Class 10 math Guide is the Last Hour Guide for all class 10 SEE Appearing Students. This Math Guide for Class 10 contains all chapter key Notes, download Pdf and Start Studying offline. To Download Pdf Notes Click on the Download Pdf Button at the Bottom of the Post.
Class 10 Math Set Chapter Key Points
- Set: A set is a collection of distinct objects. The objects in a set are called elements, and sets are denoted by capital letters. For example, the set of all natural numbers can be denoted by N.
- Types of Sets: There are several types of sets that are important to understand, including:
- Finite Set: A finite set has a definite number of elements.
- Infinite Set: An infinite set has an infinite number of elements.
- Empty Set: An empty set has no elements and is denoted by ∅.
- Universal Set: A universal set contains all possible elements of a given situation or problem.
- Subset: A subset is a set that contains only elements from another set. It is denoted by ⊆.
- Power Set: The power set of a set is the set of all possible subsets of that set.
- Operations on Sets: There are several operations that can be performed on sets, including:
- Union: The union of two sets A and B is the set of all elements that belong to either A or B, or both. It is denoted by A ∪ B.
- Intersection: The intersection of two sets A and B is the set of all elements that belong to both A and B. It is denoted by A ∩ B.
- Complement: The complement of a set A with respect to a universal set U is the set of all elements in U that are not in A. It is denoted by A’.
- Difference: The difference of two sets A and B is the set of all elements in A that are not in B. It is denoted by A – B.
- Venn Diagrams: Venn diagrams are graphical representations of sets and their relationships. They consist of circles or ovals that represent sets and overlapping areas that represent their relationships.
- De Morgan’s Laws: De Morgan’s Laws are important formulas in set theory that help to simplify expressions involving sets. They state that:
- The complement of the union of two sets is equal to the intersection of their complements: (A ∪ B)’ = A’ ∩ B’
- The complement of the intersection of two sets is equal to the union of their complements: (A ∩ B)’ = A’ ∪ B’
Tax and Money Exchange chapter
- Simple Interest: Simple interest is the interest that is earned only on the principal amount. It is calculated using the following formula:
Simple Interest = (Principal x Rate x Time)/100
where Principal is the initial amount of money, Rate is the rate of interest per year, and Time is the time period in years.
- Compound Interest: Compound interest is the interest that is earned not only on the principal amount but also on the interest that has accumulated over time. It is calculated using the following formula:
Amount = Principal x (1 + Rate/100)^Time
where Amount is the total amount of money after a certain period of time.
- Discount: A discount is a reduction in the price of an item. It is calculated using the following formula:
Discount = Marked Price x Rate of Discount/100
where Marked Price is the original price of the item, and Rate of Discount is the percentage of the discount.
- Value Added Tax (VAT): Value Added Tax is a tax that is added to the value of a product at each stage of production and distribution. It is calculated as a percentage of the selling price of the product.
- Goods and Services Tax (GST): Goods and Services Tax is a tax that is levied on the supply of goods and services. It is calculated as a percentage of the selling price of the product.
- Foreign Exchange: Foreign exchange is the exchange of one currency for another. The exchange rate is the rate at which one currency can be exchanged for another.
- Currency Conversion: Currency conversion is the process of converting one currency to another. It is calculated using the exchange rate between the two currencies.
Mensuration Chapter Key Notes
- Perimeter: The perimeter of a two-dimensional shape is the total distance around its outer boundary. It is calculated by adding the lengths of all the sides of the shape.
- Area: The area of a two-dimensional shape is the amount of surface it covers. The formula for the area of different shapes are:
- Rectangle: Area = Length x Breadth
- Square: Area = Side^2
- Triangle: Area = 1/2 x Base x Height
- Parallelogram: Area = Base x Height
- Trapezium: Area = 1/2 x (Sum of parallel sides) x Height
- Circle: Area = π x Radius^2
- Volume: The volume of a three-dimensional shape is the amount of space it occupies. The formula for the volume of different shapes are:
- Cube: Volume = Side^3
- Cuboid: Volume = Length x Breadth x Height
- Cylinder: Volume = π x Radius^2 x Height
- Sphere: Volume = 4/3 x π x Radius^3
- Surface Area: The surface area of a three-dimensional shape is the sum of the areas of all its faces. The formula for the surface area of different shapes are:
- Cube: Surface Area = 6 x Side^2
- Cuboid: Surface Area = 2 x (Length x Breadth + Breadth x Height + Height x Length)
- Cylinder: Surface Area = 2 x π x Radius x Height + 2 x π x Radius^2
- Sphere: Surface Area = 4 x π x Radius^2
- Pythagoras Theorem: Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. The formula for Pythagoras Theorem is:
Hypotenuse^2 = Base^2 + Perpendicular^2
- Trigonometry: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The three basic trigonometric ratios are:
- Sine: sinθ = Perpendicular/Hypotenuse
- Cosine: cosθ = Base/Hypotenuse
- Tangent: tanθ = Perpendicular/Base
Understanding and applying these concepts and formulas is important for success in various areas of mathematics, including geometry and engineering.
