In this post, Digital Teacher explains the SSC Class 10 Maths textbook in Telangana, from “Real Numbers,” to Chapter 14 of “Statistics”. It discusses the definition, properties, and conditions of real numbers such as integers, decimals, and fractions. It also underlines their classification and importance, allowing for arithmetic operations and representation on the number line.
Students looking for Telangana 10th-class courses may get the whole Telangana (TS) SSC lessons here. The Telangana TS 10th Class Syllabus is accessible for all courses. The Class 10 TS maths syllabus is divided into 1 to 14 units.
Telangana (TS) Board Class 10 Syllabus for Mathematics
The following are the Telangana State TS Maths Class 10 lessons. Since we have covered every significant topic for the exam, students can check here. The table below lists the chapter names for the Class 10 Mathematics course, which includes Real Numbers, Sets, Polynomials, and more.
Unit 1 Real Numbers |
Unit 2 Sets |
Unit 3 Polynomials |
Unit 4 Pair of Linear Equations in Two Variables |
Unit 5 Quadratic Equations |
Unit 6 Progressions |
Unit 7 Coordinate Geometry |
Unit 8 Similar Triangles |
Unit 9 Tangents and Secants to a Circle |
Unit 10 Mensuration |
Unit 11 Trigonometry |
Unit 12 Applications of Trigonometry |
Unit 13 Probability |
Unit 14 Statistics |
In the sections below, we will try to explain each class 10th Mathematics unit in detail so that you can understand what you need to study on the syllabus. Now, read it!
Class 10 Mathematics TS Units in Detailed:
- Real Numbers: This unit covers the fundamental properties of real numbers, including the Euclidean algorithm, rational and irrational numbers, and decimal expansions.
Definition: (Real numbers) encompass both rational and irrational numbers within the number system. They can be both positive or negative and are represented by the symbol “R”. All natural numbers, decimals, and fractions fall under this category.
Examples:
- Integers: 17, -8
- Decimal Numbers: 3.14
- Fractional Numbers: 1/3
- Irrational Constants: √2
Here is a Detailed Explanation: (Real Numbers Example)
Real numbers encompass both rational and irrational numbers within the number system. Rational numbers can be expressed as fractions of integers, while irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. Real numbers allow for various arithmetic operations and can be represented on the number line. In contrast, imaginary numbers, which are also used in mathematics, cannot be plotted on the number line and are denoted by multiples of the imaginary unit “i”. Examples such as 17 (integer), -8 (integer), 3.14 (decimal number), 1/3 (fractional number), and √2 (irrational constant) illustrate the diversity of real numbers, ranging from whole numbers to fractions and transcendental constants like √2 and π. This comprehensive understanding of real numbers is essential in various mathematical contexts, from basic arithmetic to advanced calculus.
Full Video Lesson in The Below (Real Numbers)
TS class 10 Mathematics Unit 2 & 3: (Sets & Polynomials)
Mathematics Unit-2: (Sets) or (Set of Real Numbers) Real numbers are classified into several categories, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Natural numbers are infinitely long counting numbers, whereas whole numbers encompass both zero and the whole natural number set. Integers are the sum of whole integers and their negatives, including values such as -∞, -4, -3, -2, -1, 0, 1, 2, 3, 4, and up to +∞. Rational numbers may be stated as p/q, such as 1/2, 5/4, and 12/6, however, irrational numbers cannot be written as fractions and have non-repeating and non-terminating decimal expansions, such as √2.
Unit 3: (Polynomials)
In Telangana Ts Class 10 Unit 3, we’ll explore polynomials. Think of them as math expressions made up of different parts like letters (which we call variables), numbers (which we call coefficients), and powers. We can add, subtract, and multiply them, but we can’t divide them by variables.
Imagine this: 𝑥 2 + 𝑥 − 12 x 2 +x−12. Here, we have three parts: 𝑥 2 x 2, 𝑥 x, and − 12 −12.
A polynomial is like a bunch of terms (like 𝑥 2 x 2 , 𝑥 x, and − 12 −12) all added, subtracted, or multiplied together. We use these terms to build expressions that represent different situations in math.
For example:
- Constants are just regular numbers, like 1, 2, 3, and so on.
- Variables are letters that stand for numbers we don’t know yet, like 𝑔 g, ℎ h, 𝑥 x, 𝑦 y, and so on.
