Every Formula, One Place
174 formulas, theorems, and shortcuts for Class 6 to 12 and JEE — searchable, free, no sign-up.
Logarithm Product Rule
log(x × y) = log x + log y
Example: log(6) = log(2×3) = log 2 + log 3 ≈ 0.778
Multiplication inside the log becomes addition outside.
Logarithm Quotient Rule
log(x / y) = log x - log y
Example: log(5) = log(10/2) = 1 - 0.301 = 0.699
Division inside becomes subtraction outside.
Logarithm Power Rule
log(x^y) = y × log x
Example: log(8) = log(2^3) = 3 × log 2 = 0.903
The exponent comes down and multiplies the log.
Reciprocal of Logarithm
log_a(b) = 1 / log_b(a)
Example: log_2(8) = 3 and log_8(2) = 1/3
Swapping base and argument flips the log to its reciprocal.
Logarithm Base Change Formula
log_b(a) = log_c(a) / log_c(b)
Example: log_2(10) = log(10)/log(2) = 1/0.301 ≈ 3.32
New log of argument over new log of base.
Antilogarithm Identity
a^(log_a(x)) = x
Example: 10^(log_10(5)) = 5. Log and exponential with same base cancel.
Log and exponential with same base are inverse operations.
Radian-Degree Conversion
π radians = 180°. To degrees→radians: × π/180. To radians→degrees: × 180/π.
Example: 90° = π/2 radians. π/3 radians = 60°.
π = 180° is the single most important conversion.
Pythagorean Trigonometric Identities
sin²θ + cos²θ = 1 sec²θ = 1 + tan²θ csc²θ = 1 + cot²θ
Example: If sinθ = 3/5, cos²θ = 1 - 9/25 = 16/25, cosθ = 4/5.
Memorise the first: sin²+cos²=1. The others come from dividing by cos²θ or sin²θ.
CAST Rule — Signs of Trig Functions by Quadrant
sinθ ≥ 0 in Q1 and Q2. cosθ ≥ 0 in Q1 and Q4. tanθ ≥ 0 in Q1 and Q3.
Example: θ=150° (Q2): sin>0, cos<0, tan<0.
All Students Take Calculus — All positive (Q1), Sin (Q2), Tan (Q3), Cos (Q4).
Angle Addition and Subtraction Formulas
sin(θ±φ) = sinθ cosφ ± cosθ sinφ cos(θ±φ) = cosθ cosφ ∓ sinθ sinφ tan(θ±φ) = (tanθ ± tanφ) / (1 ∓ tanθ tanφ)
Example: sin(75°) = sin(45°+30°) = (√6+√2)/4
sin addition: same-cross-cross-same (sincos ± cossin).
Double Angle Formulas
sin(2θ) = 2sinθ cosθ cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ tan(2θ) = 2tanθ / (1 - tan²θ)
Example: sinθ=3/5, cosθ=4/5: sin(2θ)=24/25, cos(2θ)=7/25.
cos(2θ) has THREE forms — pick matching what's given.
Triple Angle Formulas
sin(3θ) = 3sinθ - 4sin³θ cos(3θ) = 4cos³θ - 3cosθ tan(3θ) = (3tanθ - tan³θ) / (1 - 3tan²θ)
Example: sin(90°)=1. Verify: 3sin30° - 4sin³30° = 3/2 - 1/2 = 1 ✓
sin3θ: '3 minus 4 cubed'. cos3θ: '4 cubed minus 3'.
Sum-to-Product Formulas
sinθ + sinφ = 2sin[(θ+φ)/2]cos[(θ-φ)/2] sinθ - sinφ = 2cos[(θ+φ)/2]sin[(θ-φ)/2] cosθ + cosφ = 2cos[(θ+φ)/2]cos[(θ-φ)/2] cosθ - cosφ = 2sin[(θ+φ)/2]sin[(φ-θ)/2]
Example: sin75° + sin15° = 2sin45°cos30° = √6/2
sin+sin → 2·sin(avg)·cos(half-diff).
General Solutions of Trig Equations
sinA = sinB ⟹ A = nπ + (-1)^n × B cosA = cosB ⟹ A = 2nπ ± B tanA = tanB ⟹ A = nπ + B
Example: sinx = sin(π/6): x = nπ + (-1)^n(π/6). n=0: x=π/6. n=1: x=5π/6.
sin uses (-1)^n. cos uses ±. tan is simplest — just add nπ.
Domain and Range of Inverse Trig Functions
arcsin x: domain [-1,1], range [-π/2,π/2] arccos x: domain [-1,1], range [0,π] arctan x: domain (-∞,∞), range (-π/2,π/2)
Example: arcsin(1)=π/2. arccos(-1)=π.
arcsin and arctan: range ±π/2. arccos: 0 to π.
Quadratic Formula
For ax²+bx+c=0: x = (-b ± √(b²-4ac)) / (2a) D = b²-4ac (discriminant)
Example: x²-5x+6=0: D=1. x=(5±1)/2 → x=3 or 2.
Negative b, plus or minus root of (b²-4ac), all over 2a.
Sum and Product of Roots (Vieta's Formulas)
For ax²+bx+c=0 with roots α,β: α+β = -b/a α×β = c/a
Example: 2x²-7x+3=0: α+β=7/2, αβ=3/2.
Sum = -b/a (mind the minus!). Product = c/a.
Discriminant Analysis for Nature of Roots
D = b²-4ac D>0 → Distinct real roots D=0 → Equal real roots D<0 → Complex (imaginary) roots
Example: x²+x+1=0: D=-3<0 → complex. x²-2x+1=0: D=0 → x=1 (repeated).
Positive D→2 real. Zero D→1 real. Negative D→no real roots.
AM-GM Inequality
If a,b≥0: (a+b)/2 ≥ √(ab). Equality when a=b.
Example: a=4,b=9: 6.5 ≥ 6 ✓
Arithmetic Mean ≥ Geometric Mean. Equality only when all equal.
Arithmetic Progression — nth Term and Sum
aₙ = a + (n-1)d Sₙ = (n/2)[2a + (n-1)d] = (n/2)(a + aₙ)
Example: AP 3,7,11: a=3,d=4. a₅=19. S₅=55.
Sum has TWO forms: use (n/2)(a+last) when last term is known.
Finding a and d from General Term
If aₙ = An+B → first term a = A+B, common difference d = A If Sₙ = An²+Bn → first term a = A+B, d = 2A
Example: aₙ=3n+2: a=5, d=3.
For aₙ=An+B: d is the coefficient of n.
Geometric Progression — nth Term and Sum
aₙ = a × r^(n-1) Sₙ = a(r^n-1)/(r-1) where r≠1 S∞ = a/(1-r) where |r|<1
Example: GP 2,6,18: a=2,r=3. a₅=162. S₄=80.
Infinite GP sum exists only when |r|<1.
Harmonic Progression and Harmonic Mean
nth term of HP: aₙ = 1/[a+(n-1)d] HM of a and b: HM = 2ab/(a+b)
Example: HP 1/3,1/5,1/7... 4th term=1/9.
For HP: flip to AP, solve, flip back.
AM-GM-HM Inequality
For positive numbers: AM ≥ GM ≥ HM. Equality when all numbers are equal. For two: (a+b)/2 ≥ √(ab) ≥ 2ab/(a+b) Also: GM² = AM × HM
Example: a=4,b=9: AM=6.5, GM=6, HM≈5.54. 6.5≥6≥5.54 ✓
AM≥GM≥HM always for positive numbers.
Sum Formulas for k, k², and k³
Σk = n(n+1)/2 Σk² = n(n+1)(2n+1)/6 Σk³ = [n(n+1)/2]²
Example: Sum 1-10: 55. Squares: 385. Cubes: 55²=3025.
Key insight: Σk³ = (Σk)². Sum of cubes = square of sum of naturals!
Binomial Theorem
(x+y)^n = ΣᵣⁿCᵣ × x^(n-r) × y^r General (r+1)th term: T_(r+1) = ⁿCᵣ × x^(n-r) × y^r
Example: (1+x)^3=1+3x+3x²+x³. T₃ of (x+y)^5: ⁵C₂x³y²=10x³y².
Power of x decreases (n,n-1,...); power of y increases (0,1,2,...).
Complex Conjugate and Basic Relations
z=a+ib → z̄=a-ib z+z̄=2Re(z) z×z̄=|z|²=a²+b²
Example: z=3+4i: z̄=3-4i. z×z̄=25=|z|².
Multiplying by conjugate always gives a real number.
