 # NCERT Solutions for Class 10- Free PDF for the academic year 2020-21

## NCERT Solutions for Class 10 Maths for all the exercises from Chapters 1 to 15 are provided here. ## Real Numbers

### Euclid’s division lemma, Fundamental Theorem of Arithmetic, Proofs of irrationality of √2, √3, √5 Decimal representation of rational numbers in terms of terminating/non-terminating recurring decimals. ## Chapter 4

### Standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0). Solutions of quadratic equations (only real roots) by factorization; and by using the quadratic formula. Relationship between discriminant and nature of roots. Situational problems based on quadratic equations related to day to day activities to be incorporated. ## Coordinate Geometry

### Concepts of coordinate geometry, graphs of linear equations. Distance formula. Section formula (internal division). Area of a triangle. ## Circles

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Tangent to a circle at, 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. ## Surface Areas and Volumes

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.

Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combinations of not more than two different solids are taken). ## Polynomials

### Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. division algorithms for polynomials with real coefficients. ## Arithmetic Progressions

### Arithmetic Progression Derivation of the nth term and sum of the first n terms of A.P. and their application in solving daily life problems. ## 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 0ㅇ and 90ㅇ. Values of the trigonometric ratios of 30ㅇ, 45ㅇ and 60ㅇ. Relationships between the ratios. Proof and applications of the identity sin2A + cos2A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles. ## 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. ## Statistics

### Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph. ## Pair of Linear Equations in Two Variables

### Pair of linear equations in two variables and graphical method of their solution, consistency/inconsistency. Algebraic conditions for the number of solutions. The solution of a pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication method. ## Triangles

### Definitions, examples, counter-examples of similar triangles. Theorems on similarity ## Some Applications of Trigonometry

Simple problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30°, 45°, 60°. ## 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 the central angle of 60°, 90° and 120° only. Plane figures involving triangles, simple quadrilaterals and circles should be taken.) 