- Exponents are little numbers that tell us how many times to use the variable, like 5 in 𝑥 5 x 5.
Let’s break down a simple example into a table:
Term Type | Example | Description |
Constant | 1 | Regular number |
Variable | x | Letter representing an unknown value |
Exponent | 5 | The little number indicates repetition |
Pair of Linear Equations in Two Variables (Unit-4)
Substitution, elimination, and graphical representation are methods used in the solution of linear equations. In Unit 4, the emphasis is on solving equation pairs involving straight lines by utilizing graphing, substitution, and elimination techniques to determine where the lines cross.
- Equation: An equation is a statement that two mathematical expressions having one or more variables are equal.
- Linear Equation: Unit 4 focuses (Linear Equation) on solving pairs of equations with each equation being a straight line, utilizing techniques such as substitution, elimination, and graphing to determine where these lines intersect, ensuring that the degree of a linear equation is always one.
Let’s Explore the Fundamentals:
- Equation: It’s like a balancing scale, indicating that two items are equal. For instance, the equation 2 𝑥 + 3 = 7 2x+3=7 states that 2 𝑥 + 3 2x+3 equals 7 7.
- Linear equations have a maximum power of 1 for the variable (e.g., 𝑥 x or 𝑦 y). They produce straight lines when graphed. The equation 3 𝑥 + 2 𝑦 = 5 is linear since the powers of 𝑥 x and 𝑦 y are both 1.
- Comprehending these fundamentals will aid us in resolving issues when determining the intersection of two lines is required.
Unit-5: Quadratic Equations
In Class 10, “Quadratic Equations,” Chapter 4, is an important component of algebra. It gives an overview of quadratic equations and demonstrates several methods for solving them, including factoring, using formulas, and completing the square. Acquiring these techniques is essential as they serve as the foundation for more complex math problems in the near future!
In This Chapter, Students Will Study These Topics:
- Quadratic equations are special equations in which the variable’s maximum power is two.
- A quadratic polynomial is a unique equation, such as 𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 ax 2 +bx+c, where 𝑎 a, 𝑏 b, and 𝑐 c are real values and 𝑎 an is not zero.
Understanding these ideas will enable you to answer issues expertly and score your tests!
6th Unit – Progressions (or) Arithmetic Progression (AP)
This lesson focuses on Arithmetic Progressions (AP), a series of consistent terms that includes both arithmetic and geometric progressions, as well as determining the nth term and the sum of the first n terms.
Here are a few basic concepts:
- Sequences are lists of numbers that follow a certain pattern. For example, 1, 2, 3, 4, 5… is a natural number sequence
- Series: It is the sum of all the numbers in a series. The natural number sequence is represented by 1+2+3+4+5…
- Progressions are sequences in which we may put out a rule or formula for determining any term.
Now, let’s explore Arithmetic Progression:
What is Arithmetic Progression-AP?
It is a sequence in which each term after the first is determined by adding a set integer to the preceding term. For example, 2, 5, 8, 11, 14, and so on, with each addition of three.
Common Difference: This is the number that we keep adding to go from one word to the next. If it’s positive, the AP is rising; if it’s zero, the AP is stable; and if it’s negative, the AP is dropping.
For example, in the sequence 2, 5, 8, 11, 14,…, the common difference is 3 since we are adding three each time.
Understanding (Arithmetic Progression) APs enable students to solve many real-world situations and prepare students for higher-level math topics.
Please let me know if you need any more clarity on any aspect of this topic! Now let’s look into 7th Unit Coordinate Geometry.
Unit-7 Coordinate Geometry
Chapter 7 of Class 10 Mathematics: Coordinate Geometry, a branch that discusses the position of points on a plane, including the Cartesian plane, distance formula, section formula, and triangle area.
- Comprehending Coordinates: A pair of integers, represented as (x, y), may be used to identify any point on a plane. In this case, ‘x’ stands for the distance from the y-axis (also known as the abscissa) and ‘y’ for the distance from the x-axis (also known as the ordinate).
- Distance Formula: We’ll understand how to use this helpful formula to determine the distance between two sites. We may calculate the length of a line segment between any two locations on a coordinate plane using this formula.
- Area Calculation: If we know the coordinates of a triangle’s vertices, we can also use the distance formula to get the triangle’s area.
