Class 9 Maths · Chapter 2
Short answer:
Class 9 Maths Chapter 2 "Introduction to Linear Polynomials" (Ganita Manjari Part 1, session 2026-27) polynomial ki basic definition — variable, coefficient, constant term, degree — se shuru hokar linear polynomials ke general form p(x) = ax + b, unka zero (x = −b/a), real-life linear growth/decay patterns, function-machine ka idea, aur y = ax + b ke straight-line graph (slope aur y-intercept) tak jaata hai. Ye Ganita Manjari NCERT solutions series ka hissa hai — step-by-step solved intext questions, full exercise, formula table, common mistakes aur important questions ke saath.
Class 9 Maths ki naya textbook Ganita Manjari (session 2026-27, NCERT ka rationalised/naya syllabus) me Chapter 2 ka naam hai "Introduction to Linear Polynomials" — purane syllabus wale "Polynomials" chapter se ye kaafi alag hai. Purane chapter me quadratic aur cubic polynomials, factor theorem, remainder theorem, aur algebraic identities sab ek saath the. Naye Ganita Manjari Part 1 me ye sab split ho gaya hai: is chapter (Chapter 2) ka scope sirf linear polynomials tak simit hai, jabki identities ab Chapter 4 "Exploring Algebraic Identities" me alag se cover hoti hain.
Chapter 2 shuru hota hai bilkul basic se — variable, coefficient, constant, aur degree kya hote hain, ye real-life examples se samjhaya jaata hai (jaise stationery kharidna, garden fencing, auto-rickshaw fare). Uske baad linear polynomial p(x) = ax + b ka formal introduction, uska zero (x = −b/a) nikalna, linear growth/decay ka pattern (jaise tank me paani bharna ya ghatna, ya height/age wale examples), function-machine ka informal idea, aur end me y = ax + b ko straight-line graph ke roop me plot karna — slope (a) aur y-intercept (b) ke saath.
Ye chapter Class 9 Maths ki poori algebra-and-graphing foundation set karta hai — Chapter 1 "Orienting Yourself — Use of Coordinates" ke basic coordinate-plane skills yahan directly use hote hain, aur linear growth ka pattern aage Chapter 8 "Sequences and Progressions" me arithmetic progressions ke roop me expand hota hai. Agar tumhe Ganita Manjari ke doosre chapters ke NCERT solutions chahiye — jaise Chapter 3 "The World of Numbers", Chapter 4 "Exploring Algebraic Identities", Chapter 5 "I'm Up and Down, and Round and Round" (Circles), Chapter 6 "Measuring Space", Chapter 7 "Probability" — ya Class 9 Maths new syllabus 2026-27 ka poora chapter list chahiye, wo bhi is series me alag se available hain.
Chapter 2 Summary — 5 Minute Revision
Chapter 2 — Introduction to Linear Polynomials — key takeaways:
- Polynomial ek algebraic expression hai jisme variable ke sabhi powers whole numbers (0, 1, 2, ...) hote hain.
- Degree = highest power jiska coefficient non-zero ho. Degree 0 = constant, Degree 1 = linear, Degree 2 = quadratic, Degree 3 = cubic (naming ke liye; is chapter ka focus sirf linear par hai).
- Linear polynomial ka general form: p(x) = ax + b, jaha a ≠ 0.
- Zero of a linear polynomial: p(x) = 0 solve karke x = −b/a milta hai — linear polynomial ka exactly ek hi zero hota hai.
- Real-life situations (auto fare, tank filling/draining, height, savings) ko linear growth (a > 0) ya decay (a < 0) ke roop me model kiya ja sakta hai.
- Function machine ka idea: p: x → ax + b, jaha x input aur p(x) output hai.
- y = ax + b ka graph hamesha ek straight line hota hai — a slope (rate of change), b y-intercept (x = 0 par value), aur line x-axis ko exactly ek point (zero) par cut karti hai.
In-Text Questions — Solutions
Batao kaunse expressions polynomials hain: (i) 3x² − 2x + 5 (ii) √x + 2 (iii) 1/x + 3 (iv) 5
Polynomial me variable ka power hamesha whole number (0, 1, 2, 3, ...) hona chahiye.
