PROJECT WORK OF MATHEMATICS CHAPTERS.pptx

mathematics

1. PROJECT WORK OF MAHEMATICS Submitted by: Siddhartha Maharjan Submitted to: Tika Ram Sir

2. SETS A set is a  well-defined collection of objects, whose elements are fixed and cannot vary . The objects in a set are called its elements or members. The elements in a set can be any types of objects. For example : A =  {1,2,3,4,5,6,7}.

3. Key concepts and formulas related to sets : n(A∪B)=n(A)+n(B)−n(A∩B) n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C) n(A−B)=n(A)−n(A∩B) n(A′)=n(U)−n(A) n(AΔB)=n(A)+n(B)−2n(A∩B)

4. TAX Tax is money that a government or government agencies collect from people based on laws or constitutions. Some formulas are: Taxable income + Tax exempt income = Total Income Basic Salary + Dearness allowance = Monthly Income Total income – Tax Exempt income = Taxable Income

5. C ommission, B onus & D ividend Commission: A commission is a percentage of the selling amount that an agent receives. The commission is usually given to the agent by the buyer, seller, or both. Dividend: A dividend is a portion of a company's or co-operative's profit given to shareholders.

6. Quick Tips : Amount of commission = commission percent of total SP Combined wage = Fixed Salary + Commission

7. House Hold Arithmetic Household arithmetic refers to the practical use of basic mathematical operations like addition, subtraction, multiplication, and division to manage daily household tasks, such as budgeting, calculating bills, measuring space, and handling expenses.

8. Some Formulas are: Water Bill=Water  Consumption×Rate  per Unit Electricity Bill=(Electricity Consumption in  kWh×Rate  per kWh)+Fixed  Charges+Taxes Telephone Bill=(Call  Charges+Data   Charges+Text  Charges)+Monthly  Subscription+Taxes Taxi Fare=Base Fare+(Distance Fare×Distance)+(Time Fare×Time)

9. Area Area is the measure of the total surface space within a boundary, typically expressed in square units . Some Formulas are: General formula of area of a triangle=1/2​ × Base×Height Area of a right angled triangle=1/2 ​ × p ​ × b Area of an equilateral triangle= ( √3 a×a )/4 Area of an isosceles triangle=1/4 ​ × b √4a×a-b×b Area of a scalene triangle= √s(s-a)(s-b)(s-c)

10. Prism Prism is a three-dimensional solid object in which the two ends are identical. It is the combination of the flat faces, identical bases and equal cross-sections . FORMULAS ARE: total surface area A = 2( l x b ) + 2( b x h) + 2( l x h ) lateral surface area = 2 h ( l + b ) Cross sectional area = ½ base X height Volume = Base X Height

11. Cylinder and Sphere A solid  geometrical  figure with straight parallel sides and a circular or oval cross section is known as a cylinder. A sphere is a geometrical figure that is perfectly round, 3-dimensional and circular like a ball. Geometrically, a sphere is defined as the set of all points equidistant from a single point in space.

12. Some Formulas : Area of a circular base = π r² CSA of a cylinder = 2 π rh TSA of a cylinder = 2 π r( r+h ) Volume of a cylinder = π r²h Surface area of a sphere = 4 π r² Volume of a sphere = 4/3 π r 3

13. CSA of a hemisphere = 2 π r² TSA of a hemisphere = 3 π r² Volume of a hemisphere = 2/3 π r 3

14. Factorization   Factorization is the process of breaking down a number, expression, or equation into its factors. The factors are numbers or expressions that can be multiplied together to get the original number, expression, or equation. 

15. Some Formulas are : (a + b) 2  = a 2  + 2ab + b (a − b) 2  = a 2 − 2ab + b (a + b) 3 =a 3  + b 3  + 3ab (a + b) (a – b) 3  = a 3  – b 3  – 3ab (a – b) a 2  – b 2 = (a + b) (a – b ) a 2  + b 2  = (a + b) 2  – 2ab a 3  – b 3  = (a – b) (a 2  + ab + b 2 ) a 3  + b 3  = (a + b) (a 2  – ab + b 2 )

16. HCF and LCM The highest common factor (HCF) is  the largest number that goes into two or more subject-numbers . The least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly.

17. Simultaneous Linear Equation Two or more linear equations that all contain the same unknown variables are called a system of simultaneous linear equations. Solving such a system means finding values for the unknown variables which satisfy all the equations at the same time.

