In this guide
RRB NTPC algebra questions are short. They test two things: whether you know a few identities well enough to rearrange them, and whether you can solve a simple equation without a sign error. There is no heavy theory here, and that is exactly why it is worth getting right. These are marks you should not drop.
This guide covers the identities, the question families built on them, and two exam habits that save time: substituting easy numbers, and testing options.
The identities to know
| Identity | Expanded form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| a² − b² | (a + b)(a − b) |
| (a + b)³ | a³ + b³ + 3ab(a + b) |
| (a − b)³ | a³ − b³ − 3ab(a − b) |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
| (a + b + c)² | a² + b² + c² + 2(ab + bc + ca) |
Why these hold: each one is just multiplication written out. (a + b)² = (a + b)(a + b) = a² + ab + ba + b². The middle term 2ab is exactly what candidates forget, which is why (a + b)² is never simply a² + b².
Rearranged forms: where the questions come from
Most questions give you a sum and a product and ask for something else. These rearrangements do the work:
- a² + b² = (a + b)² − 2ab
- a² + b² = (a − b)² + 2ab
- a³ + b³ = (a + b)³ − 3ab(a + b)
- a² + b² + c² = (a + b + c)² − 2(ab + bc + ca)
The x + 1/x family
Put b = 1/x. Since x × 1/x = 1, the product term becomes 1 and the identities simplify:
| If x + 1/x = t, then | Value |
|---|---|
| x² + 1/x² | t² − 2 |
| x³ + 1/x³ | t³ − 3t |
| (x − 1/x)² | t² − 4 |
And the other way round: if x − 1/x = k, then x² + 1/x² = k² + 2.
Why t³ − 3t: (x + 1/x)³ = x³ + 1/x³ + 3 × x × 1/x × (x + 1/x) = x³ + 1/x³ + 3t. Move 3t across.
When a + b + c = 0
If a + b + c = 0, then a³ + b³ + c³ = 3abc.
Why: the full identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). If the first bracket is zero, the whole right side is zero. Check with 1, 2 and −3: 1 + 8 − 27 = −18, and 3 × 1 × 2 × (−3) = −18.
Linear equations
- One variable: move terms across, changing sign each time, then divide.
- Two variables: add or subtract the equations to remove one variable (elimination), or express one variable from the simpler equation and substitute.
- Word problems: name one unknown, write everything else in terms of it, and form one equation.
Two time-savers
- Substitute easy numbers. If a question says a + b = 7 and ab = 12, then a and b are 3 and 4. Calculate directly. For identity questions with no conditions, try a = 1, b = 0 or x = 1 and compare with the options.
- Test the options in equations. Put each option in and see which balances. With four options this is often faster than solving.
Worked questions at NTPC level
Q1. If x + 1/x = 5, find x² + 1/x².
t² − 2 = 25 − 2 = 23.
Q2. If x + 1/x = 3, find x³ + 1/x³.
t³ − 3t = 27 − 9 = 18.
Q3. If a + b = 5 and ab = 6, find a³ + b³.
(a + b)³ − 3ab(a + b) = 125 − 90 = 35.
Check: a and b are 2 and 3; 8 + 27 = 35.
Q4. If x + y + z = 0, find (x³ + y³ + z³) ÷ xyz.
x³ + y³ + z³ = 3xyz, so the value is 3.
Q5. Solve 3x + 2y = 16 and 2x − y = 6.
From the second, y = 2x − 6. Substitute: 3x + 4x − 12 = 16, so 7x = 28.
x = 4, y = 2. Check: 12 + 4 = 16 and 8 − 2 = 6.
Q6. A booking clerk sells 40 tickets, some at ₹25 and the rest at ₹40, and collects ₹1,300. How many ₹40 tickets were sold?
Let x tickets be at ₹40. Then 25(40 − x) + 40x = 1,300.
1,000 + 15x = 1,300, so x = 20.
Check: 20 × 25 + 20 × 40 = 500 + 800 = 1,300.
Common mistakes
- Dropping the middle term. (a + b)² has 2ab in it.
- Forgetting to subtract 2 after squaring x + 1/x. For x − 1/x you add 2 instead.
- Sign errors across the equals sign. Move one term at a time and rewrite the line.
- Stopping at one variable. If the question asks for x + y, give the sum, not x alone.
- Using a + b + c = 0 when it is not given. The 3abc result needs that condition.
Practice set
- If x + 1/x = 3, find x² + 1/x².
- If a + b = 9 and ab = 20, find a² + b².
- Solve 5x + 3 = 28.
- Solve 2x + y = 11 and x − y = 1.
- If a − b = 3 and ab = 10, find a² + b².
- If x + 1/x = 4, find x³ + 1/x³.
- If x − 1/x = 2, find x² + 1/x².
- If a + b + c = 6 and ab + bc + ca = 11, find a² + b² + c².
Answers:
- 7. 9 − 2.
- 41. 81 − 40.
- x = 5. 5x = 25.
- x = 4, y = 3. Adding gives 3x = 12.
- 29. (a − b)² + 2ab = 9 + 20. Check: 5 and 2.
- 52. 64 − 12.
- 6. k² + 2 = 4 + 2.
- 14. 36 − 22. Check: 1, 2 and 3.
What to do next
- Write the identity table from memory twice this week, including the x + 1/x rows.
- Solve 20 identity questions, and check each answer by substituting real numbers.
- Do 10 two-variable equations by elimination and 10 by testing options, then keep whichever is faster for you.
- Revise simplification, since many algebra slips are really arithmetic slips.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Railway Recruitment Boards website .
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