Free Formula Sheet · Samacheer Kalvi
12th Standard Mathematics
Complete Formula Sheet — All Chapters
Chapter 01
Applications of Matrices and Determinants
Determinant & Inverse
2×2 Determinant
|A| = ad − bc for A = [[a,b],[c,d]]
Inverse of Matrix A
A⁻¹ = (1/|A|) · adj(A)
Exists only when |A| ≠ 0 (non-singular)
A · adj(A)
= |A| · I
Cofactor Cᵢⱼ
Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ
Key Determinant Properties
|AB|
= |A| · |B|
|Aᵀ|
= |A|
|A⁻¹|
= 1 / |A|
|kA| for n×n
= kⁿ · |A|
|Aⁿ|
= |A|ⁿ
Rank ρ(A)
= r if r×r sub-determinant ≠ 0, all (r+1)×(r+1) = 0
Cramer’s Rule & System of Equations
Cramer’s Rule
x = Δₓ/Δ | y = Δᵧ/Δ | z = Δ_z/Δ
Matrix Method AX = B
X = A⁻¹ B
Consistent (Rouché–Capelli)
ρ(A) = ρ([A|B]) → consistent
ρ(A) ≠ ρ([A|B]) → inconsistent
ρ(A) ≠ ρ([A|B]) → inconsistent
Unique Solution
ρ(A) = ρ([A|B]) = n (number of unknowns)
Chapter 02
Complex Numbers
Basic Forms & Operations
Standard Form
z = a + ib | i² = −1
Modulus |z|
|z| = √(a² + b²)
Conjugate z̄
z̄ = a − ib
z · z̄
= |z|² = a² + b²
Argument θ
arg(z) = tan⁻¹(b/a)
Polar Form
z = r(cosθ + i sinθ) = re^(iθ)
De Moivre’s Theorem & Roots of Unity
De Moivre’s Theorem
(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)
Euler’s Formula
e^(iθ) = cosθ + i sinθ
nth Roots of Unity
zₖ = cos(2kπ/n) + i sin(2kπ/n) k = 0,1,…,n−1
Product of Moduli / Args
|z₁z₂| = |z₁||z₂|
arg(z₁z₂) = arg(z₁) + arg(z₂)
arg(z₁z₂) = arg(z₁) + arg(z₂)
Powers of i: i¹ = i | i² = −1 | i³ = −i | i⁴ = 1 | i^(4k) = 1 | i^(4k+1) = i | i^(4k+2) = −1 | i^(4k+3) = −i
Chapter 03
Theory of Equations
Vieta’s Formulas — Roots & Coefficients
| Equation | Sum of Roots | Sum of Products (pairs) | Product of All Roots |
|---|---|---|---|
| Quadratic ax²+bx+c=0 (α,β) | −b/a | — | c/a |
| Cubic ax³+bx²+cx+d=0 (α,β,γ) | −b/a | c/a | −d/a |
| Quartic ax⁴+bx³+cx²+dx+e=0 (α,β,γ,δ) | −b/a | c/a | e/a |
Key Theorems
Remainder Theorem
Remainder of p(x) ÷ (x−a) = p(a)
Factor Theorem
(x − a) is factor ⟺ p(a) = 0
Descartes’ Rule of Signs
Max +ve roots = sign changes in p(x)
Max −ve roots = sign changes in p(−x)
Max −ve roots = sign changes in p(−x)
Complex / Irrational Roots
Always appear in conjugate pairs for real-coefficient polynomials
Chapter 04
Inverse Trigonometric Functions
Domain & Principal Value Range
| Function | Domain | Range (Principal Value) |
|---|---|---|
| sin⁻¹ x | [−1, 1] | [−π/2, π/2] |
| cos⁻¹ x | [−1, 1] | [0, π] |
| tan⁻¹ x | (−∞, ∞) | (−π/2, π/2) |
| cosec⁻¹ x | (−∞,−1] ∪ [1,∞) | [−π/2, π/2] − {0} |
| sec⁻¹ x | (−∞,−1] ∪ [1,∞) | [0, π] − {π/2} |
| cot⁻¹ x | (−∞, ∞) | (0, π) |
Key Identities
sin⁻¹x + cos⁻¹x
= π/2 for x ∈ [−1,1]
tan⁻¹x + cot⁻¹x
= π/2 for x ∈ ℝ
sec⁻¹x + cosec⁻¹x
= π/2 for |x| ≥ 1
sin⁻¹(−x)
= −sin⁻¹x
cos⁻¹(−x)
= π − cos⁻¹x
tan⁻¹x + tan⁻¹y (xy < 1)
= tan⁻¹[(x+y) / (1−xy)]
tan⁻¹x − tan⁻¹y (xy > −1)
= tan⁻¹[(x−y) / (1+xy)]
2 tan⁻¹x
= sin⁻¹[2x/(1+x²)] = cos⁻¹[(1−x²)/(1+x²)]
Chapter 05
Two Dimensional Analytical Geometry – II
Circle
Standard Form — centre (h,k), radius r
(x−h)² + (y−k)² = r²
General Form x²+y²+2gx+2fy+c=0
Centre: (−g, −f) | r = √(g²+f²−c)
