🚆 RRB Railway – Mathematics – Algebra
(Study Notes for IIT‑JEE / Competitive Exam Level)
Why Algebra?
The RRB Railway exam expects you to solve a wide variety of algebraic problems in a short time. A solid grasp of the fundamentals, quick‑fire formulas, and “trick” strategies will save you valuable minutes on the test day.
📚 Table of Contents
| Section | Topics Covered |
|---|---|
| 1 | Linear equations & systems |
| 2 | Quadratic equations |
| 3 | Arithmetic Progression (AP) |
| 4 | Geometric Progression (GP) |
| 5 | Binomial Theorem & Identities |
| 6 | Permutations & Combinations |
| 7 | Rational & Trigonometric Equations |
| 8 | Exponents & Logarithms |
| 9 | Number Theory (GCD, LCM, divisibility) |
| 10 | Tricky Points & Exam‑specific Tips |
| 11 | Hard Practice Question & Detailed Solution |
1. Linear Equations & Systems
| Equation | Standard form | Key Insight |
|---|---|---|
| (ax + by = c) | (x) and (y) coefficients are integers | Use elimination or substitution |
| (mx + n = p) | Single‑variable linear | Solve for (x = \frac{p-n}{m}) |
| System of two equations | (\begin{cases}a_1x + b_1y = c_1\ a_2x + b_2y = c_2\end{cases}) | Multiply one equation by a factor so that the coefficient of a variable cancels when added/subtracted. |
Derivation – Elimination Method
- Multiply first equation by (b_2) and second by (b_1).
[ \begin{aligned} &b_2(a_1x + b_1y) = b_2c_1 \ &b_1(a_2x + b_2y) = b_1c_2 \end{aligned} ] - Subtract to eliminate (y):
[ (b_2a_1 - b_1a_2)x = b_2c_1 - b_1c_2 ] - Solve for (x).
- Back‑substitute to find (y).
Quick Checks
| Check | What to verify |
|---|---|
| Consistency | If ( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}) → No solution. |
| Infinite solutions | If all three ratios equal. |
| Unique solution | Otherwise. |
2. Quadratic Equations
| General Form | (ax^2 + bx + c = 0) |
|---|---|
| Discriminant | (\Delta = b^2 - 4ac) |
| Roots | (x = \frac{-b \pm \sqrt{\Delta}}{2a}) |
Derivation – Completing the Square
- Divide by (a) (if (a \neq 1)).
- Move (c) to RHS.
- Add ((\tfrac{b}{2a})^2) to both sides.
- Factor left side as a perfect square.
- Take square root, solve for (x).
Special Cases
| Scenario | Formula |
|---|---|
| Double root | (\Delta = 0 \Rightarrow x = -\frac{b}{2a}) |
| Integer roots | Rational root theorem: possible roots divide (c/a). |
| Sum & Product | (x_1 + x_2 = -\frac{b}{a}), (x_1x_2 = \frac{c}{a}). |
3. Arithmetic Progression (AP)
| Definition | Consecutive terms differ by constant (d). |
|---|---|
| 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)) |
| Sum of n terms from (k)th to (l)th | (S_{k\to l} = \frac{l-k+1}{2}\bigl[2a_k + (l-k)d\bigr]) |
Derivation – Pairing Technique
- Pair first and last terms: (a_1 + a_n = 2a_1 + (n-1)d).
- Number of pairs = (n/2).
- Multiply and divide by 2.
4. Geometric Progression (GP)
| Definition | Consecutive terms multiply by constant (r). |
|---|---|
| n‑th term | (a_n = a_1 r^{,n-1}) |
| Sum of first n terms | (S_n = a_1\frac{r^n-1 |