**NCERT Solutions for Class 10 Maths** made available by Goprep help students build their fundamentals in the subject. Prepared by our expert faculty, NCERT Class 10 Maths Solutions assist students in CBSE Class 10 Maths board exam preparations.

So, students who are looking for NCERT 10 Maths Solutions can access our chapter-wise solutions to solve each question in the textbook with ease. We have organized NCERT Class 10 solutions in a detailed and methodical way so as to help you get to the logic behind every solution.

Moreover, these solutions are sure to help students have an adequate practice of complex Maths topics to approach board exams confidently. Further, you can use the links given below to browse through our chapter-wise questions and solutions for NCERT Class 10 Maths book.

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Class 10thGoprep's subject experts have listed down the list of important topics and formulas for all the chapters below. It is highly recommended that you study the chapter-wise topics and formulas as these are expected to carry the maximum weightage in CBSE Class 10 Maths board exam.

Real numbers comprises of natural numbers, whole numbers, rational & irrational numbers, and integers. In class 10 NCERT maths book, you will also learn about advanced topics, including properties of positive integers- euclid’s division algorithm and fundamental theorem of arithmetic. Besides, you will also get to know several theorems of rational and irrational numbers.

**Topics**

- Euclid’s Division Lemma
- The Fundamental Theorem of Arithmetic
- Irrational Numbers
- Rational Numbers and Their Decimal Expansions

**Important Formulas of Real Numbers**

**1. Euclid’s Division Lemma: **For two positive integers *a *and *b, *there exist whole numbers q and r respectively, which forms the equation shown below

**2. Euclid’s Division Algorithm: **To find the HCF of two positive integers *a *and *b *where a > b, you have to consider the following steps.

**Step 1**: Use division lemma to find the values of q and r (a= bq + r, where 0 ≤ r < b).

**Step 2**: HCF of *a *and *b* = b, if r = 0. In case, r ≠ 0 then apply Euclid’s lemma to *b *and *r. *

**Step 3**: Follow the process until the remainder is zero. Take divisor as HCF (a, b). Also, HCF (a, b) = HCF (b,r)

Polynomials are expressions that contain variables and coefficients. A polynomial can be linear, quadratic, cubic, etc. based on their highest degree. This year, you will study the zeroes of a polynomial, relationship between zeros and coefficients of a polynomial, and division algorithm for polynomials. *Note: *In total, there are 7 units in NCERT Class 10 Maths textbook. Out of these, algebra carries the highest weightage as seen in CBSE class 10 maths previous year question papers.

**Topics**

- Geometrical Meaning of the Zeros of a Polynomial
- Relationship between Zeros and Coefficients of a Polynomial
- Division Algorithm for Polynomials

**Important Formulas of Polynomials**

1. If α and β are the zeroes of the quadratic polynomial ax^{2} + bx + c, then

α + β = -b/a, αβ = c/a

2. If α, β and γ are the zeroes of the cubic polynomial ax^{3 }+ bx^{2 }+ cx + d, then

- α + β + γ = -b/a
- αβ+ β γ + γ α = c/a
- αβ γ = -d/a

**Division Algorithm**: For a given polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x), such that

p(x) = g(x) q(x) + r(x), where r(x) = 0

A linear equation comprises of two variables and appears as a straight line on a graph. In this chapter of NCERT class 10 maths solutions, you have to represent different situations algebraically and graphically using a pair of equations. Also, you will get to learn about how to reduce equations to a pair of linear equations.

**Topics**

- Pair of Linear Equations in Two Variables
- Graphical Method of Solution of a Pair of Linear Equations
- Algebraic Methods of Solving a Pair of Linear Equations
- Equations Reducible to a Pair of Linear Equations in Two Variables

**Important Formulas of Pair of Linear Equations in Two Variables**

For a pair of linear equations p (x) = a_{1}x + b_{1}y + c_{1} = 0 and q (x) = a_{2}x + b_{2}y + c_{2} = 0, the following situations may arise:

i). a_{1}/a_{2} ≠ b_{1}/b_{2} (In this case, the pair of linear equations is said to be consistent)

ii). a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2} (In this case, the pair of linear equations is said to be inconsistent)

iii). a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2} (In this case, the pair of linear equations is said to be consistent and dependent)

A quadratic equation is represented by an equation having the highest degree as 2. In NCERT class 11, you will revisit a few concepts of quadratic equations from previous classes. You will also learn to solve a quadratic equation by various methods including, factorisation, completing the square, and using the quadratic formula.