Length of Tangent from (x₁,y₁)
L = √(x₁²+y₁²+2gx₁+2fy₁+c)
Tangent at Point (x₁,y₁)
xx₁+yy₁+g(x+x₁)+f(y+y₁)+c = 0
Conics — Standard Forms
| Conic | Equation | Focus | Directrix | e |
|---|---|---|---|---|
| Parabola | y² = 4ax | (a, 0) | x = −a | 1 |
| Ellipse | x²/a² + y²/b² = 1 | (±ae, 0) | x = ±a/e | < 1 |
| Hyperbola | x²/a² − y²/b² = 1 | (±ae, 0) | x = ±a/e | > 1 |
Conic Key Relations
Ellipse — b²
b² = a²(1 − e²)
Hyperbola — b²
b² = a²(e² − 1)
c² relations
Ellipse: c² = a²−b²
Hyperbola: c² = a²+b²
Hyperbola: c² = a²+b²
Chapter 06
Applications of Vector Algebra
Vector Products
Dot (Scalar) Product
a⃗ · b⃗ = |a⃗||b⃗| cosθ = a₁b₁ + a₂b₂ + a₃b₃
Cross Product Magnitude
|a⃗ × b⃗| = |a⃗||b⃗| sinθ
Area of parallelogram = |a⃗ × b⃗|
Scalar Triple Product [a⃗ b⃗ c⃗]
a⃗ · (b⃗ × c⃗) = Volume of parallelepiped
Vector Triple Product
a⃗ × (b⃗ × c⃗) = (a⃗·c⃗)b⃗ − (a⃗·b⃗)c⃗
Lines & Planes in 3D
Line — Vector Form
r⃗ = a⃗ + λb⃗
Plane — Vector Form
r⃗ · n̂ = d
Symmetric Line (Cartesian)
(x−x₁)/l = (y−y₁)/m = (z−z₁)/n
Plane (Cartesian)
ax + by + cz + d = 0
Distance — Point to Plane
d = |ax₁+by₁+cz₁+d| / √(a²+b²+c²)
Shortest Distance — Skew Lines
SD = |(a₂−a₁)·(b₁×b₂)| / |b₁×b₂|
Chapter 07
Applications of Differential Calculus
Standard Derivatives Table
| f(x) | f′(x) | f(x) | f′(x) |
|---|---|---|---|
| xⁿ | nxⁿ⁻¹ | eˣ | eˣ |
| aˣ | aˣ ln a | ln x | 1/x |
| sin x | cos x | cos x | −sin x |
| tan x | sec²x | cot x | −cosec²x |
| sec x | sec x tan x | cosec x | −cosec x cot x |
| sin⁻¹x | 1/√(1−x²) | cos⁻¹x | −1/√(1−x²) |
| tan⁻¹x | 1/(1+x²) | cot⁻¹x | −1/(1+x²) |
Rules & Applications
Chain Rule
dy/dx = (dy/du) · (du/dx)
Product Rule
d(uv)/dx = u·v′ + v·u′
Quotient Rule
d(u/v)/dx = (v·u′ − u·v′) / v²
L’Hôpital’s Rule (0/0 or ∞/∞)
lim f/g = lim f′/g′
Tangent at (x₁,y₁)
y − y₁ = m(x − x₁) | m = dy/dx|(x₁,y₁)
Normal at (x₁,y₁)
y − y₁ = (−1/m)(x − x₁)
Rolle’s Theorem
f(a)=f(b) → ∃ c ∈ (a,b): f′(c) = 0
Lagrange’s MVT
f′(c) = [f(b) − f(a)] / (b − a)
Second Derivative Test
f″ < 0 → max | f″ > 0 → min
Concavity
f″ > 0 → concave up | f″ < 0 → concave down
Chapter 08
Differentials and Partial Derivatives
Differentials & Approximation
Differential of y
dy = f′(x) dx
Linear Approximation
f(x + Δx) ≈ f(x) + f′(x) · Δx
Absolute Error
Δy ≈ dy = f′(x) · Δx
Relative / Percentage Error
Δy/y × 100% ≈ (dy/y) × 100%
Partial Derivatives & Euler’s Theorem
∂f/∂x
Differentiate w.r.t. x, treat y as constant
Euler’s Theorem — Homogeneous of degree n
x · ∂f/∂x + y · ∂f/∂y = n · f
Total Differential df
df = (∂f/∂x)dx + (∂f/∂y)dy
Clairaut’s Theorem (Symmetry)
∂²f / ∂x∂y = ∂²f / ∂y∂x
Chapter 09
Applications of Integration
Standard Integrals
| f(x) | ∫ f(x) dx | f(x) | ∫ f(x) dx |
|---|---|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C | 1/x | ln|x| + C |
| eˣ | eˣ + C | aˣ | aˣ/ln a + C |
| sin x | −cos x + C | cos x | sin x + C |
| tan x | ln|sec x| + C | cot x | ln|sin x| + C |
| sec²x | tan x + C | cosec²x | −cot x + C |
| 1/√(1−x²) | sin⁻¹x + C | 1/(1+x²) | tan⁻¹x + C |
| 1/√(a²−x²) | sin⁻¹(x/a) + C | 1/(a²+x²) | (1/a) tan⁻¹(x/a) + C |
