CBSE Class 10 Maths Notes

CBSE Class 10 Maths Revision Notes is undoubtedly one of the most important formal education levels. The class and complexity of studies instill in them a sense of seriousness and responsibility. Students tend to mature and stabilize. The most notable characteristic of class 10 Board examinations is that they are the first nationalized external exam for students. Class 10 Maths is one of the toughest subjects to tackle so GeeksforGeeks compiled CBSE Class 10 Maths Notes.

Our CBSE Class 10 Maths NCERT Notes are based on the latest syllabus for Class 10 Boards to help to understand all the concepts of Class 10 Maths chapter-wise. Class 10 Maths NCERT notes comprise all the formulae, theorems, concepts, proofs descriptions, definitions, and graphs, that a student needs to practice and score good marks in their board examinations.

These subject-specific revision notes cover some more important topics like Introduction to Trigonometric Ratios of a Triangle, Heights and Distances, Surface Areas and Volume, etc. Our experts have also covered Class 10 Maths Solutions like NCERT Solutions for Class 10 Maths, and RD Sharma Class 10 Solutions.

If youâ€™re looking for CBSE Class 10 Term 1 Maths Answer Key 2021 Basic Paper to solve maths problems, you can click here

Deleted Chapters/Topics from NCERT Class 10th Maths Textbook (2023-2024):

The most recent CBSE Class 10th Mathematics syllabus has been changed and reduced by 30% for the upcoming annual assessment in the academic year 2023-2024, you can find the list of all deleted chapters in the table below:

Chapter 1: Real Numbers

Any number, barring complex numbers, is a real number. Real numbers include positive and negative integers, irrational numbers, and fractions. To put it another way, a real number is any number found in the actual world. Numbers may be found everywhere. Natural numbers are used to count items, integers to measure temperature, rational numbers to represent fractions, and irrational numbers to calculate the square root of a number, among other things.

The chapter Real Numbers include both irrational and rational numbers with all natural numbers, whole numbers, integers, etc, and discuss their characteristics.

Important Points Explained In Cbse Class 10 Chapter 1- Real Numbers Are:

• Euclidâ€™s Division Algorithm (lemma): According to Euclidâ€™s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r such that a = bq + r, where 0 â‰¤ r â‰¤ b. (Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder.)
• Fundamental theorem of arithmetic said that the Composite Numbers are equal to the Product of Primes.
• HCF and LCM by prime factorization method:
• HCF = Product of the smallest power of each common factor in the numbers
• LCM = Product of the greatest power of each prime factor involved in the number
• HCF (a,b) Ã— LCM (a,b) = a Ã— b

Chapter 2: Polynomials

Polynomials are algebraic expressions that consist of coefficients and variables or are indeterminate. An arithmetic operation such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable can be performed on polynomials.

This particular chapter presents the idea of the degree of polynomials, how a polynomial with degree 1 is a linear polynomial, degree 2 is a quadratic polynomial, and degree 3 is a cubic polynomial. Moreover, the most important topics discussed in this chapter are the zeroes of a polynomial and the relationship between zeroes and coefficients of quadratic polynomials only.

Some Important Formulas Of Polynomials Learned In CBSE Class 10 Maths Note Chapter 2 Polynomials:

