Class 10 Maths · Chapter 2
Short answer:
Is chapter me 2 exercises hain — Exercise 2.1 me sirf 1 question hai (graph dekh kar zeroes ki count batana) aur Exercise 2.2 me 2 questions hain, dono multi-part (6 + 6 sub-parts) — zeroes nikalna aur unka coefficients ke saath relation verify karna, aur diye gaye sum/product se quadratic polynomial banana. Chapter sirf zeroes ka geometrical meaning aur zeroes-coefficients relationship tak limited hai — polynomial division (long division / division algorithm) ab is chapter me nahi hai, woh rationalisation me hata diya gaya.
Polynomial ke zeroes woh x-values hain jahan uska graph x-axis ko cross ya touch karta hai — aur quadratic polynomial ke liye ye zeroes coefficients se ek seedhe formula se juden hote hain. Class 9 me tumne polynomial ki basic definition (degree, terms, types) padhi thi. Ab is chapter me do naye ideas aate hain: pehla, polynomial p(x) ka graph y = p(x) khींचne par uske zeroes graph ke x-axis crossings hote hain — aur graph ka shape (degree ke hisaab se) batata hai zeroes kitne ho sakte hain. Doosra, agar quadratic polynomial ko factor kiye bina bhi uske zeroes ka sum aur product chahiye ho, to seedha coefficients se nikal sakte ho: sum α+β = −b/a, product αβ = c/a. Ye chapter Chapter 4 (Quadratic Equations) ki neev hai.
Chapter 2 Summary — 5 Minute Revision
1. Polynomial — ek recap
Polynomial p(x) ek expression hai jisme x ki non-negative integer powers hoti hain, jaise p(x) = ax² + bx + c (quadratic, degree 2) ya p(x) = ax + b (linear, degree 1).
2. Zero of a polynomial
k, p(x) ka zero kehlata hai agar p(k) = 0. Geometrically, y = p(x) ka graph banao — jahan-jahan graph x-axis ko chhoo raha hai ya cross kar raha hai, wahi points hi p(x) ke zeroes hain (x-coordinate).
| Polynomial type | Degree | Graph shape | Max zeroes |
|---|---|---|---|
| Linear (ax + b) | 1 | seedhi line | 1 |
| Quadratic (ax² + bx + c) | 2 | parabola | 2 |
| Cubic | 3 | curve | 3 |
↔ Table ko side me swipe karein
General rule: koi bhi polynomial ke zeroes uski degree se zyada nahi ho sakte. Quadratic ke paas at most 2 zeroes honge — 2, 1 (repeated), ya 0 (agar graph x-axis ko chhoota hi nahi).
3. Quadratic polynomial ke graph ki 3 possibilities
- Parabola x-axis ko 2 alag points pe cut kare → 2 distinct real zeroes
- Parabola x-axis ko 1 point pe touch kare (vertex hi touch point ho) → 1 repeated zero (dono zeroes equal)
- Parabola x-axis ko touch hi na kare → koi real zero nahi
4. Relationship between zeroes and coefficients (quadratic)
Agar α aur β kisi quadratic polynomial p(x) = ax² + bx + c (a ≠ 0) ke zeroes hain, to:
α + β = −b/a (sum of zeroes)
α × β = c/a (product of zeroes)
Derivation ka idea: agar α, β zeroes hain, to p(x) = a(x − α)(x − β) likh sakte ho. Ise expand karo:
a(x − α)(x − β) = a[x² − (α+β)x + αβ] = ax² − a(α+β)x + aαβ
Ise diye hue ax² + bx + c se compare karo:
−a(α+β) = b ⟹ α+β = −b/a
aαβ = c ⟹ αβ = c/a
5. Ulta kaam — sum aur product se polynomial banana
Agar sum α+β = S aur product αβ = P diye ho, to quadratic polynomial:
p(x) = x² − Sx + P (ya koi bhi k[x² − Sx + P], k ≠ 0)
Kyunki infinite polynomials same zeroes rakh sakte hain (bas constant multiple alag hoga) — isliye answer "k gunaa" allowed hai.
Poore Class 10 Maths ke handwritten colour notes
IITian & district toppers ke banaye short notes — revision-ready, diagram ke saath. Board se pehle poora syllabus 3 din me revise.
Exercise Questions — Solutions (Q1–Q13)
Exercise 2.1, Q1. Neeche kuch polynomials p(x) ke graphs y = p(x) diye gaye hain. Har case me p(x) ke zeroes ki sankhya batao. (i) graph x-axis ko kahin bhi touch nahi karta (ii) graph x-axis ko exactly 1 point pe touch karta hai (iii) graph x-axis ko 3 points pe cut karta hai (iv) graph x-axis ko 2 points pe cut karta hai (v) graph x-axis ko 4 points pe cut karta hai (vi) graph x-axis ko 3 points pe cut karta hai
Zeroes ki sankhya = jitni baar graph x-axis ko cross ya touch karta hai.
