Algebraic Expressions, Identities aur Factorisation pe poori practice, ek hi post mein: terms, factors aur coefficient, like aur unlike terms, expressions ko add-subtract karna, monomial aur polynomial ka multiply (distributive property), chaar important identities, factorisation ke methods (common factor, regrouping, identity se, x² + px + q wala) aur polynomial ko monomial se divide karna. Pehle chhota recap (saare naye words wahin samjhaaye hain), phir 10 MCQs, fill in the blanks, true/false, short aur long answer, HOTS, aur end mein answer key. Har sawaal ka solution Hinglish mein step by step hai. Yeh post apne current chapter Book 1 Ch 6: We Distribute, Yet Things Multiply ke saath padhiye: chapter se samjho, yahan se practice karo. 🧮
🧠 Concept Recap: Pehle Yeh Yaad Kar Lo
📖 Pehle chapter padho: Book 1 Ch 6: We Distribute, Yet Things Multiply (hub). Algebra wale puzzles aur number tricks ke liye: Book 2 Ch 6: Algebra Play (hub).
📏 Pehle yeh words aur signs samjho (har word yahin samjhaaya hai, aage yahi use honge):
- Variable = letter (jaise a, b, x, y, m) jo kisi bhi number ki jagah aa sakta hai. Constant = fixed number, jaise 7 ya −3.
- ⚠️ Algebra ka likhne ka rule: number aur letter, ya do letters, saath-saath likhe hon to unke beech multiply chhupa hai. 3a matlab 3 × a; 2xy matlab 2 × x × y; ab matlab a × b. Numbers ke beech multiply hamesha × se likha hai (3 × 4, kabhi “34” nahi).
- Power (upar likha chhota number) = batata hai ki letter ya number kitni baar khud se multiply hua. a² matlab a × a (power 2, “a square”), a³ matlab a × a × a (power 3, “a cube”). Akela a matlab a¹ (power 1, sirf ek a); 1 likhte nahi, par hota hai. Jaise 5² = 5 × 5 = 25 aur 5¹ = 5. Dhyaan: (−3)² = (−3) × (−3) = 9, kyunki minus × minus = plus.
- Sum, difference, product = add karne ka result sum (2 + 3 = 5, to 2 aur 3 ka sum 5); subtract karne ka result difference (7 − 3 = 4, to difference 4); multiply karne ka result product (2 × 3 = 6, to product 6).
- ± (plus-minus sign) = “plus ya minus”: ek formula mein do cases ek saath likhe hain. a² ± 2ab + b² matlab do alag expressions: a² + 2ab + b² (plus wala case) aur a² − 2ab + b² (minus wala case).
- Minus wale (negative) numbers = 0 se chhote numbers, jaise −4. Multiply ke rules: plus × plus = plus, minus × minus = plus, plus × minus = minus. Bracket ke aage minus ho to bracket kholte waqt andar ka har sign ulta ho jaata hai: −(a − b) = −a + b.
- Expression = numbers, letters aur +, − se bana statement jisme “=” nahi hota, jaise 3x + 5 ya 4a² − 2ab + 7.
- Term = expression ke woh parts jo + ya − se alag hote hain. 4a² − 2ab + 7 mein 3 terms: 4a², −2ab, 7 (term ke saath uska sign bhi jaata hai).
- Factor = jo cheezein multiply hokar term banaati hain: 4a² = 4 × a × a, to factors 4, a, a. Coefficient = term ka number wala factor (sign ke saath): −2ab ka coefficient −2.
- Like terms = jinke letter wale factors bilkul same hon (power bhi same), jaise 5xy aur −3yx (xy = yx). Unlike terms: 3x² aur 3x (power alag). Sirf like terms hi add/subtract hote hain: 5xy − 3xy = 2xy.
- Monomial = 1 term (7ab), binomial = 2 terms (x + 5), trinomial = 3 terms (x² + 3x + 2), polynomial = ek ya zyada terms wala expression (jisme letters ki power 0, 1, 2, 3 … ho).
