Introduction to Linear Polynomials (Linear Polynomials se Pehchaan): Class 9 Maths Chapter 2 Hinglish mein

Ganita Manjari Part 1 · Hinglish notes, simulations aur practice · Official NCERT chapter PDF ↗

Published 12 Oct 2026 · Updated 12 Oct 2026

96 min
1 simulation
43 practice questions
31 MCQs

Is chapter mein aap kya seekhenge

Kisi algebraic expression ke terms, variables, coefficients aur constant term pehchano, aur ek variable wale polynomial ki degree nikalo.
Linear polynomials ko pehchano, unhe input-output rules ki tarah use karke value nikalo, aur rozmarra ki situations se bane linear equations solve karo.
Constant difference dekhkar linear pattern pehchano aur uske nth term ka rule likho.
V = 200 + 40t jaise rules se linear growth aur linear decay ka model banao, aur batao ki aisa model kahan kaam karna band kar deta hai.
Do measurements se linear relationship y = ax + b nikalo aur samjhao ki situation mein a aur b ka kya matlab hai.
Do points ki madad se y = ax + b ka graph banao, aur usse slope, y-intercept aur kaun si lines parallel hain, yeh padho.

NCERT Mathematics Class 9, Chapter 2, pp. 16–40 par based original companion text. Yeh NCERT se juda nahi hai aur NCERT ne ise endorse nahi kiya hai.

2.1 Introduction

School ke cricket match mein scorer ko dhyaan se dekho. Woh har batter ke liye naya calculation nahi banati. Uske dimaag mein ek hi rule hai: ek four ke 4 runs, ek six ke 6, aur wickets ke beech daudkar liye gaye runs upar se jod do. Algebra se hum aisa rule ek hi baar likh sakte hain, jismein letters un numbers ki jagah lete hain jo har innings mein badalte hain.

Scorer ka rule ek algebraic expression hai. Is chapter mein hum aise expressions ko families mein baatenge, aur zyaada time sabse simple aur kaam ki family par lagaayenge: linear polynomials (रैखिक बहुपद). Hum inhe rules ki tarah padhenge, badhte patterns mein chhupe hue dhoondhenge, steady change ko samjhane ke liye use karenge, aur aakhir mein inhe seedhi lines ki tarah draw karenge.

Example 1NCERT Example 1 · p. 16

Ek batter ka score

Inter-house match mein Tanvi $f$ fours aur $s$ sixes maarti hai. Woh wickets ke beech daudkar 11 runs bhi banati hai. Uske total score ka expression likho. Agar woh 3 fours aur 2 sixes maare, to uska score kitna hoga?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Chalo $4 f + 6 s + 11$ ke parts ke naam rakhte hain. Jo pieces jode jaate hain, woh iske terms (पद) hain. Letters $f$ aur $s$ variables (चर) hain, kyunki unki values ek innings se doosri innings mein badalti (vary karti) hain; pehle tumne inhe letter-numbers kaha hoga. Jo number kisi variable se multiply hota hai, woh uska coefficient (गुणांक) hai, aur jis term mein koi variable nahi, woh constant term hai.

Table 2.1Algebraic expression ke parts

Part	$4 f + 6 s + 11$ mein	Yeh naam kyun
term	$4 f$ , $6 s$ aur $11$	plus signs se jude teen pieces
variable	$f$ aur $s$	alag-alag innings mein inki values alag hoti hain
coefficient	$f$ ke liye 4, $s$ ke liye 6	har ek kisi variable se multiply hota hai
constant term	$11$	ismein koi variable nahi, isliye yeh kabhi nahi badalta

Example 2NCERT Example 2 · p. 16

School trip ke liye seats

Ek school trip ke liye $b$ buses book karta hai. Har bus mein $s$ seats hain, aur har bus mein 2 seats teachers ke liye rakhi jaati hain. Students ke liye bachi seats ka expression likho. $b = 3$ aur $s = 40$ hone par kitni seats hain?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Is expression mein bhi do variables hain, lekin iska pehla term, $b s$ , do variables ka product hai. $4 f + 6 s + 11$ ka koi bhi term do variables ko aapas mein multiply nahi karta, aur isse dono expressions ka behaviour badal jaata hai. Agar tum $b$ aur $s$ dono ko double karo, to term $b s$ chaar guna ho jaata hai, jabki $f$ aur $s$ double hone par $4 f$ aur $6 s$ sirf double hote hain.

