ALEGEBRA TRICKS
Algebra Tricks: Zero Level se SSC CGL Level tak
Part 1: Foundation (Zero Level)
Basic Identities (Yaad rakhna zaroori hai)
Ye 6 identities algebra ki jaan hain — inhe ratto ki tarah yaad karo:
- (a+b)² = a² + 2ab + b²
- (a−b)² = a² − 2ab + b²
- a² − b² = (a+b)(a−b)
- (a+b)² = (a−b)² + 4ab → (bahut kaam ki trick, aage dekhenge)
- (a−b)² = (a+b)² − 4ab
- a³ + b³ = (a+b)(a² − ab + b²) aur a³ − b³ = (a−b)(a² + ab + b²)
Example: Agar a + b = 7 aur ab = 12, to a² + b² nikaalo.
- a² + b² = (a+b)² − 2ab = 49 − 24 = 25
Ye pattern SSC me har exam me 2-3 questions me directly aata hai.
Trick #1: (a+b)² aur (a−b)² ka connection
Agar sirf a+b aur ab diya ho, to a−b nikaalne ke liye:
(a−b)² = (a+b)² − 4ab
Example: a+b = 9, ab = 20. a−b = ?
- (a−b)² = 81 − 80 = 1
- a−b = 1
Same tareeke se agar a−b aur ab diya ho, to a+b nikaal sakte ho.
Part 2: Intermediate Level (a + 1/a type questions — SSC ka favourite)
Ye topic SSC CGL, CHSL, MTS — har exam me guaranteed aata hai. Iska ek hi core formula system hai, bas usko extend karna aata hona chahiye.
Master Formula Chain
Agar x + 1/x = k diya ho:
| Chahiye | Formula |
| x² + 1/x² | k² − 2 |
| x − 1/x | √(k² − 4) |
| x² − 1/x² | (x+1/x) × (x−1/x) |
| x³ + 1/x³ | k³ − 3k |
| x³ − 1/x³ | (x−1/x)³ + 3(x−1/x) |
| x⁴ + 1/x⁴ | (x²+1/x²)² − 2 |
Example: x + 1/x = 5, to x³ + 1/x³ = ?
- = 5³ − 3(5) = 125 − 15 = 110
Example (reverse type): Agar x − 1/x = 3, to x² + 1/x² = ?
- Trick: x² + 1/x² = (x−1/x)² + 2 = 9 + 2 = 11
Yaad rakhne ka trick:
- “+2” wala formula tab jab tumhare paas already “+” wala expression tha aur uska square kar rahe ho, tab “−2ab” hoga (kyunki square karne se extra 2ab ban jaata hai jo ab nahi chahiye)
- Simple rule: “+” ko square karo to “−2” karna, “−” ko square karo to “+2” karna
Trick #2: Speed Method — Square karke seedha answer
Jab bhi x + a/x = k type ka sawaal ho (a koi bhi number ho, sirf 1 nahi), to seedha square kar do:
Example: x + 3/x = 4, to x² + 9/x² = ?
- Square karo: x² + 9/x² + 2(x)(3/x) = 16
- x² + 9/x² + 6 = 16
- x² + 9/x² = 10
Ye method universal hai — 1 ki jagah koi bhi number ho, same tareeke se hoga.
Part 3: Advanced Level (SSC CGL Tier 1 & 2 favourite types)
Trick #3: a + b + c = 0 wala special case
Agar teen numbers ka sum zero ho (a+b+c=0), to ek golden result yaad rakho:
a³ + b³ + c³ = 3abc
Ye formula bina derivation ke seedha use karo — SSC me ye pattern bahut baar puchha jaata hai, khaaskar disguised form me.
Example: a=5, b=−3, c=−2 (sum=0 check: 5−3−2=0 ✓) a³+b³+c³ = 3×5×(−3)×(−2) = 3×30 = 90
Verify: 125 − 27 − 8 = 90 ✓ Match ho gaya, formula sahi hai.
Disguised SSC version: “If x+y+z=0, find value of (x³+y³+z³)/(xyz)”
- Directly = 3 (kyunki x³+y³+z³ = 3xyz)
Trick #4: Componendo-Dividendo (Ratio-Algebra mix, CGL Tier 2 me common)
Agar a/b = c/d, to:
(a+b)/(a−b) = (c+d)/(c−d)
Ye tab use hota hai jab equation ka form fraction jaisa ho aur direct substitution lamba ho jaaye. Values ki jagah is shortcut se directly relation nikal lo — calculation time 70% tak kam ho jaata hai.
Trick #5: Symmetric expressions ka fast calculation
Jab question me a², b², c², ab, bc, ca sab mix ho (jaise a²+b²+c²−ab−bc−ca), to ye pehchano:
a² + b² + c² − ab − bc − ca = ½[(a−b)² + (b−c)² + (c−a)²]
Isse hamesha ek non-negative value milegi — is property se options bhi jaldi eliminate ho jaate hain (negative option seedha galat).
Trick #6: Value-substitution trick (Sabse tez tareeka jab variable diya ho)
Agar question me “if a = 2, b = 3, find (a+b)³ − a³ − b³ − 3ab(a+b)” jaisa kuch ho — Seedha values daal do, formula expand mat karo. Competitive exam me time sabse bada resource hai. Agar answer ek clean identity hai (jaise ye poora expression hamesha 0 hota hai), to values dalke turant 0 pata chal jaayega bina expand kiye.
General rule: Jab bhi complex algebraic expression + specific numeric values diye ho, pehle check karo — kya values daal ke seedha calculate karna expand karne se fast hoga? 90% cases me haan.
Part 4: SSC CGL Exam Strategy — Algebra Section
- Identity-based questions (60% weightage): Upar diye formulas ratto — inme se koi na koi seedha use hoga.
- Time trick: Agar options diye hain aur equation solve karne me time lag raha hai, to options ko equation me substitute karke check karo (backward approach) — MCQ me ye sabse fast hai.
- Common mistake to avoid: (a+b)² ko a²+b² samajh lena — SSC ismein sabse zyada trap karta hai. Hamesha 2ab wala term yaad rakho.
- Practice pattern: Pichle 5 saal ke SSC CGL papers me algebra ke questions dekho — 80% questions upar diye 6 tricks me se kisi ek pattern par based honge.
Quick Revision Table
| Given | Find | Shortcut |
| a+b, ab | a²+b² | (a+b)²−2ab |
| a+b, ab | a−b | √[(a+b)²−4ab] |
| x+1/x=k | x²+1/x² | k²−2 |
| x+1/x=k | x³+1/x³ | k³−3k |
| a+b+c=0 | a³+b³+c³ | 3abc |
| a/b=c/d | (a+b)/(a−b) | (c+d)/(c−d) |
Practice tip: Har formula ko kam se kam 5 alag numbers ke saath khud solve karo bina dekhe — tabhi exam hall me reflex ban payeg
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