- Distributive property = bracket ke bahar wala har term bracket ke andar ke har term se multiply hota hai: a × (b + c) = ab + ac. Example: 3 × (10 + 2) = 30 + 6 = 36 = 3 × 12.
- Value nikaalna (substitute) = letter ki jagah number rakh kar answer nikaalna: x = 2 pe 3x + 1 matlab 3 × 2 + 1 = 7.
- Identity = aisi equality jo letters ki har value pe sach ho, jaise a + a = 2a. (Equation jaisa x + 1 = 5 sirf x = 4 pe sach hai, isliye woh identity nahi.)
- Factorisation = expression ko factors ke product (multiply) ke form mein likhna, yaani distributive property ka ulta: 2a + 4 = 2 × (a + 2) = 2(a + 2).
- Common factor = jo factor har term mein ho: 6x² + 9x mein 3 aur x dono common, to common factor 3x.
- Divide (÷) = multiply ka ulta. Monomial ka divide: numbers ko divide karo aur same letters ki power subtract karo: a⁵ ÷ a² = a³ (5 − 2 = 3), kyunki (a × a × a × a × a) ÷ (a × a) mein upar aur neeche ke 2 a cancel ho jaate hain, 3 a bachte hain.
- Odd number, multiple, counting number = counting numbers: 1, 2, 3, 4, 5 … (jinse hum cheezein count karte hain). Odd number: jo 2 se poora divide nahi hota, jaise 1, 3, 5, 7, 9 (2 se divide hone wale 2, 4, 6 even numbers hain). Multiple: kisi number ko counting number se multiply karke jo milta hai; 8 ke multiples: 8, 16, 24, 32 … (8 × 1, 8 × 2, 8 × 3 …).
- → (arrow) = “ban jaata hai” ya “iska matlab”. ≠ = “not equal”.
Worked example (pehle ek aasaan sawaal): 2x × (3x + 4) = 2x × 3x + 2x × 4 = 6x² + 8x. Check x = 1 pe: left side 2 × (3 + 4) = 2 × 7 = 14; right side 6 + 8 = 14 ✅. Har answer ko ek chhoti value rakh kar check karne ki habit daalo!
1. Expressions ko add, subtract aur multiply karna
| Operation | Rule | Example |
|---|---|---|
| Add / subtract | Sirf like terms ke coefficients add/subtract karo; letters waise hi rehte hain | 7ab − 2ab + ab = 6ab |
| Expression subtract karna | Jo subtract karna hai uske har term ka sign ulta karke add karo | (5x + 2) − (3x − 4) = 5x + 2 − 3x + 4 = 2x + 6 |
| Monomial × monomial | Coefficients multiply; same letters ki powers add | (2a²) × (3a³b) = 6a⁵b (a² × a³ = a⁵, kyunki 2 + 3 = 5) |
| Monomial × polynomial | Distributive property: bahar wala har term se multiply | 3x(x − 4) = 3x² − 12x |
| Binomial × binomial | Pehle bracket ka har term doosre bracket ke har term se (4 products), phir like terms milaao | (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10 |
2. Chaar important identities
| Identity | Number se check (a = 5, b = 2, x = 3) | Mental maths mein use |
|---|---|---|
| (a + b)² = a² + 2ab + b² | 7 × 7 = 49 aur 25 + 20 + 4 = 49 ✅ | 103² = 10000 + 600 + 9 = 10609 |
| (a − b)² = a² − 2ab + b² | 3 × 3 = 9 aur 25 − 20 + 4 = 9 ✅ | 98² = 10000 − 400 + 4 = 9604 |
| (a + b)(a − b) = a² − b² | 7 × 3 = 21 aur 25 − 4 = 21 ✅ | 52 × 48 = 2500 − 4 = 2496 |
