CBSE Class 12 Mathematics Syllabus

What conditions are required for matrix multiplication AB to be defined?

The number of columns of A must equal the number of rows of B. If A is m × n and B is n × p, then AB is m × p.

Is matrix multiplication commutative?

No. In general AB ≠ BA; matrix multiplication is associative and distributive over addition but not commutative.

What is a symmetric matrix and a skew-symmetric matrix?

Symmetric: A' = A (aij = aji). Skew-symmetric: A' = −A (aij = −aji), so all diagonal elements are 0.

How can any square matrix be expressed using symmetric and skew-symmetric parts?

A = ½(A + A') + ½(A − A'), where ½(A + A') is symmetric and ½(A − A') is skew-symmetric.

What is a minor M_ij of an element in a determinant?

The determinant of the submatrix obtained by deleting the i-th row and j-th column of that element.

What is the cofactor A_ij of element a_ij?

A_ij = (−1)^(i+j) M_ij, where M_ij is the minor of a_ij.

How is the area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) found using a determinant?

Area = ½ |Δ| where Δ = the 3×3 determinant with rows (x1, y1, 1), (x2, y2, 1), (x3, y3, 1).

How does multiplying every element of one row (or column) of a determinant by k affect its value?

The value of the determinant is multiplied by k.

What is the value of a determinant if two rows (or columns) are identical?

The determinant equals 0.

What is the adjoint (adj A) of a square matrix A?

The transpose of the matrix of cofactors of A.

State the relation A(adj A) = (adj A)A.

A(adj A) = (adj A)A = |A| I, where I is the identity matrix of the same order.

What is the formula for the inverse of a non-singular matrix A?

A⁻¹ = (1/|A|) adj A, valid when |A| ≠ 0.

When is a square matrix A invertible (non-singular)?

A is invertible if and only if |A| ≠ 0.

For |A| of order n, what is |adj A| in terms of |A|?

|adj A| = |A|^(n−1).

In the matrix method, how is the system AX = B solved when A is invertible?

X = A⁻¹B, where A is the coefficient matrix, X the variable matrix, and B the constants matrix.

When does the system AX = B have a unique solution, and what is it called when |A| = 0?

If |A| ≠ 0, there is a unique solution X = A⁻¹B (consistent). If |A| = 0, the system may be inconsistent (no solution) or have infinitely many solutions.

State the condition for a function f to be continuous at x = c.

f is continuous at c if lim(x→c) f(x) = f(c); i.e., the left limit, right limit, and value at c are all equal and finite.

State the chain rule for differentiation.

If y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx).

What is the derivative of a function defined parametrically as x = f(t), y = g(t)?

dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

State Rolle's Theorem.

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) with f'(c) = 0.

State the Mean Value Theorem (Lagrange).

If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) with f'(c) = [f(b) − f(a)]/(b − a).

What does the sign of f'(x) tell about a function on an interval?

f'(x) > 0 means f is increasing; f'(x) < 0 means f is decreasing on that interval.

State the second derivative test for local extrema at a critical point c (f'(c) = 0).

If f''(c) > 0, c is a point of local minimum; if f''(c) < 0, c is a local maximum; if f''(c) = 0, the test fails.

What is ∫ xⁿ dx for n ≠ −1?

∫ xⁿ dx = x^(n+1)/(n+1) + C.

What is the integration by parts formula?

∫ u v dx = u ∫v dx − ∫ [ (du/dx) ∫v dx ] dx (ILATE order to choose u).

State the Fundamental Theorem of Calculus (evaluation part).

If F is an antiderivative of f, then ∫ from a to b of f(x) dx = F(b) − F(a).

What is the property ∫ from a to b of f(x) dx in terms of ∫ from 0 to a of f(a−x)?

∫ from 0 to a of f(x) dx = ∫ from 0 to a of f(a − x) dx.

How is the area bounded by a curve y = f(x), the x-axis, and lines x = a, x = b found?

Area = ∫ from a to b of |f(x)| dx (take absolute value or split where the curve is below the axis).

What is the order and degree of a differential equation?

Order = order of the highest derivative present. Degree = power of the highest-order derivative, after the equation is made polynomial (free of radicals/fractions) in derivatives.

State the integrating factor (I.F.) method for a linear differential equation dy/dx + Py = Q.

I.F. = e^(∫P dx); solution: y·(I.F.) = ∫ Q·(I.F.) dx + C.