Every step clearly derived with given values and applied formulas.
Accurate syllabus coverage with deleted exercises omitted.
Common pitfalls, step-marking highlights, and shortcut checks.
📖 NCERT Class 9 Maths Chapter-wise Solutions (Complete Index)
Click on any chapter or exercise below to access full handwritten step-by-step solutions and free downloadable PDFs:
Chapter 1: Number Systems (संख्या पद्धति)
5 Exercises
Rational and irrational numbers, real numbers and decimal expansions, operations on real numbers, laws of exponents.
Chapter 2: Polynomials (बहुपद)
4 Exercises
Polynomials in one variable, zeroes of a polynomial, Factor Theorem, algebraic identities.
Chapter 3: Coordinate Geometry (निर्देशांक ज्यामिति)
2 Exercises
Cartesian plane, coordinates of a point, plotting points, quadrants and axes conventions.
Chapter 4: Linear Equations in Two Variables (दो चरों वाले रैखिक समीकरण)
2 Exercises
Standard form ax + by + c = 0, solution of linear equations, representation on Cartesian plane.
Chapter 5: Introduction to Euclid’s Geometry (यूक्लिड की ज्यामिति का परिचय)
1 Exercise
Euclid’s definitions, axioms and postulates, equivalent versions of Euclid’s fifth postulate.
MCQ Test
Chapter 6: Lines and Angles (रेखाएँ और कोण)
2 Exercises
Basic terms, intersecting & parallel lines, pairs of angles, parallel lines and a transversal.
Chapter 7: Triangles (त्रिभुज)
3 Exercises
Congruence of triangles, criteria for congruence (SAS, ASA, SSS, RHS), properties of triangles.
Chapter 8: Quadrilaterals (चतुर्भुज)
1 Exercise
Properties of parallelograms, Mid-point theorem and its converse, conditions for quadrilaterals.
MCQ Test
Chapter 9: Circles (वृत्त)
3 Exercises
Angle subtended by chords, perpendicular from centre to chord, equal chords and distance, cyclic quadrilaterals.
Chapter 10: Heron’s Formula (हीरोन का सूत्र)
1 Exercise
Area of triangle using semi-perimeter s = (a+b+c)/2 and formula √[s(s-a)(s-b)(s-c)].
Chapter 11: Surface Areas and Volumes (पृष्ठीय क्षेत्रफल एवं आयतन)
4 Exercises
Surface area and volume of right circular cones, spheres, and hemispheres.
Chapter 12: Statistics (सांख्यिकी)
1 Exercise
Graphical representation of data: bar graphs, histograms (with varying base widths), frequency polygons.
✍️ Topper’s Exemplary Handwritten Solutions (Model Step-by-Step Layout)
Below are authentic step-by-step solutions demonstrating the exact answer-presentation format that scores full marks in CBSE Class 9 examinations:
Exercise 1.5, Q5 (High-Frequency Exam Question)
Rationalize the denominator of: (1 / {7) – {6}}
To rationalize a binomial surd of the form (a – b), multiply both the numerator and the denominator by its rationalizing factor (conjugate), which is ({7} + {6}).
Applying standard identity (a – b)(a + b) = a² – b²:
Denominator = ({7})² – ({6})² = 7 – 6 = 1.
Numerator = 1 × ({7} + {6}) = {7} + {6}.
Therefore:
= {{7} + {6}}{1} = {7} + {6}
{7} + {6}
Always write out the identity (a-b)(a+b) = a² – b² explicitly in the right margin. CBSE examiners allocate 0.5 marks specifically for stating the applied algebraic identity.
Exercise 2.4, Q5 (Cubic Factorisation)
Factorise completely: p(x) = x³ – 2x² – x + 2
Let x = 1:
p(1) = (1)³ – 2(1)² – (1) + 2 = 1 – 2 – 1 + 2 = 0.
Since p(1) = 0, by the Factor Theorem, (x – 1) is a factor of p(x).
x³ – 2x² – x + 2 = x²(x – 2) – 1(x – 2)
= (x – 2)(x² – 1)
Now expand the quadratic factor using (a² – b²) = (a – b)(a + b):
x² – 1 = (x – 1)(x + 1).
(x – 1)(x + 1)(x – 2)
When factoring cubic polynomials, always check if terms can be grouped directly by pairs (here x²(x-2) – 1(x-2)) before spending 5 minutes doing long division!
Frequently Asked Questions (Class 9 Maths NCERT)
There are exactly 12 chapters in the rationalized 2026-27 curriculum. Old Chapter 9 (Areas of Parallelograms & Triangles), Chapter 11 (Constructions), and Chapter 15 (Probability) have been completely removed by NCERT.
Yes. Over 85% of CBSE and State Board annual exam question papers are directly based on NCERT examples, in-text proofs, and exercise problems. Practicing NCERT with step-by-step presentation guarantees top scores.
You can access complete PDF downloads and online readers for all Class 9 subjects on our Class 9 NCERT Solutions Hub and Class 9 Books Hub.