| (x + a)(x + b) = x² + (a + b)x + ab | 8 × 5 = 40 aur 9 + 21 + 10 = 40 ✅ | 103 × 104 = (100 + 3) × (100 + 4): x = 100, a = 3, b = 4, to x² + (a + b)x + ab = 10000 + 7 × 100 + 12 = 10000 + 700 + 12 = 10712 |
3. Factorisation ke methods
| Method | Kab use karein | Example |
|---|---|---|
| Common factor bahar nikaalo | Har term mein koi factor common ho | 6x² + 9x = 3x(2x + 3) |
| Regrouping | 4 terms hon aur do-do terms ke groups mein common factor dikhe | ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y) |
| Identity se | a² ± 2ab + b² (± matlab plus ya minus: a² + 2ab + b² = (a + b)² aur a² − 2ab + b² = (a − b)²) ya a² − b² jaisa pattern dikhe | x² − 49 = (x + 7)(x − 7) |
| x² + px + q | Yahan p = x wale term ka coefficient aur q = constant (akela number). Do numbers dhundho jinka product q aur sum p ho | x² + 5x + 6 mein p = 5, q = 6. Numbers 2 aur 3, kyunki 2 × 3 = 6 (= q) aur 2 + 3 = 5 (= p). To x² + 5x + 6 = (x + 2)(x + 3). Check: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✅ |
Check ka rule: factorisation ke baad factors ko wapas multiply karo; original expression milna chahiye. Ya ek chhoti value (jaise x = 1) rakh kar dono side ki value milaao.
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📝 Section A: 10 MCQs
Har sawaal ke 4 options mein se ek sahi chuno. Pehle khud socho, phir “Solution dekho” kholo.
Q1. Expression 5x²y − 3xy + 7 mein kitne terms hain?
Solution (step by step) dekho
Sahi answer: (b) 3
Step 1: terms + ya − se alag hote hain: 5x²y, −3xy aur 7.
Step 2: gino: 3 terms. (Isliye yeh ek trinomial hai.)
Q2. Term −7ab² ka coefficient kya hai?
Solution (step by step) dekho
Sahi answer: (b) −7
Step 1: coefficient = term ka number wala factor, sign ke saath.
Step 2: −7ab² = −7 × a × b × b, to coefficient −7. (ab² letter wala part hai, coefficient nahi.)
Q3. Neeche mein se kaunsi pair like terms ki hai?
Solution (step by step) dekho
Sahi answer: (b) 4ab aur −9ba
Step 1: like terms mein letters aur unki powers bilkul same hone chahiye.
Step 2: (a) x²y vs xy²: powers alag ❌. (b) ab aur ba: a × b = b × a, same ✅. (c) x vs x² ❌. (d) m vs n ❌.
Q4. (3a²b) × (−4ab³) = ?
Solution (step by step) dekho
Sahi answer: (b) −12a³b⁴
Step 1 (coefficients): 3 × (−4) = −12 (plus × minus = minus).
Step 2 (letters, powers add): a² × a = a² × a¹ = a³ (akela a matlab a¹, to 2 + 1 = 3); b × b³ = b¹ × b³ = b⁴ (1 + 3 = 4).
Answer: −12a³b⁴.
Q5. (x + 5)(x − 5) = ?
Solution (step by step) dekho
Sahi answer: (a) x² − 25
Step 1: yeh (a + b)(a − b) = a² − b² wali identity hai, a = x, b = 5.
Step 2: x² − 5² = x² − 25.
Check (x = 6): (6 + 5) × (6 − 5) = 11 × 1 = 11 aur 36 − 25 = 11 ✅
Q6. Identity use karke 102² = ?
Solution (step by step) dekho
Sahi answer: (a) 10404
Step 1: 102 = 100 + 2, to (a + b)² use karo: a = 100, b = 2.
Step 2: a² = 10000, 2ab = 2 × 100 × 2 = 400, b² = 4.
Step 3: 10000 + 400 + 4 = 10404.
Q7. 6x²y + 9xy² ka poora factorisation kya hai?