(i) 3x² − 2x + 5 → powers 2, 1, 0 — sab whole numbers → POLYNOMIAL ✔ (degree 2)
(ii) √x + 2 = x^(1/2) + 2 → power 1/2, whole number nahi → POLYNOMIAL NAHI ✘
(iii) 1/x + 3 = x⁻¹ + 3 → power −1 (negative) → POLYNOMIAL NAHI ✘
(iv) 5 = 5x⁰ → power 0 → POLYNOMIAL ✔ (constant polynomial, degree 0)
p(x) = 7x − 4 me coefficient of x aur constant term batao.
Standard form ax + b se compare karo:
Coefficient of x (a) = 7
Constant term (b) = −4
p(x) = 9 ka degree kya hai? Isko classify karo.
9 ko 9x⁰ likh sakte hain, toh highest power of x = 0.
Degree = 0 → ye ek CONSTANT polynomial hai (linear nahi, kyunki x ka coefficient 0 hai)
Agar p(x) = 3x − 6, toh p(2), p(0), aur p(−1) nikalo.
p(2) = 3(2) − 6 = 6 − 6 = 0
p(0) = 3(0) − 6 = −6
p(−1) = 3(−1) − 6 = −3 − 6 = −9
p(x) = 5x + 10 ka zero nikalo.
Zero nikalne ke liye p(x) = 0 rakho:
5x + 10 = 0 ⟹ 5x = −10 ⟹ x = −2
Zero x = −2 hai.
Ek water tank me 200 L paani hai aur 15 L/min ki rate se fill ho raha hai. x minute baad water level ka polynomial likho aur 6 minute baad level nikalo.
Starting value = 200 (constant term), rate = 15 L/min (coefficient, growth hai isliye positive):
p(x) = 15x + 200
p(6) = 15(6) + 200 = 90 + 200 = 290 L
Ek function machine input me 3 add karti hai, phir result ko 2 se multiply karti hai. Polynomial likho aur input 4 ke liye output nikalo.
Machine: x → (x + 3) → 2(x + 3)
p(x) = 2(x + 3) = 2x + 6
Ye linear hai (a = 2, b = 6). Input x = 4 par:
p(4) = 2(4) + 6 = 8 + 6 = 14
p(x) = −4x + 9 ka slope aur y-intercept nikalo. Ye growth represent karta hai ya decay?
Slope (a) = −4, y-intercept (b) = 9
Slope negative hai, isliye jaise-jaise x badhta hai, p(x) ghat-ta hai — ye decay represent karta hai.
Exercise Questions — Solutions (Q1–Q14)
Q1. Kin polynomials ko linear kaha ja sakta hai? Reason bhi do.
(a) 2x + 3 (b) x² − 1 (c) 7 (d) −x/2 + 5
(a) 2x + 3 (b) x² − 1 (c) 7 (d) −x/2 + 5
Linear polynomial ka matlab hai degree exactly 1 ka polynomial, form ax + b jisme a ≠ 0.
(a) 2x + 3 → highest power of x = 1, a = 2 ≠ 0 → LINEAR ✔
(b) x² − 1 → highest power = 2 → QUADRATIC (linear nahi) ✘
(c) 7 = 7x⁰ → degree 0, coefficient of x hai hi 0 → CONSTANT polynomial, linear nahi ✘
(d) −x/2 + 5 → a = −1/2 ≠ 0, degree 1 → LINEAR ✔
Isliye (a) aur (d) linear polynomials hain.
Q2. Ek linear polynomial likho jiska zero 5 ho.
Zero x = 5 chahiye, matlab p(5) = 0 hona chahiye.
p(x) = x − 5 lo
p(5) = 5 − 5 = 0 ✔
Toh p(x) = x − 5 ek valid answer hai (aisi infinite polynomials ban sakti hain, jaise 2x − 10, −3x + 15, etc. — sab ka zero 5 hi hoga).
Q3. p(x) = 2x − 3 diya hai. Check karo x = 3/2 iska zero hai ya nahi.
p(3/2) = 2 × (3/2) − 3 = 3 − 3 = 0
Kyunki p(3/2) = 0, isliye x = 3/2, p(x) ka zero hai.
Q4. Ek taxi ka base fare ₹25 hai aur uske baad ₹12 per km lagta hai.
(i) x km ke liye cost C(x) ka linear polynomial likho.
