Algebra identities
- (a + b)²a² + 2ab + b²
- (a − b)²a² − 2ab + b²
- a² − b²(a + b)(a − b)
- (a + b + c)²a² + b² + c² + 2ab + 2bc + 2ca
- (a + b)³a³ + 3a²b + 3ab² + b³
- (a − b)³a³ − 3a²b + 3ab² − b³
- a³ + b³(a + b)(a² − ab + b²)
- a³ − b³(a − b)(a² + ab + b²)
- a³ + b³ + c³ − 3abc(a + b + c)(a² + b² + c² − ab − bc − ca)
- (a + b + c)³a³ + b³ + c³ + 3(a + b)(b + c)(c + a)
- a⁴ + b⁴(a² + b²)² − 2a²b²
- a⁴ − b⁴(a² + b²)(a + b)(a − b)
- 1² + 2² + … + n²n(n + 1)(2n + 1) / 6
- 1³ + 2³ + … + n³[n(n + 1) / 2]²
- x² − (a + b)x + abQuadratic with roots a and b
- Quadratic rootsx = [−b ± √(b² − 4ac)] / (2a)
- DiscriminantD = b² − 4ac (D > 0: real distinct, D = 0: equal, D < 0: imaginary)
- Sum of rootsFor ax² + bx + c = 0 ⇒ −b / a
- Product of rootsFor ax² + bx + c = 0 ⇒ c / a
Number system basics
- Prime factor form N = p₁ᵃ · p₂ᵇ · p₃ᶜ …
- Total factors If N = p₁ᵃ · p₂ᵇ · p₃ᶜ … then total factors = (a + 1)(b + 1)(c + 1)…
- Sum of factors Product of (1 + p + p² + … + pᵃ) over all prime factors pᵃ.
- Factorial n! = 1 · 2 · 3 · … · n
- Trailing zeros in n! ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
- Sum 1 to n 1 + 2 + … + n = n(n + 1) / 2
- Sum of first n evens 2 + 4 + … + 2n = n(n + 1)
- Sum of first n odds 1 + 3 + … + (2n − 1) = n²
- Sum of first n squares 1² + 2² + … + n² = n(n + 1)(2n + 1) / 6
- Sum of first n cubes 1³ + 2³ + … + n³ = [n(n + 1) / 2]²
- AP nth term Tₙ = a + (n − 1)d
- AP sum Sₙ = n/2 · [2a + (n − 1)d] = n(a + l)/2
- GP nth term Tₙ = arⁿ⁻¹
- GP sum (n terms) Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1
- Infinite GP sum S = a/(1 − r), |r| < 1
- HCF × LCM For two numbers: HCF × LCM = product of the numbers.
- HCF from primes Product of common primes with minimum power.
- LCM from primes Product of all primes with maximum power.
- Even / odd form Even = 2k, odd = 2k + 1, where k is any integer.
- Perfect square test In prime factorization all powers even ⇒ number is a perfect square.
Percentage & profit–loss
- Percentage value P% of X = (P × X) / 100
- Fraction to percent x / y ⇒ (x ÷ y) × 100%
- Percent increase new value = old × (1 + P / 100)
- Percent decrease new value = old × (1 − P / 100)
- Successive x% and y% net% ≈ x + y + (x·y / 100)
- Growth (compound) After n years: value = P(1 + R/100)ⁿ
- Depreciation After n years: value = P(1 − R/100)ⁿ
- Profit / loss profit = SP − CP, loss = CP − SP
- Profit percent profit% = (profit / CP) × 100
- Loss percent loss% = (loss / CP) × 100
- SP for profit% SP = CP × (1 + P / 100)
- SP for loss% SP = CP × (1 − L / 100)
- CP from profit% CP = SP × 100 / (100 + P)
- CP from loss% CP = SP × 100 / (100 − L)
- Discount discount = MP − SP
- Discount percent discount% = (discount / MP) × 100
- Successive discounts equivalent% = x + y − (x·y / 100)
- A is x% more than B A = B(1 + x/100)
- A is x% less than B A = B(1 − x/100)
- Reverse percentage If A is x% more than B, then B is [x·100 / (100 + x)]% less than A.
