NCERT Solutions
Class 12 Mathematics
13 chapters · 22 important questions
Relations and Functions
Covers types of relations (reflexive, symmetric, transitive, equivalence) and types of functions (one-one, onto, bijective). Composition of functions and invertible functions are key topics.
Key Topics
Important Questions
Show that the relation R in the set A = {1,2,3,4,5} defined as R = {(a,b): |a−b| is even} is an equivalence relation.
Let f: R→R defined by f(x) = 3x+5. Show f is bijective and find f⁻¹.
Inverse Trigonometric Functions
Defines inverse trig functions, their domains and ranges, and principal values. Properties and identities of inverse trig functions are frequently asked.
Key Topics
Important Questions
Find the principal value of sin⁻¹(−√3/2).
Prove: tan⁻¹(1/2) + tan⁻¹(1/3) = π/4.
Matrices
Covers matrix types, operations (addition, multiplication, transpose), and properties. Symmetric and skew-symmetric matrices are important concepts.
Key Topics
Important Questions
Express the matrix A = [[2,−2,−4],[−1,3,4],[1,−2,−3]] as the sum of a symmetric and skew-symmetric matrix.
Determinants
Covers evaluation of determinants, properties, Cramer's rule, and area of triangles. Adjoint and inverse of a matrix using determinants is a key exam topic.
Key Topics
Important Questions
Using properties of determinants, prove: |a+b+2c, a, b; c, b+c+2a, b; c, a, c+a+2b| = 2(a+b+c)³
Continuity and Differentiability
Covers continuity, differentiability, and differentiation of composite, implicit, and parametric functions. Chain rule, logarithmic differentiation, and Rolle's/Mean Value theorems are included.
Key Topics
Important Questions
Find dy/dx if y = (sin x)^x + x^(sin x).
If x = a(cos θ + θ sin θ) and y = a(sin θ − θ cos θ), find d²y/dx².
Application of Derivatives
Covers rate of change, increasing/decreasing functions, maxima and minima, tangents and normals. Optimization word problems are exam favourites.
Key Topics
Important Questions
Find the intervals in which f(x) = 2x³ − 9x² + 12x − 5 is increasing or decreasing.
Find the dimensions of a rectangle with maximum area inscribed in a circle of radius r.
Integrals
Covers integration as anti-differentiation and definite integrals. Methods: substitution, partial fractions, integration by parts. Properties of definite integrals are frequently used in board exams.
Key Topics
Important Questions
Evaluate: ∫(x² + 1)/(x⁴ + x² + 1) dx
Evaluate: ∫₀^π x/(1 + sin x) dx
Application of Integrals
Covers finding areas under curves and between curves using definite integrals. Sketching the region before integrating is expected in board answers.
Key Topics
Important Questions
Find the area enclosed by the ellipse x²/4 + y²/9 = 1.
Find the area of the region bounded by y² = 4x and the line x = 3.
Differential Equations
Covers order and degree of differential equations, formation, and solution methods — variable separable, homogeneous, and linear first-order DEs.
Key Topics
Important Questions
Solve: (x + y)dy + (x − y)dx = 0
Solve: dy/dx + 2y = e^(2x) sin x
Vector Algebra
Covers vectors and their types, dot product, cross product, and applications. Scalar and vector triple products are also included.
Key Topics
Important Questions
If a⃗ × b⃗ = c⃗ × d⃗ and a⃗ × c⃗ = b⃗ × d⃗, show that (a⃗ − d⃗) is parallel to (b⃗ − c⃗).
Three Dimensional Geometry
Covers direction cosines and ratios, equations of lines and planes in 3D, and angle between lines/planes. Distance of a point from a plane is a common numerical.
Key Topics
Important Questions
Find the shortest distance between the lines r⃗ = (1+λ)î + (2−λ)ĵ + (1+λ)k̂ and r⃗ = 2î − ĵ − k̂ + μ(2î + ĵ + 2k̂).
Linear Programming
Covers formulation of LPP, graphical method of solution, feasible region, and corner point theorem. Word problems on manufacturing, diet, and transportation are common.
Key Topics
Important Questions
Maximise Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0, y ≥ 0.
A factory makes tennis rackets and cricket bats. Formulate as LPP and solve graphically.
Probability
Covers conditional probability, Bayes' theorem, random variables, and Binomial distribution. Bayes' theorem problems are very commonly asked in board exams.
Key Topics
Important Questions
A card from a pack of 52 cards is lost. From the remaining 51 cards, two are drawn and both are found to be spades. Find the probability that the missing card is a spade.
Five cards are drawn from a pack of 52. Find the probability of getting 3 diamonds and 2 spades.