Algebra Chapter Key Notes
- Algebraic Expressions: Algebraic expressions are mathematical expressions that consist of variables, constants, and operators. They can be simplified, combined, and evaluated using the following operations:
- Addition: a + b
- Subtraction: a – b
- Multiplication: a x b or ab
- Division: a/b or a ÷ b
- Exponentiation: a^n
- Linear Equations: Linear equations are equations that can be written in the form ax + b = c, where x is the variable and a, b, and c are constants. They can be solved using the following formula:
x = (c – b)/a
- Quadratic Equations: Quadratic equations are equations that can be written in the form ax^2 + bx + c = 0, where x is the variable and a, b, and c are constants. They can be solved using the following formula:
x = (-b ± √(b^2 – 4ac))/2a
- Factorisation: Factorisation is the process of expressing a polynomial as a product of its factors. It is an important concept in algebra and can be done using the following methods:
- Common factor method
- Grouping method
- Quadratic formula
- Identities: Identities are equations that are true for all values of the variables. Some important algebraic identities are:
- (a + b)^2 = a^2 + 2ab + b^2
- (a – b)^2 = a^2 – 2ab + b^2
- (a + b)(a – b) = a^2 – b^2
- a^2 – b^2 = (a + b)(a – b)
- Graphs: Graphs are visual representations of algebraic equations. They can be used to solve and interpret algebraic problems, and are especially useful in understanding linear and quadratic equations.
Geometry Chapter Key Notes
- Lines and Angles: In geometry, a line is a straight path that extends infinitely in both directions, while an angle is the measure of the space between two intersecting lines. Some important concepts and formulas related to lines and angles include:
- Types of angles: acute angle (less than 90 degrees), right angle (exactly 90 degrees), obtuse angle (greater than 90 degrees), straight angle (exactly 180 degrees), reflex angle (greater than 180 degrees), and full angle (exactly 360 degrees).
- Types of lines: parallel lines (lines that never intersect), intersecting lines (lines that cross at one point), perpendicular lines (lines that intersect at a right angle), and transversal lines (lines that intersect two or more parallel lines).
- Angle properties: vertically opposite angles are equal, adjacent angles add up to 180 degrees, alternate angles are equal, corresponding angles are equal, and interior angles of a triangle add up to 180 degrees.
- Triangles: A triangle is a polygon with three sides and three angles. Some important concepts and formulas related to triangles include:
- Types of triangles: scalene (all sides and angles are different), isosceles (two sides and two angles are equal), and equilateral (all sides and angles are equal).
- Area of a triangle: A = 1/2 x base x height
- Pythagorean theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Congruent triangles: Triangles are congruent if they have the same size and shape. This can be proved using various methods such as SSS (side-side-side), SAS (side-angle-side), and ASA (angle-side-angle).
- Circles: A circle is a two-dimensional shape with all points on its boundary at an equal distance from its center. Some important concepts and formulas related to circles include:
- Radius: The distance from the center of a circle to any point on its boundary.
- Diameter: The distance across a circle, passing through its center.
- Circumference: The distance around the boundary of a circle. C = 2πr or C = πd, where r is the radius and d is the diameter.
- Area of a circle: A = πr^2
- Constructions: Constructions are techniques used to create geometric shapes and figures using only a compass and a straightedge. Some important constructions include:
- Bisecting a line segment
- Constructing an angle
- Constructing a perpendicular bisector of a line segment
- Constructing a triangle given its sides or angles
Trigonometry Chapter Key Notes
- Trigonometric ratios: Trigonometric ratios are used to find the relationships between the angles and sides of a right triangle. The three basic trigonometric ratios are:
- Sine (sin) = opposite/hypotenuse
- Cosine (cos) = adjacent/hypotenuse
- Tangent (tan) = opposite/adjacent
- Pythagorean identity: The Pythagorean identity is a trigonometric identity that relates the values of sine, cosine, and tangent in a right triangle. The identity is:
- sin²θ + cos²θ = 1
- Trigonometric identities: Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Some important trigonometric identities include:
- Reciprocal identities: sin(θ) = 1/csc(θ), cos(θ) = 1/sec(θ), tan(θ) = 1/cot(θ)
- Quotient identities: tan(θ) = sin(θ)/cos(θ), cot(θ) = cos(θ)/sin(θ)
- Pythagorean identities: sin²(θ) + cos²(θ) = 1, 1 + tan²(θ) = sec²(θ), 1 + cot²(θ) = csc²(θ)
- Trigonometric functions of special angles: The values of sine, cosine, and tangent for certain angles can be found without the use of a calculator. Some important angles and their trigonometric function values are:
- 0°: sin(0°) = 0, cos(0°) = 1, tan(0°) = 0
- 30°: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = √3/3
- 45°: sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1
- 60°: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3
- 90°: sin(90°) = 1, cos(90°) = 0, tan(90°) is undefined
Statistics Chapter Key Notes:
- Mean: The mean of a set of data is the sum of all the values in the data set divided by the total number of values. The formula for calculating the mean is:
- Mean = (sum of all values) / (total number of values)
- Median: The median of a set of data is the middle value when the values are arranged in numerical order. If there are an even number of values, the median is the average of the two middle values.
- Mode: The mode of a set of data is the value that occurs most frequently.
- Range: The range of a set of data is the difference between the largest and smallest values.
- Range = (largest value) – (smallest value)
- Variance: Variance measures how spread out the data is from the mean. The formula for variance is:
- Variance = sum of [(value – mean)²] / (total number of values)
- Standard Deviation: Standard deviation is a measure of how spread out the data is from the mean. It is the square root of the variance.
- Standard Deviation = √(variance)
- Interquartile Range (IQR): The IQR is the range of the middle 50% of the data. It is calculated by finding the difference between the upper and lower quartiles.
- IQR = Q3 – Q1
- Quartiles: Quartiles divide a set of data into four equal parts. The first quartile (Q1) is the value below which 25% of the data falls, the second quartile is the median, and the third quartile (Q3) is the value below which 75% of the data falls.