For example: Coordinate Geometry
Let’s think about the point R(4, 3). In this instance, ‘4’ denotes the object’s separation from the y-axis, and ‘3’ is the object’s separation from the x-axis. Knowing Coordinate Geometry helps us see and understand mathematical topics by allowing us to accurately describe and evaluate objects and figures on a graph!
Please do comment! if you want any further explanation or examples!
Similar Triangles (Unit 8th)
Students in Class 10 Math’s eighth unit study the idea of Similar triangles, which is an important test topic. The lesson explores the requirements for triangle similarity along with its theorems.
Similar Triangles:
Similar triangles, such as ∆ABC and ∆PQR, have the same form but may differ in size. For this to be true:
- (i) Angles A, B, and C must be equal (A = P, B = Q, and C = R).
- (ii) Their corresponding sides are proportional, suggesting that the length ratios of the corresponding sides stay constant.
Example with Answer: (Similar Triangles)
Q) For example, Consider two triangles, ∆DEF and ∆GHI, where ∆DEF ~ ∆GHI. If DE = 4 cm, EF = 6 cm, and GH = 8 cm, what is the length of side HI?
Ans) To find the length of side HI in triangle ∆GHI, given that triangle ∆DEF is similar to triangle ∆GHI, we can use the properties of similar triangles.
Since triangles ∆DEF and ∆GHI are similar, their corresponding sides are proportional.
We can set up the proportions: GH/DE=HI/EF
Substituting the given values: 4 /8 = 6 /HI
Now, let’s solve for HI:
8/4 = HI/6
4×HI=6×8
4×HI=48
HI= 48/4
HI=12 cm
So, the length of side HI is 12 cm.
Unit-9 Tangents and Secants to a Circle
In Chapter 9 of Class 10 Math, we look at Tangents and Secants, focusing on their properties and practical applications. Let us put it in simpler terms:
Understand Tangents and Secants:
- Tangent: Imagine a line that only touches a circle at one place. That is what we call a tangent.
- Secant: Draw a line that cuts through the circle and intersects it at two different points. That is a secant.
Tangents to a circle:
- When a tangent touches a circle, it only touches it at one place, known as the point of contact.
- The tangent is always perpendicular to the radius of the circle at the point of contact.
- If you draw tangents from an external point to a circle, they will be the same length.
- We may also determine the area of a circle segment using an angle and radius formula.
For Example:
Q) Consider a circle with a radius 5 cm. A tangent is drawn to the circle from a point outside the circle, and it intersects the circle at point P. If the distance from the point to the centre of the circle is 8 cm, what is the length of the tangent segment, i.e., the length of line segment PT?
Ans) To find the length of the tangent segment PT, we can use the property that the length of the tangent segment from an external point to a circle is equal to the radius of the circle drawn perpendicular to the tangent line.
Given that the distance from the point to the center of the circle is 8 cm and the radius of the circle is 5 cm, we can use the Pythagorean theorem to find the length of the tangent segment PT.
Let’s denote the length of PT as 𝑥 x. According to the Pythagorean theorem:
x2=(8+5)2−5(square)
x2=132−5(square)
x2=169−25
x2=144
Taking the square root of both sides:
x= 144
x=12
10th Class Mathematics: Mensuration (Unit-10)
In Ts Class 10 Mathematics, we look at mensuration, which is the process of determining the areas and volumes of geometric forms including triangles, quadrilaterals, circles, and solids, as well as their properties such as area, length, volume, and surface area.
Mensuration:
Defining Mensuration: Mensuration is a field of geometry concerned with quantifying the area, volume, and size of different forms and figures in both two and three dimensions (3D and 2D).
Shapes in Two(2D) and (3D) Three Dimensions:
Two categories of forms are encountered in the field of mensuration:
- 2D forms are flat shapes like squares, circles, triangles, and rectangles that only have length and breadth.
- 3D Three-dimensional forms have three dimensions: length, breadth, and height. Cones, spheres, cylinders, and cubes are a few examples.
Mensuration Formulas:
We examine a variety of formulae for calculating the characteristics of 2D and 3D shapes. These formulae assist us in determining areas, volumes, surface areas, and other characteristics required for problem-solving with geometric forms.
For example:
Mensuration formulae may be used to compute the volume and surface area of a rectangular box with dimensions of 10 cm long, 5 cm wide, and 3 cm high.