Solution (step by step) dekho
Sahi answer: (a) 3xy(2x + 3y)
Step 1 (common factor): numbers 6 aur 9 ka common factor 3; letters mein x aur y dono terms mein hain. To common factor 3xy.
Step 2: 6x²y ÷ 3xy = 2x aur 9xy² ÷ 3xy = 3y.
Answer: 3xy(2x + 3y). Check: 3xy × 2x = 6x²y, 3xy × 3y = 9xy² ✅
Q8. x² + 7x + 12 ke factors kya hain?
Solution (step by step) dekho
Sahi answer: (a) (x + 3)(x + 4)
Step 1: x² + px + q mein p = 7, q = 12 (recap dekho). Do numbers chahiye jinka product (multiply ka result) 12 aur sum (add ka result) 7 ho.
Step 2: 3 × 4 = 12 aur 3 + 4 = 7 ✅ (2 × 6 ka sum 8, 1 × 12 ka sum 13).
Answer: (x + 3)(x + 4). (Option d se −7x aata, +7x nahi.)
Q9. 49m² − 25 = ?
Solution (step by step) dekho
Sahi answer: (b) (7m + 5)(7m − 5)
Step 1: 49m² = (7m)² aur 25 = 5². To yeh a² − b² hai, a = 7m, b = 5.
Step 2: a² − b² = (a + b)(a − b) = (7m + 5)(7m − 5).
Check (m = 1): 49 − 25 = 24 aur 12 × 2 = 24 ✅
Q10. (12a³b²) ÷ (4ab) = ?
Solution (step by step) dekho
Sahi answer: (a) 3a²b
Step 1: numbers: 12 ÷ 4 = 3.
Step 2: letters, powers subtract: a³ ÷ a = a³ ÷ a¹ = a² (akela a matlab a¹, to 3 − 1 = 2); b² ÷ b¹ = b¹ = b (2 − 1 = 1).
Answer: 3a²b. Check: 4ab × 3a²b = 12a³b² ✅
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✍️ Section B: Fill in the Blanks
Blank (_____) mein sahi word ya number bharo.
Q1. (a − b)² = a² − _____ + b²
Answer aur reason
Answer: 2ab
(a − b)² = (a − b)(a − b) = a² − ab − ab + b² = a² − 2ab + b². Check a = 5, b = 2: 3 × 3 = 9 aur 25 − 20 + 4 = 9 ✅
Q2. 3x + 5x − 2x = _____
Answer aur reason
Answer: 6x
Teeno like terms hain, sirf coefficients: 3 + 5 − 2 = 6. To 6x.
Q3. 4y(3y − 2) = 12y² − _____
Answer aur reason
Answer: 8y
Distributive property: 4y × 3y = 12y² aur 4y × 2 = 8y. To 12y² − 8y.
Q4. (x + 3)(x + 4) = x² + _____ x + 12
Answer aur reason
Answer: 7
(x + a)(x + b) = x² + (a + b)x + ab: a + b = 3 + 4 = 7 aur ab = 3 × 4 = 12.
Q5. 99 × 101 = 100² − 1² = _____
Answer aur reason
Answer: 9999
99 × 101 = (100 − 1) × (100 + 1) = 100² − 1² = 10000 − 1 = 9999.
Q6. a² + 2ab + b² ko factor form mein _____ likhte hain.
Answer aur reason
Answer: (a + b)²
Yahi (a + b)² wali identity ulti padhi: a² + 2ab + b² = (a + b)(a + b) = (a + b)².
Q7. Sirf ek term wale expression ko _____ kehte hain.
Answer aur reason
Answer: monomial
1 term = monomial (jaise 7ab), 2 terms = binomial, 3 terms = trinomial.
Q8. 15p²q ÷ 5pq = _____
Answer aur reason
Answer: 3p
15 ÷ 5 = 3; p² ÷ p = p; q ÷ q = 1. To 3p. Check: 5pq × 3p = 15p²q ✅
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✅❌ Section C: True / False
Batao statement sahi (True) hai ya galat (False). False ho to use sahi karke likho.