Euler's Formula and Polar Form
e^(iθ) = cosθ + i sinθ Polar form: z = r(cosθ + i sinθ) = re^(iθ) e^(iπ) + 1 = 0
Example: e^(iπ) = cos π + i sin π = -1.
Euler's formula unites e, sin, cos in one equation.
De Moivre's Theorem
z^n = r^n[cos(nθ) + i sin(nθ)] nth roots of z: r^(1/n)[cos{(θ+2πk)/n} + i sin{(θ+2πk)/n}], k=0,1,...,n-1
Example: (cos60°+i sin60°)^3 = cos180°+i sin180° = -1.
Raise modulus to power, multiply argument by power.
Cube Roots of Unity
1, ω, ω² where ω=(-1+i√3)/2 1+ω+ω²=0, 1×ω×ω²=1 ω³=1, ω^4=ω (cycle period 3)
Example: ω^3=1 so ω^100=ω^(99+1)=ω.
Sum=0, product=1. Powers of ω cycle with period 3.
Matrix Operations
Trace: tr(A)=Σaᵢᵢ Addition: [aᵢⱼ]+[bᵢⱼ]=[aᵢⱼ+bᵢⱼ] Multiplication: (AB)ᵢⱼ=Σaᵢₖbₖⱼ Matrix mult is NOT commutative (AB≠BA in general)
Example: Trace of [[1,2],[3,4]]=5.
Trace=diagonal sum. Multiplication requires inner dimensions to match.
Adjugate and Inverse of Matrix
A×adj(A)=|A|×I A^(-1)=(1/|A|)×adj(A) — exists only when |A|≠0 (AB)^(-1)=B^(-1)×A^(-1) (reverse order!)
Example: 2×2: A=[[a,b],[c,d]], A^(-1)=(1/(ad-bc))[[d,-b],[-c,a]].
Inverse of product reverses order.
Determinant Properties
|A|=|A^T| Interchanging rows/cols changes sign Proportional rows → det=0 |AB|=|A||B| Det of skew-symmetric matrix of odd order = 0
Example: det[[1,2,3],[1,2,3],[0,0,1]]=0 (identical rows).
Identical rows→det=0. Swap rows→sign changes.
Cramer's Rule
For AX=B: x=Δ₁/Δ, y=Δ₂/Δ, z=Δ₃/Δ Δ=coefficient determinant Δₖ=replace k-th column with RHS values
Example: x+y=3, x-y=1: Δ=-2, Δ₁=-4. x=2, y=1.
Replace the column of the unknown you want with the RHS values.
Distance, Section, and Polar Formulas
Distance: d=√[(x₁-x₂)²+(y₁-y₂)²] Internal division m:n: ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)) Polar: x=rcosθ, y=rsinθ
Example: (1,3) and (5,7) in 1:3: ((5+3)/4,(7+9)/4)=(2,4).
mx₂+nx₁: multiply each by the OTHER ratio.
Centroid and Area of Triangle
Centroid: G=((x₁+x₂+x₃)/3,(y₁+y₂+y₃)/3) Area=(1/2)|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|
Example: (0,0),(4,0),(0,3): G=(4/3,1). Area=6.
Centroid=average. Area='cross' pattern with coordinates.
Equations of Straight Lines — All Forms
Slope-intercept: y=mx+c Point-slope: y-y₁=m(x-x₁) Two-point: (y-y₁)/(y₂-y₁)=(x-x₁)/(x₂-x₁) Intercept: x/a+y/b=1 General: ax+by+c=0
Example: Through (1,2),(3,6): m=2. y=2x.
Six forms — choose matching what's given.
Distance from Point to Line; Parallel Lines
Point (x₁,y₁) to line ax+by+c=0: d=|ax₁+by₁+c|/√(a²+b²) Between parallels ax+by+c₁=0 and ax+by+c₂=0: d=|c₁-c₂|/√(a²+b²)
Example: (3,4) to 3x-4y+5=0: |9-16+5|/5=2/5.
Plug point into line equation, absolute value, divide by √(a²+b²).
Standard Circle Equations
Standard: (x-h)²+(y-k)²=r² (centre (h,k), radius r) General: x²+y²+2gx+2fy+c=0 (centre (-g,-f), radius √(g²+f²-c)) Parametric: x=h+rcosθ, y=k+rsinθ
Example: x²+y²-6x+4y+4=0: centre(3,-2), r=3.
In general form: centre is (-g,-f) — NEGATIVE of g,f.
Tangents to Circle
Tangent at (a,b) on x²+y²=r²: ax+by=r² With slope m: y=mx±r√(1+m²) Length from (a,b): √(a²+b²-r²)
Example: x²+y²=25, tangent at (3,4): 3x+4y=25.
Point form T=0: replace x² with ax, y² with by.
Parabola y²=4ax — Properties
y²=4ax (rightward): focus (a,0), directrix x=-a Latus rectum=4a Parametric: x=at², y=2at Eccentricity=1
Example: y²=12x: a=3. Focus(3,0), directrix x=-3, LR=12.
y²=4ax opens right. x²=4ay opens up.
Ellipse x²/a²+y²/b²=1 — Properties
e=√(1-b²/a²)∈(0,1) Foci:(±ae,0), Directrices:x=±a/e Latus rectum:2b²/a Parametric: x=acosθ, y=bsinθ
Example: x²/25+y²/9=1: a=5,b=3. e=4/5. Foci(±4,0).
e<1 for ellipse. Larger denominator=major axis direction.
Hyperbola x²/a²-y²/b²=1 — Properties
e=√(1+b²/a²)>1 Foci:(±ae,0), Asymptotes:y=±(b/a)x Latus rectum:2b²/a Parametric: x=asecθ, y=btanθ
Example: x²/9-y²/16=1: a=3,b=4. e=5/3. Foci(±5,0). Asymptotes:y=±(4/3)x.
Hyperbola e>1. Asymptotes are lines it approaches but never touches.
Important Standard Limits
lim(x→0)[sinx/x]=1=lim[tanx/x] lim(x→0)[(eˣ-1)/x]=1 lim(x→0)[ln(1+x)/x]=1 lim(x→0)(1+x)^(1/x)=e lim(x→∞)(1+1/x)^x=e lim(x→a)[(x^n-a^n)/(x-a)]=na^(n-1)
Example: lim[sin(3x)/x]=3×lim[sin(3x)/(3x)]=3.
sinx/x→1 is the fundamental limit. Scale argument and coefficient together.
L'Hospital's Rule
If f(x)/g(x) gives 0/0 or ∞/∞: lim[f(x)/g(x)]=lim[f'(x)/g'(x)] Apply repeatedly until determinate.
Example: lim[sinx/x]: 0/0→lim[cosx/1]=1.
L'Hospital: differentiate numerator and denominator SEPARATELY.
Rules of Differentiation
(fg)'=f'g+fg' (Product) (f/g)'=[gf'-fg']/g² (Quotient) [f(g(x))]'=f'(g(x))×g'(x) (Chain Rule)
Example: d/dx[x²sinx]=2xsinx+x²cosx.
Product: 'diff-first×second PLUS first×diff-second'. Chain: outer at inner × inner'.
Derivatives of Common Functions
(x^n)'=nx^(n-1) (eˣ)'=eˣ (ln x)'=1/x (sinx)'=cosx, (cosx)'=-sinx (tanx)'=sec²x, (cotx)'=-csc²x (arcsinx)'=1/√(1-x²) (arctanx)'=1/(1+x²)
Example: d/dx[x³+sinx+eˣ]=3x²+cosx+eˣ.
arcsin: 1/√(1-x²). arctan: 1/(1+x²) — no square root!
Conditions for Local Maxima/Minima
f'(p)=0 and f''(p)<0 → local MAX at x=p f'(p)=0 and f''(p)>0 → local MIN at x=p
Example: f(x)=x³-3x: f'=3x²-3=0→x=±1. f''(1)=6>0→min. f''(-1)=-6<0→max.
Negative second derivative→maximum (top). Positive→minimum (bottom).
Standard Indefinite Integrals
∫(ax+b)^n dx=(ax+b)^(n+1)/(a(n+1)) ∫1/(ax+b)dx=ln|ax+b|/a ∫eˣdx=eˣ ∫sinxdx=-cosx ∫cosxdx=sinx ∫sec²xdx=tanx ∫tanxdx=ln|secx|
Example: ∫sin(2x+1)dx=-cos(2x+1)/2.
For f(ax+b): integrate normally, divide by 'a'.