(ii) 8 km ke liye cost nikalo.
(iii) Agar ek customer ne ₹133 pay kiye, toh usne kitni distance travel ki?
(i) x km ke liye cost C(x) ka linear polynomial likho.
(ii) 8 km ke liye cost nikalo.
(iii) Agar ek customer ne ₹133 pay kiye, toh usne kitni distance travel ki?
(i) Fixed cost = 25 (constant term), rate = 12 per km (coefficient of x):
C(x) = 12x + 25
(ii) x = 8 par:
C(8) = 12 × 8 + 25 = 96 + 25 = ₹121
(iii) C(x) = 133 rakho:
12x + 25 = 133 ⟹ 12x = 108 ⟹ x = 9
Toh customer ne 9 km travel ki.
Q5. Ek tank me shuru me 500 L paani hai aur 20 L/min ki rate se drain ho raha hai. Volume ka polynomial V(x) likho aur pata karo tank kab (kitne minute me) khali hoga.
Paani kam ho raha hai (decay), toh slope negative hoga:
V(x) = 500 − 20x
Tank khali hone par V(x) = 0:
500 − 20x = 0 ⟹ 20x = 500 ⟹ x = 25
Tank 25 minute me khali ho jayega.
Q6. y = 2x − 4 ka graph banane ke liye table banao (x = −1, 0, 1, 2, 3 ke liye), slope aur y-intercept batao, aur line rising hai ya falling.
x = −1: y = 2(−1) − 4 = −6
x = 0: y = 2(0) − 4 = −4 (y-intercept = −4)
x = 1: y = 2(1) − 4 = −2
x = 2: y = 2(2) − 4 = 0 (yehi zero/x-intercept hai)
x = 3: y = 2(3) − 4 = 2
Slope a = 2 (positive) hai, isliye line left se right rising hai — ye linear growth pattern show karta hai. y-intercept = −4 (x = 0 par value), zero x = 2 par hai.
Q7. p(x) = kx − 8 diya hai. Agar x = 2 iska zero hai, toh k ka value nikalo.
Zero ka matlab p(2) = 0:
k(2) − 8 = 0 ⟹ 2k = 8 ⟹ k = 4
Toh k = 4, aur polynomial p(x) = 4x − 8 banta hai.
Q8. In expressions ko classify karo — polynomial hai ya nahi, aur agar hai toh degree batao:
(i) x³ + 2x − 1 (ii) 5x − 9 (iii) 4 (iv) x² + 3x (v) 1/(x + 1)
(i) x³ + 2x − 1 (ii) 5x − 9 (iii) 4 (iv) x² + 3x (v) 1/(x + 1)
Polynomial hone ke liye variable ka power ek non-negative integer hona chahiye.
(i) x³ + 2x − 1 → highest power 3 → polynomial, degree 3 (cubic)
(ii) 5x − 9 → highest power 1 → polynomial, degree 1 (linear)
(iii) 4 = 4x⁰ → polynomial, degree 0 (constant)
(iv) x² + 3x → highest power 2 → polynomial, degree 2 (quadratic)
(v) 1/(x+1) = (x+1)⁻¹ → negative power hai → polynomial NAHI hai
Q9. Ek pattern ka pehla term (x = 0 par) 12 hai aur har agla term pichle se 4 zyada hai. Polynomial p(x) likho aur 10th term nikalo (x = 9 par, kyunki x = 0 pehla term hai).
Constant amount 4 se badh raha hai (growth), starting value 12:
p(x) = 4x + 12
10th term ke liye x = 9 (kyunki x=0 pehla term hai):
p(9) = 4(9) + 12 = 36 + 12 = 48
10th term = 48.
Q10. Ramesh ki height 8 saal ki age me 120 cm thi aur har saal 5 cm badh rahi hai. h(x) likho (x = age 8 ke baad ke saal) aur age 13 par uski height nikalo. Ye growth hai ya decay?
h(x) = 5x + 120
Age 13 matlab x = 13 − 8 = 5:
h(5) = 5(5) + 120 = 25 + 120 = 145 cm
Slope a = 5 (positive) hai, isliye ye linear growth hai.
Q11. Function machine p: x → 3x − 7 diya hai. Agar p(x) = 20, toh input x kya hoga?