18. Ways to solve simultaneous linear equations Substitution method : Substitution method involves substituting the value of any one of the variables from one equation into the other equation . Elimination method :The elimination method is  an algebraic technique for solving systems of equations by removing a variable from the equations .

19. And . . . Graphical method :The procedure of solving a system of linear equations by drawing the graph is known as the graphical method.

20. I NDICES An index, or power, is the small floating number that appears after a number or letter. Indices show how many times a number or letter has been multiplied by itself .   3 x 2

21. Some Laws of Indices are: First Index Law: a m  × a n  = a m   +   n   Second Index Law: a m  / a n  = a m   –   n   Third Index Law: a  = 1 (where a ≠ 0) Fourth Index Law: (a m ) n  = a m   ×   n   Fifth Index Law: (a × b) m  = a m  × b m   Sixth Index Law: (a / b) m  = a m  / b m   Negative Indices: a -n  = 1 / a n  (where a≠0) Square Roots: √a = a 1/2

22. Sequence and Series : A sequence is a list of numbers or objects in a specific order, while a series is the sum of all the terms in a sequence.  A sequence is seperated by comma while series is seperated by “+” or “–” sign.

23. Sequence and Series Formulas : General Term (nth Term ) in arithmetic progression = a n  = a + ( n-1)d General Term (nth Term) in geometric progression = a n  = ar (n-1 ) Sn = n/2 [2a + (n-1)d]

24. Triangles A triangle is a closed plane figure bounded by three line segments. Types of triangles: Congruent triangles: Congruent triangles are triangles having corresponding sides and angles to be equal. Similar triangles: Similar triangles are triangles that have the same shape, but their sizes may vary

25. Tests to prove congruencey : side-side-side (SSS ) side-angle-side (SAS ) angle-side-angle (ASA ) angle-angle-side (AAS ) right angle-hypotenuse-side (RHS) 

26. Q uadrilatera l A quadrilateral is a four-sided polygon with four angles, and its sides can be of varying lengths and arrangements.

27. Types of Quadrilaterals : The source mentions specific types of quadrilaterals such as squares, rectangles, parallelograms, rhombus and trapezoids . Parallelograms as Quadrilaterals : A quadrilateral with opposite sides parallel is called a parallelogram. Properties and theorems related to parallelograms are discussed extensively. Trapeziums as Quadrilaterals: A quadrilateral with one pair of opposite sides parallel is called a trapezium. Construction of Quadrilaterals: The source provides instructions on how to construct different types of quadrilaterals, including squares, rectangles, parallelograms, rhombuses, and trapeziums, given specific conditions. General Properties and Theorems: The sum of the internal angles of a quadrilateral is equal to 360° . Mid-Points of a Quadrilateral A quadrilateral formed by joining the mid-points of the sides of a quadrilateral is a parallelogram.

28. Construction In mathematics , construction is  the process of accurately drawing geometric shapes, lines, and angles using only a compass and a scale, essentially creating precise figures without relying on numerical calculations; it's considered a "pure" form of geometric manipulation where only visual relationships are used to create the desired shapes. 

29. C IRCLE A circle is a plane figure traced out by a moving point in a plane under the geometrical condition that it remains equidistant from some point. Parts of a Circle: The source refers to parts of a circle, including: radius, circumference, chord, diameter, arc, semi-circle, sector, segment .

30. THEOREMS OF CIRCLE Theorem 1 : A perpendicular drawn from the centre of a circle to a chord bisects the chord . Theorem 2 : A line segment joining the centre of a circle to the mid-point of its chord is perpendicular to the chord . Theorem 3 : The perpendicular bisector of a chord of a circle passes through the centre of that circle . Theorem 4 : The equal chords of a circle are equidistant from the centre of the circle . Theorem 5 : (Converse of theorem 4) In any circle, the chords that are equidistant from the centre are equal to each other .

31. Area and Circumference: Circumference (C) = 2πr = πd Area (A) = πr2 square unit

32. S TATISTICS Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. Collection, classification, tabulation, representation, reasoning, testing and drawing inferences are all things done in the statistical method.

33. Key Terms: Frequency : The number of times a particular data value occurs. Class interval: A range of values within which data is grouped. Cumulative frequency: The sum of frequencies up to a certain point in a data set.