Inverse Trig Integrals
∫1/√(a²-x²)dx=arcsin(x/a) ∫1/(a²+x²)dx=(1/a)arctan(x/a) ∫√(a²-x²)dx=(x/2)√(a²-x²)+(a²/2)arcsin(x/a)
Example: ∫1/(4+x²)dx=(1/2)arctan(x/2).
√(a²-x²)→arcsin. 1/(a²+x²)→arctan.
Integration by Parts (ILATE)
∫f(x)g(x)dx=f(x)∫g(x)dx - ∫[f'(x)×∫g(x)dx]dx ILATE priority: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential Special: ∫eˣ[f(x)+f'(x)]dx=eˣf(x)+C
Example: ∫xsinx dx: u=x, v=sinx. = -xcosx+sinx+C.
ILATE: pick u from higher in list. Log and inverse trig: always differentiate them.
Properties of Definite Integrals
∫ₐᵇf(x)dx=∫ₐᵇf(a+b-x)dx (King Property) ∫₋ₐᵃf(x)dx=0 if f is ODD ∫₋ₐᵃf(x)dx=2∫₀ᵃf(x)dx if f is EVEN ∫₀ⁿᵀf(x)dx=n×∫₀ᵀf(x)dx (for periodic T)
Example: ∫₋π^πsinxdx=0 (sin is ODD). ∫₋₁¹x²dx=2/3 (x² is EVEN).
King Property: replace x with (a+b-x). Extremely useful for complex-looking definite integrals.
Dot and Cross Products
a⃗·b⃗=|a||b|cosθ=a₁b₁+a₂b₂+a₃b₃ |a⃗×b⃗|=|a||b|sinθ Perpendicular: a⃗·b⃗=0 Parallel: a⃗×b⃗=0 Area of parallelogram=|a⃗×b⃗|
Example: a⃗=(2,3), b⃗=(1,-1): a·b=2-3=-1.
Dot=scalar (gives number). Cross=vector (gives area/direction).
Scalar Triple Product
[a⃗ b⃗ c⃗]=a⃗·(b⃗×c⃗)=det of 3×3 matrix Volume of parallelepiped=|[a⃗ b⃗ c⃗]| Volume of tetrahedron=|[a⃗ b⃗ c⃗]|/6 Coplanar ↔ [a⃗ b⃗ c⃗]=0
Example: Unit vectors i,j,k: [i j k]=1. Tetrahedron volume=1/6.
STP=volume of box. Zero means flat (coplanar).
3D Geometry — Lines and Planes
Line: (x-x₁)/l=(y-y₁)/m=(z-z₁)/n Plane: ax+by+cz+d=0 Point to plane: |ax₁+by₁+cz₁+d|/√(a²+b²+c²) Skew lines distance: |(b⃗-a⃗)·(p⃗×q⃗)|/|p⃗×q⃗|
Example: 2x+3y+6z=0. Distance from (1,1,1): 11/7.
Same formula as 2D point-to-line but with three terms.
Basic Probability and Bayes
P(A)=favourable/total P(A∪B)=P(A)+P(B)-P(A∩B) P(A|B)=P(A∩B)/P(B) Bayes: P(B|A)=P(B)P(A|B)/P(A)
Example: Bag: 3 red, 2 blue. P(red)=3/5.
Conditional probability: restricted universe.
Binomial Distribution
P(X=x)=C(n,x)×p^x×q^(n-x), q=1-p Mean=np Variance=npq SD=√(npq)
Example: 5 coins, P(3 heads)=C(5,3)(0.5)⁵=10/32=5/16.
'n choose x' ways to arrange x successes.
Mean, Median, Mode for Grouped Data
Mean=A+(Σfᵢdᵢ/Σfᵢ) where dᵢ=xᵢ-A Median: M=l+(n/2-cf)/f×h Mode: Mo=l+(f₁-f₀)/(2f₁-f₀-f₂)×h Empirical: Mode=3Median-2Mean
Example: Mean=10, Median=9: Mode=27-20=7.
Mode=3Median-2Mean (empirical). Works for mildly skewed data.
Section Formula — External Division
Point P dividing AB externally in m:n: x=(mx₂-nx₁)/(m-n), y=(my₂-ny₁)/(m-n)
Example: Ext div 3:1 between (0,0) and (4,4): x=12/2=6. Point(6,6).
Same as internal but use minus signs in both numerator and denominator.
Angle Between Two Lines
tanθ=|(m₁-m₂)/(1+m₁m₂)| Parallel: m₁=m₂ Perpendicular: m₁×m₂=-1
Example: y=2x+1 (m₁=2) and y=-x+3 (m₂=-1): m₁m₂=-2≠-1, not perp.
Perpendicular: product of slopes = -1.
Reflection of a Point about a Line
Reflection of (x₁,y₁) about ax+by+c=0: (x-x₁)/a=(y-y₁)/b=-2(ax₁+by₁+c)/(a²+b²) Foot of perpendicular: use -1 instead of -2
Foot=one step to line. Reflection=two steps (double distance).
Pair of Lines Through Origin
ax²+2hxy+by²=0 (combined equation) tanθ=2√(h²-ab)/(a+b) Perpendicular pair: a+b=0 Lines coincide: h²=ab
No constant term=both lines through origin. Perpendicular: sum of x² and y² coefficients=0.
Circle: Intercepts and Tangent Forms
x-intercept=2√(g²-c), y-intercept=2√(f²-c) Tangent at (x₁,y₁): xx₁+yy₁=a² (T=0) Length of tangent from (x₁,y₁): √(x₁²+y₁²-a²)
Example: From (5,0) to x²+y²=9: tangent length=√(25-9)=4.
T=0 trick: replace x² with xx₁, y² with yy₁.
Tangent to Parabola y²=4ax
Slope form: y=mx+a/m At point t: ty=x+at² Normal at t: y+tx=2at+at³
Example: y²=4x(a=1), tangent at t=2: 2y=x+4.
Tangent slope form y=mx+a/m — the extra a/m distinguishes it.
Tangent to Ellipse
Slope form: y=mx±√(a²m²+b²) Point form at (x₁,y₁): xx₁/a²+yy₁/b²=1 Normal at (x₁,y₁): a²x/x₁-b²y/y₁=a²-b²
Point form: same T=0 trick — replace x² with xx₁, y² with yy₁, divide by a² and b².
Tangent to Hyperbola
Slope form: y=mx±√(a²m²-b²) [requires a²m²>b²] Point form at (x₁,y₁): xx₁/a²-yy₁/b²=1
Hyperbola tangent at (x₁,y₁): same as ellipse but MINUS sign.
Definition of Limit and LHL/RHL
lim(x→a)f(x)=M if LHL=RHL=M LHL=lim(h→0)f(a-h) RHL=lim(h→0)f(a+h) If LHL≠RHL, limit does not exist
Left approach and right approach must agree for limit to exist.
Types of Discontinuity
Missing Point: limit exists but f(p) undefined Isolated: limit≠f(p) Jump: LHL≠RHL (both finite) Infinite: LHL or RHL→∞ Oscillatory: sin(1/x) as x→0
Example: f(x)=sinx/x: missing point at x=0. Define f(0)=1 to make continuous.
For continuity: limit exists AND equals function value AND function is defined.
Rolle's Theorem
If f is continuous on [a,b], differentiable on (a,b), f(a)=f(b): ∃c∈(a,b) with f'(c)=0
Example: f(x)=x²-4x+3 on [1,3]: f'(c)=0→c=2.
Same height at ends→must have a flat spot somewhere in between.
Lagrange's Mean Value Theorem (LMVT)
If f is continuous on [a,b], differentiable on (a,b): ∃c∈(a,b) with f'(c)=(f(b)-f(a))/(b-a)
Example: f(x)=x² on [1,3]: f'(c)=4→c=2.
Instantaneous speed=average speed at some moment. Like checking a speeding car.
Taylor/Maclaurin Series
eˣ=1+x+x²/2!+x³/3!+... ln(1+x)=x-x²/2+x³/3-... (|x|≤1) sinx=x-x³/3!+x⁵/5!-... cosx=1-x²/2!+x⁴/4!-...
sinx: odd powers only. cosx: even powers only. eˣ: all powers.
Leibnitz Theorem (Differentiation Under Integral)
If F(x)=∫_{g(x)}^{h(x)}f(t)dt: dF/dx=h'(x)f(h(x))-g'(x)f(g(x))
Example: F(x)=∫₀^{x²}sint dt → F'(x)=2x sin(x²).