• The general Polynomial Formula is, F (x) = anxn + bxn-1 + an-2xn-2 + â€¦â€¦.. + rx + s
• When n is a natural number: an â€“ bn = (a â€“ b)(an-1 + an-2b +â€¦+ bn-2a + bn-1)
• When n is even (n = 2a): xn + yn = (x + y)(xn-1 â€“ xn-2y +â€¦+ yn-2x â€“ yn-1)
• When n is odd number: xn + yn = (x + y)(xn-1 â€“ xn-2y +â€¦- yn-2x + yn-1)
• Algebraic Polynomial Identities:
• (a+b)2 = a2 + b2 + 2ab
• (a-b)2 = a2 + b2 â€“ 2ab
• (a+b) (a-b) = a2 â€“ b2
• (x + a)(x + b) = x2 + (a + b)x + ab
• (x + a)(x â€“ b) = x2 + (a â€“ b)x â€“ ab
• (x â€“ a)(x + b) = x2 + (b â€“ a)x â€“ ab
• (x â€“ a)(x â€“ b) = x2 â€“ (a + b)x + ab
• (a + b)3 = a3 + b3 + 3ab(a + b)
• (a â€“ b)3 = a3 â€“ b3 â€“ 3ab(a â€“ b)
• (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
• (x + y â€“ z)2 = x2 + y2 + z2 + 2xy â€“ 2yz â€“ 2xz
• (x â€“ y + z)2 = x2 + y2 + z2 â€“ 2xy â€“ 2yz + 2xz
• (x â€“ y â€“ z)2 = x2 + y2 + z2 â€“ 2xy + 2yz â€“ 2xz
• x3 + y3 + z3 â€“ 3xyz = (x + y + z)(x2 + y2 + z2 â€“ xy â€“ yz -xz)
• x2 + y2 =Â½ [(x + y)2 + (x â€“ y)2]
• (x + a) (x + b) (x + c) = x3 + (a + b +c)x2 + (ab + bc + ca)x + abc
• x3 + y3= (x + y) (x2 â€“ xy + y2)
• x3 â€“ y3 = (x â€“ y) (x2 + xy + y2)
• x2 + y2 + z2 -xy â€“ yz â€“ zx = Â½ [(x-y)2 + (y-z)2 + (z-x)2]
• Division algorithm for polynomials: If p(x) and g(x) are any two polynomials with g(x) â‰  0, then we can find polynomials q(x) and r(x) such that

p(x) = q(x) Ã— g(x) + r(x)

where r(x) = 0 or degree of r(x) < degree of g(x). Here p(x) is divided, g(x) is divisor, q(x) is quotient and r(x) is remainder.

Chapter 3: Pair of Linear Equations in Two Variables

An equation of the form ax+by+c, where a, b, and c are real numbers and a, b are not equal to zero, is termed as the linear equation in two variables. However, in a pair of linear equations in two variables, there exist two such equations whose solution is a point on the line denoting the equation.

In this chapter notes, the introduction to the pair of linear equations in one variable and the graphical method of their solution, consistency/inconsistency, and algebraic conditions for the number of solutions are covered. Solutions of a pair of linear equations in two variables algebraically â€“ by substitution and by elimination, simple situational problems, and simple problems on equations reducible to linear equations are also explained in this chapter.

Important Points Explained Here In Cbse Class 10 Chapter 3- Pair Of Linear Equations In Two Variables Are:

An equation which can be put in the form

ax + by + c = 0

where a, b and c are Pair of Linear Equations in Two Variables, and a and b are not both zero, is called a linear equation in two variables x and y.

Chapter 3 of CBSE Class 10 Maths Notes covers the following topics:

In chapter 4 of Maths NCERT Notes, weâ€™ll cover the Quadratic Equations. The degree 2 polynomial equations in one variable are called Quadratic equations. The general form of a quadratic equation is ax2 + bx + c where a, b, c, âˆˆ R and a â‰  0, where â€˜aâ€™ is called the leading coefficient and â€˜câ€™ is called the absolute term of the complete equation.

This chapter helps to understand the concept of the standard form of quadratic equations, various methods of solving quadratic equations (by factorization, by completing the square), and the nature of roots

Important Formulas And Their Conclusions Discussed In Cbse Class 10 Chapter 4- Quadratic Equations Are:

