As the saying goes, “*Mathematics is the music of Reason*” and this is exactly what maths is doing which is building up your reasoning skills. It’s not that you are just preparing maths for your exam but maths is preparing you for a bigger exam i.e. LIFE. NCERTBOOKS.GURU has compiled the most accurate and comprehensive solutions for **class 10 Maths NCERT solutions**.

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**NCERT Solutions for Class 10 Maths PDF Free Download Chapter-wise**

- Chapter 1 Real Numbers
- Chapter 2 Polynomials
- Chapter 3 Pair of Linear Equations in Two Variables
- Chapter 4 Quadratic Equations
- Chapter 5 Arithmetic Progression
- Chapter 6 Triangles
- Chapter 7 Coordinate Geometry
- Chapter 8 Introduction to Trigonometry
- Chapter 9 Some Applications of Trigonometry
- Chapter 10 Circles
- Chapter 11 Constructions
- Chapter 12 Area Related to Circles
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability

### NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Euclid’s division lemma, Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results – irrationality of √2, √3, √5, decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.

### NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

Zeros of a polynomial. The relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.

### NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables

Pair of linear equations in two variables and their graphical solution. Geometric representation of different possibilities of solutions/inconsistency.Algebraic conditions for a number of solutions. A solution of a pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication method. Simple situational problems must be included. Simple problems on equations reducible to linear equations.

### NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations

Standard form of a quadratic equation ax^{2 }+ bx + c = 0, (a ≠ 0). A solution of the quadratic equations (only real roots) by factorization, by completing the square and by using the quadratic formula. The relationship between discriminant and the nature of roots.Situational problems based on quadratic equations related to day to day activities to be incorporated.

### NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions

The motivation for studying Arithmetic Progression Derivation of the nth term and some of the first n terms of A.P. and their application in solving daily life problems.

### NCERT Solutions for Class 10 Maths Chapter 6 Triangles

Definitions, examples, counter examples of similar triangles.

- (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
- (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
- (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
- (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
- (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
- (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
- (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
- (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
- (Prove) In a triangle, if the square on one side is equal to some of the squares on the other two sides, the angles opposite to the first side is a right angle.

### NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

**Lines (In two-dimensions) **Review: Concepts of coordinate geometry, graphs of linear equations. Distance formula. Section formula (internal division). Area of a triangle.

### NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry

**1. Introduction to Trigonometry **Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios whichever are defined at 0^{o} and 90^{o}. Values (with proofs) of the trigonometric ratios of 30^{o}, 45^{o} and 60^{o}. Relationships between the ratios.

**2. Trigonometric Identities **Proof and applications of the identity sin^{2}A + cos^{2}A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.

### NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry

**Heights and Distances: Angle of elevation, Angle of Depression **Simple problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30^{o}, 45^{o}, 60^{o}.

### NCERT Solutions for Class 10 Maths Chapter 10 Circles

Tangent to a circle at, the point of contact.

- (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- (Prove) The lengths of tangents drawn from an external point to a circle are equal.

### NCERT Solutions for Class 10 Maths Chapter 11 Constructions

- Division of a line segment in a given ratio (internally).
- Tangents to a circle from a point outside it.
- Construction of a triangle similar to a given triangle.

### NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles

Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the above said plane figures. (In calculating the area of a segment of a circle, problems should be restricted to a central angle of 60°, 90°, and 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken).

### NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes

1. Surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.2. Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with the combination of not more than two different solids be taken).

### NCERT Solutions for Class 10 Maths Chapter 14 Statistics

Mean, median and mode of grouped data (a bimodal situation to be avoided). Cumulative frequency graph.

### NCERT Solutions for Class 10 Maths Chapter 15 Probability

The classical definition of probability. Simple problems on single events (not using set notation).

All the 15 chapters have been covered in the most systematic manner following the latest standards followed by **NCERT** i.e. National Council for education and Research Training.

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