OMEGA EDUCARE | Maharashtra Std. 12 Mathematics | Learning Hub

OMEGA EDUCARE • LEARNING HUB

Std. 12
Mathematics

Maharashtra State Board Mathematics explained as a self-learning resource — concepts, definitions, methods, formulae and applications are explained so students can learn from the page rather than merely read chapter headings.

MAHARASHTRA STATE BOARD • STD. XII • MATHEMATICS & STATISTICS • ARTS & SCIENCE

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How to Learn Std. 12 Mathematics

At HSC level, every formula should be connected to the mathematical idea behind it.

Understand

Learn the definition, meaning and conditions before memorising a result.

Connect

Connect logic, matrices, trigonometry, vectors, geometry, calculus and probability.

Apply

Use the correct method in derivations, proofs, numerical problems and graphical questions.

Check

Check domains, signs, constants, limits and whether the final answer satisfies the question.

MAHARASHTRA STATE BOARD • STD. XII

Mathematics — Complete 15-Chapter Learning Hub

The current Maharashtra Arts & Science Mathematics and Statistics structure is organised into 7 Part-I chapters and 8 Part-II chapters.

01 • Mathematical Logic

Statement

A statement has a definite truth value: true or false.

Connectives

AND, OR, NOT, implication and biconditional combine statements.

Tautology / contradiction

A tautology is always true; a contradiction is always false.

Quantifiers

Universal means 'for every'; existential means 'there exists'.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

02 • Matrices

Matrix and order

A matrix arranges elements in rows and columns; order is rows × columns.

Elementary transformations

Standard row or column operations help simplify matrices and solve systems.

Adjoint

The adjoint is the transpose of the cofactor matrix.

Inverse

A⁻¹ = adj(A)/|A| when |A| ≠ 0.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

03 • Trigonometric Functions

Trigonometric equations

Find all angles satisfying the equation while respecting periodicity.

General solutions

sinθ=sinα ⇒ θ=nπ+(−1)ⁿα
cosθ=cosα ⇒ θ=2nπ±α
tanθ=tanα ⇒ θ=nπ+α

Sine rule

a/sinA=b/sinB=c/sinC=2R

Cosine rule

a²=b²+c²−2bc cosA

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

04 • Pair of Straight Lines

Homogeneous equation

ax²+2hxy+by²=0 can represent a pair of lines through the origin.

Angle between lines

tanθ=2√(h²−ab)/(a+b) where applicable.

Perpendicular pair

a+b=0

Slope method

If the two lines have slopes m₁ and m₂, their angle is obtained from the slope-angle relation.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

05 • Vectors

Magnitude

|a|=√(a₁²+a₂²+a₃²)

Dot product

a·b=|a||b|cosθ

Cross product

|a×b|=|a||b|sinθ

Scalar triple product

[abc]=a·(b×c); absolute value gives parallelepiped volume.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

06 • Line and Plane

Vector equation of line

r=a+λb

Cartesian line

(x−x₁)/l=(y−y₁)/m=(z−z₁)/n

Plane equation

ax+by+cz+d=0

Point-plane distance

d=|ax₁+by₁+cz₁+d|/√(a²+b²+c²)

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

07 • Linear Programming

Objective function

The linear expression to be maximised or minimised.

Constraints

Linear inequalities describe restrictions on the variables.

Feasible region

The common region satisfying all constraints.

Optimal solution

For a standard two-variable LPP, test the appropriate corner points of the feasible region.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

08 • Differentiation

Derivative

The derivative measures instantaneous rate of change and gives tangent slope.

Chain rule

d[f(g(x))]/dx=f′(g(x))g′(x)

Implicit differentiation

Differentiate both sides with respect to x when y is defined implicitly.

Second derivative

d²y/dx²=d/dx(dy/dx)

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

09 • Applications of Derivatives

Increasing / decreasing

f′(x)>0 indicates increasing behaviour and f′(x)<0 decreasing behaviour on an interval.

Critical points

Candidates occur where f′(x)=0 or f′(x) does not exist, subject to the domain.