(Î±, Î²) = [-b Â± âˆš(b2 â€“ 4ac)]/2ac

• Roots of the quadratic equation: x = (-b Â± âˆšD)/2a, where D = b2 â€“ 4ac is known as the Discriminant of a quadratic equation. The discriminant of a quadratic equation decides the nature of roots.
• Nature of Roots of Quadratic Equation
• D > 0, roots are real and distinct (unequal).
• D = 0, roots are real and equal (coincident) i.e. Î± = Î² = -b/2a.
• D < 0, roots are imaginary and unequal i.e Î± = (p + iq) and Î² = (p â€“ iq). Where â€˜iqâ€™ is the imaginary part of a complex number.
• Sum of roots: S = Î±+Î²= -b/a = coefficient of x/coefficient of x2
• Product of roots: P = Î±Î² = c/a = constant term/coefficient of x2
• Quadratic equation in the form of roots: x2 â€“ (Î±+Î²)x + (Î±Î²) = 0
• The quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have;
• One common root if (b1c2 â€“ b2c1)/(c1a2 â€“ c2a1) = (c1a2 â€“ c2a1)/(a1b2 â€“ a2b1)
• Both roots common if a1/a2 = b1/b2 = c1/c2
• In quadratic equation ax+ bx + c = 0 or [(x + b/2a)2 â€“ D/4a2]
• If a > 0, minimum value = 4ac â€“ b2/4a at x = -b/2a.
• If a < 0, maximum value 4ac â€“ b2/4a at x= -b/2a.
• If Î±, Î², Î³ are roots of cubic equation ax3 + bx2 + cx + d = 0, then, Î± + Î² + Î³ = -b/a, Î±Î² + Î²Î³ + Î»Î± = c/a, and Î±Î²Î³ = -d/a

Chapter 5: Arithmetic Progressions

The Arithmetic progression is explained as the sequence of numbers where the difference between any two subsequent numbers is a constant.

In this chapter, students will learn about the concepts of Arithmetic Progression and its Derivation of the nth term, the Sum of the first n terms of A.P, and their real-life application in solving everyday-life problems.

Major Formulas Covered In Cbse Class 10 Chapter 5- Arithmetic Progressions Are:

• nth term of AP: an = a + (n – 1) d, where an is the nth term.
• Sum of nth terms of AP: Sn= n/2 [2a + (n – 1)d]

Chapter 6: Triangles

Geometrically, a triangle is defined as a three-sided polygon consisting of three edges and three vertices. The most important and applied property of a triangle is its Angle sum property which means the sum of the internal angles of a triangle is equal to 180 degrees only.

This chapter from geometry is all about the definitions, examples, and examples of similar triangles. Moreover, the criteria for triangle similarity and some related theorems are also learned in this chapter.

Major Topics Explained In Cbse Class 10 Chapter 6- Triangles Are:

• Criteria for Triangle Similarity
1. Angle angle angle (AAA Similarity)
2. Side angle Side (SAS) Similarity
3. Side-side side (SSS) Similarity
• Basic Proportionality Theorem: According to this theorem, when a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio.
• Converse of Basic Proportionality Theorem: According to this theorem, in a pair of triangles when the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.

Chapter 7: Coordinate Geometry

Coordinate Geometry is defined as the link between geometry and algebra using graphs along with curves and lines. In this way, it provides geometric aspects in Algebra and leads to solving geometric problems.

The topics covered in this chapter are the basics of Coordinate Geometry, and graphs of linear equations. Distance formula and Section formula only.

Important Formulas Covered In Cbse Class 10 Chapter 7- Coordinate Geometry Are:

• Distance Formula: For a line having two-point A(x1, y1) and B(x2, y2), then the distance of these points is given as:

AB= âˆš[(xâˆ’ x1)+ (yâˆ’ y1)2]

• Section Formula: For any point p divides a line AB with coordinates A(x1, y1) and B(x2, y2), in ratio m:n, then the coordinates of the point p are given as:

P={[(mx2 + nx1) / (m + n)] , [(my2 + ny1) / (m + n)]}

• Midpoint Formula: The coordinates of the mid-point of a line AB with coordinates A(x1, y1) and B(x2, y2), are given as:

P = {(x1 + x2)/ 2, (y1+y2) / 2}

• Area of a Triangle: Consider the triangle formed by the points A(x1, y1) and B(x2, y2) and C(x3, y3) then the area of a triangle is given as-

âˆ†ABC = Â½ |x1(y2 âˆ’ y3) + x2(y3 â€“ y1) + x3(y1 â€“ y2)|

Chapter 8: Introduction to Trigonometry

Trigonometry can be defined as calculations including triangles to study their lengths, heights, and angles. Trigonometry and its functions have an enormous number of uses in our daily life.