34. Measures of Central Tendencies : Arithmetic mean (average): The sum of all values divided by the number of values. For individual series: Mean = (Sum of all items) / (Total number of items). For discrete series: Mean = Σfx / Σf (where fx is the product of each item and its frequency). Median: The middle value in an ordered data set. For individual data, the median is the ((N+1)/2) th value, where N is the number of data points. Mode: The most frequently occurring value in a data set. Quartiles: Values that divide the data into four equal parts. Q1 (first quartile): The 25th percentile. Q3 (third quartile): The 75th percentile.

35. PROBABILITY Probability is used to estimate different events that can happen in daily life. It deals with uncertain forecast and possibilities . Calculating Probability: The probability of an event P(E) is calculated as: P(E) = (Number of favourable outcomes) / (Total number of possible outcomes)

36. T RIGONOMETRY trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine ( cos ), tangent (tan), cotangent (cot), secant (sec), and cosecant ( cosec ).

37. Fundamental Trigonometric Ratios: In a right-angled triangle ABC, with reference angle θ, the three basic trigonometric ratios are defined as: Sine of θ (sin θ) = Opposite side / Hypotenuse = p/h Cosine of θ ( cos θ) = Adjacent side / Hypotenuse = b/h Tangent of θ (tan θ) = Opposite side / Adjacent side = p/b Other trigonometric ratios include: Cosecant of θ (cosec θ) = 1/sin θ = Hypotenuse / Opposite side Secant of θ (sec θ) = 1/ cos θ = Hypotenuse / Adjacent side Cotangent of θ (cot θ) = 1/tan θ = Adjacent side / Opposite side

38. Questions : (SETS) : The survey of patients in a hospital showed that 150 patients have high blood pressure and 120 have diabetes . If 80 of them have both diseases , find the no. of patients who have at least one of them. (TAX):Calculate the tax paid by a man who earns Rs.3,10,000 a year if his tax free allowance amounts to Rs.2,50,000 and the tax rate is 15%.

39. (COMMISSION AND BONUS ):A car was sold through an agent with a commission of 3.5% . If the agent got Rs.64,890 as a commission, how much did the car owner receive? (HOUSEHOLD ARITHMETICS ):The rate of electricity charge up to 20 units is Rs.3 per unit from 21 to 30 units . Find the charge of consumption of 28 units with Rs.50 service charge.

40. (AREA ):If the sides of a triangle are in the ratio of 2:3:4 and it’s perimeter is 36cm , find the area of the triangle . (PRISM ):The length and breadth of a cuboid are in the ratio of 5:2 and it’s height is 12cm. If the volume of the cuboid is 480 cm³, find the TSA of the cuboid.

41. (CYLINDER AND SPHERE):The circumference of the base of a cylinder is 44cm. If the sum of it’s radius and height of the cylinder is 27 cm , find the TSA of the cylinder. (FACTORIZATION):If x+ =2 , find the value of x 2 + and + .  

42. (HCF AND LCM):Find HCF of y +2xy – 3x , -4 , -2 + 8 – 4x . (SIMULTANEOUS LINEAR EQUATIONS ):Solve the following linear equations with substitution method : 2x -3y -7 = 0 and 3x + 4y = 2  

43. (INDICES): Find the value of . (SEQUENCE AND SERIES ):Find the general term and write the following series : 1,4,9,16,25, … … … 10 terms  

44. (TRIANGLES ):Two angles , a and b are complementary and a is one fourth of b . Find the values of a and b. (QUADRILATERALS ):If the diagonals of a parallelogram are perpendicular to each other , prove that it is a rhombus .

45. (CONSTRUCTIONS): Construct a trapeium ABCD with the following measurements. AB//CD , AB = 7cm , BC = 5cm , CD = 8cm and BCD = 60 . (CIRCLE ):The diameter of a circle is 10 cm. Find the length of the chord which is at the distance of 4 cm from the center of the circle.  

46. (STATISTICS):x,x+2,x+5,2x+1,3x+4 and 4x are in ascending order . If it’s median is 30 , find the value of x . (PROBABILITY):A card is drawn from a pack of 52 playing cards . What is the probability that the card so drawn is a king or a queen?

47. (TRIGONOMETRY):Express the trigonometric ratio sin A in terms of cos A.

48. THANK YOU!!!