Example 3NCERT Example 3 · p. 17

Chaudai se do guna lamba plot

Ek kisaan sabziyon ke liye ek rectangular plot banata hai jiski length uski width ki double hai. Agar width $x$ metres hai, to plot ke perimeter aur area ke expressions likho. $x = 5$ hone par dono nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Score aur seats dono mein do letters chahiye the, lekin plot ko sirf ek letter, $x$ , chahiye. Iske area mein ek power bhi aa jaati hai: $2 x^{2}$ ka matlab hai $2 \times x \times x$ . Is chapter ka baaki hissa ek baar mein ek hi letter par tika rahega, jaise $6 x$ aur $2 x^{2}$ mein.

Quick check

3 mein se 1

b s - 2 b

mein doosre term mein

b

ka coefficient kya hai?

s

2

1

- 2

Answer dekho

Answer (D) $- 2$

Sign term ke saath chalta hai: doosra term

- 2 b

hai, isliye uska coefficient

- 2

hai. Letter

s

pehle term,

b s

, ka hissa hai.

Ek variable wale polynomials

Expression $2 x^{3} - x^{2} + 7 x - 5$ lo, yeh $x$ mein ek polynomial (बहुपद) hai. Har term ek number guna $x$ ki whole-number power hai: $2 x^{3}$ , $- x^{2}$ , $7 x$ aur $- 5$ . Har power ke aage wala number uska coefficient hai; dhyaan do ki $- x^{2}$ ka coefficient $- 1$ hai, bhale hi 1 likha nahi gaya. Jis term mein $x$ nahi hai woh constant term hai, aur $x$ ki sabse badi power, yahan 3, degree (घात) hai. Fig. 2.1 is expression ko ek-ek idea karke kholta hai.

5/5Sabse badi power 3 hai, isliye degree 3 hai.

Fig. 2.1

2 x^{3} - x^{2} + 7 x - 5

ke terms, coefficients aur powers.

Step 1 mein polynomial 2x cube, minus x square, plus 7x, minus 5 ek expression ki tarah likha hai. Step 2 chaaron terms ke charon taraf box banata hai; constant term minus 5 ka box orange hai. Step 3 terms ke neeche coefficients ki ek row jodta hai: 2, minus 1 (gola lagaya hai, kyunki 1 likha nahi jaata), 7, aur minus 5 jispar constant likha hai. Step 4 ek row jodta hai jismein har term mein x ki power hai: 3, 2, 1 aur 0. Step 5 power 3 par gola lagata hai aur ek banner dikhata hai ki sabse badi power 3 hai, isliye degree 3 hai.

Definition

Ek variable wala polynomial

Aisa expression jiska har term ek number guna ek hi variable ki whole-number power (0, 1, 2, 3, …) ho. Mathematicians ise univariate polynomial bhi kehte hain, 'uni' (ek) aur 'variate' (variable) se.

Example $2 x^{3} - x^{2} + 7 x - 5$ , $y^{4} - 9$ aur $3 m + 2$ ek variable wale polynomials hain.

Definition

Polynomial ki degree

Polynomial mein aane wali variable ki sabse badi power (jiska coefficient zero na ho).

Example $2 x^{3} - x^{2} + 7 x - 5$ ki degree 3 hai; $3 m + 2$ ki degree 1 hai.