Tangent / normal

mₜ=dy/dx
mₙ=−1/mₜ when defined.

Maxima / minima

Derivative tests help locate and classify local extreme values.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

10 • Indefinite Integration

Antiderivative

F is an antiderivative of f when F′(x)=f(x).

Power rule

∫xⁿdx=xⁿ⁺¹/(n+1)+C, n≠−1

Logarithmic integral

∫dx/x=ln|x|+C

Integration by parts

∫u dv=uv−∫v du

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

11 • Definite Integration

Fundamental theorem

∫ₐᵇf(x)dx=F(b)−F(a) where F′=f.

Basic properties

∫ₐᵃf=0 and ∫ₐᵇf=−∫ᵇₐf

Additivity

∫ₐᶜf+∫ᶜᵇf=∫ₐᵇf

Symmetry

Odd functions integrate to zero on [−a,a]; even functions give twice the integral from 0 to a.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

12 • Application of Definite Integration

Area under a curve

A=∫ₐᵇy dx when y is non-negative.

Area between curves

A=∫ₐᵇ(upper−lower)dx

With respect to y

A=∫(right−left)dy

Geometrical meaning

A definite integral represents accumulated area when the integrand represents a height or width.

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

13 • Differential Equations

Order and degree

Order is the highest derivative; degree is its power after the equation is polynomial in derivatives.

Variable separation

dy/dx=f(x)g(y) ⇒ dy/g(y)=f(x)dx

First-order linear form

dy/dx+Py=Q

Integrating factor

I.F.=e^(∫Pdx)

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

14 • Probability Distributions

Random variable

A random variable assigns a numerical value to each outcome.

Probability mass function

p(x)=P(X=x), p(x)≥0, Σp(x)=1

Expected value

E(X)=Σxp(x)

Variance / S.D.

Var(X)=E(X²)−[E(X)]²
σ=√Var(X)

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

15 • Bernoulli Trials and Binomial Distribution

Bernoulli trial

Each trial has two outcomes, with constant success probability p and failure probability q=1−p.

Binomial probability

P(X=r)=ⁿCᵣpʳqⁿ⁻ʳ

Mean

E(X)=np

Variance / S.D.

Var(X)=npq
σ=√(npq)

Study Tip: Understand the idea first, then solve a board-style problem and verify the result.

Remember: Check conditions, signs, domain and constants before applying a result.

COMPLETE FORMULA BANK

Std. 12 Mathematics — Chapter-wise Formulae

This is the full revision bank for the 15-chapter Maharashtra Std. XII Mathematics & Statistics structure used on this Learning Hub page. It is intentionally much more detailed than the earlier quick-reference box.

01 • Mathematical Logic

Basic logical forms

Negation: ¬p (or p′).
Conjunction: p ∧ q — true only when both p and q are true.
Disjunction: p ∨ q — false only when both p and q are false.
Implication: p → q ≡ ¬p ∨ q.
Biconditional: p ↔ q ≡ (p → q) ∧ (q → p).

Important equivalences

Double negation: ¬(¬p) ≡ p.
De Morgan: ¬(p ∧ q) ≡ ¬p ∨ ¬q.
De Morgan: ¬(p ∨ q) ≡ ¬p ∧ ¬q.
Commutative: p ∨ q ≡ q ∨ p; p ∧ q ≡ q ∧ p.
Associative: (p ∨ q) ∨ r ≡ p ∨ (q ∨ r); similarly for ∧.
Distributive: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r).
Distributive: p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).
Idempotent: p ∨ p ≡ p; p ∧ p ≡ p.
Identity: p ∨ F ≡ p; p ∧ T ≡ p.
Domination: p ∨ T ≡ T; p ∧ F ≡ F.
Complement: p ∨ ¬p ≡ T; p ∧ ¬p ≡ F.

Implication and quantifiers

Converse of p → q: q → p.
Inverse of p → q: ¬p → ¬q.
Contrapositive: ¬q → ¬p; it is logically equivalent to p → q.
Negation of ∀x P(x): ∃x ¬P(x).
Negation of ∃x P(x): ∀x ¬P(x).
Duality: interchange ∨ with ∧ and T with F in a logical statement.