3x − 7 = 20 ⟹ 3x = 27 ⟹ x = 9
Input x = 9 hone par output 20 milta hai.
Q12. p(x) = 2x + 1 aur q(x) = x − 4 diye hain. (p + q)(x) nikalo aur check karo ye bhi linear hai ya nahi. Uska zero bhi nikalo.
(p + q)(x) = (2x + 1) + (x − 4) = 3x − 3
Ye bhi degree-1 hai (a = 3 ≠ 0), isliye linear hai.
Zero: 3x − 3 = 0 ⟹ x = 1
Q13. Blanks bharo: Ek linear polynomial p(x) = ax + b (a ≠ 0) ka graph hamesha ek ______ line hota hai aur x-axis ko exactly ______ point pe cut karta hai.
Blanks: straight line, aur x-axis ko exactly ek (1) point pe cut karta hai — wahi point uska zero (−b/a) hota hai.
Q14 (HOTS). Ek candle 20 cm lambi hai aur constant rate se jal rahi hai. 4 minute jalne ke baad uski length 12 cm reh jaati hai. Length L(x) ka polynomial likho aur pata karo candle poori tarah jalne me kitna time lagega.
Do points diye hain: L(0) = 20 aur L(4) = 12.
Rate (slope) = (12 − 20) / (4 − 0) = −8/4 = −2 cm per min
L(x) = −2x + 20
Candle khatam hone par L(x) = 0:
−2x + 20 = 0 ⟹ x = 10
Candle 10 minute me poori tarah jal jayegi.
Important Equations — Ek Nazar Me
Class 9 Maths Chapter 2 — Introduction to Linear Polynomials: Key Formulas
| Concept | Formula / Rule |
|---|---|
| Polynomial (general, one variable) | a₀ + a₁x + a₂x² + ... + aₙxⁿ, jaha exponents whole numbers (0,1,2,...) hain |
| Degree of a polynomial | Highest power of x jiska coefficient non-zero ho |
| Constant polynomial | Degree 0, form: p(x) = b (b ek fixed real number) |
| Linear polynomial | Degree 1, general form: p(x) = ax + b, jaha a ≠ 0 |
| Quadratic / Cubic (naming only) | Degree 2 → quadratic; Degree 3 → cubic (is chapter me sirf classify karna hai, deep solve nahi) |
| Value of polynomial at x = k | p(k) = a·k + b (x ki jagah k substitute karo) |
| Zero of a linear polynomial | ax + b = 0 ⟹ x = −b/a (exactly ek zero hota hai) |
| Linear growth | a > 0 → quantity constant rate se badhti hai (positive slope) |
| Linear decay | a < 0 → quantity constant rate se ghatti hai (negative slope) |
| Function notation | p: x → ax + b (input x, output p(x) = ax + b) |
| Slope-intercept form of graph | y = ax + b — a = slope (rate of change), b = y-intercept (value at x = 0) |
| Graph shape | Linear polynomial ka graph hamesha ek straight line hota hai, x-axis ko exactly 1 point (zero) par cut karta hai |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Har algebraic expression ko "linear" bol dena bina degree check kiye — jaise x² + 3x ko linear samajh lena, jabki uski highest power 2 hai isliye ye quadratic hai, linear nahi.
- Zero nikalte waqt sign ki galti karna — ax + b = 0 se x = b/a likh dena (galat), jabki sahi step x = −b/a hai (minus sign miss karna common mistake hai).
- Constant polynomial (jaise p(x) = 7) ko linear polynomial samajh lena — actually iska degree 0 hota hai kyunki coefficient of x = 0 hai, aur linear ke liye a ≠ 0 zaroori hai.
- y = ax + b me slope aur y-intercept ko ulta samajh lena — 'a' ko constant term aur 'b' ko rate of change bol dena, jabki 'a' slope hai aur 'b' y-intercept.
- Real-life word problems (growth/decay) me x kya represent kar raha hai (jaise 'minutes elapsed' vs 'total time') clearly define na karna — isse substitution galat ho jaata hai aur final answer wrong aata hai.
- √x, 1/x, ya x⁻¹ jaisi terms ko polynomial ka valid term maan lena — inme power whole number nahi hoti (1/2 ya −1), isliye ye expressions polynomial hote hi nahi hain, aur inhe identify karte waqt ye galti bahut common hai.