The most important topic covered in this chapter is the Trigonometric ratios of an acute angle of a right-angled triangle. Along with the Proof of their existence, Values of the trigonometric ratios of 30 degrees, 45 degrees and 60 degrees, and Relationships between the ratios.

Some Major Formulas That Are Learned In Cbse Class 10 Chapter 8- Introduction To Trigonometry Are:

• If in a circle of radius r, an arc of length l subtends an angle of Î¸ radians, then l = r Ã— Î¸.
• Radian Measure = Ï€/180 Ã— Degree Measure
• Degree Measure = 180/Ï€ Ã— Radian Measure
• Trigonometric ratios:
• sin Î¸ = (Perpendicular (P)) / (Hypotenuse (H)).
• cos Î¸ = (Base (B)) / ( Hypotenuse (H)).
• tan Î¸ = (Perpendicular (P)) / (Base (B)).
• cosec Î¸ = (Hypotenuse (H)) / (Perpendicular (P)).
• sec Î¸ = (Hypotenuse (H)) / (Base (B)).
• cot Î¸ = (Base (B)) / (Perpendicular (P)).
• Reciprocal Trigonometric Ratios:
• sin Î¸ = 1 / (cosec Î¸)
• cosec Î¸ = 1 / (sin Î¸)
• cos Î¸ = 1 / (sec Î¸)
• sec Î¸ = 1 / (cos Î¸)
• tan Î¸ =  1 / (cot Î¸)
• cot Î¸ = 1 / (tan Î¸)
• Trigonometric Ratios of Complementary Angles:
• sin (90Â° â€“ Î¸) = cos Î¸
• cos (90Â° â€“ Î¸) = sin Î¸
• tan (90Â° â€“ Î¸) = cot Î¸
• cot (90Â° â€“ Î¸) = tan Î¸
• sec (90Â° â€“ Î¸) = cosec Î¸
• cosec (90Â° â€“ Î¸) = sec Î¸
• Trigonometric Identities
• sin2 Î¸ + cos2 Î¸ = 1 â‡’ sin2 Î¸ = 1 – cos2 Î¸ â‡’ cos2 Î¸ = 1 – sin2 Î¸
• cosec2 Î¸ – cot2 Î¸ = 1 â‡’ cosec2 Î¸ = 1 + cot2 Î¸ â‡’ cot2 Î¸ = cosec2 Î¸ – 1
• sec2 Î¸ – tan2 Î¸ = 1 â‡’ sec2 Î¸ = 1 + tan2 Î¸ â‡’ tan2 Î¸ = sec2 Î¸ – 1

Chapter 9: Some Applications of Trigonometry

Trigonometry has a lot of practical applications in real life. This part of geometry discusses the line of sight, angle of deviation, angle of elevation, and angle of depression. Using trigonometry and trigonometric ratios the height of a building, or a mountain, from a viewpoint and the elevation angle can be determined easily.

As we have learned the basics of trigonometry in the previous chapter, so now it’s time to learn their practical applications. Hence, the topics studied in this chapter will help to understand learners’ use of trigonometry. This chapter also helps with the practical concepts of trigonometry like the line of sight, angle of depression, and angle of elevation and to determine height or distance.

Important Terms That Are Explained In Cbse Class 10 Chapter 9- Some Applications Of Trigonometry Are:

• Line of Sight – The Line of Sight is the line formed by our vision as it passes through an item when we look at it.
• Horizontal Line – The distance between the observer and the object is measured by a horizontal line.
• Angle of Elevation – The angle formed by the line of sight to the top of the item and the horizontal line is called an angle of elevation. It is above the horizontal line, i.e. when we gaze up at the item, we make an angle of elevation.
• Angle of Depression – When the spectator must look down to perceive the item, an angle of depression is formed. When the horizontal line is above the angle, the angle of depression is formed between it and the line of sight.

Chapter 10: Circles

A circle is a geometrical shape that is defined as the locus of points that move in a plane so that its distance from a fixed point is always constant. This fixed point is the Centre of the circle while the fixed distance from it is called the radius of the circle

In this chapter Circles, students will learn about tangents and the different cases when lines touch or bisect circles on a given plane. Also, the concept of point of contact and important theorems related to the same are discussed in this chapter.