Switching circuits

Series connection corresponds to AND (∧).
Parallel connection corresponds to OR (∨).
NOT operation corresponds to complement/inversion.

02 • Matrices

Basic operations

If A=[aᵢⱼ] is m×n and B is m×n, then A±B=[aᵢⱼ±bᵢⱼ].
For scalar k: kA=[kaᵢⱼ].
If A is m×n and B is n×p, then AB is m×p.
(AB)ᵀ=BᵀAᵀ.
(A+B)ᵀ=Aᵀ+Bᵀ.

Determinant and inverse

For A=[[a,b],[c,d]], |A|=ad−bc.
adj(A) is the transpose of the cofactor matrix.
A⁻¹=adj(A)/|A|, provided |A|≠0.
AA⁻¹=A⁻¹A=I.
AX=B ⇒ X=A⁻¹B when A is non-singular.

3×3 determinant

|a b c; d e f; g h i| = a(ei−fh) − b(di−fg) + c(dh−eg).

Elementary transformations

Row operations: Rᵢ↔Rⱼ; Rᵢ→kRᵢ (k≠0); Rᵢ→Rᵢ+kRⱼ.
Corresponding column operations: Cᵢ↔Cⱼ; Cᵢ→kCᵢ; Cᵢ→Cᵢ+kCⱼ.

Useful determinant facts

Interchanging two rows/columns changes the sign of a determinant.
Multiplying one row/column by k multiplies the determinant by k.
Adding a multiple of one row/column to another does not change the determinant.
A determinant with two identical or proportional rows/columns is zero.

03 • Trigonometric Functions

Core identities

sin²θ+cos²θ=1.
1+tan²θ=sec²θ.
1+cot²θ=cosec²θ.
tanθ=sinθ/cosθ; cotθ=cosθ/sinθ.
secθ=1/cosθ; cosecθ=1/sinθ.

Compound angles

sin(A+B)=sinA cosB+cosA sinB.
sin(A−B)=sinA cosB−cosA sinB.
cos(A+B)=cosA cosB−sinA sinB.
cos(A−B)=cosA cosB+sinA sinB.
tan(A+B)=(tanA+tanB)/(1−tanA tanB).
tan(A−B)=(tanA−tanB)/(1+tanA tanB).

Multiple and half angles

sin2A=2sinA cosA.
cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A.
tan2A=2tanA/(1−tan²A).
sin²(A/2)=(1−cosA)/2.
cos²(A/2)=(1+cosA)/2.
tan(A/2)=sinA/(1+cosA)=(1−cosA)/sinA.

Triangle formulae

Sine rule: a/sinA=b/sinB=c/sinC=2R.
Cosine rule: a²=b²+c²−2bc cosA; similarly cyclically.
Area of triangle: Δ=½bc sinA=½ca sinB=½ab sinC.
Δ=abc/(4R).
Heron's formula: Δ=√[s(s−a)(s−b)(s−c)], s=(a+b+c)/2.

Inverse trigonometric functions

sin⁻¹x principal range: [−π/2, π/2].
cos⁻¹x principal range: [0,π].
tan⁻¹x principal range: (−π/2,π/2).
sin(sin⁻¹x)=x; cos(cos⁻¹x)=x; tan(tan⁻¹x)=x within their domains.
tan⁻¹x+tan⁻¹y = tan⁻¹((x+y)/(1−xy)) with the appropriate branch adjustment.
tan⁻¹x−tan⁻¹y = tan⁻¹((x−y)/(1+xy)) with the appropriate branch adjustment.

04 • Pair of Straight Lines

Homogeneous pair through origin

ax²+2hxy+by²=0 represents a pair of straight lines through the origin when it factors into two real linear factors.
If y=mx, then bm²+2hm+a=0 gives the slopes m₁,m₂.
m₁+m₂=−2h/b; m₁m₂=a/b, when b≠0.