Q1. (a + b)² = a² + b²
Answer aur reason
❌ False
False. Sahi identity: (a + b)² = a² + 2ab + b². Number se dekho: a = 2, b = 3 pe left side 5 × 5 = 25, par a² + b² = 4 + 9 = 13. 2ab = 12 missing hai.
Q2. 3xy aur −5yx like terms hain.
Answer aur reason
✅ True
True. xy = yx (multiply mein order se farq nahi padta), to letter wala part same hai: 3xy − 5xy = −2xy.
Q3. Jo equality letters ki har value pe sach ho, use identity kehte hain.
Answer aur reason
✅ True
True. Jaise (a + b)(a − b) = a² − b² har a, b pe sach hai. Equation jaisa 2x = 10 sirf x = 5 pe sach hai.
Q4. (x − 4)(x + 4) = x² − 8
Answer aur reason
❌ False
False. (x − 4)(x + 4) = x² − 4² = x² − 16. Check x = 5: 1 × 9 = 9 aur 25 − 16 = 9 ✅ (25 − 8 = 17 nahi aata).
Q5. 2x(3x + 1) = 6x² + 2x
Answer aur reason
✅ True
True. 2x × 3x = 6x² aur 2x × 1 = 2x. Check x = 1: 2 × 4 = 8 aur 6 + 2 = 8 ✅
Q6. x² + 9 ko (x + 3)(x − 3) mein factorise kar sakte hain.
Answer aur reason
❌ False
False. (x + 3)(x − 3) = x² − 9 hota hai, x² + 9 nahi. Check x = 1: x² + 9 = 10, par (1 + 3) × (1 − 3) = 4 × (−2) = −8.
Q7. Binomial ko binomial se multiply karne pe (like terms milaane se pehle) 4 products aate hain.
Answer aur reason
✅ True
True. (a + b)(c + d) = ac + ad + bc + bd: pehle bracket ke 2 terms × doosre bracket ke 2 terms = 4 products. Kabhi like terms milne se kam terms bachte hain, jaise (x + 2)(x + 5) = x² + 7x + 10.
Q8. (−3x)² = −9x²
Answer aur reason
❌ False
False. (−3x)² = (−3x) × (−3x) = 9x², kyunki minus × minus = plus. Check x = 1: (−3) × (−3) = 9.
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🟢 Section D: Short Answer Questions
2 se 3 marks wale sawaal. Har step likhne ki habit daalo.
Q1. Add karo: (3x² − 5x + 2) aur (−x² + 7x − 6).
Step-by-step solution
Step 1 (like terms ek saath): x² wale: 3x² − x² = 2x². x wale: −5x + 7x = 2x. Constants: 2 − 6 = −4.
Answer: 2x² + 2x − 4.
Check (x = 1): pehla 3 − 5 + 2 = 0, doosra −1 + 7 − 6 = 0, sum 0; answer 2 + 2 − 4 = 0 ✅
Q2. 7a + b − 5 mein se 4a − 3b + 2 subtract karo.
Step-by-step solution
Step 1: (7a + b − 5) − (4a − 3b + 2). Doosre bracket ke har term ka sign ulta: 7a + b − 5 − 4a + 3b − 2.
Step 2 (like terms): 7a − 4a = 3a; b + 3b = 4b; −5 − 2 = −7.
Answer: 3a + 4b − 7. Check (a = 1, b = 1): 3 − 3 = 0 aur 3 + 4 − 7 = 0 ✅
Q3. Multiply karo: (2x + 3)(x − 4).
Step-by-step solution
Step 1 (4 products): 2x × x = 2x²; 2x × (−4) = −8x; 3 × x = 3x; 3 × (−4) = −12.
Step 2 (like terms): −8x + 3x = −5x.
Answer: 2x² − 5x − 12. Check (x = 2): (4 + 3) × (2 − 4) = 7 × (−2) = −14 aur 8 − 10 − 12 = −14 ✅
Q4. Identity use karke 97² nikaalo (seedha multiply mat karo).
