NDA formula sheet for Algebra! Notes

Chapter 1: Exponents & Logarithms

Exponent rules and logarithmic properties form the baseline operations required in calculus, algebra simplification, and exponential decay models in NDA Mathematics.

Fundamental Laws of Exponents

For any positive bases \(a,b\) and real exponents \(m,n\):

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power Rule: \((a^m)^n = a^{mn}\)
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)

Core Logarithm Identities

Logarithms convert multiplicative operations into additive ones, defined as \(\log_b(x) = y \iff b^y = x\) where base \(b > 0, b eq 1\) and \(x > 0\):

• Product Property: \(\log_b(xy) = \log_b x + \log_b y\)
• Quotient Property: \(\log_b(\frac{x}{y}) = \log_b x - \log_b y\)
• Power Property: \(\log_b(x^k) = k \log_b x\)
• Change of Base Formula: \(\log_b x = \frac{\log_c x}{\log_c b}\)

Chapter 2: Quadratic Equations & Discriminant Analysis

Quadratic equations appear frequently in the NDA exam, both as direct polynomial problems and inside calculus applications.

Standard Form and Roots

Given the general quadratic equation \(ax^2 + bx + c = 0\) with \(a eq 0\), the roots are given by the Quadratic Formula:

Nature of Roots via Discriminant (\(\Delta\))

The discriminant determines the fundamental nature of the roots:

• If \(\Delta > 0\): The equation has two distinct real roots.
• If \(\Delta = 0\): The equation has exactly one real repeated root (double root).
• If \(\Delta < 0\): The equation has two non-real complex conjugate roots.

Chapter 3: Polynomials & Binomial Theorem

Understanding polynomial factorization and the expansion of algebraic powers allows rapid evaluation of binomial coefficients.

Binomial Theorem

For any non-negative integer \(n\), the expansion of a binomial power is given by:

Where the combination coefficient is defined as:

Special Factorization Formulas

• Difference of Squares: \(a^2 - b^2 = (a-b)(a+b)\)
• Sum and Difference of Cubes: \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)
• Cube of a Binomial: \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Chapter 4: Sequences & Series (AP & GP)

Progressions form a regular section of the NDA paper, evaluating common differences, ratios, and finite summation formulas.

Arithmetic Progression (AP)

An AP is a sequence where consecutive terms differ by a constant difference \(d\).

• General Term (\(n\)-th term): \(a_n = a_1 + (n-1)d\)
• Sum of First \(n\) Terms: \(S_n = \frac{n}{2} [2a_1 + (n-1)d] = \frac{n}{2}(a_1 + a_n)\)

Geometric Progression (GP)

A GP is a sequence where consecutive terms have a constant ratio \(r\).

• General Term (\(n\)-th term): \(a_n = a_1 r^{n-1}\)
• Sum of First \(n\) Terms (\(r eq 1\)): \(S_n = \frac{a_1(1-r^n)}{1-r}\)

Chapter 5: Complex Numbers & De Moivre's Theorem

Complex numbers extend the real line into a two-dimensional complex plane, introducing polar form and exponential notation.

Representations & Conjugates

A complex number \(z\) can be written in rectangular form \(z = a + bi\), polar form \(r(\cos\theta + i\sin\theta)\), or Euler's exponential form \(re^{i\theta}\):

• Modulus: \(\vert{}z\vert{} = \sqrt{a^2 + b^2}\)
• Complex Conjugate: \(\bar{z} = a - bi\)
• Multiplication in Polar Form: \(z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}\)

De Moivre's Theorem

For any integer \(n\) and real angle \(\theta\):

Chapter 6: Matrices & Determinants

Matrix algebra is critical for solving systems of linear equations and spatial transformations.

2x2 Matrix Inversion & Determinants

For a square matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\):

• Determinant: \(\det(A) = ad - bc\)
• Inverse Matrix (when \(\det(A) eq 0\)): \(A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

Chapter 7: Inequalities, Absolute Value & AM-GM Inequality

Inequalities and bounds provide essential tools for optimization and domain analysis in algebraic functions.

Absolute Value Inequalities

For a positive constant \(c > 0\):

• \(\vert{}x\vert{} < c \implies -c < x < c\)
• \(\vert{}x\vert{} > c \implies x < -c \text{ or } x > c\)

Arithmetic Mean - Geometric Mean (AM-GM) Inequality

For non-negative real numbers \(a, b \geq 0\), the arithmetic mean is always greater than or equal to the geometric mean:

Equality holds if and only if \(a = b\).

NDA Exam Pattern & Weightage Analysis

Understanding chapter weightage and previous year question (PYQ) distributions allows NDA candidates to optimize their preparation strategy.

Chapter Weightage & PYQ Analysis

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