NCERT Solutions for Class 10 Maths – Chapter-wise Questions & Answers (Free PDF, Hindi & English)

Table of Contents

Chapter 1 – Real Numbers

Start with Euclid’s Division Lemma to compute HCF via repeated division and see how it powers many proofs. The Fundamental Theorem of Arithmetic ensures unique prime factorisation, which helps deduce results like terminating vs non-terminating recurring decimals (check prime factors of the denominator: only 2 and/or 5 → terminating). Typical board questions ask you to prove irrationality (e.g., √2, 3+2√5), to find HCF/LCM using prime trees, and to convert fractions to decimals (and back). The provided solutions show clean, line-by-line reasoning, highlight where to justify each step (lemma/theorem reference), and include bilingual links so you can revise in the language you prefer.

Chapter 2 – Polynomials

Recall degree, coefficients, and the Remainder & Factor Theorems. You’ll relate zeros and coefficients for linear/quadratic/cubic cases, use division algorithm to divide one polynomial by another, and construct quadratics given zeros. Board-style tasks also ask to find a polynomial when sum and product of zeros are known, or to verify identities via factorisation. The solutions use minimal algebraic steps with clear justifications, and show how sign conventions affect sum/product relations—an easy place to lose marks without a tidy working.

Chapter 3 – Pair of Linear Equations in Two Variables

Build intuition graphically (intersecting/parallel/coincident lines) and solve algebraically via substitution, elimination, and cross-multiplication. Word problems span mixtures, numbers, money, time–work, and boats–streams. The key is framing: define variables, form two independent equations, and choose the method that keeps arithmetic light. Our solutions annotate each step (why we multiplied, or which term we eliminated), and include quick checks to avoid sign slips—especially in cross-multiplication.

Chapter 4 – Quadratic Equations

Identify quadratics from contexts, convert statements to equations, and solve using factorisation, completing the square, or the formula. Discriminant tells you the nature of roots (real/distinct/equal/complex). We also show parameter-based questions (for what ‘k’ are roots equal?) and sum/product of roots techniques for quick checks. Each solution is annotated with the method choice and the minimal algebra needed to avoid errors under time pressure.

Chapter 5 – Arithmetic Progressions

Master the two pillars: a_n = a + (n−1)d and S_n = n/2 [2a + (n−1)d]. Word problems will ask for missing terms, how many terms meet a condition, or total cost/earnings over time. Our solutions show how to choose the correct variable (term number vs number of terms), label knowns, and justify transitions. We also include quick tests to detect when an AP model is inappropriate—saving you time in mixed-section papers.

Chapter 6 – Triangles

Similarity rules (AAA, SAS, SSS) lead to proportional sides and equal corresponding angles. From the Basic Proportionality Theorem (Thales) to Pythagoras and its converse, you’ll prove results and apply them to find lengths/areas. Expect questions mixing algebra with geometry—e.g., using ratios from similarity to compute unknown segments. Our proofs are sentence-structured with reasons in-line so you can mirror the style in the exam.

Chapter 7 – Coordinate Geometry

Distance formula grounds many checks (equal sides, radii). Section formula lets you divide a segment internally at a given ratio. The area of triangle formula detects collinearity (area 0). Our solutions keep arithmetic organised and show when to square early to avoid radicals, plus sign-care tips for ratios to reduce rework.

Chapter 8 – Introduction to Trigonometry

Define the six ratios on a right triangle, learn values at standard angles (0°, 30°, 45°, 60°, 90°), identities like sin²θ + cos²θ = 1, and complementary relationships (sin(90°−θ)=cosθ, etc.). We include value tables, proofs of identities, and quick mental hints to remember special angles—then apply them to simple heights/lengths questions that don’t require diagrams.

Chapter 9 – Applications of Trigonometry

Real-life height and distance problems using a single right triangle or two linked triangles. The steps: sketch, mark angles of elevation/depression, assign variables, pick the correct ratio (tan most often), and solve. Our solutions stress unit consistency and when to add/subtract angles if observations are from different points.

Chapter 10 – Circles

Core facts: a tangent is perpendicular to the radius at the point of contact; from an external point, lengths of tangents are equal. Expect diagram-based reasoning with algebraic equalities. Our solutions show how to convert equal tangent lengths into segment equations and close the proof quickly.

Chapter 11 – Constructions

Use similarity to construct a triangle similar to a given one with a specified ratio; draw tangents to a circle from a point outside it. The solutions break every compass-and-straightedge step into short bullets and include figure checks so you don’t miss a mark on precision.

Chapter 12 – Areas Related to Circles

Compute circumference and area swiftly, then handle sectors/segments using central angle ratios. For composite figures, split into basic shapes and sum/subtract systematically. Our solutions show common angle–radian conversions and neat fraction forms to keep arithmetic clean.

Chapter 13 – Surface Areas and Volumes

Apply nets and formulas for cuboid/cube, cylinder, cone, sphere/hemisphere, and the frustum. Combination problems are common—identify each solid, compute required areas/volumes, and add/subtract correctly. We’ve included a quick-reference formula band and unit checks to avoid last-minute slips.

Chapter 14 – Statistics

Work with grouped data to find mean (assumed mean/step-deviation), median (via cumulative frequency), and mode (modal class). The solutions explain when each method is efficient and how to verify with rough estimates. We also show how small class-interval mistakes derail answers and how to keep cf tables consistent.

Chapter 15 – Probability

Classical (theoretical) probability on equally likely outcomes: coins, dice, cards, and simple bags of coloured balls. You’ll use complementary events (P(Aᶜ)=1−P(A)) and union for exclusive events. Our worked steps show outcome listing or counting via multiplication principle and include quick sanity checks so answers are guaranteed between 0 and 1.

CBSE Class 10 Maths – Unit-wise Weightage (Indicative)

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