Important Theorems Covered In Cbse Class 10 Chapter 10- Circles Are:

Theorem 10.1 The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Theorem 10.2 The lengths of tangents drawn from an external point to a circle are equal.

Different cases for the number of Tangents from a Point on a Circle:

1. There is no tangent to a circle passing through a point lying inside the circle.
2. There is one and only one tangent to a circle passing through a point lying on the circle.
3. There are exactly two tangents to a circle through a point lying outside the circle.

Chapter 11: Constructions

Construction helps to understand the approach to constructing different types of triangles for different given conditions using a ruler and compass of required measurements.

Here in the Chapter Construction of Class 10 the major subtopics covered are constructing the line segment in a given ratio internally and drawing a tangent to a circle from a point outside the circle.

Important Constructions Explained In Cbse Class 10 Chapter 11- Constructions Are:

Construction 11.1: Construction for the division of a line segment in a given ratio.

Construction 11.2: Construction of a triangle similar to a given triangle as per the given scale factor.

Construction 11.3: Construction of the tangents to a circle from a point outside it.

Chapter 12: Areas Related to Circles

The area related to circles is the amount of space covered by a circle, which is defined in different ways. Some areas related to a circle are, the area of the circle itself, the Area of the sector, the area of the segment, the area of the triangle or parallelogram, etc. located in a circle.

This chapter included subtopics like the area of a circle; the area of sectors and segments of a circle. Particularly the problems based on the areas and circumference of the circles and related plane figures are explained in depth.

Important Formulas Covered In Cbse Class 10 Chapter 12- Areas Related To Circles Are:

• Circumference of the circle = 2 Ï€ r
• Area of the circle = Ï€ r2
• Area of the sector of angle, Î¸ = (Î¸/360) Ã— Ï€ r2
• Length of an arc of a sector of angle, Î¸ = (Î¸/360) Ã— 2 Ï€ r
• Distance moved by a wheel in one revolution = Circumference of the wheel.
• The number of revolutions = Total distance moved / Circumference of the wheel.

Chapter 13: Surface Areas and Volumes

Surface area and volume are the measures calculated for a three-dimensional geometrical shape like a cube, cuboid, sphere, etc. The surface area of any given object is the area occupied by the surface of the object while volume is the amount of space available in an object.

This chapter covers some most important topics of mensuration like the Surface Areas and Volumes of the combination of any 3-D figures (cube, cuboid, sphere, hemispheres, and right circular cylinders, cones). Along with the practical problems related to the conversion of one type of metallic 3-D solid to another. And various similar everyday life problems related to the surface areas and volumes of 3-D figures.

Important Formulas Covered In Cbse Class 10 Chapter 13- Surface Areas And Volumes Are:

1. Total Surface Area (TSA): The whole area covered by the object’s surface is called the Total Surface area. Following is the list of the total surface areas of some important geometrical figures-
• TSA of a Cuboid = 2(l x b) +2(b x h) +2(h x l)
• TSA of a Cube = 6a2
• TSA of a Right circular Cylinder = 2Ï€r(h+r)
• TSA of a Right circular Cone = Ï€r(l+r)
• TSA of a Sphere = 4Ï€r2
• TSA of a Right Pyramid = LSA + Area of the base
• TSA of a Prism = LSA Ã— 2B
• TSA of a Hemisphere = 3 Ã— Ï€ Ã— r2
2. Lateral/Curved Surface Area: The curved surface area is the area of only the curved component, or in the case of cuboids or cubes, it is the area of only four sides, excluding the base and top. It’s called the lateral surface area for forms like cylinders and cones.
• CSA of a Cuboid = 2h(l+b)
• CSA of a Cube = 4a2
• CSA of a Right circular Cylinder = 2Ï€rh
• CSA of a Right circular Cone = Ï€rl
• LSA of a Right Pyramid = Â½ Ã— p Ã— l
• LSA of a Prism = p Ã— h
• LSA of a Hemisphere = 2 Ã— Ï€ Ã— r2
3. Volume: The volume of an object or material is the amount of space it takes up, measured in cubic units. There is no volume in a two-dimensional object, only area. A circle’s volume cannot be calculated since it is a 2D figure, while a sphere’s volume can be calculated because it is a 3D figure.
• Volume of a Cuboid = l x b x h
• Volume of a Cube = a3
• Volume of a Right circular Cylinder = Ï€r2h
• Volume of a Right circular Cone = 1/3Ï€r2h
• Volume of a Sphere = 4/3Ï€r3
• Volume of a Right Pyramid = â…“ Ã— Area of the base Ã— h
• Volume of a Prism = B Ã— h
• Volume of a Hemisphere = â…” Ã— (Ï€r3)

