Basic Mathematics - BCA 1st Semester

Unit I: Divisibility, Prime Factorization, Fractions, VBODMAS, and Algebraic Formula

Topics Covered:

• Test for Divisibility of Numbers
• General Properties of Divisibility
• Division and Remainder Rules
• Prime Factorization
• Difference between HCF and LCM
• Definition and Comparison of Fractions
• Insertion of Fractions between Two Given Fractions
• Operation Order Sequence (VBODMAS)
• Algebraic Formula
• Percentage and Inter-conversion
• Average, Ratio, and Proportion
• Binomial Theorem and Expansions

Test for Divisibility of Numbers

Rules to check if a number can be divided by another number without a remainder. Example: A number is divisible by 2 if its last digit is even.

General Properties of Divisibility

Properties that govern divisibility, such as transitivity (if a divides b and b divides c, then a divides c).

Division and Remainder Rules

When dividing a number, the result may have a remainder. For example, dividing 7 by 3 gives 2 as a quotient and 1 as a remainder.

Prime Factorization

Expressing a number as a product of its prime numbers. Example: 28 = 2 × 2 × 7.

Difference between HCF and LCM

HCF (Highest Common Factor): The largest number that divides two or more numbers. Example: HCF of 12 and 16 is 4.
LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers. Example: LCM of 12 and 16 is 48.

Definition and Comparison of Fractions

Fractions represent parts of a whole. Example: ½, ⅓. You can compare fractions by converting them to have the same denominator.

Insertion of Fractions between Two Given Fractions

Finding fractions between two fractions, like finding ¾ between ½ and 1.

Operation Order Sequence (VBODMAS)

The order of operations in solving math problems: V (Vinculum), B (Brackets), O (Order or Exponents), D (Division), M (Multiplication), A (Addition), S (Subtraction). Example: Solve 3 + 6 × (5 + 4) ÷ 3 - 7 using VBODMAS.

Algebraic Formula

Common formulas, like the quadratic formula:
𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0, solution is:
𝑥 = (−𝑏 ± √(𝑏² − 4𝑎𝑐)) / (2𝑎).

Percentage and Inter-conversion

Percent means "per hundred." Conversion between fractions, decimals, and percentages: Example: 50% = 0.5 = ½.

Average

Average (mean) = (Sum of all numbers) ÷ (Number of numbers). Example: Average of 2, 4, 6 is (2 + 4 + 6) / 3 = 4.

Ratio and Proportion

Ratio compares two quantities, e.g., 3:5. Proportion states that two ratios are equal, e.g., 3/5 = 6/10.

Binomial Theorem

Formula for expanding expressions like (𝑎 + 𝑏)^𝑛: (𝑎 + 𝑏)^𝑛 = Σ_𝑘=0^𝑛 (𝑛 choose 𝑘) 𝑎^𝑛−𝑘 𝑏^𝑘. Example: (𝑥 + 𝑦)² = 𝑥² + 2𝑥𝑦 + 𝑦².

Unit II: Sequence, Series, and Progression

Topics Covered:

• Definition of Sequence, Series, and Progression:
• Sequence: An ordered list of numbers. Example: 1, 2, 3, 4.
• Series: The sum of the terms of a sequence. Example: 1 + 2 + 3 + 4.
• Progression: A type of sequence with a specific pattern (AP, GP, etc.).
• Arithmetic Progression (AP):
• Sequence with a constant difference between terms. Example: 3, 7, 11, 15 (difference of 4).
• nth term of AP: a_n = a₁ + (n - 1)d.
• Sum of n terms of AP: S_n = n/2 (2a₁ + (n - 1)d).
• Arithmetic Mean (AM):
• The mean between two numbers in an AP.
• Formula: AM = (a + b) / 2.
• Geometric Progression (GP):
• Sequence where each term is multiplied by a constant ratio. Example: 2, 4, 8, 16 (ratio 2).
• nth term of GP: a_n = a₁ r^{n - 1}.
• Sum of n terms of GP: S_n = a₁ (1 - rⁿ) / (1 - r) (if r ≠ 1).
• Geometric Mean (GM):
• The mean between two numbers in a GP.
• Formula: GM = √(ab).
• Harmonic Progression (HP):
• Sequence formed by taking the reciprocals of an AP. Example: 1, ½, ⅓, ¼.
• Harmonic Mean (HM):
• The mean between two numbers in an HP.
• Formula: HM = 2ab / (a + b).
• Relations between AM, GM, and HM:
• AM ≥ GM ≥ HM

Unit III: Matrices and Determinants

Topics Covered:

• Definition and Types of Matrices:
• Matrix: A rectangular array of numbers arranged in rows and columns.
• Example of a 2×2 matrix:

                  [
                  1  2
                  3  4
                  ]
                  

• Operations on Matrices:
• Addition: Matrices of the same size can be added by adding corresponding elements.
• Multiplication: Multiply row elements by column elements.
• Symmetric and Skew-Symmetric Matrices:
• Symmetric Matrix: A = A^T (transpose of the matrix equals the matrix).
• Skew-Symmetric: A = -A^T.
• Row and Column Operations:
• Changing rows or columns by adding, subtracting, or multiplying elements.
• Inverse of a Matrix:
• A matrix A has an inverse if A × A^-1 = I (identity matrix).
• Found by Elementary Row Operations.
• Determinants:
• A scalar value that can be computed from a square matrix.
• Example for a 2×2 matrix: det(A) = | a b | = ad - bc

                  | c  d |
                  

• Minors and Co-factors:
• Minor: Determinant of a smaller matrix formed by removing one row and one column.
• Co-factor: Sign-adjusted minor.

Unit IV: Differentiation

Topics Covered:

• Basic Formulae of Differentiation:
• Derivative of xⁿ:

                  d
                  dx
                  xn = nxn-1
                  

• Derivative of sin(x):

                  d
                  dx
                  sin(x) = cos(x)
                  

• First Principle of Differentiation:

•                 f'(x) = lim
                  h→0
                  (f(x+h) - f(x)) / h
                  

• Product, Quotient, and Chain Rule:
• Product Rule:

                  d
                  dx
                  (uv) = u'v + uv'
                  

• Quotient Rule:

                  d
                  dx
                  (u/v) = (u'v - uv') / v²
                  

• Chain Rule:

                  d
                  dx
                  f(g(x)) = f'(g(x)) * g'(x)
                  

• Derivatives of Exponential, Logarithmic, and Inverse Trigonometric Functions:

•                 d
                  dx
                  ex = ex
                  

•                 d
                  dx
                  ln(x) = 1/x
                  

• Differentiation using Trigonometrical Transformations
• Implicit Differentiation
• Differentiation using Logarithms

Unit V: Integration

Topics Covered:

• Basic Formulae and Standard Results of Integration:

•                 ∫xn dx = (xn+1)/(n+1) + C (if n ≠ -1)
                  

• Methods of Integration:
• Substitution Method: Change of variable to simplify integration.
• Integration by Parts:

                  ∫uv' dx = uv - ∫u'vdx
                  

• Definite Integral and Properties:

•                 Integral over a fixed range [a, b]: ∫ab f(x) dx
                  

• Integration using Trigonometric Identities
• Integration by Partial Fractions