Board-Style Important Questions
- Linear polynomial ki definition ek example ke saath likho.
- Polynomial p(x) = 7x − 21 ka zero nikalo.
- p(x) = kx + 3 diya hai. Agar x = −1 iska zero hai, toh k ka value nikalo.
- Ek mobile recharge plan me ₹50 fixed cost hai aur ₹0.50 per minute call ka charge hai. Cost ka linear polynomial likho aur 100 minutes call ke liye total cost nikalo.
- y = −2x + 6 ka graph ka table banao (kam se kam 3 points), aur batao ye line x-axis aur y-axis ko kaha cut karti hai.
- Ek plant ki height lagaate waqt 15 cm thi aur har hafte 3 cm badh rahi hai. (i) Height ka polynomial h(x) likho (x = weeks). (ii) 8 hafte baad height nikalo. (iii) Plant ki height 60 cm kab hogi, x nikalo. (iv) Batao ye growth hai ya decay, aur kyu.
Aksar Poochhe Jaane Wale Sawaal
Class 9 Maths chapter 2 me kya sirf linear polynomials cover hote hain, ya quadratic/cubic bhi detail me aate hain?
Is chapter ("Introduction to Linear Polynomials") ka scope sirf linear polynomials (ax + b form) tak hai — inhi ka zero nikalna, growth/decay pattern, aur straight-line graph is chapter ka core hai. Quadratic aur cubic polynomials sirf classification/degree-identify karne tak mention hote hain (jaise 'x² + 3x quadratic hai'), inki deep factorization ya algebraic identities is chapter me nahi aati — wo Chapter 4 'Exploring Algebraic Identities' me cover hoti hai. Agar Chapter 4 ki extra questions chahiye, wo alag se dekh lena.
Ganita Manjari Class 9 Maths NCERT solutions ka poora set kahan milega, aur kitne chapters hain?
Session 2026-27 se Class 9 Maths ki naya textbook 'Ganita Manjari' hai (purani 'Mathematics' book completely retire ho chuki hai). Abhi sirf Ganita Manjari Part 1 published hai, jisme 8 chapters hain: (1) Orienting Yourself — Use of Coordinates, (2) Introduction to Linear Polynomials, (3) The World of Numbers, (4) Exploring Algebraic Identities, (5) I'm Up and Down, and Round and Round (Circles), (6) Measuring Space — Perimeter and Area, (7) Probability, (8) Sequences and Progressions. Har chapter ke solutions available hain.
Class 9 Maths Ganita Manjari Part 2 kab aayega?
Abhi tak (session 2026-27 ke dauraan) Ganita Manjari Part 2 release nahi hua hai aur official release date bhi announce nahi hui hai. Multiple secondary sources ke hisaab se Part 2 me Linear Equations in Two Variables, Euclid's Geometry, Lines & Angles, Triangles, Quadrilaterals, Surface Area & Volume, Statistics jaise topics expected hain — par ye sirf expected list hai, official confirmation nahi.
Class 9 Maths ke new syllabus 2026-27 ka chapter list PDF kahan se milega?
Sabse reliable source NCERT ki official site ncert.nic.in par 'GANITA MANJARI Textbook of Mathematics for GRADE 9 Part I' listing hai. Wahan se chapter-wise PDF download ho sakti hai. Abhi 8 chapters ka hi Part 1 available hai (jaisa upar list kiya gaya).
Linear polynomial ka graph banane ke liye Chapter 1 ke coordinates ka concept kaam aata hai kya?
Haan — y = ax + b jaisi linear polynomial ka graph banane ke liye Chapter 1 'Orienting Yourself — Use of Coordinates' me sikhaya gaya x-axis, y-axis, aur ordered pairs (x, y) plot karne ka basic concept directly use hota hai. Isliye Chapter 1 pehle clear hona helpful hai.
Linear growth/decay pattern ka Chapter 8 (Sequences and Progressions) se kya connection hai?
Linear growth/decay me quantity constant amount se badhti/ghatti hai (jaise p(x) = ax + b) — yehi pattern Chapter 8 'Sequences and Progressions' me arithmetic progression (AP) ke roop me detail me explore hota hai. Chapter 2 iska informal, real-life introduction deta hai.
Class 9 Maths — Saare Chapters
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