Understanding mensuration gives us the tools we need to tackle real-world measuring and geometry issues, making it an important aspect of 10th-class mathematics.
If you want more information or examples, please continue reading! the following table!
Shape | Parameter | Formula |
---|---|---|
Square | Area | Area = side^{2} |
Rectangle | Area | Area = length × width |
Perimeter | Perimeter = 2 × (length + width) | |
Triangle | Area | Area = 0.5 × base × height |
Perimeter | Perimeter = side_{1} + side_{2} + side_{3} | |
Circle | Area | Area = π × radius^{2} |
Circumference | Circumference = 2 × π × radius |
Moving on to our next topic, let’s look into the interesting area of trigonometry.
Unit-11 Trigonometry (TS Class 10 Mathematics Syllabus)
Introduction to Trigonometry:
The study of triangles and their relationships is known as trigonometry, and it is covered in this section. It concentrates on triangles with one 90-degree angle, or right-angled triangles. Trigonometric functions, such as sine, cosine, and tangent, may be determined by analyzing side ratios. These functions are important in a variety of domains, including physics, engineering, astronomy, and navigation.
Trigonometric Ratios: These ratios define the relationship between the angles and sides of a right triangle. The main trigonometric ratios are:
- Sine (sin): Opposite/Hypotenuse
- Cosine (cos): Adjacent/Hypotenuse
- Tangent (tan): Opposite/Adjacent
Pythagorean Identity: This fundamental identity relates the lengths of the sides of a right triangle: (sin2θ+cos2θ=1)
Trigonometry Formulas:
Throughout this unit, we’ll explore a range of trigonometric formulas and identities, including:
- Sine Rule
- Cosine Rule
- Area of a Triangle Using Trigonometry
- Angle of Elevation and Depression
- Trigonometric Identities
Unit 12 – Applications of Trigonometry
This section focuses on using trigonometry practically in everyday circumstances, such as measuring heights and distances. Trigonometry may be used to calculate heights, lengths, and distances with a few particular techniques.
Here’s how it (Trigonometry Formulas) works:
Formula | Explanation | Example in Real Life |
sin(θ)=opposite/hypotenuse | Ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. | Finding the height of a tree using the angle of elevation and distance from the base. |
cos(θ)=adjacent/hypotenuse | Ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. |
Calculating the distance from the base of a tower when the height and angle of elevation are known. |
tan(θ)=opposite/adjacent | Ratio of the length of the opposite side to the adjacent side in a right-angled triangle. | Determining the height of a building by knowing the distance from it and the angle of elevation. |
csc(θ)= 1/sin(θ) | Reciprocal of sine. | Used in specific calculations where sine values are small. |
sec(θ)= 1/cos(θ) | Reciprocal of cosine. | Useful in certain engineering calculations. |
cot(θ)= 1/tan(θ) | Reciprocal of tangent. | Often used in trigonometric simplifications. |
sin2(θ)+cos2(θ)=1 | Pythagorean identity. | Verifying calculations in trigonometric problems. |
tan(θ)=sin(θ)/cos(θ) | Relationship between sine and cosine. | Used in simplifying trigonometric expressions. |
- Path of Sight: Imagine drawing an imagined line from your eyes to the point you’re looking at on an object. That is the “line of sight.”
- The angle of Altitude: When you stare up at anything above your eye level, the angle between your line of sight and the ground is referred to as the “angle of altitude.” You crane your head up to view it.
- Height of Depression: When you’re looking down at anything below eye level, the angle between your line of sight and the ground is known as the “length of depression.” You lean your head downward to view it.
These concepts can in handy in everyday situations, such as determining the height of a structure or the distance between two objects.
Unit 13: Probability and Unit 14: Statistics
TS Class 10th Unit 13: Probability
This chapter addresses many probability ideas, both experimental and theoretical. RD Sharma’s answers offer well-curated thorough answers for Class 10 Maths Chapter 13 – Probability. Regular practice improves academic performance and develops vital skills such as time management and problem-solving, both of which are critical for board exam success.
RD Sharma Solutions for TS Class 10 Mathematics Chapter 13
- Enhances academic performance and nurtures time management and problem-solving skills.
- Designed by Digital Teacher faculty for conceptual clarity.
