Topics
Sets and Relations
- Fundamental Concepts of Ordered Pairs and Relations
- Representation of a Set
- Intervals
- Classification of Sets
- Relations of Sets
Functions
- Functions
- Types of Functions
- Representation of Function
- Graph of a Function
- Fundamental Functions
- Algebra of Functions
- Composite Function
- Inverse Functions
- Some Special Functions
Complex Numbers 33
- Introduction of Complex Number
- Imaginary Number
- Concept of Complex Numbers
- Conjugate of a Complex Number
- Algebraic Operations of Complex Numbers
- Square Root of a Complex Number
- Solution of a Quadratic Equation in Complex Number System
- Cube Root of Unity
Sequences and Series
- Sequence, Series, and Progression
- Geometric Progression (G. P.)
- General Term Or the nth Term of a G.P.
- Sum to' n' Terms of a Geometric Progression
- Sum to' n' Terms of a Geometric Progression
- Recurring Decimals
- Harmonic Progression (H. P.)
- Types of Means
- Special Series (Sigma Notation)
Locus and Straight Line
- Locus
- Line
- Equations of Lines in Different Forms
- Equation of a Straight Line
Determinants
- Determinants
- Properties of Determinants
- Application of Determinants
- Determinant Method (Cramer’s Rule)
- Consistency of Three Linear Equations in Two Variables
- Area of a Triangle Using Determinants
- Collinearity of Three Points
Limits
- Definition of Limit of a Function
- Algebra of Limits
- Evaluation of Limits
- Direct Method
- Factorization Method
- Rationalization Method
- Limits of Exponential and Logarithmic Functions
Continuity
- Continuous and Discontinuous Functions
- Continuity of a Function at a Point
- Definition of Continuity
- Continuity from the Right and from the Left
- Properties of Continuous Functions
- Continuity in the Domain of the Function
- Examples of Continuous Functions Whereever They Are Defined
Differentiation
- The Meaning of Rate of Change
- Definition of Derivative and Differentiability
- Derivative by the Method of First Principle
- Rules of Differentiation (Without Proof)
- Applications of Derivatives
Partition Values
- Partition Values
- Deciles
- Percentiles
- Relations Among Quartiles, Deciles and Percentiles
- Graphical Location of Partition Values
Measures of Dispersion
- Measures of Dispersion
- Quartiles and Range in Statistics
- Measures of Dispersion > Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Skewness
- Skewness
- Asymmetric Distribution (Positive Skewness)
- Asymmetric (Negative Skewness)
- Measures of Skewness
- Karl Pearson’S Coefficient of Skewness (Pearsonian Coefficient of Skewness)
- Features of Pearsonian Coefficient
- Bowley’s Coefficient of Skewness
Bivariate Frequency Distribution and Chi Square Statistic
- Bivariate Frequency Distribution
- Classification and Tabulation of Bivariate Data
- Marginal Frequency Distributions
- Conditional Frequency Distributions
- Categorical Variables
- Contingency Table
- Chi-Square Statistic ( χ2 )
Correlation
- Correlation
- Concept of Covariance
- Properties of Covariance
- Concept of Correlation Coefficient
- Scatter Diagram
- Interpretation of Value of Correlation Coefficient
Permutations and Combinations
- Introduction of Permutations and Combinations
- Fundamental Principles of Counting
- Concept of Addition Principle
- Concept of Multiplication Principle
- Concept of Factorial Function
- Permutations
- Permutations When All Objects Are Distinct
- Permutations When Repetitions Are Allowed
- Permutations When All Objects Are Not Distinct
- Circular Permutations
- Properties of Permutations
- Combination
- Properties of Combinations
Probability
- Concept of Probability
- Elementary Types of Events in Probability
- Elementary Properties of Probability
- Addition Theorem of Probability
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
Linear Inequations
- Linear Inequations
- Method of Solving a Linear Inequality
- Representation of Inequalities
- Graphical Solution of Linear Inequality of Two Variable
- Solution of System of Linear Inequalities in Two Variables
Commercial Mathematics
- Percentage
- Profit and Loss
- Simple and Compound Interest (Entrance Exam)
- Concept of Depreciation
- Partnership
- Shares
Definition
Partnership: A partnership is an arrangement where parties, known as business, partners, agree to cooperate to advance their mutual interests.