Angle and perpendicularity

tanθ=2√(h²−ab)/(a+b), when a+b≠0.
Perpendicular pair condition: a+b=0.
Coincident lines occur when the quadratic has equal roots.

Pair through a point

A general pair through (x₁,y₁) can be expressed using translated coordinates X=x−x₁, Y=y−y₁ in the corresponding homogeneous pair.
For a pair of lines represented by two linear factors L₁=0 and L₂=0, the combined equation is L₁L₂=0.

05 • Vectors

Basic vector formulae

If a=a₁i+a₂j+a₃k, then |a|=√(a₁²+a₂²+a₃²).
Unit vector along a: â=a/|a|.
Vector from A to B: AB⃗=b−a.
Section formula internally: if AP:PB=m:n, position vector of P=(n a+m b)/(m+n).

Dot product

a·b=a₁b₁+a₂b₂+a₃b₃.
a·b=|a||b|cosθ.
cosθ=(a·b)/(|a||b|).
a·b=0 ⇒ a⊥b (for non-zero vectors).
Projection of a on b: (a·b)/|b|.
Vector projection of a on b: [(a·b)/|b|²]b.

Cross product

a×b=| i j k; a₁ a₂ a₃; b₁ b₂ b₃ |.
|a×b|=|a||b|sinθ.
a×b=−(b×a).
a×a=0.
Area of parallelogram=|a×b|; area of triangle=½|a×b|.

Triple products

Scalar triple product: [abc]=a·(b×c).
[abc]=b·(c×a)=c·(a×b).
[abc]=−[bac].
Volume of parallelepiped=|a·(b×c)|.
Coplanarity condition: a·(b×c)=0.

06 • Line and Plane

Line in 3D

Vector form: r=a+λb.
Cartesian form: (x−x₁)/l=(y−y₁)/m=(z−z₁)/n.
Direction cosines l,m,n satisfy l²+m²+n²=1.
Line through two points A and B has direction vector B−A.

Angle between lines

cosθ=(b₁·b₂)/(|b₁||b₂|).
For direction ratios (l₁,m₁,n₁) and (l₂,m₂,n₂): cosθ=(l₁l₂+m₁m₂+n₁n₂)/(√(l₁²+m₁²+n₁²)√(l₂²+m₂²+n₂²)).

Plane

Plane equation: ax+by+cz+d=0.
Normal vector to the plane is n=ai+bj+ck.
Plane through point (x₁,y₁,z₁): a(x−x₁)+b(y−y₁)+c(z−z₁)=0.
Plane through three points can be obtained from a determinant equation.

Distances and angles

Distance of P(x₁,y₁,z₁) from ax+by+cz+d=0 is |ax₁+by₁+cz₁+d|/√(a²+b²+c²).
Angle between planes: cosθ=|a₁a₂+b₁b₂+c₁c₂|/(√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²)).
Planes are perpendicular when a₁a₂+b₁b₂+c₁c₂=0.
Planes are parallel when their normal vectors are parallel.

07 • Linear Programming

Standard structure

Objective function: Z=ax+by, to be maximised or minimised.
Constraints are linear inequalities such as a₁x+b₁y≤c₁.
Non-negativity restrictions commonly are x≥0, y≥0.
Feasible region = common region satisfying all constraints.

Graphical method

Plot each constraint boundary as a straight line.
Use a test point to determine the required half-plane.
Find all corner points of the feasible region.
Evaluate Z at each relevant corner point; the largest/smallest value gives the optimum when the standard conditions apply.

08 • Differentiation

Definition and rules

f′(x)=lim(h→0)[f(x+h)−f(x)]/h.
d(c)/dx=0.
d(xⁿ)/dx=nxⁿ⁻¹.
d(ku)/dx=k du/dx.
d(u±v)/dx=u′±v′.

Product and quotient

d(uv)/dx=u(dv/dx)+v(du/dx).
d(u/v)/dx=[v u′−u v′]/v².

Chain rule

d[f(g(x))]/dx=f′(g(x))g′(x).