Upper limit: derivative×f at upper. MINUS lower limit: derivative×f at lower.
Completing the Square for Integration
For ∫dx/(ax²+bx+c): complete the square, sub t=x+b/(2a), reduce to standard form.
Example: ∫dx/(x²+4x+5): x²+4x+5=(x+2)²+1. t=x+2→∫dt/(t²+1)=arctan(x+2)+C.
Complete the square: add (b/2a)² inside the quadratic. Sub t=x+b/2a.
Walli's Formula
∫₀^(π/2) sinⁿx dx = ∫₀^(π/2) cosⁿx dx Even n: [(n-1)!!/ n!!]×π/2 Odd n: [(n-1)!!/ n!!]
Example: ∫₀^(π/2) sin⁴x dx=(1×3)/(2×4)×π/2=3π/16.
Even→multiply odds over evens, add π/2. Odd→multiply evens over odds, no π/2.
Orthogonality and Radical Axis
Two circles orthogonal if: 2g₁g₂+2f₁f₂=c₁+c₂ Radical axis: S₁-S₂=0 Family through intersection: S₁+kS₂=0
Radical axis: subtract one circle from the other — squared terms cancel leaving a straight line.
Chord of Contact
From (x₁,y₁) to x²+y²=a²: T=0 (xx₁+yy₁=a²) Chord length=2LR/√(R²+L²), L=tangent length, R=radius Area of triangle from tangents: RL³/(R²+L²)
Chord of contact uses T=0. Results involve L (tangent length) and R (radius).
Algebraic Identities — Complete Reference
(a+b)²=a²+2ab+b² (a-b)²=a²-2ab+b² (a+b)(a-b)=a²-b² (a+b)³=a³+3a²b+3ab²+b³ a³+b³=(a+b)(a²-ab+b²) a³-b³=(a-b)(a²+ab+b²) (a+b+c)²=a²+b²+c²+2(ab+bc+ca) a³+b³+c³-3abc=(a+b+c)(a²+b²+c²-ab-bc-ca)
Example: If a+b+c=0: a³+b³+c³=3abc.
If three numbers sum to zero, their cubes sum to three times their product!
Roots Under Special Coefficient Conditions
For ax²+bx+c=0: c=0 → one root is 0 b=0 → roots are ±√(-c/a) a=c → roots are reciprocals a+b+c=0 → one root is 1, other is c/a a-b+c=0 → one root is -1, other is -c/a
Example: x²-5x+4=0: a+b+c=0, so x=1 is a root. Other=c/a=4.
MOST USEFUL: Check a+b+c=0 first. If yes, x=1 is always a root!
Location of Roots of Quadratic
For f(x)=ax²+bx+c (a>0): Both roots>k: D≥0, f(k)>0, vertex>k k lies between roots: f(k)<0 One root in (k₁,k₂): f(k₁)×f(k₂)<0
If k is BETWEEN roots→parabola dips below x at k→f(k)<0.
Pythagoras Theorem
In right triangle with hypotenuse c: a²+b²=c² Common triplets: (3,4,5),(5,12,13),(8,15,17),(7,24,25)
Example: Ladder 13m, foot 5m from wall: height=√(169-25)=12m.
Hypotenuse²=sum of other two squares. Hypotenuse is ALWAYS longest.
Triangle Similarity — Tests
AA: two equal angles SSS: all sides proportional SAS: two sides proportional + included angle equal Ratio of areas=k² when sides in ratio k
Example: Similar triangles sides 3:5. Areas=9:25.
Linear ratio→square for area. Sides 3:5→areas 9:25.
BPT — Basic Proportionality Theorem
Line parallel to one side divides other two in same ratio. DE||BC in △ABC: AD/DB=AE/EC
Example: AD=3,DB=2,AE=6: EC=4.
Parallel line=proportional division. Converse also true.
Triangle — Area Formulas
A=(1/2)×base×height Heron's: A=√[s(s-a)(s-b)(s-c)], s=(a+b+c)/2 A=(1/2)ab sinC A=abc/(4R) (R=circumradius) A=r×s (r=inradius)
Example: 3-4-5 triangle: s=6. A=√(6×3×2×1)=6.
Five ways to find area. Heron's best when only sides given.
Circle Theorems — Key Results
Angle in semicircle=90° Central angle=2×inscribed angle (same arc) Angles in same segment are equal Tangent⊥radius at contact Two tangents from external point are equal
Example: Arc AB→80° central. Inscribed angle=40°.
Central angle=2×inscribed. Tangent always perpendicular to radius.
Heron's Formula and Triangle Properties
s=(a+b+c)/2 Area=√[s(s-a)(s-b)(s-c)] Inradius r=Area/s Circumradius R=abc/(4×Area) Median: mₐ=(1/2)√(2b²+2c²-a²)
Example: 7,8,9: s=12. A=√(12×5×4×3)=12√5.
r=Area/s. R=abc/4A. Median formula: double two squares, subtract one, root, halve.
Regular Polygon Properties
Exterior angle=360°/n Interior angle=180(n-2)/n Sum of interior angles=180(n-2)° Diagonals=n(n-3)/2
Example: Pentagon(n=5): exterior=72°, interior=108°, diagonals=5.
Exterior=360/n. Interior=180-exterior. Diagonals=n(n-3)/2.
Vector Cross Product
a⃗×b⃗: determinant with i,j,k in first row |a⃗×b⃗|=|a||b|sinθ î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ Area of parallelogram=|a⃗×b⃗|
Example: a⃗=(1,0,0),b⃗=(0,1,0): a×b=k̂.
Cross product=area vector. Zero means parallel. Use 3×3 determinant.
Rationalizing Surds — Conjugate Trick
1/(√a-√b) = (√a+√b)/(a-b) Multiply by conjugate to remove surd from denominator.
Example: 1/(√5-2)=(√5+2)/1=√5+2≈4.236.
Conjugate of (a-√b) is (a+√b). Multiply top and bottom by it.
Surd Equality Rule
If a+√b=c+√d (rational a,c and surd parts), then a=c AND b=d. Rational parts equal; irrational parts equal.
Example: If x+√(y+1)=3+√5: x=3, y=4.
Rational=Rational, Irrational=Irrational. Never cancel rational with irrational.
Factorial, nPr, nCr
n! = n×(n-1)×...×1, 0!=1 nPr=n!/(n-r)! (ordered) nCr=n!/[(n-r)!×r!] (unordered) nCr=nC(n-r)
Example: 5P3=60. 5C3=10.
P=Position matters. C=Choice only. nCr=nPr/r!
Partitioning and Stars & Bars
n identical items into r distinct slots (can be empty): (n+r-1)C(r-1) n identical into r slots (each ≥1): (n-1)C(r-1) n distinct into r slots (any number): r^n
Example: 5 balls into 3 boxes (empty allowed): 7C2=21.
Identical→Stars & Bars. Distinct→r choices for each item=r^n.
Circular Arrangements
n items around a circle: (n-1)! ways Necklace (flippable): (n-1)!/2 Derangements: D(n)=n!×(1/0!-1/1!+1/2!-...+(−1)^n/n!)
Example: 4 people circular: 3!=6. D(3)=2.
Circular: fix one, arrange rest=(n-1)!. Derangement≈n!/e (≈37% of permutations).
Simple Interest
SI=(P×T×R)/100 Amount=P+SI
Example: P=₹2000,R=5%,T=3yr: SI=₹300, Amount=₹2300.
SI=PTR/100. In SI, interest always on original principal.
Compound Interest
Amount=P×(1+R/100)^N CI=Amount-P Half-yearly: P×(1+R/200)^(2N) For same P,R,T>1: CI>SI always
Example: P=₹1000,R=10%,N=2: Amount=₹1210. CI=₹210 vs SI=₹200.
CI multiplies every year; SI adds. After 2+ years, CI always wins.
CI-SI Difference Shortcut
CI-SI (2 years)=P×(R/100)² From difference: P=(CI-SI)×(100/R)²
Example: P=₹10000,R=10%: CI-SI=10000×0.01=₹100.
2-year CI-SI=P×(R%)². The extra CI is interest on first year's interest.
Profit, Loss, and Discount
Profit%=(Profit/CP)×100 Loss%=(Loss/CP)×100 Discount%=(Discount/MP)×100 SP=CP×(100+P%)/100 or ×(100-L%)/100
Example: CP=₹400,P=15%: SP=₹460.
Profit/Loss%→always over CP. Discount%→always over MP. Different bases!
