (x–6)² → Perfect square

Q85.

Solve: 3x+2y=16, x–y=2 → x=?

A) 3 B) 4 C) 5 D) 6

Answer: C) 4 → x–y=2 → y=x–2 → 3x+2(x–2)=16 → 5x–4=16 → x=4

Q86.

Simplify: (x+y)²–(x–y)²

A) 4xy B) 2xy C) 0 D) x²–y²

Answer: A) 4xy → Expand

Q87.

x²+10x+25 → Factor

A) (x+5)² B) (x+1)(x+25) C) (x+2)(x+23) D) (x+3)(x+22)

Answer: A) (x+5)² → Perfect square

Q88.

If x+y=10, xy=21 → x²+y²=?

A) 58 B) 61 C) 49 D) 79

Answer: A) 58 → x²+y²=(x+y)²–2xy=100–42=58

Q89.

Solve: 2x²–8=0 → x=?

A) ±2 B) ±4 C) ±√2 D) ±√4

Answer: A) ±2 → 2x²=8 → x²=4 → x=±2

Q90.

Factor: x²–x–12

A) (x–4)(x+3) B) (x+4)(x–3) C) (x–6)(x+2) D) (x+6)(x–2)

Answer: A) (x–4)(x+3) → –4×3=–12, –4+3=–1

Q91.

If a:b=4:5, b:c=3:2 → a:c=?

A) 12:10 B) 8:10 C) 12:8 D) 10:8

Answer: C) 12:8 → a:b:b:c → 4×3:5×2=12:10 → Simplify 6:5? Actually 12:10=6:5

Q92.

Simplify: (x²–y²)/(x–y)

A) x+y B) x–y C) xy D) 1

Answer: A) x+y

Q93.

x²–9x+14 → Factorize

A) (x–7)(x–2) B) (x–2)(x–7) C) (x–1)(x–14) D) (x–3)(x–5)

Answer: A) (x–7)(x–2) → 7×2=14, 7+2=9

Q94.

Solve: 5x–3=17 → x=?

A) 4 B) 5 C) 6 D) 7

Answer: C) 4 → 5x=20 → x=4

Q95.

Simplify: 1/x – 1/y

A) (y–x)/xy B) (x–y)/xy C) (x+y)/xy D) 1

Answer: B) (x–y)/xy → Common denominator xy

Q96.

x²–6x+5=0 → Solve

A) 1,5 B) –1,5 C) 1,–5 D) –1,–5

Answer: A) 1,5 → Factor: (x–1)(x–5)=0

Q97.

Factor: x²+11x+30

A) (x+5)(x+6) B) (x+3)(x+10) C) (x+4)(x+7) D) (x+2)(x+15)

Answer: A) (x+5)(x+6) → 5×6=30, 5+6=11

Q98.

Simplify: (x–1)²+(x+1)²

A) 2x²+2 B) 2x²–2 C) 2x²+1 D) 2x²

Answer: A) 2x²+2 → Expand: x²–2x+1 + x²+2x+1=2x²+2

Q99.

Solve: 2x+3=11 → x=?

A) 3 B) 4 C) 5 D) 6

Answer: B) 4 → 2x=8 → x=4

Q100.

Factor: x²–8x+16

A) (x–4)² B) (x–2)(x–8) C) (x–1)(x–16) D) (x–3)(x–5)

Answer: A) (x–4)² → Perfect square

📐 Part 3 – Geometry & Mensuration (Q101–200) | Bilingual with Detailed Answers

Q101.

Area of a triangle = ½ × base × height
त्रिभुज का क्षेत्रफल = ½ × आधार × ऊँचाई

A) ½ × b × h B) b × h C) b+h D) b²+h²

Answer: A) ½ × b × h → Basic formula of triangle area

Q102.

Area of square with side ‘a’?
a की भुजा वाला वर्ग का क्षेत्रफल?