Partnership deed: An agreement that contains the details of every aspect and terms that the partners agree upon before starting a partnership firm is termed as partnership deed.
Capital: When starting a business enterprise, money is required for an office, raw materials, etc. This amount is called the capital.
Notes
Partnership:
- A partnership is an agreement between two or more persons to share profits and losses of the firm. According to Section 4 of the Indian Partnership Act, 1932, “Partnership is the relation between persons who have agreed to share profits of a business carried on by all or any one of them acting for all.”
- A partnership is an arrangement where parties, known as business, partners, agree to cooperate to advance their mutual interests. The partners in a partnership may be individuals, businesses, interest-based organizations, schools, governments, or combinations.
- An agreement that contains the details of every aspect and terms that the partners agree upon before starting a partnership firm is termed as partnership deed.
- At least two persons are required to form a partnership business. However, according to the Companies Act, 1956, a firm cannot have more than 10 partners in case of banking business and not more than 20 persons in case of any other business.
- Organizations may partner to increase the likelihood of each achieving their mission and to amplify their reach. A partnership may result in issuing and holding equity or maybe only governed by a contract.
- When starting a business enterprise, money is required for an office, raw materials, etc. This amount is called the capital. Often, two or more people put in money for the capital. In other words, these people start a business by investing in the partnership.
- In a business partnership, all partners have a joint account in a bank. The profit made or the loss incurred is shared by the partners in proportion to the money each one has invested.
Example
Jhelum and Atharva invested 2100 and 2800 rupees respectively and started a business. They made a profit of 3500 rupees. How should it be shared?
Let us find out the proportion of investments.
2100: 2800 = `2100/2800 = 3/4` = 3: 4.
∴ The proportion of investments is 3: 4.
The profit must also be shared in the same proportion.
Let Jhelum’s profit be 3x and that of Atharva, 4x. Then,
3x + 4x = 3500 as total profit is 3500.
∴ 7x = 3500
∴ x = 500
Jhelum’s share = 3x = 1500 rupees and Atharva’s share = 4x = 2000 rupees.
Example
Chinmaya and Sam invested a total of 130000 rupees in business in proportion 3: 2 respectively. What amount did each of them invest? If their total profit was 36000 rupees, what is the share of each?
The proportion of Chinmaya’s and Sam’s investment is 3: 2.
The profit is shared in the same proportion as the investment, hence, the proportion of profit is 3: 2.
Let Chinmaya’s investment be 3y and Sam’s 2y.
3y + 2y = Total investment.
∴ 5y = 130000
`5/(5y) = 130000/5`..... (dividing by 5)
∴ y = 26000
∴ Chinmaya’s investment = 3y = 3 × 26000 = 78,000
Sam’s investment = 2y = 2 × 26000 = 52000 rupees
Let Chinmaya’s profit be 3x and Sam’s 2x.
3x + 2x = Total Profit
5x = 36000
`5/(5x) = 36000/5`
∴ x = 7200
∴ Chinmaya’s profit = 3x = 3 × 7200 = 21600
Sam’s profit = 2x = 2 × 7200 = 14400 rupees
Example
Abdul, Sejal, and Soham each gave Sayali 30 rupees, 70 rupees, and 50 rupees respectively. Sayali put in 150 rupees and bought paper, colours, etc. Together they made greeting cards and sold them all. If they made a total profit of 420 rupees, what was each one’s share in the profit?
The capital invested by all four was 300 rupees. Of this Sayali had invested 150 rupees, that is, half of the capital. The total profit was 420 rupees. So, Sayali’s profit was half of that, i.e., 210 rupees. The remaining 210 was shared by Abdul, Sejal, and Soham.
Abdul, Sejal, and Soham’s investment is 30, 70, and 50 rupees. The proportion is 30: 70: 50 i.e. 3: 7: 5. Their share of the profit is altogether 210 rupees.
Let their individual profit be 3k, 7k, 5k.
Then, 3k + 7k + 5k = 210
∴ 15k = 210
∴ k = 14
Abdul’s profit = 3k = 3 × 14 = 42 rupees.
Sejal’s profit = 7k = 7 × 14 = 98 rupees.
Soham’s profit = 5k = 5 × 14 = 70 rupees.