Here, l is the length, b is the breadth, h is the height, r is the radius, a is the side, p is the perimeter of the base, B is the area of the base of the respective geometrical figure.

Chapter 14: Statistics

Statistics is the study of the representation, collection, interpretation, analysis, presentation, and organization of data. In other words, it is a mathematical way to collect and summarize data. The representation of data is different along with the frequency distribution.

This chapter covers subtopics like mean (average), median, and mode from a grouped information set. Another section in this chapter helps to learn the representation of data graphically and to understand trends and their correlations.

Important Formulas Learned In Cbse Class 10 Chapter 14- Statistics:

• Different methods to calculate the Mean of a grouped data,
• Direct method: X = âˆ‘fi xi / âˆ‘fi
• Assumed Mean Method: X = a + âˆ‘fi di / âˆ‘fi  (where di = xi – a)
• Step Deviation Method: X = a + âˆ‘fi ui / âˆ‘fi Ã— h
• Mode of the grouped data = a + âˆ‘fi ui / âˆ‘fi Ã— h
• Median of the grouped data = l + (n/2 – cf) / f Ã— h

Chapter 15: Probability

The Probability in this class includes basic probability theory, which is also used in the probability distribution, to learn the possibility of outcomes for a random experiment and to find the probability of a single event to occur, when the total number of possible outcomes.

Important Formulas Covered In CBSE Class 10 Math Chapter 15- Probability Are:

Empirical Probability = Number of Trials which expected outcome come / Total Number of Trials

Theoretical Probability = Number of favourable outcomes to E / Total Number of possible outcomes of the experiment

FAQs on CBSE Class 10 Maths Notes

Question 1: How CBSE Class 10 Maths Notes are helpful for students?

Students have the ability to construct solid and dependable conceptions by referring to these Class 10 Maths Notes. When attempting a problem, there’s a good chance a pupil won’t be able to finish it because of doubts. Also the NCERT solutions for class 10 maths are presented in a way that makes even the most difficult theories understandable, allowing students to solve all levels of sums successfully.

Question 2: How to score full marks in Class 10th Maths?

Following are the useful tips to score good marks in Class 10th Maths:

1. Understand the syllabus, paper pattern & marking scheme.
2. First try to solve and understand the concepts explained in NCERT exercises and exemplar questions, before referring to any other textbook.
3. Learn how to answer problems step-by-step to score better.
4. Don’t forget to practise the case-study based questions.
5. Try to solve both Standard and Basic Sample Papers
6. In order to learn all the formula used in Class 10th board exams, write down them all in one page.

Question 3: List down some important formulae for Maths Class 10 boards.

Class 10 math covers a number of key ideas that are necessary for understanding higher-level mathematics. Math formulae are necessary to solve problems quickly and accurately. The first step is to understand how a formula came to be and the concept that governs it. Then you can memorise them and use the formulas to answer questions. The following are some of the key formulas mentioned in NCERT Notes for class 10 math.

• (a + b)3 = a3 + b3 + 3ab(a + b)
• (a â€“ b)3 = a3 – b3 – 3ab(a â€“ b)
• (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
• an = a + (n – 1) d
• S2= n/2 [2a + (n – 1)d]
• sin2 Î¸ + cos2 Î¸ = 1
• cosec2 Î¸ – cot2 Î¸ = 1
• sec2 Î¸ – tan2 Î¸ = 1
• Volume of Sphere = 4/3 Ã—Ï€ r3
• Surface Area of Sphere = 4Ï€r2
• Total surface area of cuboid = 2(lÃ—b + bÃ—h + lÃ—h)

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