Mathematics

Chapter 1 of Class 10 Maths, "Real Numbers", covers the Fundamental Theorem of Arithmetic — which states every composite number has a unique prime factorisation — and uses it to prove that numbers such as √2 and √3 are irrational.

• 1The Fundamental Theorem of Arithmetic: every composite number has a unique prime factorisation (apart from the order of factors).
• 2HCF of two numbers equals the product of the smallest powers of all common prime factors; LCM equals the product of the greatest powers of all prime factors involved.
• 3For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b (this does NOT extend to three numbers).
• 4If a prime p divides a², then p divides a (Theorem 1.2) — the key lemma used to prove irrationality.
• 5√2 and √3 are proved irrational using proof by contradiction together with the Fundamental Theorem of Arithmetic.

NCERT Class 10 Maths Chapter 2 covers Polynomials — their degrees, zeroes, geometrical meaning on graphs, and the relationship between zeroes and coefficients of quadratic and cubic polynomials.

• 1Polynomials of degree 1, 2, and 3 are called linear, quadratic, and cubic polynomials respectively
• 2A real number k is a zero of polynomial p(x) if p(k) = 0; geometrically, zeroes are x-coordinates where the graph y = p(x) crosses the x-axis
• 3A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes
• 4For a quadratic ax² + bx + c with zeroes α and β: sum of zeroes = −b/a and product of zeroes = c/a
• 5For a cubic ax³ + bx² + cx + d with zeroes α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, and αβγ = −d/a

NCERT Class 10 Maths Chapter 3 covers Pair of Linear Equations in Two Variables, teaching students to solve such pairs using graphical, substitution, and elimination methods, and to determine whether a pair is consistent, inconsistent, or dependent based on the ratio of coefficients.

• 1A pair of linear equations can be consistent (has at least one solution) or inconsistent (has no solution); a dependent pair is always consistent with infinitely many solutions.
• 2Graphically, two lines either intersect at one point (unique solution), coincide (infinite solutions), or are parallel (no solution).
• 3The substitution method involves expressing one variable in terms of the other and substituting into the second equation to solve.
• 4The elimination method multiplies equations by suitable constants to make coefficients of one variable equal, then adds or subtracts to eliminate that variable.
• 5If a1/a2 ≠ b1/b2, the pair is consistent with a unique solution; if a1/a2 = b1/b2 = c1/c2, it is dependent (infinitely many solutions); if a1/a2 = b1/b2 ≠ c1/c2, it is inconsistent (no solution).

NCERT Class 10 Maths Chapter 4, Quadratic Equations, covers equations of the form ax² + bx + c = 0 (a ≠ 0) and teaches three methods to find their roots: factorisation, completing the square (leading to the quadratic formula), and the discriminant to determine the nature of roots.

• 1A quadratic equation in standard form is ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
• 2A quadratic equation has at most two roots; its roots are the same as the zeroes of the quadratic polynomial ax² + bx + c.
• 3Roots can be found by factorisation: split the middle term, express as a product of two linear factors, and set each factor to zero.
• 4The quadratic formula gives roots as x = (−b ± √(b²−4ac)) / 2a, valid when b²−4ac ≥ 0.
• 5The discriminant (b²−4ac) determines the nature of roots: >0 means two distinct real roots, =0 means two equal real roots, <0 means no real roots.

An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number called the common difference (d) to the preceding term; the nth term is given by an = a + (n − 1)d and the sum of the first n terms by Sn = n/2 [2a + (n − 1)d].

• 1An AP is defined as a list of numbers where each term is obtained by adding a fixed common difference d to the preceding term; d can be positive, negative, or zero.
• 2The general form of an AP is a, a+d, a+2d, a+3d, … where a is the first term and d is the common difference.
• 3The nth term (general term) of an AP is given by an = a + (n−1)d; the last term of a finite AP is sometimes denoted l.
• 4The sum of the first n terms is Sn = n/2 [2a + (n−1)d], or equivalently Sn = n/2 (a + l) when the last term l is known.
• 5A list is an AP if and only if the difference between every pair of successive terms is the same, i.e., ak+1 − ak is constant for all k.

NCERT Class 10 Maths Chapter 6 covers similarity of triangles, including the Basic Proportionality Theorem (Thales Theorem) and three criteria for similarity — AAA, SSS, and SAS — with an application to proving the Pythagoras Theorem.

• 1Two figures are similar if they have the same shape but not necessarily the same size; all congruent figures are similar but not vice versa.
• 2Two polygons are similar if their corresponding angles are equal and their corresponding sides are in the same ratio (scale factor).
• 3Basic Proportionality Theorem (Thales Theorem): a line parallel to one side of a triangle divides the other two sides in the same ratio; the converse is also true.
• 4AAA Similarity Criterion: if corresponding angles of two triangles are equal, their corresponding sides are proportional and the triangles are similar.
• 5SSS Similarity Criterion: if sides of one triangle are proportional to sides of another, their corresponding angles are equal and the triangles are similar.