- Price vs quantity Price increases by x% ⇒ quantity for same cost = original × 100 / (100 + x).
Simple & compound interest
- Simple interest SI = (P · R · T) / 100
- Amount (SI) A = P + SI
- Rate (SI) R = (100 · SI) / (P · T)
- Time (SI) T = (100 · SI) / (P · R)
- Amount (CI – yearly) A = P (1 + R/100)ᵀ
- Compound interest CI = A − P
- CI – half‑yearly A = P (1 + R/200)²ᵀ
- CI – quarterly A = P (1 + R/400)⁴ᵀ
- Different rates each year A = P (1 + R₁/100)(1 + R₂/100)(1 + R₃/100)…
- CI − SI (2 years) CI − SI = P (R² / 100²)
- CI − SI (3 years) CI − SI = P (R² / 100²)(1 + R/100)
Mensuration (2D & 3D)
- Rectangle Perimeter = 2(l + b), Area = l·b, Diagonal = √(l² + b²)
- Square Perimeter = 4a, Area = a², Diagonal = a√2
- Triangle (base, height) Area = ½ · base · height
- Triangle (Heron) Area = √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2
- Equilateral triangle Area = (√3/4)a², Height = (√3/2)a
- Parallelogram Area = base × height
- Trapezium Area = ½ (sum of parallel sides) × height
- Circle Circumference = 2πr, Area = πr²
- Sector (θ°) Area = (θ/360)πr², Arc length = (θ/360)·2πr
- Ring / annulus Area = π(R² − r²)
- Cuboid Volume = l·b·h, TSA = 2(lb + bh + hl)
- Cube Volume = a³, TSA = 6a², Diagonal = a√3
- Cylinder Volume = πr²h, CSA = 2πrh, TSA = 2πr(h + r)
- Cone Volume = (1/3)πr²h, Slant height l = √(r² + h²), CSA = πrl
- Sphere Volume = (4/3)πr³, Surface area = 4πr²
- Hemisphere Volume = (2/3)πr³, CSA = 2πr², TSA = 3πr²
- Frustum of cone Volume = (1/3)πh(R² + r² + Rr)
Ratio, proportion & averages
- Ratio a : b Represents a/b; multiplying both terms by same factor keeps ratio same.
- Proportion a : b = c : d ⇒ a/b = c/d ⇒ ad = bc
- Mean proportional Between a and b: x = √(ab)
- Third proportional a : b = b : x ⇒ x = b² / a
- Fourth proportional a : b = c : x ⇒ x = (b·c) / a
- Simple average Average = (sum of observations) / (number of observations)
- Combined average If a₁ with n₁ items, a₂ with n₂ items ⇒ avg = (n₁a₁ + n₂a₂)/(n₁ + n₂)
- Weighted average avg = (Σxᵢwᵢ) / (Σwᵢ)
- Mixture mean price Cheaper : dearer = (D − M) : (M − C)
- Partnership Profit share ∝ capital × time invested
- Speed vs time For fixed distance: speed ∝ 1 / time
Time, work & speed
- Work relation Work = Rate × Time; rate for unit work = 1 / time.
- A & B together 1/T = 1/T₁ + 1/T₂ ⇒ T = (T₁T₂)/(T₁ + T₂)
- A, B, C together 1/T = 1/T₁ + 1/T₂ + 1/T₃
- Efficiency vs time Time ∝ 1/efficiency; if efficiency ratio a : b, time ratio b : a.
- Filling pipe Rate = 1 / time to fill; empty pipe rate is taken as negative.
- Speed basics Speed = Distance / Time; Distance = Speed × Time.
- Unit conversion 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h.
- Average speed (equal distance) Vavg = 2xy / (x + y)
- Train crosses pole Time = train length / speed
- Train crosses platform Time = (train length + platform length) / speed
- Two trains opposite Relative speed = S₁ + S₂
- Two trains same direction Relative speed = |S₁ − S₂|
- Downstream speed vdown = vboat + vstream
- Upstream speed vup = vboat − vstream
- Boat speed vboat = (vdown + vup) / 2
- Stream speed vstream = (vdown − vup) / 2
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