Step-by-step solution
Step 1: 97 = 100 − 3, to (a − b)² = a² − 2ab + b², a = 100, b = 3.
Step 2: a² = 10000; 2ab = 2 × 100 × 3 = 600; b² = 9.
Step 3: 10000 − 600 + 9 = 9409.
Check: 97 × 97 = 9409 ✅
Q5. Regrouping se factorise karo: 6xy − 4y + 9x − 6.
Step-by-step solution
Step 1 (groups banao): (6xy − 4y) + (9x − 6).
Step 2 (har group ka common factor): 6xy − 4y = 2y(3x − 2); 9x − 6 = 3(3x − 2).
Step 3: ab (3x − 2) dono mein common: 2y(3x − 2) + 3(3x − 2) = (3x − 2)(2y + 3).
Check: (3x − 2)(2y + 3) = 6xy + 9x − 4y − 6 ✅ (same 4 terms).
Q6. Divide karo: (10x³ − 15x² + 5x) ÷ 5x.
Step-by-step solution
Step 1 (har term alag se divide): 10x³ ÷ 5x = 2x²; 15x² ÷ 5x = 3x; 5x ÷ 5x = 1.
Step 2: signs waise hi: 2x² − 3x + 1.
Check: 5x × (2x² − 3x + 1) = 10x³ − 15x² + 5x ✅
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📘 Section E: Long Answer Questions
4 se 5 marks wale sawaal. Solution mein har step ka reason bhi likha hai.
Q1. (a) Ek square ki side (a + b) hai. Uske area ko 4 parts mein divide karke dikhao ki (a + b)² = a² + 2ab + b². (b) Is identity se 205² nikaalo. (c) a = 3, b = 2 rakh kar identity check karo.
Step-by-step solution
(a) Step 1: square ki side a + b ko a aur b mein divide karo (upar aur left dono taraf). Square 4 parts mein bat gaya (upar recap ki picture dekho).
Step 2: parts ke area: side a wala square = a × a = a²; side b wala square = b²; do rectangles, har ek ki length a aur breadth b, area ab + ab = 2ab.
Step 3: poore square ka area = (a + b)² = sab parts ka sum = a² + 2ab + b² ✅
(b) 205 = 200 + 5: a² = 40000; 2ab = 2 × 200 × 5 = 2000; b² = 25. Sum 40000 + 2000 + 25 = 42025.
(c) left side (3 + 2)² = 5 × 5 = 25; right side 9 + 2 × 3 × 2 + 4 = 9 + 12 + 4 = 25 ✅
Q2. Factorise karo aur har ek ko multiply karke check karo: (a) 4x² − 12xy + 9y² (b) x² − 2x − 15 (c) 3m³ − 48m
Step-by-step solution
(a) Step 1: 4x² = (2x)², 9y² = (3y)², aur beech wala 12xy = 2 × 2x × 3y. Yeh a² − 2ab + b² pattern hai, a = 2x, b = 3y.
Step 2: answer (2x − 3y)². Check: (2x − 3y)(2x − 3y) = 4x² − 6xy − 6xy + 9y² = 4x² − 12xy + 9y² ✅
(b) Step 1: do numbers chahiye jinka product −15 aur sum −2 ho. −5 × 3 = −15 aur −5 + 3 = −2 ✅
Step 2: answer (x − 5)(x + 3). Check: x² + 3x − 5x − 15 = x² − 2x − 15 ✅
(c) Step 1 (common factor): 3m³ − 48m = 3m(m² − 16).
Step 2 (identity a² − b²): m² − 16 = (m + 4)(m − 4). Answer 3m(m + 4)(m − 4).
Check (m = 5): 3 × 125 − 48 × 5 = 375 − 240 = 135 aur 15 × 9 × 1 = 135 ✅
Q3. Simplify karo: (x + 2y)² − (x − 2y)². Phir x = 3, y = 1 rakh kar answer check karo.