NCERT Class 10 Maths Chapter 7 covers Coordinate Geometry, teaching students how to find the distance between two points using the Distance Formula and how to find the coordinates of a point dividing a line segment in a given ratio using the Section Formula.

• 1Distance Formula: the distance between P(x₁,y₁) and Q(x₂,y₂) is √[(x₂–x₁)²+(y₂–y₁)²], derived from the Pythagoras theorem.
• 2Distance of a point P(x,y) from the origin O(0,0) is √(x²+y²).
• 3Section Formula: the point dividing segment AB — where A is (x₁,y₁) and B is (x₂,y₂) — internally in ratio m₁:m₂ has coordinates ((m₁x₂+m₂x₁)/(m₁+m₂), (m₁y₂+m₂y₁)/(m₁+m₂)).
• 4Midpoint Formula (special case of Section Formula with ratio 1:1): midpoint of AB is ((x₁+x₂)/2, (y₁+y₂)/2).
• 5Three points are collinear if the sum of the distances between the two pairs of adjacent points equals the distance between the outermost pair.

NCERT Class 10 Maths Chapter 8 introduces trigonometry as the study of relationships between the sides and angles of a right triangle, defining six trigonometric ratios — sin, cos, tan, cosec, sec, and cot — and establishing key identities such as sin²A + cos²A = 1.

• 1Trigonometry studies relationships between sides and angles of a triangle; the word comes from Greek 'tri' (three), 'gon' (sides), 'metron' (measure).
• 2Six trigonometric ratios are defined for an acute angle A in a right triangle: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent, with cosec, sec, and cot as their respective reciprocals.
• 3The values of trigonometric ratios depend only on the angle, not on the size of the right triangle, because similar triangles have proportional sides.
• 4Standard angle values: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1; cos decreases from 1 to 0 as the angle increases from 0° to 90°.
• 5Three Pythagorean identities hold for all valid acute angles: sin²A + cos²A = 1, sec²A = 1 + tan²A, and cosec²A = 1 + cot²A.

NCERT Class 10 Maths Chapter 9 covers Some Applications of Trigonometry, teaching students how to use trigonometric ratios to find heights of towers, buildings, and other objects, and distances between them, using the concepts of angle of elevation and angle of depression.

• 1The line of sight is the line drawn from the observer's eye to the object being viewed.
• 2The angle of elevation is formed when the line of sight is above the horizontal level (observer looks up).
• 3The angle of depression is formed when the line of sight is below the horizontal level (observer looks down).
• 4Trigonometric ratios such as tan, sin, and cot are used in right-angled triangles to calculate unknown heights and distances.
• 5A tower 15 m away with an angle of elevation of 60° has a height of 15√3 m, calculated using tan 60° = √3.

NCERT Class 10 Maths Chapter 10 covers Circles, focusing on tangents to a circle — proving that the tangent at any point is perpendicular to the radius at the point of contact, and that the two tangents drawn from an external point to a circle are equal in length.

• 1A tangent to a circle intersects it at exactly one point, called the point of contact; a secant intersects at two points.
• 2Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact (OP ⊥ XY).
• 3Theorem 10.2: The lengths of the two tangents drawn from an external point to a circle are equal (PQ = PR).
• 4From a point inside a circle, no tangent can be drawn; from a point on the circle, exactly one tangent exists; from a point outside, exactly two tangents can be drawn.
• 5The centre of a circle lies on the angle bisector of the angle formed between the two tangents drawn from an external point.

NCERT Class 10 Maths Chapter 11 covers areas related to circles, teaching students how to calculate the area and arc length of a sector using the formula (θ/360) × πr², and the area of a segment by subtracting the corresponding triangle area from the sector area.

• 1A sector is the region enclosed by two radii and an arc; a segment is the region between a chord and its arc — both come in minor and major variants.
• 2Area of a sector with radius r and central angle θ (in degrees) = (θ/360) × πr².
• 3Length of an arc of a sector with radius r and angle θ = (θ/360) × 2πr.
• 4Area of a segment = Area of the corresponding sector − Area of the corresponding triangle (formed by the two radii and the chord).
• 5Area of the major sector = πr² − Area of the minor sector; area of the major segment = πr² − Area of the minor segment.

NCERT Class 10 Maths Chapter 12 covers surface areas and volumes of solids formed by combining basic shapes such as cuboids, cones, cylinders, spheres, and hemispheres, teaching students to calculate both the combined surface area (using curved surface areas of visible parts) and the combined volume (by summing the volumes of constituent solids).

