Mathematics I · Class 12

Chapters

1. Relations and Functions
   • A pairing that need not follow any pattern
   • Reading a relation's behaviour: reflexive, symmetric, transitive
   • When 'related' means 'the same': equivalence relations and the classes they build
   • Equivalence relations at work: building from what a relation already is
   • Two questions every function answers: is it one-one, is it onto?
   • Why a finite set only ever needs one check, not two
   • Combining two functions into one, and whether the order matters
   • Undoing a function exactly: invertibility and why it needs both properties
2. Inverse Trigonometric Functions
   • One ratio, many angles
   • Restricting sine and cosine
   • Restricting tangent and cotangent
   • Restricting secant and cosecant, where the domain itself has a gap
   • The complete table, and a warning the notation forces on you
   • Reading an angle back out of its own principal branch
   • Rewriting a compound expression in its simplest form
   • Two angles that add to one, stated once and used everywhere
3. Matrices
   • Lining up numbers so they mean something
   • A ladder of shapes, each one a stricter version of the last
   • When two tables of numbers are really the same one
   • Adding, subtracting and scaling, one entry at a time
   • The rules these operations never break
   • When two tables combine into one, and what you get
   • What multiplication refuses to promise
   • Flipping a matrix across its own diagonal
   • Matrices that mirror themselves, and the split every matrix admits
   • Undoing a matrix, and what this chapter cannot yet show you
4. Determinants
   • A single number that decides everything
   • Building the determinant, one order at a time
   • Does the choice of row or column matter?
   • Minors and cofactors: reading a determinant entry by entry
   • Why a row's cofactors from a different row always cancel
   • The determinant as area, and when three points collapse to a line
   • The adjoint, a matrix assembled to reverse another
   • Singular, non-singular, and the inverse formula you can derive yourself
   • What an inverse obeys: products, and a shortcut that skips the adjoint entirely
   • Solving a system when the coefficient matrix is not singular
   • When the coefficient matrix is singular, telling infinitely many apart from none
5. Continuity and Differentiability
   • When a graph can be trusted not to jump
   • Continuity across a domain, and what its edges are allowed to ask
   • When a limit refuses to settle at a real number
   • The algebra that keeps continuity intact
   • When a rate of change actually exists
   • The chain rule, and what it unlocks for a relation you cannot solve for y
   • Differentiating the inverse trigonometric functions, and where the formula stops applying
   • Building the exponential and logarithmic functions, and differentiating them outright
   • Logarithmic differentiation: turning a product or a power into a sum
   • Differentiating a curve traced by a third variable
   • The second derivative: how fast the rate of change is itself changing
6. Application of Derivatives
   • The rate hiding inside a changing quantity
   • When two changing quantities are yoked together
   • The derivative as a verdict on cost and revenue
   • Reading the sign of a derivative like a map
   • Turning points: where growth stops and reverses
   • When the slope test alone is not enough
   • The best value on a bounded stage
   • Designing the best shape from one equation