Hum kyun kehte hain ki constant term ki power 0 hai? Kisi bhi non-zero number ki power 0 karne par 1 aata hai, isliye $- 5 = - 5 \times x^{0}$ . Isi wajah se jo polynomial sirf ek number ho, jaise $6$ , uski degree 0 hoti hai.

Jo power nahi dikhti, uska coefficient 0 hota hai. $3 x^{3} + 2 x - 1$ mein koi $x^{2}$ term nahi hai, isliye $x^{2}$ ka coefficient 0 hai; hum polynomial ko $3 x^{3} + 0 x^{2} + 2 x - 1$ bhi likh sakte hain, value badle bina.

Kam degree wale polynomials ke apne naam hain, jo Table 2.2 mein diye hain. Lekin ek variable wala har expression polynomial nahi hota. $\frac{1}{x} = x^{- 1}$ mein power $- 1$ hai, aur $\sqrt{x}$ mein power aadha hai. Dono mein se koi bhi power whole number nahi hai, isliye dono mein se koi bhi expression polynomial nahi hai.

Table 2.2Degree se polynomials ke naam

Degree	Naam	Examples
0	constant polynomial	$6$ , $- \frac{3}{4}$
1	linear polynomial	$5 m - 2$ , $9 - x$
2	quadratic polynomial	$t^{2} - 4 t$ , $3 - 2 x^{2}$
3	cubic polynomial	$y^{3} + 1$ , $4 x - x^{3}$

Quick check

4 mein se 1

6 - 5 y^{2} + y^{4} - y

ki degree kya hai?

1

2

4

6

Answer dekho

Answer (C) $4$

y

ki powers 0, 2, 4 aur 1 hain, isliye degree 4 hai. Yeh 6 nahi hai: woh to constant term hai, aur yeh pehle likhe gaye term ki power bhi nahi hai.

Practice · 5 questions

Practice 2.1: answers ke saath

1
Short answerEasyNCERT Exercise Set 2.1, Q1
Har polynomial ki degree likho.
1. $7 + x - 4 x^{2}$
  Answer dekho
  Answer 2
  $x$ ki sabse badi power 2 hai.
2. $5 p - 2 p^{4} + p^{3}$
  Answer dekho
  Answer 4
  Powers 1, 4 aur 3 hain; sabse badi 4 hai.
3. $12$
  Answer dekho
  Answer 0
  Non-zero constant ki degree 0 hoti hai, kyunki $12 = 12 p^{0}$ .
4. $- 3 + 9 w$
  Answer dekho
  Answer 1
  $9 w$ ki power 1 hai aur $- 3$ ki power 0, isliye degree 1 hai.
2
Short answerMediumNCERT Exercise Set 2.1, Q2
Har kind ka ek polynomial likho. Kai answers ho sakte hain.
1. Variable $z$ mein degree 1, aur constant term 6.
  Solution dikhao
  AnswerJaise, $4 z + 6$ .
  $a \neq 0$ ke saath koi bhi $a z + 6$ chalega, jaise $4 z + 6$ ya $6 - z$ .
2. Degree 2, aur $x$ wala koi term nahi (sirf ek $x^{2}$ term aur ek constant).
  Solution dikhao
  AnswerJaise, $3 x^{2} - 5$ .
  $x^{2}$ ke liye ek non-zero coefficient aur ek constant chuno, aur $x$ ko chhod do: $3 x^{2} - 5$ ya $x^{2} + 1$ .
3. Degree 3, aur theek do terms.
  Solution dikhao
  AnswerJaise, $x^{3} + 8$ ya $2 t^{3} - t$ .
  Ek term variable ka cube hona chahiye; doosra koi bhi chhoti power ho sakta hai, jaise $x^{3} + 8$ ya $2 t^{3} - t$ .
3
Short answerEasyNCERT Exercise Set 2.1, Q3
Polynomial $2 x^{5} - x^{4} + 9 x^{2} - x + 3$ ko dekho.
1. $x^{4}$ ka coefficient kya hai?
  Answer dekho
  Answer -1
  $- x^{4} = - 1 \times x^{4}$ , isliye coefficient $- 1$ hai.
2. $x^{2}$ ka coefficient kya hai?
  Answer dekho
  Answer 9
  Term $9 x^{2}$ hai.
3. $x^{3}$ ka coefficient kya hai?
  Answer dekho
  Answer 0
  Koi $x^{3}$ term nahi hai, isliye uska coefficient 0 hai.
Short answerEasyNCERT Exercise Set 2.1, Q4
$m^{4} + 3 m^{2} - 7$ mein $m$ ka coefficient batao.
Answer dekho
Answer 0
Polynomial mein koi $m$ term nahi hai: yeh $m^{4} + 0 m^{3} + 3 m^{2} + 0 m - 7$ hai. Isliye $m$ ka coefficient 0 hai.
5
Short answerMediumNCERT Exercise Set 2.1, Q5
Har polynomial ka constant term nikalo.
1. $12 - 5 k + k^{3}$
  Answer dekho
  Answer 12
  Jis term mein $k$ nahi hai woh 12 hai.
2. $(k - 2) (k + 3)$
  Answer dekho
  Answer -6
  $(k - 2) (k + 3) = k^{2} + 3 k - 2 k - 6 = k^{2} + k - 6$ , isliye constant term $- 6$ hai.

