Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board Syllabus 2025-26

• Universal quantifier (∀)
• Existential quantifier (∃)

• Concept of Statements

1. Conjunction (∧)
2. Disjunction (∨)
3. Conditional statement (Implication) (→)
4. Biconditional (Double implication) (↔) or (⇔)
5. Negation (∼)

• Idempotent law
• Associative law
• Commutative law
• Distributive law
• Identity law
• Complement law
• Involution law
• DeMorgan’s laws

• Two switches in series
• Two switches in parallel

• Concept of Statements

1. Conjunction (∧)
2. Disjunction (∨)
3. Conditional statement (Implication) (→)
4. Biconditional (Double implication) (↔) or (⇔)
5. Negation (∼)

• Universal quantifier (∀)
• Existential quantifier (∃)

• Negation of conjunction
• Negation of disjunction
• Negation of implication
• Negation of biconditional

• Idempotent law
• Associative law
• Commutative law
• Distributive law
• Identity law
• Complement law
• Involution law
• DeMorgan’s laws

• Two switches in series
• Two switches in parallel

•  Inverse of a nonsingular matrix by elementary transformation
•  Inverse of a square matrix by adjoint method

• Method of Inversion
• Method of Reduction

• Consistent System
• Inconsistent System
• Solution of a system of linear equations using the inverse of a matrix

• Trigonometric equation
• Solution of Trigonometric equation
• Principal Solutions
• The General Solution

• Polar co-ordinates
• Relation between the polar co-ordinates and the Cartesian co-ordinates
• Solving a Triangle
• The Sine rule
• The Projection rule
• Applications of the Sine rule, the Cosine rule and the Projection rule

• Introduction of Inverse Trigonometric Functions

• Degree of a term
• Homogeneous Equation

• The necessary conditions for a general second degree equation
  ax² + 2hxy + by² + 2gx + 2fy + c = 0

1. abc + 2fgh - af² - bg² - ch² = 0
2. h² - ab ≥ 0

• Equation of a line through a given point and parallel to a given vector `vec b`
• Equation of a line passing through two given points

• Magnitude of a Vector

• Zero Vector
• Unit Vector
• Co-initial and Co-terminus Vectors
• Equal Vectors
• Negative of a Vector
• Collinear Vectors
• Free Vectors
• Localised Vectors

• Addition of Two Vectors
  - Parallelogram Law
  - Triangle Law of addition of two vectors
• Subtraction of two vectors
• Scalar multiplication of a vector

• Co-ordinates of a point in space
• Co-ordinates of points on co-ordinate axes
• Co-ordinates of points on co-ordinate planes
• Distance of P(x, y, z) from co-ordinate planes
• Distance of any point from origin
• Distance between any two points in space
• Distance of a point P(x, y, z) from coordinate axes

• Vector addition using components
• Components of a vector in two dimensions space
• Components of a vector in three-dimensional space

• Section formula for internal division
• Midpoint formula
• Section formula for external division

• Finding angle between two vectors
• Projections
• Direction Angles and Direction Cosine
• Direction ratios
• Relation between direction ratios and direction cosines

• Angle between two vectors
• Geometrical meaning of vector product

• Equation of a line passing through a given point and parallel to given vector
• Equation of a line passing through given two points

• Introduction of Distance of a Point from a Line
• Distance between two parallel lines

• Distance between skew lines
• Distance between parallel lines

• Passing through a point and perpendicular to a vector
• Passing through a point and parallel to two vectors
• Passing through three non-collinear points
• In normal form
• Passing through the intersection of two planes

• Convex Sets
• Graphical representation of linear inequations in two variables
• Graphical solution of linear inequation

• Meaning of Linear Programming Problem
• Mathematical formulation of a linear programming problem
• Familiarize with terms related to Linear Programming Problem

• Graphical method of solution for problems in two variables
• Feasible and infeasible regions and bounded regions
• Feasible and infeasible solutions
• Optimum feasible solution

• Interchange of any two rows or any two columns
• Multiplication of the elements of any row or column by a non-zero scalar
• Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)

• Rule of Differentiation
• Introduction

• Velocity
• Acceleration
• Jerk

• First and Second Derivative test
• Determine critical points of the function
• Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
• Find the absolute maximum and absolute minimum value of a function

• ∫ tan x dx = log | sec x |  + C
• ∫ cot x dx = log | sin x | + C
• ∫ sec x dx = log | sec x + tan x | + C
• ∫ cosec x dx = log | cosec x – cot x | + C

• `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
• Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
• Integrals of some more types

1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`

If ∫ f(x) dx = g(x) + c, then

`int_a^b f(x) dx = [g(x) + c]_a^b = g(b) - g(a)`.

• Formation of Differential equations from Physical Situations
• Formation of Differential Equations from Geometrical Problems

• Population Growth and Growth of bacteria
• Ratio active Decay
• Newton's Law of Cooling
• Surface Area

• Probability distribution of a random variable

• Discrete random variable
• Continuous random variable
• Probability Mass Function
• Cumulative Distribution Function or Distribution Function
• Cumulative Distribution Function from Probability Mass function
• Probability Mass Function from Cumulative Distribution Function 

• Probability density function
• Cumulative distribution function

• Polar co-ordinates
• Relation between the polar co-ordinates and the Cartesian co-ordinates
• Solving a Triangle
• The Sine rule
• The Projection rule
• Applications of the Sine rule, the Cosine rule and the Projection rule

• Introduction of Inverse Trigonometric Functions

Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

Acute Angle Between the Lines represented by ax²+2hxy+by²=0

• Section formula for internal division
• Midpoint formula
• Section formula for external division

• Position Vector
• Direction Cosines and Direction Ratios of a Vector

• Direction cosines of a line passing through two points.

• Introduction of Distance of a Point from a Line
• Distance between two parallel lines

• Coplanar
• Skew lines
• Distance between two skew lines
• Distance between parallel lines

• Equation of a line through a given point and parallel to a given vector `vec b`
• Equation of a line passing through two given points

• Definition of related terminology such as constraints, objective function, optimization.

• Different types of linear programming (L.P.) problems

1. Manufacturing problem
2. Diet Problem
3. Transportation problem

• Graphical method of solution for problems in two variables
• Feasible and infeasible regions and bounded regions
• Feasible and infeasible solutions
• Optimum feasible solution

left hand derivative and right hand derivative (need and concept)

• Derivative of Functions Which Expressed in Higher Order Derivative Form

• Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations

• First and Second Derivative test
• Determine critical points of the function
• Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
• Find the absolute maximum and absolute minimum value of a function

• ∫ tan x dx = log | sec x |  + C
• ∫ cot x dx = log | sin x | + C
• ∫ sec x dx = log | sec x + tan x | + C
• ∫ cosec x dx = log | cosec x – cot x | + C

• `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
• Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
• Integrals of some more types

1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`

Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

1. `int_a^a f(x) dx = 0`
2. `int_a^b f(x) dx = - int_b^a f(x) dx`
3. `int_a^b f(x) dx = int_a^b f(t) dt`
4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
   where a < c < b, i.e., c ∈ [a, b]
5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
   = 0,  if f(x) is odd function

• Area of the Region Bounded by a Curve & X-axis Between two Ordinates
• Area of the Region Bounded by a Curve & Y-axis Between two Abscissa 
• Circle-line, elipse-line, parabola-line

• Independent Events

• Probability distribution of a random variable