**Topics**

- Quadratic Equations
- Solution of a Quadratic Equation by Factorisation
- Solution of a Quadratic Equation by Completing the Square
- Nature of Roots

**Important Formulas of Quadratic Equations**

**Quadratic formula**: To find the roots of a quadratic equation ax2 + bx + c = 0, you have to apply the values of a, b and c in the following formula.

(-b ± √b^{2} - 4ac)/ 2a, where b^{2} - 4ac, c ≥ 0

It will be the first time you will study Arithmetic Progression (AP). AP is a list of numbers where you can obtain successive terms by adding a fixed number to the preceding terms. In this chapter, there will be questions asking you to find the nth term of an AP, sum of first (n) terms of an AP, and the last term of the finite AP using relevant formulas.

**Topics**

- Arithmetic Progressions
- nth Term of an AP
- Sum of First n Terms of an AP

**Important Formulas of Arithmetic Progression**

1. An AP can expressed in the form of a, a + d, a + 2d, a + 3d, . . .

2. The nth term of an AP with first term *a* and common difference *d* can be represented as

A_{n} = a + (n – 1) d

3. To find the sum of the first *n* terms of an AP, use the following equation

S_{n} = ½. n [2a + (n - 1)d]

4. To find the sum of all terms where the value of the last term of the finite AP is given, use the following equation

S_{n} = ½. n [a + l]

As you are already familiar with Triangles and its Properties, you will learn to judge whether two triangles are similar or not, using Basic Proportionality Theorem. Further, you will be able to figure out the similarity of two triangles using different criteria, some of which you have already learned in the previous NCERT maths books.

**Topics**

- Similar Figures
- Similarity of Triangles
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Pythagoras Theorem

Coordinate Geometry involves locating the position of a point on a plane using a pair of coordinate axes. This year, you will learn to find the distance between the two points using Distance Formula. To find the coordinates of the point dividing a line segment, you will have to apply Section Formula. Also, the Area of Triangle will come to use for finding the coordinates of points forming a triangle.

**Topics**

- Distance Formula
- Section Formula
- Area of a Triangle

**Important Formulas of Coordinate Geometry**

1. To find the distance between two points P (x_{1}, y_{1}) and Q (x_{2}, y_{2}), you can apply the following formula.

√(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

2. The distance of a point P (x,y) from the origin can be found by-

√x^{2} + y^{2}

3. The coordinates of the point P (x,y) which divides the line segment with points A (x_{1}, y_{1}) and B (x_{2}, y_{2}) internally in the ratio m_{1} : m_{2} are

[(m_{1}x_{2} + m_{2} x_{1})/ (m_{1} + m_{2}), (m_{1}y_{2} + m_{2} y_{1})/ (m_{1} + m_{2})]

4. The midpoint of the line segment having end-points P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) is given by

[(x_{1} + x_{2})/ 2, (y_{1} + y_{2})/ 2]

5. To find the area of triangle with points (x_{1} , y_{1} ), (x_{2} , y_{2} ) and (x_{3} , y_{3} ), you can use the following expression

½. [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})

The concept of Trigonometry is applied to study the relationships between the sides and angles of a triangle. For the first time, you will learn to find the Trigonometric Ratios of the Angle and prove two equations equal using Trigonometric Identities.

**Topics**

- Trigonometric Ratios
- Trigonometric Ratios of Some Specific Angles
- Trigonometric Ratios of Complementary Angles
- Trigonometric Identities

**Important Formulas of Introduction to Trigonometry**

1. In the right-angled triangle ABC, where ∠B = 90°

- sin A = Perpendicular the (side opposite to angle A)/ hypotenuse
- cos A = base (side adjacent to angle A)/ hypotenuse
- tan A = Perpendicular (the side opposite angle A)/ base (side adjacent to angle A)

2. Some more relations

- cosec A = 1/ sin A
- sec A = 1/cos A
- tan A= sin A/cos A

3. sin (90° – A) = cos A, cos (90° – A) = sin A;

tan (90° – A) = cot A, cot (90° – A) = tan A;

sec (90° – A) = cosec A, cosec (90° – A) = sec A

4. sin2 A + cos2 A = 1,

sec2 A – tan2 A = 1 for 0° ≤ A < 90°,

cosec2 A = 1 + cot2 A for 0° < A ≤ 90º

After covering the previous chapter of NCERT class 10 maths, you will now study how trigonometry is used in our surroundings. While finding heights and distances in questions, you will have to examine a few terms before solving them. These include the Line of Sight, the Angle of Elevation, the Angle of Depression. The height or length of an object or distance between two objects can be found out using trigonometric ratios.