2.2 Linear Polynomials

Ab se hum degree 1 wale polynomials par focus karenge. Aisa har polynomial $a x + b$ ki tarah likha ja sakta hai, jahan $a$ aur $b$ numbers hain aur $a \neq 0$ . Agar $a$ zero hota, to $x$ gayab ho jaata aur sirf ek constant bachta. Isliye $3 x + 2$ , $5 - 4 t$ aur $0.5 m$ sab linear polynomials hain.

Definition

Linear polynomial

Degree 1 ka polynomial. Ise hamesha $a x + b$ ki tarah likh sakte hain, jahan $a \neq 0$ .

Example $7 y - 1$ , $4 - x$ aur $0.5 n$ linear hain; $x^{2} + 1$ aur $8$ nahi hain.

Jab bhi koi cheez har step par same amount se badalti hai, wahan linear polynomials aa jaate hain. Yeh rahe rozmarra ke do cases.

Example 4NCERT Example 4 · p. 19

Ek litre mein kitne kilometre

Arjun ka scooter ek litre petrol mein lagbhag 45 km chalta hai. $L$ litres mein woh kitni distance tay kar sakta hai, likho. $L = 1, 2, 3$ aur $4$ ke liye distance nikalo. Har extra litre se woh kitna aage ja sakta hai? Aur extra 1.5 litres se?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 5NCERT Example 5 · p. 19

Project file ki binding

School ke paas ek dukaan project files bind karti hai. Woh cover aur binding ke ₹25 leti hai, aur har printed page ke ₹2 alag se. $p$ pages ka bill likho aur 1 se 5 pages ki table banao. Riya ka bill ₹85 aaya. Usne kitne pages print karaye?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Fig. 2.2 ke bills ko line mein rakho: 27, 29, 31, 33, 35. Har bill ko agle bill se ghatao, to har baar 2 milta hai, kyunki har extra page ka wahi ₹2 lagta hai. Linear polynomials hamesha aisa hi behave karte hain: unhe barabar steps mein badhte inputs do, to outputs bhi barabar steps mein badalte hain. Numbers ki aisi list jismein saare jumps barabar hon, linear pattern kehlati hai, aur Section 2.3 mein hum aisi lists padhenge.

Example 3 wale plot ke areas alag tarah se behave karte hain. Jab width 1, 2, 3, 4, 5 hoti hai, to area $2 x^{2}$ 2, 8, 18, 32, 50 hota hai, aur jumps badhte jaate hain. Barabar jumps ek test hai jo hum use kar sakte hain: agar jumps barabar nahi hain, to rule linear nahi ho sakta.