Step-by-step solution
Step 1 (pehla bracket): (x + 2y)² = x² + 2 × x × 2y + (2y)² = x² + 4xy + 4y².
Step 2 (doosra bracket): (x − 2y)² = x² − 4xy + 4y².
Step 3 (subtract, har sign ulta): x² + 4xy + 4y² − x² + 4xy − 4y² = 8xy (x² aur 4y² cancel).
Doosra method: a² − b² = (a + b)(a − b), a = x + 2y, b = x − 2y: (a + b) = 2x, (a − b) = 4y, product 8xy ✅
Check (x = 3, y = 1): x + 2y = 3 + 2 = 5 aur x − 2y = 3 − 2 = 1, to 5² − 1² = 25 − 1 = 24 aur 8 × 3 × 1 = 24 ✅
Q4. Ek rectangle ka area (x² + 9x + 20) square cm hai aur length (x + 5) cm hai. (a) Breadth nikaalo. (b) Perimeter ka expression likho. (c) x = 3 pe length, breadth aur area nikaal kar check karo.
Step-by-step solution
(a) Step 1: area = length × breadth, to breadth = area ÷ length.
Step 2: x² + 9x + 20 ko factorise karo: 4 × 5 = 20 aur 4 + 5 = 9, to (x + 4)(x + 5).
Step 3: (x + 4)(x + 5) ÷ (x + 5) = x + 4. Breadth = (x + 4) cm.
(b) perimeter = 2 × (length + breadth) = 2 × (x + 5 + x + 4) = 2 × (2x + 9) = (4x + 18) cm.
(c) x = 3: length 3 + 5 = 8 cm, breadth 3 + 4 = 7 cm, area 8 × 7 = 56. Expression se: 9 + 27 + 20 = 56 ✅. Perimeter 4 × 3 + 18 = 30 aur 2 × (8 + 7) = 30 ✅
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🧩 Section F: HOTS (Soch Wale Sawaal)
HOTS matlab Higher Order Thinking Skills: aise sawaal jinme formula se aage thoda mind lagana padta hai.
Q1. Agar a + b = 9 aur ab = 20 ho, to a² + b² ki value kya hogi? (a aur b alag-alag nikaale bina.)
Soch aur solution dekho
Soch: (a + b)² = a² + 2ab + b², to a² + b² = (a + b)² − 2ab.
Step 1: (a + b)² = 9 × 9 = 81.
Step 2: 2ab = 2 × 20 = 40.
Step 3: a² + b² = 81 − 40 = 41.
Check: a = 4, b = 5 (sum 9, product 20): 16 + 25 = 41 ✅
Q2. Bina lamba multiply kiye 63² − 37² nikaalo. Kaunsi identity kaam aayi?
Soch aur solution dekho
Soch: a² − b² = (a + b)(a − b).
Step 1: a + b = 63 + 37 = 100; a − b = 63 − 37 = 26.
Step 2: 100 × 26 = 2600.
Check: 63 × 63 = 3969, 37 × 37 = 1369, aur 3969 − 1369 = 2600 ✅
Q3. Dikhao ki do consecutive odd numbers (ek odd number aur uske turant baad wala odd number, jaise 5 aur 7) ke squares ka difference hamesha 8 ka multiple hota hai. (Odd number = 1, 3, 5, 7 …, jo 2 se poora divide nahi hote; 8 ka multiple = 8, 16, 24 … yaani 8 × koi counting number; counting number = 1, 2, 3 …. Hint: odd numbers ko 2n − 1 aur 2n + 1 likho, n koi bhi counting number. Jaise n = 3 pe 5 aur 7.)
Soch aur solution dekho
Step 1: (a + b)² aur (a − b)² identity se: (2n + 1)² = 4n² + 4n + 1 aur (2n − 1)² = 4n² − 4n + 1.
Step 2 (subtract): 4n² + 4n + 1 − 4n² + 4n − 1 = 8n.