Two Successive Discounts
Effective discount=a+b-ab/100 % Buy x get y free: discount%=y/(x+y)×100
Example: 30%+20%: effective=30+20-6=44%. Buy 3 get 1 free: 25%.
Add discounts, subtract their product/100. Always less than the sum!
Same SP at Equal P% and L%
If sold at same SP, one at P% profit and other at P% loss: Always a LOSS. Loss%=P²/100
Example: Sold at ₹300 each, 20% profit and 20% loss: Loss%=4%.
Same selling price + equal P% and L% = ALWAYS a loss!
Set Operations and De Morgan's Laws
n(A∪B)=n(A)+n(B)-n(A∩B) De Morgan's: (A∪B)'=A'∩B', (A∩B)'=A'∪B' Power set: 2^n subsets if |A|=n
Example: 30 like cricket, 25 like football, 10 like both. n(C∪F)=45. Neither=50-45=5.
De Morgan: flip ∪↔∩ and distribute complement.
Alligation Rule
Cheaper C and dearer D mixed at mean M: Qty(cheaper)/Qty(dearer)=(D-M)/(M-C)
Example: Milk ₹30/L + water ₹0/L → ₹20/L: ratio=10/20=1:2.
Cross subtract: draw X, put mean in centre, subtract diagonally.
Repeated Replacement (Mixture)
After n operations removing b from total a and replacing: Original fraction remaining=((a-b)/a)^n
Example: 100L milk, remove 20L replace water 3 times: milk=100×(0.8)³=51.2L.
Each replacement keeps (a-b)/a fraction. Raise to n.
Volume and Surface Area — All 3D Shapes
Cube: V=a³, TSA=6a² Cuboid: V=lbh, TSA=2(lb+bh+hl) Cylinder: V=πr²h, CSA=2πrh Cone: V=(1/3)πr²h, CSA=πrl, l=√(r²+h²) Sphere: V=(4/3)πr³, SA=4πr² Hemisphere: V=(2/3)πr³, TSA=3πr² Frustum: V=(π/3)h(R²+Rr+r²)
Example: Sphere r=3: V=36π, SA=36π.
Cone and pyramid=1/3 of cylinder/prism. Sphere SA=4 circles of radius r.
2D Areas — Complete Reference
Triangle=(1/2)bh Rhombus/Kite=(1/2)d₁d₂ Trapezoid=(1/2)(sum of parallels)×h Circle=πr² Sector=(θ/360)×πr² Segment=Sector area-Triangle area Equilateral △=(√3/4)a²
Example: Trapezoid: parallels 8,12; h=5: A=50.
Rhombus and kite: half product of diagonals. Circle sector: fraction of full circle.
Perfect Squares — Form 3n or 4n
Square of any natural number is 3n or 3n+1 (NEVER 3n+2). Also always 4n or 4n+1 (NEVER 4n+2 or 4n+3).
Example: If remainder on dividing by 3 is 2 → NOT a perfect square!
Use to eliminate MCQ options instantly. Remainder 2 mod 3 = impossible square.
Last Digits of Perfect Squares
Perfect square can ONLY end in 0,1,4,5,6,9. NEVER ends in 2,3,7,8. If ends in 5, second-last digit MUST be 2.
Example: 1237 ends in 7 → impossible square. 625: ends 5, second-last 2 ✓. 25²=625.
2,3,7,8 at the end = instant NO. Fastest MCQ elimination.
HCF × LCM = Product of Two Numbers
For two positive integers a,b: HCF(a,b)×LCM(a,b)=a×b Warning: does NOT extend to three numbers.
Example: 12 and 18: HCF=6, LCM=36. 6×36=216=12×18 ✓
HCF and LCM product equals number product. Only for exactly 2 numbers.
Number of Factors of N
N=x^a×y^b×z^c: Total factors=(a+1)(b+1)(c+1) Even factors=a×(b+1)(c+1) [where 2^a is power of 2] Odd factors=(b+1)(c+1) [ignore 2^a entirely]
Example: 72=2³×3²: factors=(4)(3)=12. Odd factors=3.
Add 1 to each exponent, multiply. Odd factors: ignore 2 and its power.
Euler's Totient φ(N)
φ(N)=N×(1-1/p₁)×(1-1/p₂)×... for each prime factor pᵢ Euler's Theorem: if gcd(M,N)=1, M^φ(N)≡1(mod N)
Example: φ(12)=12×(1/2)×(2/3)=4. Co-prime numbers: {1,5,7,11}.
For each prime p dividing N, multiply N by (p-1)/p.
Fermat's Little Theorem
For prime p and gcd(M,p)=1: M^(p-1)≡1(mod p) Equivalently M^p≡M(mod p)
Example: 2^100÷101 (101 prime): 2^100≡1(mod101). Remainder=1.
Exponent (prime-1) reduces to 1. Use to shrink huge powers mod prime.
Cyclicity — Last Digit of a^n
Digit 1,5,6: cycle 1 (always same) Digit 4,9: cycle 2 Digit 2,3,7,8: cycle 4 Divide power by cyclicity, use remainder to find position.
Example: 7^53: cycle=4. 53mod4=1. 7^1=7. Last digit=7.
0,1,5,6: never change. 4,9: 2-cycle. 2,3,7,8: 4-cycle.
Divisibility Rules
÷2: last digit even ÷3: digit sum ÷3 ÷4: last 2 digits ÷4 ÷5: ends in 0 or 5 ÷6: even AND digit sum ÷3 ÷8: last 3 digits ÷8 ÷9: digit sum ÷9 ÷11: alternating sum of digits=0 or multiple of 11
Example: 73194: 4-9+1-3+7=0 → divisible by 11 ✓
2,4,8→last 1,2,3 digits. 3,9→digit sums. 11→alternating sum.
Highest Power of Prime in n!
Highest power of prime p in n!: [n/p]+[n/p²]+[n/p³]+... Trailing zeros in n! = highest power of 5 (since 2s>5s)
Example: Highest power of 5 in 100!: 20+4+0=24 trailing zeros.
Divide n by prime repeatedly, add quotients.
Speed, Distance, Time — Core
D=S×T, S=D/T, T=D/S 1 km/h=5/18 m/s 1 m/s=18/5=3.6 km/h Same distance avg speed=HM=2s₁s₂/(s₁+s₂) Same time avg speed=AM=(s₁+s₂)/2
Example: 60+40 avg (same dist)=48 not 50.
Same distance→HM (lower). Same time→AM (middle).
Time and Work
If A takes 'a' days, rate=1/a per day. A+B together: ab/(a+b) days A,B,C together: abc/(ab+bc+ca) days
Example: A=12d, B=18d: together=216/30=7.2d.
Together=product÷sum. Rates add like fractions.
Boats and Streams
Downstream=boat+current Upstream=boat-current Boat speed=(downstream+upstream)/2 Stream speed=(downstream-upstream)/2
Example: Downstream=18, upstream=10: boat=14, stream=4.
Current helps downstream (+), fights upstream (-). Average=still water speed.
Variance and Standard Deviation
Variance σ²=Σxₖ²/n-x̄² Standard deviation=√(Variance) Key: Variance=mean of squares MINUS square of mean σ²=E(x²)-[E(x)]²
Example: 2,4,6,8: x̄=5. V=(4+16+36+64)/4-25=5. SD=√5.
Mean of squares minus square of mean. Easy to remember as E(x²)-[E(x)]².
Circular Track and Meeting Times
Track L, speeds a>b: Same direction: first meet=L/(a-b) Opposite direction: first meet=L/(a+b) First meet at START=LCM(L/a,L/b)
Example: 600m track, 10+6 m/s: opposite=600/16=37.5s.
Opposite→add speeds. Same→subtract speeds.
Fraction-to-Percentage (Must Know)
1/2=50%, 1/3≈33.33%, 1/4=25%, 1/5=20% 1/6≈16.67%, 1/8=12.5%, 1/9≈11.11%, 1/10=10% 3/4=75%, 2/3≈66.67%, 1/7≈14.29%
Example: 37.5% of 80: recognise 37.5%=3/8. Answer=3/8×80=30. Much faster than 80×0.375.
Convert % to its fraction equivalent — fraction×number always beats long decimal multiplication.
Assumed Mean Shortcut for Average
Choose any convenient assumed mean (A). Deviation of each element: dᵢ = xᵢ - A True mean = A + (Σdᵢ / n) For frequency data: x̄ = A + Σfᵢdᵢ / Σfᵢ
Example: Find avg of 197, 203, 200, 196, 204. Assume A=200. Deviations: -3,+3,0,-4,+4. Sum=0. Mean=200+0=200.