Standard derivatives

d(eˣ)/dx=eˣ.
d(aˣ)/dx=aˣ ln a, a>0, a≠1.
d(ln x)/dx=1/x.
d(logₐx)/dx=1/(x ln a).
d(sin x)/dx=cos x.
d(cos x)/dx=−sin x.
d(tan x)/dx=sec²x.
d(cot x)/dx=−cosec²x.
d(sec x)/dx=sec x tan x.
d(cosec x)/dx=−cosec x cot x.

Inverse trigonometric derivatives

d(sin⁻¹x)/dx=1/√(1−x²).
d(cos⁻¹x)/dx=−1/√(1−x²).
d(tan⁻¹x)/dx=1/(1+x²).
d(cot⁻¹x)/dx=−1/(1+x²) under the standard principal convention.
d(sec⁻¹x)/dx=1/(|x|√(x²−1)).
d(cosec⁻¹x)/dx=−1/(|x|√(x²−1)).

Parametric and higher derivatives

If x=f(t), y=g(t), then dy/dx=(dy/dt)/(dx/dt).
d²y/dx² = [d/dt(dy/dx)]/(dx/dt).
Leibniz notation: y′=dy/dx; y″=d²y/dx².

09 • Applications of Derivatives

Tangent and normal

Slope of tangent at x=a: m=f′(a).
Tangent: y−f(a)=f′(a)(x−a).
Slope of normal: −1/f′(a), when f′(a) is finite and non-zero.
Normal: y−f(a)=−(x−a)/f′(a).

Increasing and decreasing

f′(x)>0 on an interval ⇒ f is increasing there.
f′(x)<0 on an interval ⇒ f is decreasing there.
Critical points are candidates where f′(x)=0 or f′(x) does not exist, subject to the domain.

Approximation and differentials

dy=f′(x)dx.
For small Δx: Δy≈dy=f′(x)Δx.
Relative error ≈ dy/y; percentage error ≈ (dy/y)×100.

Rolle and LMVT

Rolle: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then some c∈(a,b) satisfies f′(c)=0.
LMVT: if f is continuous on [a,b] and differentiable on (a,b), then some c satisfies f′(c)=[f(b)−f(a)]/(b−a).

Maxima and minima

If f′ changes + to − at c, f has a local maximum at c.
If f′ changes − to + at c, f has a local minimum at c.
Second derivative test: f′(c)=0 and f″(c)<0 ⇒ local maximum; f″(c)>0 ⇒ local minimum.
If f″(c)=0, the second derivative test is inconclusive.

10 • Indefinite Integration

Basic rules

∫k dx=kx+C.
∫xⁿdx=xⁿ⁺¹/(n+1)+C, n≠−1.
∫dx/x=ln|x|+C.
∫(u±v)dx=∫u dx±∫v dx.
∫k f(x)dx=k∫f(x)dx.

Standard integrals

∫eˣdx=eˣ+C.
∫aˣdx=aˣ/ln a+C.
∫sin x dx=−cos x+C.
∫cos x dx=sin x+C.
∫sec²x dx=tan x+C.
∫cosec²x dx=−cot x+C.
∫sec x tan x dx=sec x+C.
∫cosec x cot x dx=−cosec x+C.
∫tan x dx=ln|sec x|+C.
∫cot x dx=ln|sin x|+C.
∫sec x dx=ln|sec x+tan x|+C.
∫cosec x dx=ln|cosec x−cot x|+C.

Quadratic denominator forms

∫dx/(x²+a²)=(1/a)tan⁻¹(x/a)+C.
∫dx/(a²−x²)=(1/2a)ln|(a+x)/(a−x)|+C.
∫dx/(x²−a²)=(1/2a)ln|(x−a)/(x+a)|+C.
∫dx/√(a²−x²)=sin⁻¹(x/a)+C.

Methods

Substitution: if u=g(x), then ∫f(g(x))g′(x)dx=∫f(u)du.
Integration by parts: ∫u dv=uv−∫v du.
Choose u by a suitable priority such as algebraic/logarithmic/trigonometric/exponential structure.
Partial fractions decompose a rational function into simpler fractions before integration.