4/4Jumps 6, 10, 14, 18 badhte jaate hain, isliye $2 x^{2}$ linear nahi hai.

Fig. 2.2Barabar jumps linear rule se aate hain; badhte jumps nahi.

Step 1 mein pages 1 se 5 aur binding bills 27, 29, 31, 33 aur 35 rupees ki table hai. Step 2 paas-paas ke bills ko solid green arcs se jodta hai, har arc par plus 2 likha hai. Step 3 plot ke liye doosri table jodta hai, rule A barabar 2 x square ke liye, widths 1 se 5 aur areas 2, 8, 18, 32 aur 50 square metres. Step 4 paas-paas ke areas ko dashed orange arcs se jodta hai jin par plus 6, plus 10, plus 14 aur plus 18 likha hai, jo barabar nahi hain.

Linear equations

Riya ke bill se $25 + 2 p = 85$ bana: ek side par linear polynomial aur doosri side par ek number. Is tarah ki equation linear equation (रैखिक समीकरण) kehlati hai. Ise solve karne ka matlab hai variable ki woh value dhoondhna jisse dono sides barabar ho jaayein. Hum dono sides mein same number jod ya ghata sakte hain, ya dono sides ko same non-zero number se multiply ya divide kar sakte hain, aur dono sides barabar hi rehti hain.

Example 6NCERT Example 6 · p. 20

Annual day ke liye kursiyan

Annual day ke liye school hall mein kursiyan do barabar blocks mein lagayi jaati hain, aur guests ke liye aage 14 extra kursiyan. Kul 150 kursiyan hain. Har block mein kitni kursiyan hain?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Fig. 2.3 yahi do moves ek balance par dikhata hai, kursiyon ke har block ke liye ek box aur 14 guest kursiyon ke liye ek bag. Dono pans se same amount hatane se, ya har pan ka aadha rakhne se, balance kabhi nahi jhukta. Yahan $2 x + 14$ linear polynomial hai aur 150 woh number hai jiske barabar use rakha gaya hai.

3/3Har side ka aadha rakho: $x = 68$ .

Fig. 2.3Dono sides ke saath same kaam karne se balance seedha rehta hai.

Step 1 mein ek seedha balance hai: left pan mein x likhe do boxes hain, kursiyon ke har block ke liye ek, aur 14 likha ek bag, aur right pan mein 150 likha ek bag hai; neeche equation 2x plus 14 barabar 150 likhi hai. Step 2 mein 14 wala bag left pan se uthaya jaata hai aur right bag 136 ho jaata hai, dono taraf take 14 off label ke saath; equation 2x barabar 136 ban jaati hai. Step 3 mein ek x box halka ho jaata hai aur right side 68 ke do bags mein bant jaati hai, jismein se ek halka hai, keep half label ke saath; equation x barabar 68 padhti hai.

Polynomials input–output rules ki tarah

Polynomial ko ek rule ki tarah bhi padh sakte hain jo ek number ko doosre number mein badal deta hai. Fig. 2.4 rule $p (x) = 3 x - 4$ ko ek ke baad ek do stages se banata hai: pehla stage jo bhi mile use teen guna karta hai, aur doosra 4 ghatata hai. Do numbers ko isse guzarkar dekho. Input 6 pehle 18 aur phir 14 banta hai, kyunki $3 \times 6 - 4 = 14$ . Input $- 2$ pehle $- 6$ aur phir $- 10$ banta hai.

5/5 $3 \times (- 2) = - 6$ aur $- 6 - 4 = - 10$ , isliye $p (- 2) = - 10$ .