Pick a round number close to the data. Small deviations are far easier than working with the full large values.
Sum of Deviations from Mean = Zero
For any dataset, the sum of deviations of each value from the arithmetic mean is always zero: (x₁-x̄) + (x₂-x̄) + ... + (xₙ-x̄) = 0
Example: Data: 3,5,7. Mean=5. Deviations: -2+0+2=0 ✓
The mean is the balance point of the data — like a seesaw. Positive and negative deviations always cancel perfectly.
Population Growth and Depreciation
Growth: P' = P × (1 + r/100)^n Depreciation: V = V₀ × (1 - r/100)^n Same structure as compound interest — growth multiplies by (1+r/100), decay by (1-r/100).
Example: Town 10,000 grows at 5% for 2yr: 10000×(1.05)²=11025. Machine ₹50,000 depreciates 10%/yr for 3yr: 50000×(0.9)³=₹36,450.
Growth: × (1+r/100). Depreciation: × (1-r/100). Same formula — just + or -.
CAGR — Compound Annual Growth Rate
CAGR = (Final Value / Initial Value)^(1/n) - 1 where n = number of years. CAGR is always ≤ AAGR (simple average annual growth rate).
Example: ₹10,000 grew to ₹14,641 in 4 years: CAGR=(1.4641)^0.25-1=0.10=10%.
CAGR = the constant annual rate that gets you from start to finish. For multi-year periods, CAGR < simple average rate.
Componendo-Dividendo
If a/b = c/d, then: Componentdo: (a+b)/b = (c+d)/d Dividendo: (a-b)/b = (c-d)/d Componendo-Dividendo: (a+b)/(a-b) = (c+d)/(c-d)
Example: x/y=3/4. Comp-Div: (x+y)/(x-y)=(3+4)/(3-4)=7/(-1)=-7.
Componendo: add denominator to numerator. Dividendo: subtract. Both simultaneously = (sum)/(difference).
Direct and Inverse Variation
Direct: x ∝ y → x=ky → x₁/x₂=y₁/y₂ (ratio constant) Inverse: x ∝ 1/y → xy=k → x₁y₁=x₂y₂ (product constant) Duplicate ratio of a:b = a²:b² Sub-duplicate ratio = √a:√b
Example: Speed varies inversely with time (same distance): 60 km/h in 2hr → 40 km/h takes 3hr.
Direct: both go same direction. Inverse: one up, other down. Constant is ratio (direct) or product (inverse).
Wilson's Theorem
For any prime p: (p-1)! ≡ -1 (mod p) equivalently: (p-1)! mod p = p-1 Corollary: (p-2)! ≡ 1 (mod p)
Example: p=7: 6!=720. 720÷7=102 R6. 6=7-1 ✓ p=31: 29! ≡ 1 (mod 31).
(p-1) factorial leaves remainder (p-1) when divided by prime p. Think: the 'last' nonzero remainder before p itself.
Remainder of Product and Sum
Remainder of (a×b×c) ÷ d: = remainder of [rem(a/d) × rem(b/d) × rem(c/d)] ÷ d Remainder of (a+b+c) ÷ d: = remainder of [rem(a/d) + rem(b/d) + rem(c/d)] ÷ d
Example: Rem[17×19×23 ÷ 7]: rem 17/7=3, rem 19/7=5, rem 23/7=2. 3×5×2=30. Rem 30/7=2.
Break the product/sum, find each piece's remainder, combine those small remainders, reduce again.
HCF and LCM of Fractions
HCF of fractions = HCF(numerators) / LCM(denominators) LCM of fractions = LCM(numerators) / HCF(denominators)
Example: HCF(2/3, 4/9): HCF(2,4)/LCM(3,9) = 2/9. LCM(2/3, 4/9): LCM(2,4)/HCF(3,9) = 4/3.
For HCF: small (HCF) over big (LCM). For LCM: big (LCM) over small (HCF). They flip — opposite of what feels natural!
LCM with Fixed Remainder
A number N leaves the same remainder R when divided by a, b, and c: N = k × LCM(a,b,c) + R for some non-negative integer k. Smallest positive N = LCM(a,b,c) + R (when k=1, if R>0) or just R (when k=0).
Example: Leaves remainder 5 when divided by 6, 8, or 12. LCM=24. N=24k+5. Smallest=5.
Find LCM first, then add the constant remainder. LCM is the engine; R is the offset.
Clock Hands — Speed and Angle
Minute hand speed: 6°/min Hour hand speed: 0.5°/min Relative speed (minute over hour): 5.5°/min Hands coincide every 720/11 ≈ 65.45 minutes (NOT every 65 minutes) Hands are opposite (180°) every 720/11 minutes as well In 12 hours: 11 coincidences, 11 opposites
Example: First coincidence after 12:00 → 720/11 ≈ 65 min 27 sec, NOT 1:05:00.
Minute gains 5.5° per minute on hour hand. 360°÷5.5=720/11 minutes per full lap.
Linear Race Formulas
Winner covers full race distance. Loser covers: race length - (beat distance) - (start advantage) Beat distance: gap the winner gives the loser Start advantage (head start): distance loser began ahead Speed ratio = distance ratio in same time.
Example: 100m race: A beats B by 10m, B given 5m head start. B ran: 100-10-5=85m while A ran 100m. Speed ratio A:B = 100:85 = 20:17.
Beat distance=winner's advantage. Start=loser's advantage. Subtract both to find how far loser actually ran.
Ways to Express N as Product of Two Factors
If N has P total factors: • N is NOT a perfect square (P even): ways = P/2 • N IS a perfect square (P odd): ways = (P+1)/2 This counts ordered pairs where smaller × larger = N.
Example: N=12, P=6 (even): 3 ways: (1,12),(2,6),(3,4). N=36, P=9 (odd): 5 ways: (1,36),(2,18),(3,12),(4,9),(6,6).
Perfect squares have ODD number of factors (one pairs with itself). That's the only difference.
Pipes and Cisterns
Inlet fills tank in x hours → rate = +1/x per hour Outlet empties in y hours → rate = -1/x per hour Both open: net rate = 1/x - 1/y Time to fill = 1/(net rate) Same formulas as Time and Work.
Example: Inlet: 6hr, Outlet: 9hr. Net=1/6-1/9=1/18. Fills in 18hr.
Inlet=positive, Outlet=negative. Add rates algebraically. Same as work-rate problems!
False Weight Gain Formula
Trader claims to sell at cost price but uses false (lighter) weight: Gain% = (Error / (True Value - Error)) × 100 where Error = True weight - False weight used.
Example: Shopkeeper uses 900g instead of 1000g. Error=100g. Gain% = 100/(1000-100) × 100 = 100/900 × 100 ≈ 11.11%.
Error ÷ (True - Error) × 100. Denominator is what the buyer actually receives.
CP of x Articles = SP of y Articles
If the Cost Price of x articles equals the Selling Price of y articles: Profit/Loss % = (y - x) / y × 100 • y > x → Profit • y < x → Loss
Example: CP of 10 = SP of 8: Profit% = (10-8)/10 × 100 = 20%. CP of 8 = SP of 10: Loss% = (8-10)/10 × 100 = -20%.
'y' is always in the denominator. y>x means you collected money for more than you gave → profit.
Weighted Average and Combined Mean
Weighted Mean: x̄ = Σwₖxₖ / Σwₖ Combined Mean of two groups: x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁+n₂) Weighted Average of speeds for equal distances = Harmonic Mean. Weighted Average of speeds for equal times = Arithmetic Mean.
Example: Class A (30 students, avg 70) + Class B (20 students, avg 80): Combined = (30×70+20×80)/50 = 3700/50 = 74.
Each value weighted by its group size. Larger groups pull the average toward their value.
Remainder Theorem
When polynomial p(x) is divided by (x-a), remainder = p(a). If p(a) = 0, then (x-a) is a factor (Factor Theorem).
Example: p(x) = x³ - 3x + 5 divided by (x-1): p(1) = 3. Remainder = 3.
Set divisor = 0, find x, plug into p(x). That value IS the remainder — no long division needed.
Polynomial Division Algorithm
p(x) = q(x) × g(x) + r(x) Degree of r(x) < degree of g(x) Structure: Dividend = Divisor × Quotient + Remainder
Example: x² + 3x + 1 ÷ (x+1): q = x+2, r = -1. Check: (x+1)(x+2) + (-1) = x²+3x+1 ✓
Identical structure to number division. Remainder's degree must always be less than divisor's degree.