(i) 0 zeroes — graph x-axis ko chhoota hi nahi
(ii) 1 zero — graph x-axis ko sirf ek jagah touch karta hai
(iii) 3 zeroes
(iv) 2 zeroes
(v) 4 zeroes
(vi) 3 zeroes
Note: exact graph shapes textbook figure me hain — yahan sirf x-axis crossings ki counting logic use hui hai, jo har case me directly zeroes ki sankhya deti hai.
Exercise 2.2, Q1(i). Find zeroes of x² − 2x − 8 aur zeroes-coefficients relationship verify karo.
Factorise karke zeroes nikalo:
x² − 2x − 8 = x² − 4x + 2x − 8 = x(x−4) + 2(x−4) = (x−4)(x+2)
Zeroes: x = 4, x = −2 ⟹ α = 4, β = −2
Verification (a=1, b=−2, c=−8):
α+β = 4+(−2) = 2 = −b/a = −(−2)/1 = 2 ✓
αβ = 4×(−2) = −8 = c/a = −8/1 = −8 ✓
Exercise 2.2, Q1(ii). Find zeroes of 4s² − 4s + 1 aur verify karo.
4s² − 4s + 1 = (2s−1)²
Zero: s = 1/2 (repeated) ⟹ α = β = 1/2
Verification (a=4, b=−4, c=1):
α+β = 1/2+1/2 = 1 = −b/a = −(−4)/4 = 1 ✓
αβ = 1/2 × 1/2 = 1/4 = c/a = 1/4 ✓
Exercise 2.2, Q1(iii). Find zeroes of 6x² − 3 − 7x aur verify karo.
Pehle standard form me likho: 6x² − 7x − 3
6x² − 9x + 2x − 3 = 3x(2x−3) + 1(2x−3) = (3x+1)(2x−3)
Zeroes: x = −1/3, x = 3/2
Verification (a=6, b=−7, c=−3):
α+β = −1/3 + 3/2 = 7/6 = −b/a = −(−7)/6 = 7/6 ✓
αβ = (−1/3)(3/2) = −1/2 = c/a = −3/6 = −1/2 ✓
Exercise 2.2, Q1(iv). Find zeroes of 4u² + 8u aur verify karo.
4u² + 8u = 4u(u+2)
Zeroes: u = 0, u = −2
Verification (a=4, b=8, c=0):
α+β = 0+(−2) = −2 = −b/a = −8/4 = −2 ✓
αβ = 0×(−2) = 0 = c/a = 0/4 = 0 ✓
Exercise 2.2, Q1(v). Find zeroes of t² − 15 aur verify karo.
t² − 15 = (t−√15)(t+√15)
Zeroes: t = √15, t = −√15
Verification (a=1, b=0, c=−15):
α+β = √15+(−√15) = 0 = −b/a = 0/1 = 0 ✓
αβ = √15×(−√15) = −15 = c/a = −15/1 = −15 ✓
Exercise 2.2, Q1(vi). Find zeroes of 3x² − x − 4 aur verify karo.
3x² − 4x + 3x − 4 = x(3x−4) + 1(3x−4) = (3x−4)(x+1)
Zeroes: x = 4/3, x = −1
Verification (a=3, b=−1, c=−4):
α+β = 4/3+(−1) = 1/3 = −b/a = −(−1)/3 = 1/3 ✓
αβ = (4/3)(−1) = −4/3 = c/a = −4/3 ✓
Exercise 2.2, Q2(i). Sum aur product diye hue hain: 1/4, −1. Aisa quadratic polynomial dhundo jiske zeroes ka sum aur product yahi ho.
p(x) = x² − Sx + P, jahan S = 1/4, P = −1
p(x) = x² − (1/4)x + (−1)
Fraction hatane ke liye 4 se multiply karo: p(x) = 4x² − x − 4
Exercise 2.2, Q2(ii). Sum = √2, Product = 1/3.
p(x) = x² − √2 x + 1/3
3 se multiply karke: p(x) = 3x² − 3√2 x + 1
Exercise 2.2, Q2(iii). Sum = 0, Product = √5.
p(x) = x² − 0·x + √5 = x² + √5
Exercise 2.2, Q2(iv). Sum = 1, Product = 1.
p(x) = x² − x + 1
Exercise 2.2, Q2(v). Sum = −1/4, Product = 1/4.
p(x) = x² − (−1/4)x + 1/4 = x² + (1/4)x + 1/4
4 se multiply karke: p(x) = 4x² + x + 1
Exercise 2.2, Q2(vi). Sum = 4, Product = 1.
p(x) = x² − 4x + 1
Important Equations — Ek Nazar Me
| Relation | Formula (ax² + bx + c, zeroes α, β) | Note |
|---|---|---|
| Sum of zeroes | α + β = −b/a | coefficient of x ka coefficient of x² se ratio, minus sign ke saath |
| Product of zeroes | αβ = c/a | constant term ka coefficient of x² se ratio, sign same rehta hai |
| Polynomial from S, P | x² − Sx + P (ya k[x² − Sx + P]) | S = sum, P = product; k koi bhi non-zero constant |
| Max zeroes | degree ke barabar | quadratic ke max 2, linear ke max 1 |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Sum ke formula me sign bhool jaana. α+β = −b/a hai, sirf b/a nahi — minus sign miss karna sabse common exam mistake hai.