**Topics**

- Heights and Distances
- Line of Sight
- Angle of Elevation
- Angle of Depression

A Circle is a collection of all points in a plane which are placed at a constant radius from the centre. In this chapter, you will study the meaning of a Tangent to a circle and related theorems.

**Topics**

- Tangent to a Circle
- Number of Tangents from a Point on a Circle

Basic Construction includes the knowledge of bisecting an angle, drawing a perpendicular bisector of a line segment, Construction of triangle etc. In this chapter, you will learn to divide a line segment in a given ratio; construct a triangle similar to a given triangle; construct a pair of tangents from an external point to a circle.

**Topics**

- Division of a Line Segment
- Construction of Tangents to a Circle

In chapter 12 “Areas related to circles” of NCERT class 10 maths book, you will first revisit the basic formulas related to a Circle, i.e. Circumference and Area of a Circle. Next, you will learn to find the Length of an Arc of a Sector of a Circle, Area of a Sector of a Circle, and Area of Segment of a Circle.

**Topics**

- Perimeter and Area of a Circle
- Areas of Sector and Segment of a Circle
- Areas of Combinations of Plane Figures

**Important Formulas of Areas Related to Circles**

1. Circumference of a circle = 2 π r

2. Area of a circle = π r^{2}

3. Length of an arc of a sector of a circle with radius *r* and angle θ is given by = θ/360. 2 π r

4. Area of a sector of a circle with radius *r* and angle θ is given by = θ/360. π r^{2}

With previous knowledge of finding surface areas and volumes of solids, you will learn to find the surface area and volume of objects formed by the combination of any two objects. This chapter also introduces you with formulae that helps you find the frustum of a cone.

**Topics**

- Surface Area of a Combination of Solids
- Volume of a Combination of Solids
- Conversion of Solid from One Shape to Another
- Frustum of a Cone

**Important Formulas of Surface Areas and Volumes**

1. Volume of a frustum of a cone = ⅓. Πh (r_{12 }+ r_{2 }+ r_{1} r_{2})

2. Curved surface area of a frustum of a cone = πl(r_{1 }+ r_{2} ), where l = √h^{2 }+ (r1 - r2)^{2}

3. Total surface area of frustum of a cone = πl(r_{1} + r_{2} ) + π(r_{1}^{2} + r_{2}^{2})

In this chapter, you will learn to find Mean for Grouped Data through three different formulae- Direct method, Assumed Mean method, and Step Deviation method. Also, the chapter introduces you to formulae to find the median and mode of grouped data. Obtaining Cumulative Frequency and representing it graphically another new edition to your previous knowledge.

**Topics**

- Mean of Grouped Data
- Mode of Grouped Data
- Median of Grouped Data
- Graphical Representation of Cumulative Frequency Distribution

**Important Formulas of Statistics**

1. The mean for grouped data can be obtained using the following methods:

- Direct method (x-bar) = ∑f
_{i}x_{i}/ ∑f_{i} - Assumed mean method (x-bar) = a + (∑f
_{i}d_{i}/ ∑f_{i}) - Step deviation method (x-bar) = a + (∑f
_{i}u_{i}/ ∑f_{i}) x h

2. The mode for grouped data can be obtained using the formula:

Mode = [l + h{(f_{1 }- f_{0})/(2f_{1} - f_{0} - f_{2})}]

3. The median for grouped data can be obtained by using the formula:

Median = [l + h {(n/2 - cf)/f}]

Having discussed experimental Probabilities in the earlier NCERT textbooks, you will study the difference between Experimental Probability and Theoretical Probability. Both these topics form the core of this chapter. Further readings will involve the Probability of a sure event, Probability of an impossible event, Probability of an event, Elementary event, and Complementary events.

**Topics**

- Probability
- Sure event
- Impossible event
- Complementary events

**Important Formulas of Probability**

**Theoretical Probability**: The theoretical probability of an event E, can be defined as

P (E) = Number of outcomes favorable to E/ Number of all possible outcomes of the experiment.