Cubic Polynomial — Root Relations (Vieta's)
For ax³ + bx² + cx + d with roots α, β, γ: α + β + γ = -b/a αβ + βγ + γα = c/a αβγ = -d/a
Example: x³ - 6x² + 11x - 6: roots 1,2,3. Sum=6, pairs=11, product=6.
Signs alternate: -, +, -. Product of roots = -d/a.
Trigonometric Ratios — SOH CAH TOA
In a right triangle with angle θ: sin θ = Opposite / Hypotenuse cos θ = Adjacent / Hypotenuse tan θ = Opposite / Adjacent cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Example: 3-4-5 right triangle: sin θ = 3/5, cos θ = 4/5, tan θ = 3/4.
SOH-CAH-TOA. The 'co' functions (cosec, sec, cot) are just reciprocals.
Trigonometric Values — Standard Angles
Angle: 0° 30° 45° 60° 90° sin: 0 1/2 1/√2 √3/2 1 cos: 1 √3/2 1/√2 1/2 0 tan: 0 1/√3 1 √3 ND
Example: sin 30° + cos 60° = 1/2 + 1/2 = 1. sin²45° + cos²45° = 1 ✓
For sin: √0/2, √1/2, √2/2, √3/2, √4/2 — square root of 0,1,2,3,4 all divided by 2. cos is the same list reversed.
Complementary Angle Identities
sin(90°-θ) = cos θ cos(90°-θ) = sin θ tan(90°-θ) = cot θ cosec(90°-θ) = sec θ sec(90°-θ) = cosec θ
Example: sin 70° = cos 20°. tan 65° = cot 25°. Simplify: sin 35°/cos 55° = sin 35°/sin 35° = 1.
Every trig function becomes its co-function when the angle changes to (90°-θ). The 'co' in cosine literally means complement.
Mean for Grouped Data — Three Methods
Direct: x̄ = Σfᵢxᵢ / Σfᵢ (xᵢ = class mark) Deviation: x̄ = A + Σfᵢdᵢ/Σfᵢ where dᵢ = xᵢ - A Step: x̄ = A + (Σfᵢuᵢ/Σfᵢ) × h where uᵢ = dᵢ/h
Example: Step deviation is fastest when all class widths are equal (most common case).
Choose A as the class mark closest to the middle of data. Small dᵢ values make arithmetic much easier.
Median and Mode for Grouped Data
Median: M = l + (n/2 - cf)/f × h l = lower limit of median class cf = cumulative freq BEFORE the median class f = freq of median class, h = class size Mode: Mo = l + (f₁-f₀)/(2f₁-f₀-f₂) × h f₁ = modal freq, f₀ = prev class, f₂ = next class
Example: Modal class 30-40 (l=30,h=10,f₁=15,f₀=8,f₂=10): Mode = 30 + (7/12)×10 ≈ 35.83.
Median: find class where cumulative freq first crosses n/2. Mode: class with the highest frequency.
Probability — Basic Formula and Complement
P(E) = n(E) / n(S) (favourable ÷ total equally likely outcomes) 0 ≤ P(E) ≤ 1 P(Ē) = 1 - P(E) (complement)
Example: Rolling a die: P(4) = 1/6. P(even) = 3/6 = 1/2. P(not 4) = 5/6.
P = Favourable ÷ Total. Complement: subtract from 1. Together they always sum to 1.
Sector, Segment, Arc, and Ring
Sector area = (θ/360) × πr² Arc length = (θ/360) × 2πr Also: Area of sector = (1/2) × arc × radius Segment area = Sector area - Triangle area Ring (annulus): π(R² - r²) = π(R+r)(R-r)
Example: r=6, θ=60°: Sector area=6π. Arc length=2π. Triangle area=9√3. Segment=6π-9√3.
Sector = pizza slice. Segment = slice minus the triangle (just the curved 'bite').
Midpoint Theorem
The line joining the midpoints of any two sides of a triangle is: 1. Parallel to the third side 2. Equal to HALF the length of the third side. Converse: A line through the midpoint of one side, parallel to another, bisects the third side.
Example: X is midpoint of AC, Y is midpoint of BC: XY ∥ AB and XY = AB/2. If AB=10, XY=5.
Midpoint connector = parallel and half. Think: the 'midline' is a scaled-down version of the base.
Angle Bisector Theorem
In △ABC, if AD bisects angle A (D on BC): BD/DC = AB/AC Exterior Angle Bisector: divides BC externally in ratio AB:AC
Example: AB=8, AC=6, BC=7. AD bisects A: BD/DC=4/3. BD=4, DC=3.
The bisector splits the opposite side in the ratio of the two adjacent sides. Adjacent → opposite.
Multiply Any 2-Digit Number by 11
AB × 11: insert the sum of the digits between them. 32×11=352 (3+2=5) 85×11=935 (8+5=13 → carry 1 → 8+1=9) 99×11=1089 (9+9=18 → carry)
Example: 77×11: 7+7=14, write 4, carry 1 to the 7 → 8. Answer: 847.
Digits split apart, sum fills the gap. Carry left if sum ≥ 10.
Squaring Numbers Ending in 5
For any number ending in 5: 1. Multiply digits before the 5 by (itself + 1) 2. Append 25 35² = 3×4 = 12 → 1225 85² = 8×9 = 72 → 7225 125² = 12×13 = 156 → 15625
Example: 65²: leading digit is 6. 6×7=42. Append 25. Answer: 4225.
First × (first+1), ALWAYS ends in 25. Works for any length number ending in 5!
Multiply by 25 and 5
N × 25 = (N × 100) ÷ 4 — append two zeros, halve twice. N × 5 = (N × 10) ÷ 2 — append a zero, halve. N × 125 = (N × 1000) ÷ 8 — append three zeros, halve three times.
Example: 68×25 = 6800÷4 = 1700. 44×125 = 44000÷8 = 5500.
25 = 100÷4. 5 = 10÷2. 125 = 1000÷8. Multiply by power of 10, then divide out.
Left-to-Right Mental Addition
Add left to right (most significant digit first). Break the second number into hundreds, tens, units. Add each part. 538 + 327: 538 + 300 = 838 838 + 20 = 858 858 + 7 = 865
Example: 623 + 159: +100=723, +50=773, +9=782.
Left to right = most important part first. Each step makes the problem simpler. You can give partial answers as you go!
Two-Digit Squaring (Round to Nearest 10)
A² = (A+d)(A-d) + d² where d = distance to nearest multiple of 10. Round A to nearest 10, multiply the pair, add d². You'll only ever need to add 1, 4, 9, 16, or 25.
Example: 77²: d=3. 80×74=5920. Add 9. Answer: 5929. 56²: d=4. 60×52=3120. Add 16. Answer: 3136.
Round to 10, pair multiplication + d-squared. For numbers near 100, round to 100 instead.
Close-Together Multiplication
When two numbers are both close to same base z: (z+a)(z+b) = z(z+a+b) + a×b Steps: pick z, find offsets a and b, multiply z by (first number + other's offset), add product of offsets.
Example: 107×111 (z=100): a=7, b=11. 100×118 + 7×11 = 11800+77 = 11877. 396×387 (z=400): a=-4, b=-13. 400×383 + (-4)(-13) = 153200+52 = 153252.
Same base = easier mental multiplication. The leftover product (a×b) is small and manageable.
Casting Out Nines — Arithmetic Verification
Digital root of N = repeatedly sum digits until single digit. For any correct addition or multiplication: Digital root(answer) = digital root(operation on individual digital roots) Catches errors 8 out of 9 times.
Example: 853×762: roots=7,6. 7×6=42→root=6. Answer 649986: 6+4+9+9+8+6=42→6 ✓
Reduce every number to one digit, do the operation on those, check against answer's one digit.
Rule of 70 and 110
Years to DOUBLE money = 70 ÷ (annual interest rate %) Years to TRIPLE money = 110 ÷ (annual interest rate %) Based on: ln(2) ≈ 0.693, ln(3) ≈ 1.099
Example: At 5%: doubles in 14 years (70÷5), triples in 22 years (110÷5). At 7%: doubles in 10 years, triples in ≈15.7 years.
Rule of 70 → doubles. Rule of 110 → triples. Memorize: 70/rate = doubling time.
Successive Percentage Changes — Net Effect
After a% change then b% change: Net % change = a + b + (a×b)/100 % (Use negatives for decreases) For two successive same-direction discounts: same formula.