Fig. 2.4Rule

p (x) = 3 x - 4

do stages wali machine ki tarah.NCERT Fig. 2.3 · p. 20

Do lambe machine boxes ek row mein khade hain, pehle par times 3 likha hai aur doosre par minus 4, aur columns ke upar x, 3x aur 3x minus 4 likha hai. Step 1 mein saare number slots khaali hain. Step 2 input 6 ko pehle slot mein daalta hai. Step 3 dikhata hai ki times 3 box se 18 nikalta hai. Step 4 dikhata hai ki minus 4 box se 14 nikalta hai. Step 5 neeche wali row mein doosra input, minus 2, chalata hai, jisse pehle minus 6 aur phir minus 10 milta hai.

Har input se theek ek output nikalta hai, aur aisi property wale rule ko function (फलन) kehte hain. Input aur uska output saath mein likhne ke liye hum $p (6) = 14$ likhte hain, jise 'p of 6 equals 14' padhte hain, ya $p (- 2) = - 10$ . Rule ka naam $p$ rakhne se hum kisi bhi number par uski value pooch sakte hain: $p (0) = - 4$ aur $p (10) = 26$ .

Area machine $A (10) = 2 \times 100 = 200$ deti hai, aur uske outputs 2, 8, 18, 32 mein 6, 10 aur 14 ke jumps hain. Isliye $p (x) = 3 x - 4$ ek linear function hai, jiske jumps barabar hain, jabki $A (x) = 2 x^{2}$ ek quadratic function hai, jiske jumps badhte jaate hain.

Quick check

4 mein se 1

Inmein se kaun sa linear polynomial hai?

5 - 3 y

y^{2} - 3

\frac{3}{y} + 1

4

Answer dekho

Answer (A) $5 - 3 y$

5 - 3 y

ki degree 1 hai.

y^{2} - 3

quadratic hai,

\frac{3}{y}

mein negative power hai isliye woh polynomial nahi hai, aur

4

ek constant hai.