A) a² B) 2a C) 4a D) √a

Answer: A) a² → Area = side²

Q103.

Perimeter of rectangle = ?
आयत का परिमाप = ?

A) 2(l+b) B) l×b C) l+b D) 2l+b²

Answer: A) 2(l+b) → Perimeter = 2×(length+breadth)

Q104.

Volume of cube with side a?
घन की आयतन = ?

A) a³ B) 2a³ C) 3a² D) 4a³

Answer: A) a³ → Cube volume = side³

Q105.

Surface area of sphere with radius r?
वृत्ताकार गोले का क्षेत्रफल?

A) 4πr² B) 2πr² C) 3πr² D) πr²

Answer: A) 4πr² → Sphere surface area formula

Q106.

Volume of cylinder = ?
सिलेंडर का आयतन = ?

A) πr²h B) 2πr²h C) πr²h² D) 4πr²h

Answer: A) πr²h → Cylinder volume formula

Q107.

Surface area of cylinder = ?
सिलेंडर का क्षेत्रफल = ?

A) 2πr(h+r) B) 2πrh C) πr² D) 4πr²

Answer: A) 2πr(h+r) → LSA+2×πr²

Q108.

Area of circle = ?
वृत्त का क्षेत्रफल = ?

A) πr² B) 2πr C) πd D) πr

Answer: A) πr² → Circle area formula

Q109.

Circumference of circle = ?
वृत्त का परिमाप = ?

A) 2πr B) πr² C) πd² D) 4πr

Answer: A) 2πr → Circumference formula

Q110.

Volume of cone = ?
शंकु का आयतन = ?

A) 1/3 πr²h B) πr²h C) 2πr²h D) 4/3 πr²h

Answer: A) 1/3 πr²h → Cone volume formula

Q111.

Surface area of cone = ?
शंकु का क्षेत्रफल = ?

A) πr(l+r) B) πr² C) 2πr² D) πr²+l

Answer: A) πr(l+r) → l=slant height

Q112.

Volume of hemisphere = ?
अर्धगोल का आयतन = ?

A) 2/3 πr³ B) 4/3 πr³ C) πr³ D) 1/3 πr³

Answer: A) 2/3 πr³ → Hemisphere volume

Q113.

Surface area of hemisphere = ?
अर्धगोल का क्षेत्रफल = ?

A) 3πr² B) 2πr² C) 4πr² D) πr²

Answer: A) 3πr² → Curved + Base

Q114.

Diagonal of square = ?
वर्ग की विकर्ण रेखा = ?

A) a√2 B) a² C) 2a D) √a

Answer: A) a√2 → Diagonal formula

Q115.

Diagonal of rectangle = ?
आयत की विकर्ण रेखा = ?

A) √(l²+b²) B) l+b C) 2√(l²+b²) D) l²+b²

Answer: A) √(l²+b²) → Pythagoras theorem

Q116.

Area of parallelogram = ?
समांतर चतुर्भुज का क्षेत्रफल = ?

A) base × height B) l+b C) ½×base×height D) 2(l+b)

Answer: A) base × height

Q117.

Area of trapezium = ?
समलंब चतुर्भुज क्षेत्रफल = ?

A) ½ × (sum of parallel sides) × height B) base×height C) 2×base D) ½×base×height²

Answer: A) ½ × (a+b) × h

Q118.

Volume of prism = ?
समान्तर छेद वाला प्रिज्म आयतन = ?

A) base area × height B) ½×base×height C) 2πr² D) πr²h

Answer: A) base area × height

Q119.

Area of rhombus = ?
समलंब वर्गाकार क्षेत्रफल = ?

A) ½ × d1 × d2 B) base×height C) d1×d2 D) πr²

Answer: A) ½ × d1 × d2 → d1,d2=diagonals

Q120.

Surface area of cube = ?
घन का क्षेत्रफल = ?

A) 6a² B) 4a² C) 2a² D) a²

Answer: A) 6a² → 6 faces