Example: +20% then +10%: net = 20+10+2 = 32%. +30% then -30%: net = 30-30-9 = -9% (LOSS of 9%!)
Add the percentages AND their product/100. The extra term = compounding effect.
Divisibility by 7 (Quick Test)
To test if N is divisible by 7: 1. Add or subtract a multiple of 7 to make the result end in 0 2. Remove the trailing 0 3. Repeat until you reach a recognisable multiple of 7
Example: 7336: 7336-56=7280 → remove 0 → 728-28=700 → 70 → 7 = 7×1 ✓ So 7336 is divisible by 7.
Make a 0, delete the 0, repeat. Goal: reach a small known multiple of 7.
Gauss's Sum — Adding 1 to N
Sum of 1 to N = N(N+1)/2 Sum of any arithmetic series = n × (first + last) / 2 Gauss's insight: pair first+last, second+second-last, etc. Each pair = N+1. There are N/2 such pairs.
Example: 1 to 100: 100×101/2 = 5050. Sum of odd numbers 1 to 99 (50 terms): 50×(1+99)/2 = 2500.
N times (N+1) divided by 2. Or: pair opposites — each pair sums to (N+1) and there are N/2 of them.
Multiply by 5 — Halve and Shift
N × 5 = N ÷ 2, then × 10 (shift decimal one place right) For even N: halve first, then append 0. For odd N: result will end in 5 — just use N×10÷2.
Example: 86×5: 86÷2=43 → 430. 47×5: 47×10=470 → 470÷2=235.
×5 = append zero then halve. Or: halve then append zero. Same result!
Law of Sines
a/sin A = b/sin B = c/sin C = 2R (a, b, c are sides opposite angles A, B, C; R = circumradius)
Example: A=30°, B=60°, a=5: b/sin60° = 5/sin30° = 10 → b = 5√3.
Each side ÷ sine of its opposite angle = circumdiameter (2R). All three give the same constant.
Law of Cosines
cos A = (b² + c² - a²) / (2bc) cos B = (a² + c² - b²) / (2ac) cos C = (a² + b² - c²) / (2ab)
Example: Triangle 3,4,5: cos A = (16+25-9)/40 = 32/40 = 4/5 → A ≈ 36.9°.
'The OTHER two sides squared, minus the TARGET side squared, all over 2 times those two sides'.
Range of a sinθ + b cosθ
a sinθ + b cosθ can be written as R sin(θ+φ) where R = √(a²+b²) and tan φ = b/a Minimum = -√(a²+b²) Maximum = +√(a²+b²)
Example: 3sinθ + 4cosθ: R = √(9+16) = 5. Range = [-5, 5].
Amplitude R = √(a²+b²) — same as the hypotenuse of sides a and b. Max = +R, Min = -R.
Binomial Coefficient Key Identities
C(n,r) = C(n,n-r) (complementary selection) Pascal: C(n,r-1) + C(n,r) = C(n+1,r) Sum: C₀+C₁+...+Cₙ = 2ⁿ Alternating: C₀-C₁+C₂-... = 0 Squares: C₀²+C₁²+...+Cₙ² = C(2n,n)
Example: All subsets of {1,2,3,4}: 2⁴=16. Pascal: C(5,2)+C(5,3)=10+10=C(6,3)=20 ✓
Put x=1 → sum = 2ⁿ. Put x=-1 → alternating sum = 0. These two substitutions give all the key results.
Exponential and Logarithm Power Series
eˣ = 1 + x + x²/2! + x³/3! + ... (all terms positive) ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ... (alternates sign, valid |x|≤1)
Example: e ≈ 1+1+0.5+0.167+0.042 = 2.718. ln(2) ≈ 1-0.5+0.333-0.25 ≈ 0.693.
eˣ: all powers, factorials in denominator, all positive. ln(1+x): integers in denominator, alternating signs.
Quadratic Inequalities
(x-m)(x-n) < 0 → m < x < n [between the roots] (x-m)(x-n) > 0 → x < m OR x > n [outside the roots]
Example: x²-5x+6 < 0: roots are 2,3. Answer: 2 < x < 3. x²-5x+6 > 0: x < 2 or x > 3.
< 0 → between roots (middle). > 0 → outside roots (sides). Draw the parabola to visualise instantly.
Arithmetic-Geometric Series (AGP)
General term: a, (a+d)r, (a+2d)r², (a+3d)r³, ... Infinite sum (|r|<1): S∞ = a/(1-r) + dr/(1-r)²
Example: Series 1, 1×(1/2), 3×(1/4), 4×(1/8): a=1, d=1, r=1/2. S∞ = 2 + 2 = 4.
AGP = AP × GP element-wise. The infinite sum formula has two terms: one for the AP part, one for the d-part.
Descartes' Rule of Signs
Max positive real roots of f(x) = number of sign changes in coefficients of f(x) Max negative real roots = number of sign changes in coefficients of f(-x) Actual count may be less by even numbers (complex roots come in pairs). Odd degree polynomial → at least one real root always.
Example: f(x)=x³-3x²+x-2: sign changes +,-,+,- = 3. Max 3 positive roots.
Count sign flips in f(x) for positive, in f(-x) for negative roots. Replace x with -x to get f(-x).
Sum of All Factors of N
For N = p^a × q^b × r^c (distinct primes): Sum of all factors = [(p^(a+1)-1)/(p-1)] × [(q^(b+1)-1)/(q-1)] × [(r^(c+1)-1)/(r-1)]
Example: N=12=2²×3: Sum = [(2³-1)/(2-1)] × [(3²-1)/(3-1)] = 7 × 4 = 28. Verify: 1+2+3+4+6+12=28 ✓
Each bracket is the sum (1 + p + p² + ... + p^e) for that prime. Multiply all brackets together.
Three-Set Venn Diagram Formula
n(A∪B∪C) = n(A)+n(B)+n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C) n(A-B) = n(A) - n(A∩B) [elements in A but not B]
Example: 30+25+20-10-8-7+3 = 53 elements in total.
Add all three, subtract pairs (each counted twice), add back triple (subtracted once too many). Inclusion-exclusion!
Train Crossing — Relative Speed
Two trains lengths L₁ and L₂, speeds S₁ and S₂: Opposite direction: time = (L₁+L₂)/(S₁+S₂) Same direction: time = (L₁+L₂)/|S₁-S₂| Crossing a stationary pole: time = own length / own speed Crossing a platform of length p: time = (L+p)/S
Example: Trains 200m+150m, 72 km/h+36 km/h opposite: t=350/(30m/s)≈11.7s.
ALWAYS add the lengths. Opposite direction → add speeds. Same direction → subtract speeds.
Frustum of a Cone
Slant height: l = √(h² + (R-r)²) Volume: V = (π/3) × h × (R² + Rr + r²) Lateral SA: π(R+r)l Total SA: π(R+r)l + πR² + πr²
Example: Frustum R=5, r=3, h=4: l=√(16+4)=√20=2√5. V=(π/3)×4×(25+15+9)=(196π/3).
Frustum looks like a bucket. Slant height uses DIFFERENCE of radii (not sum). Volume has three terms: R², Rr, r².
Rectangular Hyperbola xy = c²
Eccentricity: e = √2 (always) Parametric: x = ct, y = c/t Tangent at (ct, c/t): x/t + ty = 2c Chord with midpoint (h,k): kx + hy = 2hk
Example: xy=4 (c=2). Tangent at (2,2): 2/2+2×2=x+y=4. Check: 2×2=4 ✓
Asymptotes are the coordinate axes themselves. e=√2 always. Product of any point's coordinates = c².
Half-Angle Formulas for Triangles
sin(A/2) = √[(s-b)(s-c) / bc] cos(A/2) = √[s(s-a) / bc] tan(A/2) = Δ / [s(s-a)] where Δ = area of triangle
Example: These let you find angles from sides without computing arccos directly.
sin(A/2): uses (s-b)(s-c). cos(A/2): uses s(s-a) — note 'a' appears in the cos formula.
Special Matrix Types
Symmetric: A^T = A Skew-symmetric: A^T = -A (all diagonal elements = 0) Orthogonal: A^(-1) = A^T or A^T × A = I Any square matrix can be written as sum of symmetric + skew-symmetric.
Example: [[0,1,-2],[-1,0,3],[2,-3,0]] is skew-symmetric (check: zeros on diagonal, opposite signs across).
Symmetric = mirror image of itself. Skew = flips sign when reflected. Skew always has zeros on the main diagonal.