Practice · 7 questions

Practice 2.2: answers ke saath

1
Short answerEasyNCERT Exercise Set 2.2, Q1
Linear polynomial $4 - 3 x$ ki value nikalo jab
1. $x = 0$
  Answer dekho
  Answer 4
  $4 - 3 (0) = 4$ .
2. $x = - 2$
  Answer dekho
  Answer 10
  $4 - 3 (- 2) = 4 + 6 = 10$ .
3. $x = 5$
  Answer dekho
  Answer -11
  $4 - 3 (5) = 4 - 15 = - 11$ .
2
Short answerEasyNCERT Exercise Set 2.2, Q2
Quadratic polynomial $2 t^{2} + 5 t - 3$ ki value nikalo jab
1. $t = 0$
  Answer dekho
  Answer -3
  $0 + 0 - 3 = - 3$ .
2. $t = - 2$
  Answer dekho
  Answer -5
  $2 (4) + 5 (- 2) - 3 = 8 - 10 - 3 = - 5$ .
3. $t = 3$
  Answer dekho
  Answer 30
  $2 (9) + 5 (3) - 3 = 18 + 15 - 3 = 30$ .
3
Short answerMediumNCERT Exercise Set 2.2, Q3
Ek water tank mein ek drum se chaar guna paani hai. Agar tank se 30 litres nikaale jaayein aur drum mein 6 litres daale jaayein, to dono mein barabar paani ho jaata hai. Shuru mein har ek mein kitna paani tha?
1. Shuru mein drum mein kitna paani tha?
  L
  Answer dekho
  Answer 12 L
  $4 x - 30 = x + 6$ se $3 x = 36$ milta hai, isliye $x = 12$ .
2. Shuru mein tank mein kitna paani tha?
  L
  Answer dekho
  Answer 48 L
  $4 \times 12 = 48$ litres. Check: $48 - 30 = 18$ aur $12 + 6 = 18$ .
Short answerMediumNCERT Exercise Set 2.2, Q4
Kapde ke do rolls ki lengths ka ratio $3 : 7$ hai. Dono milakar 150 m lambe hain. Har roll kitna lamba hai?
Solution dikhao
Answer45 m aur 105 m.
Maan lo lengths $3 k$ aur $7 k$ hain. Unka total $3 k + 7 k = 10 k = 150$ hai, isliye $k = 15$ . Rolls $3 \times 15 = 45$ m aur $7 \times 15 = 105$ m ke hain. Check: $45 + 105 = 150$ aur $45 : 105 = 3 : 7$ .
5
Short answerMediumNCERT Exercise Set 2.2, Q5
School play mein adult ticket ₹50 ka aur child ticket ₹20 ka hai. Kul 54 tickets bike aur collection ₹1,620 hua. Har tarah ke kitne tickets bike?
1. Adult tickets
  tickets
  Answer dekho
  Answer 18 tickets
  $50 a + 20 (54 - a) = 1620$ se $30 a + 1080 = 1620$ milta hai, isliye $30 a = 540$ aur $a = 18$ .
2. Child tickets
  tickets
  Answer dekho
  Answer 36 tickets
  $54 - 18 = 36$ . Check: $50 \times 18 + 20 \times 36 = 900 + 720 = 1620$ .
Short answerMediumNCERT Exercise Set 2.2, Q6
Ek quiz mein ₹1,500 ka prize fund pehli aur doosri team mein baanta jaata hai. Pehli team ko doosri team ke hisse ke double se ₹300 zyaada milte hain. Har team ka hissa nikalo.
Solution dikhao
AnswerDoosri team ₹400, pehli team ₹1,100.
Pehli team ko $2 x + 300$ milte hain, isliye $x + (2 x + 300) = 1500$ . Phir $3 x = 1200$ aur $x = 400$ . Pehli team ko $2 \times 400 + 300 = 1100$ rupees milte hain. Check: $400 + 1100 = 1500$ .
7
Short answerMediumNCERT Exercise Set 2.2, Q7
Ek isosceles triangle mein base dono barabar sides mein se har ek se 5 cm chhota hai. Perimeter 70 cm hai. Sides ki lengths nikalo.
1. Har barabar side
  cm
  Answer dekho
  Answer 25 cm
  $s + s + (s - 5) = 70$ se $3 s - 5 = 70$ milta hai, isliye $3 s = 75$ aur $s = 25$ .
2. Base
  cm
  Answer dekho
  Answer 20 cm
  $25 - 5 = 20$ cm. Check: $25 + 25 + 20 = 70$ .

2.3 Linear patterns ko explore karna

Rangoli designs aksar beech se bahar ki taraf badhte hain. Fig. 2.5 square tiles se bana ek simple design dikhata hai. Stage 1 mein ek centre tile hai aur uski chaaron sides par ek-ek tile. Har naya stage banane ke liye artist har arm ke end par ek aur tile jodti hai.

5/54 tiles wali chaar arms aur centre: $4 \times 4 + 1 = 17$ .

Fig. 2.5Har stage 4 tiles jodta hai, har arm ke end par ek.NCERT Fig. 2.4 · p. 21

Ek dark centre tile ke charon taraf square tiles ki plus shape badhti hai, aur right mein ek panel stage number, tiles ki ginti aur ab tak ki ginti ki list dikhata hai. Step 1 stage 1 dikhata hai: centre aur har side par ek tile, 5 tiles. Steps 2, 3 aur 4 har arm ke end par ek tile jodte hain, jo nayi tiles ki tarah orange mein dikhti hain, aur 9, 13 aur 17 tiles bante hain; list 5, 9, 13, 17 badhti hai aur paas-paas ke numbers ke beech plus 4 likha hai. Step 5 chaaron arms ko alag rangon mein dikhata hai aur samjhata hai: 4 tiles wali 4 arms aur 1 centre tile, 4 guna 4 plus 1 barabar 17.

