NDA Algebra Formula Sheet, Theorems & Quick Revision Notes

Welcome to this comprehensive LibreTexts-style study module for NDA Algebra. Designed specifically for UPSC NDA (National Defence Academy) aspirants, this guide consolidates key theoretical concepts, fundamental algebraic formulas, and problem-solving shortcuts based on recent exam trends and PYQ weightage.

1. Exponents and Logarithms

Exponents and logarithms form the foundational calculation tools in NDA mathematics. Understanding power rules and base-change properties enables rapid simplification during the exam.

1.1 Fundamental Rules of Exponents

For any positive base \(a\) and real exponents \(m\) and \(n\):

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

1.2 Fundamental Properties of Logarithms

Logarithmic identities are heavily tested in combination with algebraic equations and calculus limit evaluations:

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

2. Quadratic Equations & Discriminant Analysis

Quadratic equations are a regular feature in the NDA Mathematics paper, often yielding 3 to 5 questions directly or as sub-steps in algebra problems.

2.1 Standard Form & Quadratic Formula

For the second-degree polynomial equation \(ax^2 + bx + c = 0\) where \(a eq 0\), the roots are given by:

2.2 Discriminant (\(\Delta\)) and Nature of Roots

The expression \(\Delta = b^2 - 4ac\) determines the qualitative nature of the roots:

• \(\Delta > 0\): Two distinct real roots.
• \(\Delta = 0\): One real repeated root (equal roots).
• \(\Delta < 0\): Two complex conjugate roots.

2.3 Vieta's Relations

If \(r_1\) and \(r_2\) are the roots of \(ax^2 + bx + c = 0\):

• Sum of Roots: \(r_1 + r_2 = -\frac{b}{a}\)
• Product of Roots: \(r_1 r_2 = \frac{c}{a}\)

3. Polynomials and Binomial Theorem

3.1 Important Special Products and Algebraic Identities

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a-b)(a+b)\)
• \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

3.2 Binomial Theorem for Positive Integer Index

The expansion of \((x+y)^n\) for any non-negative integer \(n\) is given by:

where the binomial coefficient is calculated as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

4. Sequences and Series (AP & GP)

Questions on Arithmetic and Geometric progressions test both formula retention and logical manipulation.

4.1 Arithmetic Progression (AP)

For an AP with first term \(a_1\) and common difference \(d\):

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

4.2 Geometric Progression (GP)

For a GP with first term \(a_1\) and common ratio \(r\):

• \(n\)-th Term: \(a_n = a_1 r^{n-1}\)
• Sum of \(n\) Terms (\(r eq 1\)): \(S_n = \frac{a_1(1-r^n)}{1-r}\)
• Sum of Infinite GP (\(|r| < 1\)): \(S_\infty = \frac{a_1}{1-r}\)

5. Complex Numbers & De Moivre's Theorem

Let \(z = a + bi\) represent a complex number where \(i = \sqrt{-1}\) (i.e., \(i^2 = -1\)).

5.1 Basic Properties

• Modulus (Magnitude): \(|z| = \sqrt{a^2 + b^2}\)
• Complex Conjugate: \(\bar{z} = a - bi\)

5.2 Euler's Formula and De Moivre's Theorem

• Euler's Form: \(e^{i\theta} = \cos\theta + i\sin\theta\)
• De Moivre's Theorem: For any integer \(n\), \[\bigl(r(\cos\theta + i\sin\theta)\bigr)^n = r^n\bigl(\cos(n\theta) + i\sin(n\theta)\bigr)\]

6. Matrices and Determinants

Matrix algebra is a high-scoring section in the NDA paper, primarily focusing on \(2 \times 2\) and \(3 \times 3\) systems.

6.1 Determinant of a \(2 \times 2\) Matrix

For matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the determinant is:

6.2 Inverse of a \(2 \times 2\) Matrix

Provided \(\det(A) eq 0\), the inverse matrix \(A^{-1}\) is calculated as:

7. NDA Exam Pattern & Chapter Weightage Analysis

In the UPSC NDA Mathematics examination (120 questions, 300 marks), Algebra carries substantial weightage. Here is a typical distribution analysis based on recent trends:

• Quadratic Equations & Complex Numbers: ~10–12 questions
• Sequences and Series: ~6–8 questions
• Matrices and Determinants: ~8–10 questions
• Binomial Theorem & Logarithms: ~4–6 questions

Focusing on conceptual clarity, speed of calculation, and eliminating options using formulas like Vieta's relations or properties of logarithms significantly boosts total accuracy in the paper.

8. Frequently Asked Questions (People Also Ask)

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