• 1The total surface area of a combined solid equals the sum of the curved surface areas of only the exposed (visible) parts of each constituent solid.
• 2The volume of a combined solid is always the direct sum of the volumes of its individual constituent solids.
• 3Common composite shapes include: cylinder with two hemispheres (oil tankers, capsules), cone surmounted by hemisphere (toys, tops), cuboid surmounted by hemisphere (decorative blocks), and cuboid surmounted by half cylinder (industrial sheds).
• 4For the playing top (cone surmounted by hemisphere, total height 5 cm, diameter 3.5 cm), the surface area to be coloured is approximately 39.6 cm².
• 5For a decorative cube-and-hemisphere block (cube edge 5 cm, hemisphere diameter 4.2 cm), the total surface area is 163.86 cm².

NCERT Class 10 Maths Chapter 13 covers Statistics, extending the study of mean, median, and mode from ungrouped data to grouped data, and introducing cumulative frequency distributions and ogives.

• 1Three methods — Direct, Assumed Mean, and Step-Deviation — are used to find the mean of grouped data and always give the same result.
• 2The mode of grouped data is calculated using the formula Mode = l + [(f1 − f0) / (2f1 − f0 − f2)] × h, where the modal class has the highest frequency.
• 3The median of grouped data is found using Median = l + [(n/2 − cf) / f] × h, after identifying the median class from the cumulative frequency table.
• 4Cumulative frequency distributions can be of the 'less than' type (using upper class limits) or 'more than' type (using lower class limits).
• 5The empirical relationship between the three measures of central tendency is: 3 Median = Mode + 2 Mean.

Chapter 14 of NCERT Class 10 Mathematics introduces theoretical (classical) probability, defined as the number of outcomes favourable to an event divided by the total number of equally likely possible outcomes, with probability values always lying between 0 and 1 inclusive.

• 1Theoretical probability P(E) = (Number of outcomes favourable to E) / (Number of all possible outcomes), assuming equally likely outcomes
• 2Probability of any event E satisfies 0 ≤ P(E) ≤ 1; an impossible event has probability 0 and a sure (certain) event has probability 1
• 3For complementary events E and not-E: P(E) + P(not E) = 1, so P(not E) = 1 – P(E)
• 4The sum of probabilities of all elementary events of an experiment is always 1
• 5A standard deck has 52 cards in 4 suits of 13 each; probability of drawing an ace is 4/52 = 1/13

Appendix A1 covers the foundations of mathematical proofs for Class 10, teaching students how to classify mathematical statements, apply deductive reasoning, negate statements, find converses, and use proof by contradiction — all illustrated through worked theorems and examples.

• 1Mathematical statements must be always true or always false; ambiguous sentences are not accepted in mathematics
• 2Deductive reasoning draws conclusions from given premises or hypotheses, regardless of whether those premises are actually true
• 3A conjecture is an intelligent guess; a single counter-example is sufficient to disprove it
• 4The negation of statement p, written ~p, is true whenever p is false and false whenever p is true; double negation ~(~p) equals p
• 5The converse of 'if p then q' is 'if q then p'; a true statement's converse is not necessarily true

Mathematical modelling is the process of converting a real-world problem into a mathematical description, solving it, and checking that the solution makes sense back in the original situation. NCERT Class 10 Appendix A2 covers five stages—understanding the problem, formulating it mathematically, solving it, interpreting the solution, and validating the model—illustrated through examples such as estimating fish population, rolling dice, and comparing instalment interest rates.

• 1A mathematical model is a mathematical description of a real-life situation; modelling converts a real problem into an equivalent mathematical problem, solves it, and interprets the result.
• 2Five stages: (1) understand the problem with simplifying assumptions, (2) formulate mathematically using variables, equations, data tables, graphs, or probabilities, (3) solve, (4) interpret, and (5) validate.
• 3If validation shows the solution does not make sense in the real situation, the assumptions from Step 1 are revised and the cycle repeats.
• 4Fish-in-a-lake mark-recapture example: mark 20 fishes, re-sample 50 and find 5 marked, so 1/10 of the population is marked, giving an estimated total of 200 fishes.
• 5Dice example: out of 36 equally likely outcomes, sum 7 has the highest probability (6/36 = 1/6), making 7 the best guess.

This page provides the free PDF of NCERT Class 10 Mathematics Answers/Hints — the official appendix giving final answers and solution hints to every textbook exercise across Chapters 1 through 14 plus two appendices.

• 1Covers all 14 chapters of the NCERT Class 10 Mathematics textbook, from Chapter 1 (Real Numbers) to Chapter 14 (Probability).
• 2Includes answers for multiple exercises per chapter — for example, Exercises 3.1, 3.2, 3.3 for Pair of Linear Equations and Exercises 5.1, 5.2, 5.3, and the optional 5.4 for Arithmetic Progressions.
• 3Appendix 1 (Mathematical Reasoning) answers are provided across six exercises: A1.1 through A1.6.
• 4Appendix 2 (Mathematical Modelling) answers are provided for exercises A2.2 and A2.3.
• 5Many answers include brief hints or key steps — not just final values — helping students understand where they went wrong, not just what the answer is.