- Solutions are accompanied by illustrative diagrams for better comprehension.
- Provides descriptive explanations in a simple, accessible manner.
Probability: Key Concepts and Formulas
Concept | Explanation | Probability Formula |
Probability | The measure of the likelihood that an event will occur. | P(E)=Number of favorable outcomes /Total number of outcomes |
Experiment | An action or process that leads to one or more outcomes. | – |
Sample Space (S) | The set of all possible outcomes of an experiment. | – |
Event (E) | A subset of the sample space. It can have one or more outcomes. | – |
Complement of an Event ( 𝐸 ′ E ′ ) | The set of all outcomes in the sample space that are not in event E. |
P(E′)=1−P(E) |
Mutually Exclusive Events | Two events that cannot occur at the same time. | P(A∪B)=P(A)+P(B) if A and B are mutually exclusive |
Conditional Probability | The probability of an event occurring given that another event has already occurred. | (P(A |
Independent Events | Two events that do not affect each other’s occurrence. | P(A∩B)=P(A)⋅P(B) if A and B are independent |
Addition Rule of Probability | The probability that either event A or event B will occur. | P(A∪B)=P(A)+P(B)−P(A∩B) |
Multiplication Rule of Probability | The probability that both events A and B will occur. | ( P(A \cap B) = P(A) \cdot P(B |
The above table method provides students such a clear and simple guide to help them understand and use important computations of probability and concepts in their classes.
Telangana TS Class 10 Mathematics Unit 14: Statistics
Statistics is important to students since it offers fundamental abilities for data analysis and interpretation, such as comprehending mean, median, mode, and standard deviation. These abilities are essential for academic performance and real-life decision-making, creating prospects for success in academics and future undertakings.
Here’s How It (Statistics Formulas) Works:
Understanding Using (Statistics ): Statistics is a branch of mathematics that focuses on collecting, analyzing, interpreting, presenting, and organizing data. It is important for students as it provides fundamental skills for data analysis and interpretation, including understanding mean, median, mode, and standard deviation.
In this unit, students will learn about various statistical measures and how to calculate them. The table below summarizes the key concepts and formulas covered in this unit, providing a valuable reference for students.
Here’s How It (Statistics Formulas) Works: Table
Concept | Explanation | Statistics Formula |
Mean (Average) | The sum of all observations is divided by the number of observations. | Mean(x)=n∑x |
Median | The middle value is when observations are arranged in ascending or descending order. | If 𝑛 n is odd: Median = Middle value Median=Middle value. If 𝑛 n is even: Median = Middle value 1 + Middle value 2 2 Median= Middle value 1+Middle value 2 / 2 |
Mode | The value that appears most frequently in a data set. | – |
Range | The difference between the highest and lowest values. | Range=Maximum value−Minimum value |
Class Interval | A group of values within which a data point falls in a frequency distribution table. | – |
Frequency | The number of times a particular value or class interval occurs. | – |
Class Mark (Midpoint) | The middle value of a class interval. | Class Mark= Lower class limit+Upper class limit / 2 |
Mean for Grouped Data | The average of data grouped in class intervals. | x=∑fi/∑fixi where 𝑓 𝑖 f i is the frequency and 𝑥 𝑖 x i is the class mark |
Median for Grouped Data | The median of data is grouped in class intervals. | Median=L+(2n−CF/f ×h where 𝐿 L = lower boundary of median class, 𝑛 n = total frequency, 𝐶 𝐹 CF = cumulative frequency of class before median class, 𝑓 f = frequency of median class, ℎ h = class width |
Mode for Grouped Data | The mode of data is grouped in class intervals. | Mode=L+ (f1−f0/2f1−f0-f2)×h where 𝐿 L = lower boundary of modal class, 𝑓 1 f 1 = frequency of modal class, 𝑓 0 f 0 = frequency of class before modal class, 𝑓 2 f 2 = frequency of class after modal class, ℎ h = class width |
The above Statistics Formula Table will help students understand and use the statistical formulas by providing information for the table.
Example:
Imagine you’re playing a game where you roll a fair six-sided die. The probability of rolling a “4” is 1/6. This means that out of all the possible outcomes (1, 2, 3, 4, 5, or 6), there’s a 1 in 6 chance of rolling a “4.”
Please do comment! if you want any further explanation or examples!
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