11 • Definite Integration

Fundamental theorem

If F′(x)=f(x), then ∫ₐᵇf(x)dx=F(b)−F(a).
∫ₐᵃf(x)dx=0.
∫ₐᵇf(x)dx=−∫ᵇₐf(x)dx.
∫ₐᶜf(x)dx+∫ᶜᵇf(x)dx=∫ₐᵇf(x)dx.

Symmetry properties

If f is even: ∫₋ₐᵃf(x)dx=2∫₀ᵃf(x)dx.
If f is odd: ∫₋ₐᵃf(x)dx=0.
∫₀ᵃf(x)dx=∫₀ᵃf(a−x)dx.

Useful transformations

∫₀ᵃf(x)dx=∫₀ᵃf(a−x)dx.
∫₋ₐᵃf(x)dx can be split into ∫₋ₐ⁰f(x)dx+∫₀ᵃf(x)dx.
Always preserve the order of limits and the sign.

12 • Application of Definite Integration

Area

Area under y=f(x) from x=a to x=b, when f≥0: A=∫ₐᵇf(x)dx.
Area between y=f(x) and y=g(x): A=∫ₐᵇ|f(x)−g(x)|dx.
If f is above g throughout: A=∫ₐᵇ[f(x)−g(x)]dx.
With respect to y: A=∫[right curve−left curve]dy.

Volume

Volume by discs/washers about the x-axis: V=π∫ₐᵇy²dx.
Volume by discs/washers about the y-axis: V=π∫ₐᵇx²dy.
Volume of a solid of revolution using shells about the y-axis: V=2π∫ₐᵇx y dx, when x and y describe the shell radius and height.

Limit of a sum

∫ₐᵇf(x)dx = lim(n→∞) Σ f(xᵢ*)Δx, with Δx=(b−a)/n.
For equal subintervals xᵢ=a+iΔx, the integral is the limit of the corresponding Riemann sum.

13 • Differential Equations

Basic terms

Order = highest order derivative present.
Degree = power of the highest-order derivative after the equation is polynomial in derivatives.
A solution is a function satisfying the differential equation.

Formation

To eliminate n arbitrary constants, differentiate n times and eliminate the constants to form a differential equation of order n.

Variable separable

If dy/dx=f(x)g(y), then dy/g(y)=f(x)dx and integrate both sides.

Homogeneous first-order form

If dy/dx=F(y/x), put y=vx.
Then dy/dx=v+x dv/dx.
If dx/dy=F(x/y), put x=vy and use dx/dy=v+y dv/dy.

Linear differential equation

Standard form: dy/dx+Py=Q.
Integrating factor: I.F.=e^(∫P dx).
Solution: y(I.F.)=∫Q(I.F.)dx+C.

Applications

For growth/decay: dy/dt=ky ⇒ y=Ce^(kt).
With y(0)=y₀: y=y₀e^(kt).
For cooling models, the rate is proportional to the difference between the object's temperature and surrounding temperature; solve the resulting first-order equation with the given initial condition.

14 • Probability Distributions

Discrete random variable

P(X=x)=p(x), with p(x)≥0.
Σp(x)=1.
Expected value: E(X)=Σxp(x).
E(X²)=Σx²p(x).
Variance: Var(X)=E(X²)−[E(X)]².
Standard deviation: σ=√Var(X).

Continuous random variable

Probability density function f(x) satisfies f(x)≥0 and ∫₋∞^∞f(x)dx=1.
P(a
Distribution function: F(x)=P(X≤x)=∫₋∞ˣf(t)dt.
Mean: E(X)=∫₋∞^∞x f(x)dx.
E(X²)=∫₋∞^∞x²f(x)dx.
Variance: Var(X)=E(X²)−[E(X)]².
Standard deviation: σ=√Var(X).

Expectation rules

E(c)=c.
E(aX+b)=aE(X)+b.
Var(aX+b)=a²Var(X).
σ(aX+b)=|a|σ(X).

15 • Bernoulli Trials and Binomial Distribution

Bernoulli trial

Each trial has two outcomes: success and failure.
Probability of success=p; probability of failure=q=1−p.
Trials are independent and p remains constant.