| 1/(x²−a²) | (1/2a) ln|(x−a)/(x+a)| + C | 1/(a²−x²) | (1/2a) ln|(a+x)/(a−x)| + C |
Area, Volume & Important Properties
Area Between Curves
A = ∫[a to b] |f(x) − g(x)| dx
Volume of Revolution (x-axis)
V = π ∫[a to b] [f(x)]² dx
Integration by Parts (ILATE)
∫u dv = uv − ∫v du
ILATE: Inverse trig → Log → Algebraic → Trig → Exponential
∫[−a to a] f(x) dx
= 2∫[0 to a]f(x)dx if f even
= 0 if f odd
= 0 if f odd
∫[0 to 2a] f(x) dx
= 2∫[0 to a]f(x)dx if f(2a−x)=f(x)
= 0 if f(2a−x)=−f(x)
= 0 if f(2a−x)=−f(x)
∫[0 to π/2] sinⁿx dx = ∫[0 to π/2] cosⁿx dx
Walli’s formula (reduction formula)
Chapter 10
Ordinary Differential Equations
Classification
Order
Highest derivative present in the ODE
Degree
Power of highest order derivative (after rationalising)
Solution Methods
Separable Variables
f(y) dy = g(x) dx → integrate both sides
Homogeneous ODE
dy/dx = f(y/x) → substitute y = vx
Linear ODE: dy/dx + Py = Q
I.F. = e^(∫P dx)
Solution: y · I.F. = ∫Q · I.F. dx + C
Solution: y · I.F. = ∫Q · I.F. dx + C
Bernoulli’s ODE: dy/dx + Py = Qyⁿ
Substitute v = y^(1−n) → reduces to linear
Exact ODE: M dx + N dy = 0
Condition: ∂M/∂y = ∂N/∂x
Solution of Exact ODE
∫M dx + ∫(terms in N free of x) dy = C
Chapter 11
Probability Distributions
Discrete Random Variable — Mean & Variance
Mean E(X)
μ = E(X) = Σ x · P(X = x)
Variance Var(X)
σ² = E(X²) − [E(X)]²
E(X²)
Σ x² · P(X = x)
Standard Deviation σ
σ = √Var(X)
Binomial Distribution B(n, p)
P(X = x)
ⁿCₓ · pˣ · qⁿ⁻ˣ | q = 1 − p | x = 0,1,…,n
Mean & Variance
μ = np | σ² = npq
Poisson Distribution P(λ)
P(X = x)
e^(−λ) · λˣ / x! | x = 0,1,2,…
Mean & Variance
μ = λ | σ² = λ
Normal Distribution N(μ, σ²)
PDF
f(x) = (1/(σ√(2π))) · e^(−(x−μ)² / (2σ²))
Standard Normal Z-score
Z = (X − μ) / σ
Properties
Mean = Median = Mode = μ | Symmetric about μ | Total area = 1
68-95-99.7 Rule
μ ± σ: 68.27% | μ ± 2σ: 95.45% | μ ± 3σ: 99.73%
Chapter 12
Discrete Mathematics
Mathematical Logic — Connectives
Conjunction (AND)
p ∧ q true only when both true
Disjunction (OR)
p ∨ q false only when both false
Implication (IF…THEN)
p → q ≡ ¬p ∨ q
Biconditional (IFF)
p ↔ q ≡ (p→q) ∧ (q→p)
Contrapositive
p → q ≡ ¬q → ¬p
Converse & Inverse
Converse: q → p | Inverse: ¬p → ¬q
De Morgan’s Laws & Boolean Algebra
De Morgan’s (Logic)
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
De Morgan’s (Boolean)
(A · B)′ = A′ + B′
(A + B)′ = A′ · B′
(A + B)′ = A′ · B′
Boolean Identity Laws
A+0=A | A+1=1 | A·0=0 | A·1=A
A+A=A | A·A=A | A+A′=1 | A·A′=0
A+A=A | A·A=A | A+A′=1 | A·A′=0
Absorption Laws
A + (A · B) = A
A · (A + B) = A
A · (A + B) = A
Graph Theory
Handshaking Theorem
Σ deg(v) = 2 × |E|
Sum of all degrees = twice the number of edges
Euler’s Formula (planar graph)
V − E + F = 2
V = vertices, E = edges, F = faces (including outer)
Complete Graph Kₙ
Number of edges = n(n−1) / 2
Eulerian Circuit
Connected + every vertex has even degree
Tautology: Always true for all truth values (e.g., p ∨ ¬p). | Contradiction: Always false (e.g., p ∧ ¬p). | Contingency: Neither tautology nor contradiction. Verify using truth tables.
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