- Product ke formula me galti se minus laga dena. αβ = c/a hai (koi minus nahi) — sum aur product ke formula ko mix-up kar dete hain students.
- Negative coefficient ke saath sign ka double-confuse ho jaana. Jaise agar b khud negative hai (b = −7), to −b/a karte waqt −(−7)/a = +7/a hoga — yahan sign error bahut hota hai.
- 'Zero of polynomial' aur 'root of equation' ko alag treat karna. Dono ek hi cheez hain jab p(x) = 0 form me dekha jaaye — bas terminology polynomial ke context me 'zero' aur equation ke context me 'root' use hoti hai.
- Quadratic ke 2 se zyada zeroes maan lena. Degree 2 ka polynomial kabhi bhi 3 zeroes nahi rakh sakta — max utne hi zeroes jitni degree.
- Sum/product se polynomial banate waqt k factor bhool jaana. x² − Sx + P sirf ek answer hai — koi bhi constant multiple (jaise 4x² − x − 4) bhi utna hi sahi hai, isliye fraction-free form banane ke liye multiply karna zaroori hai lekin ye 'ek hi correct answer' nahi hai.
Board-Style Important Questions
- 1 mark: Ek quadratic polynomial ka graph diya gaya hai jo x-axis ko do points pe cut karta hai. Uske zeroes ki sankhya batao.
- 1 mark: Agar quadratic polynomial ax² + bx + c ke zeroes α aur β hain, to α+β aur αβ coefficients ke terms me likho.
- 2 marks: Quadratic polynomial 4u² + 8u ke zeroes nikalo aur unka sum-product relationship verify karo.
- 2 marks: Aisa quadratic polynomial dhundo jiske zeroes ka sum √2 aur product 1/3 ho.
- 3 marks: Quadratic polynomial 6x² − 7x − 3 ke zeroes factorisation se nikalo aur coefficients ke saath unka relation verify karo.
Quick Quiz — Score Check Karein
Q1. Class 10 Maths Ch2 (Polynomials) me polynomial p(x) ka 'zero' kya hota hai?
Q2. Geometrically, polynomial p(x) ke zeroes graph y = p(x) me kahan milte hain?
Q3. Quadratic polynomial ax² + bx + c ke max kitne real zeroes ho sakte hain?
Q4. Agar quadratic ke zeroes α aur β hain, to sum α+β kis formula se milta hai (ax²+bx+c)?
Q5. Product of zeroes α×β kis formula se milta hai?
Q6. x² − 2x − 8 ke zeroes factorisation se nikalo.
Q7. 4s² − 4s + 1 ke zeroes kya hain?
Q8. 6x² − 7x − 3 (i.e. 6x² − 3 − 7x) ke zeroes kya hain?
Q9. t² − 15 ke zeroes hain — inka sum kya nikalta hai (verification ke liye)?
Q10. Agar quadratic polynomial ke zeroes ka sum 1/4 aur product −1 diya ho, to fraction-free polynomial kya banega (Ex 2.2 Q2(i))?
Aksar Poochhe Jaane Wale Sawaal
Zero of a polynomial ka matlab kya hota hai?
Koi bhi value k jisko p(x) me substitute karne par p(k) = 0 aa jaye, wahi k polynomial p(x) ka zero kehlata hai. Graph pe ye woh point hota hai jahan curve x-axis ko touch/cross karta hai.
Quadratic polynomial ke max kitne zeroes ho sakte hain?
2. Kisi bhi polynomial ke zeroes uski degree se zyada nahi ho sakte — quadratic (degree 2) ke max 2 real zeroes honge, ho sakta hai wo 2 distinct ho, ek repeated ho, ya real zero ho hi na.
Sum aur product ke formula me sign kaise yaad rakhein?
'Sum me sign ulta, product me sign seedha' — α+β = −b/a (minus lagega) aur αβ = c/a (koi minus nahi). Polynomial ax²+bx+c likhte waqt yaad rakho b ke saath minus jud jaata hai.
Division algorithm ye chapter me hai kya?
Nahi — rationalised NCERT syllabus me polynomial division (division algorithm, quotient-remainder wala part) is chapter se hata diya gaya hai. Ab sirf zeroes aur unka coefficients ke saath relationship padhaya jaata hai.
Agar sirf sum aur product diye ho to quadratic polynomial kaise banaye?
Formula x² − (sum)x + (product) use karo. Agar coefficients fraction me aa rahe hain to poore expression ko kisi convenient number se multiply kar sakte ho — jawab still correct rahega kyunki constant multiple se zeroes change nahi hote.
'Zero of polynomial' aur 'root of equation' same hain kya?
Practically haan — jab polynomial p(x) ko zero ke barabar rakh kar equation p(x) = 0 banate ho, to us equation ke roots hi p(x) ke zeroes hote hain. Sirf naming context-dependent hai.
Class 10 Maths — Saare Chapters
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