Binomial probability mass function

If X is the number of successes in n Bernoulli trials: P(X=r)=ⁿCᵣpʳqⁿ⁻ʳ, r=0,1,...,n.
ⁿCᵣ=n!/[r!(n−r)!].
Σᵣ₌₀ⁿ P(X=r)=1.

Mean, variance and S.D.

E(X)=np.
Var(X)=npq.
σ=√(npq).
Mode is commonly the integer part of (n+1)p when (n+1)p is not an integer; when (n+1)p is an integer, there are two modes, (n+1)p and (n+1)p−1.

Useful probabilities

P(X=0)=qⁿ.
P(X=n)=pⁿ.
P(X≥1)=1−qⁿ.
P(X≤r)=Σₖ₌₀ʳ ⁿCₖpᵏqⁿ⁻ᵏ.
P(X≥r)=Σₖ₌ᵣⁿ ⁿCₖpᵏqⁿ⁻ᵏ.

⭐ OMEGA Formula Rule

Formulae are not meant to be memorised blindly. Each formula should be read with its condition, meaning and appropriate method of application.

BOARD COVERAGE

15 Chapters — Maharashtra Std. 12 Mathematics

The list below is a navigation map; the detailed cards above are the actual learning material.

Mathematical Logic

Matrices

Trigonometric Functions

Pair of Straight Lines

Vectors

Line and Plane

Linear Programming

Differentiation

Applications of Derivatives

Indefinite Integration

Definite Integration

Application of Definite Integration

Differential Equations

Probability Distributions

Bernoulli Trials and Binomial Distribution

PROBLEM-SOLVING METHOD

How to Approach an HSC Mathematics Problem

Use a repeatable process rather than trying to recall a formula randomly.

Identify

Write what is given and what the question asks.

Choose

Select the theorem, formula, identity or method that connects the given quantities.

Solve

Write the working clearly and keep algebraic signs under control.

Verify

Check restrictions, substitutions, constants and whether the answer satisfies the original condition.

Learn Mathematics, Don't Just Memorise It.

Understand the idea. Choose the method. Solve carefully. Verify the result.

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Std. 12Mathematics

How to Learn Std. 12 Mathematics

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Mathematics — Complete 15-Chapter Learning Hub

01 • Mathematical Logic

02 • Matrices

03 • Trigonometric Functions

04 • Pair of Straight Lines

05 • Vectors

06 • Line and Plane

07 • Linear Programming

08 • Differentiation

09 • Applications of Derivatives

10 • Indefinite Integration

11 • Definite Integration

12 • Application of Definite Integration

13 • Differential Equations

14 • Probability Distributions

15 • Bernoulli Trials and Binomial Distribution

Std. 12 Mathematics — Chapter-wise Formulae

01 • Mathematical Logic

Basic logical forms

Important equivalences

Implication and quantifiers

Switching circuits

02 • Matrices

Basic operations

Determinant and inverse

3×3 determinant

Elementary transformations

Useful determinant facts

03 • Trigonometric Functions

Core identities

Compound angles

Multiple and half angles

Triangle formulae

Inverse trigonometric functions

04 • Pair of Straight Lines

Homogeneous pair through origin

Angle and perpendicularity

Pair through a point

05 • Vectors

Basic vector formulae

Dot product

Cross product

Triple products

06 • Line and Plane

Line in 3D

Angle between lines

Plane

Distances and angles

07 • Linear Programming

Standard structure

Graphical method

08 • Differentiation

Definition and rules

Product and quotient

Chain rule

Standard derivatives

Inverse trigonometric derivatives

Parametric and higher derivatives

09 • Applications of Derivatives

Tangent and normal

Increasing and decreasing

Approximation and differentials

Rolle and LMVT

Maxima and minima

10 • Indefinite Integration

Basic rules

Standard integrals

Quadratic denominator forms

Methods

11 • Definite Integration

Fundamental theorem

Symmetry properties

Useful transformations

12 • Application of Definite Integration

Std. 12
Mathematics