Step 3: 8n = 8 × n, to hamesha 8 ka multiple ✅ (identity a² − b² se bhi: (a + b)(a − b) = 4n × 2 = 8n).
Examples: 7² − 5² = 49 − 25 = 24 = 8 × 3; 11² − 9² = 121 − 81 = 40 = 8 × 5 ✅
Q4. Riya ne homework mein yeh 3 steps likhe. Har step mein mistake dhundho aur sahi answer likho: (i) (x + 3)² = x² + 9 (ii) 2(x + 4) = 2x + 4 (iii) (4x² + 8x) ÷ 4x = x + 8x
Soch aur solution dekho
(i) beech wala term bhool gayi: (x + 3)² = x² + 2 × x × 3 + 9 = x² + 6x + 9. Check x = 1: 4 × 4 = 16 aur 1 + 6 + 9 = 16 ✅
(ii) distributive property mein 2 ko 4 se bhi multiply karna tha: 2(x + 4) = 2x + 8. Check x = 1: 2 × 5 = 10 aur 2 + 8 = 10 ✅
(iii) sirf pehle term ko divide kiya, doosre ko nahi. Polynomial ko monomial se divide karte waqt har term divide hota hai: 4x² ÷ 4x = x aur 8x ÷ 4x = 2. Sahi answer x + 2. Check x = 1: (4 + 8) ÷ 4 = 12 ÷ 4 = 3 aur 1 + 2 = 3 ✅
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🔑 Answer Key (Ek Nazar Mein)
Pehle saare sawaal khud karo, phir yahan milaao. Jahan mistake ho, upar us sawaal ka step-by-step solution dobara padho.
| Section | Answers |
|---|---|
| A. MCQ | 1: (b) · 2: (b) · 3: (b) · 4: (b) · 5: (a) · 6: (a) · 7: (a) · 8: (a) · 9: (b) · 10: (a) |
| B. Fill in the Blanks | 1: 2ab · 2: 6x · 3: 8y · 4: 7 · 5: 9999 · 6: (a + b)² · 7: monomial · 8: 3p |
| C. True / False | 1: F · 2: T · 3: T · 4: F · 5: T · 6: F · 7: T · 8: F |
| D. Short Answer | 1: 2x² + 2x − 4 · 2: 3a + 4b − 7 · 3: 2x² − 5x − 12 · 4: 9409 · 5: (3x − 2)(2y + 3) · 6: 2x² − 3x + 1 |
| E. Long Answer | 1: (b) 205² = 42025; (c) 25 = 25 · 2: (a) (2x − 3y)² (b) (x − 5)(x + 3) (c) 3m(m + 4)(m − 4) · 3: 8xy (x = 3, y = 1 pe 24) · 4: (a) x + 4 (b) 4x + 18 (c) 8 cm, 7 cm, 56 square cm |
| F. HOTS | 1: 41 · 2: 2600 (a² − b² identity) · 3: (2n + 1)² − (2n − 1)² = 8n · 4: (i) x² + 6x + 9 (ii) 2x + 8 (iii) x + 2 |
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👩🏫 CharuMam Ka Tip
Algebra ke har answer ko x = 1 ya x = 2 rakh kar check karo: dono side ki value same aani chahiye. Identities ko area wali picture se yaad rakho: (a + b)² mein do rectangles (2ab) hote hain, isliye beech wala term kabhi mat bhoolo. Factorisation mein sabse pehle common factor dhundho, phir identity ya “product q, sum p” wala method lagao, aur end mein factors ko multiply karke wapas check karo. 🧮
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👉 Aage Kya?
Pehle chapter We Distribute, Yet Things Multiply ke posts padh lo (link neeche): wahan distributive property aur identities ko area wali pictures se samjhaaya hai. Algebra ke puzzles aur number tricks ke liye Book 2 Ch 6: Algebra Play bhi dekho. Phir yeh practice dobara karo aur score milao. Agla practice unit: Direct and Inverse Proportions, jald aa raha hai!
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