Table 2.3Rangoli ke har stage par chahiye tiles

Stage $n$	1	2	3	4	5	6	7
Tiles	5	9	13	17	21	25	29

Stage $n$ ka rule nikalne ke do achhe tareeke hain. Growth view: stage 1 mein 5 tiles hain aur har agla stage 4 jodta hai, isliye stage $n$ mein $5 + 4 (n - 1)$ tiles hain. Structure view: stage $n$ mein $n$ tiles wali chaar arms hain, aur centre, isliye usmein $4 n + 1$ tiles hain. Dono rules ek jaise hone chahiye, aur algebra yeh confirm karta hai: $5 + 4 (n - 1) = 5 + 4 n - 4 = 4 n + 1$ .

Ginti 5, 9, 13, 17, … har baar wahi 4 se badhti hai, aur iske peeche ka rule, $4 n + 1$ , degree 1 ka polynomial hai. Yeh do baatein hamesha saath aati hain: neeche wali property prove karti hai ki degree 1 ka har rule barabar jumps banata hai.

PropertyBarabar steps andar, barabar steps bahar

Kisi bhi linear polynomial $p (n) = a n + b$ ke liye, lagaataar whole numbers par values mein hamesha $a$ ka fark hota hai: har $n$ ke liye $p (n + 1) - p (n) = a$ .

Rangoli check

3 mein se 1

Rangoli ke stage 20 mein kitni tiles chahiye?

tiles

Answer dekho

Answer 81 tiles

4 \times 20 + 1 = 81

Try karke dekho

Apna khud ka badhta pattern design karo

Tumhe chahiye

lagbhag 20 coins, buttons ya bottle caps
ek kaagaz ki sheet

5 coins, buttons ya bottle caps lekar apni pasand ki kisi bhi shape ka stage 1 banao.
Har naya stage banane ke liye theek 3 cheezein jodkar ise badhao.
Stages 2, 3 aur 4 banao aur ginti table mein likho.
Stage $n$ ka rule do tareekon se likho: growth se, aur apne design ki shape se.

Stage	1	2	3	4
Cheezein	5

Tumhari entries apne aap save hoti hain.

Example 7NCERT Example 7 · p. 22

Metro card par rides

Rehana ke metro card par ₹250 hain. Uske roz ke route par har ride ₹30 ki hai. $n$ rides ke baad kitne bachenge? Kitni rides ke baad ₹70 bachenge?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 8NCERT Example 8 · p. 23

Parcel bhejna

Ek courier 3 kg tak ke kisi bhi parcel ke ₹60 leta hai. 3 kg se upar har kilogram ke ₹25 aur lagte hain, aur weight ko upar ki taraf poore kilograms mein round kiya jaata hai. 1 se 7 kg ki table banao. $n$ kg ke parcel ke charge ka rule nikalo, jahan $n \geq 3$ . 12 kg ka parcel kitne ka padega?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Charges 60, 60, 60, 85, 110, 135, 160 bilkul shuru se linear pattern nahi banate, kyunki pehle jumps 0 hain. Lekin 3 kg se aage har jump 25 hai, aur rule $25 n - 15$ fit hota hai. Hum rule ko ulta bhi chala sakte hain: jis parcel ka charge ₹210 hai uska weight 9 kg hai, kyunki $25 n - 15 = 210$ se $25 n = 225$ milta hai.

Definition

Linear pattern

Numbers ki aisi list jismein har term pichhle term mein ek hi fixed number jodne se banta hai. Woh fixed number negative bhi ho sakta hai.

Example5, 9, 13, 17, … (4 jodo) aur 250, 220, 190, … ( $- 30$ jodo).

Compare

Linear hai ya nahi?

Linear patterns

7, 11, 15, 19, … (4 ke jumps)
40, 34, 28, 22, … ( $- 6$ ke jumps)
2.5, 3, 3.5, 4, … (0.5 ke jumps)

Linear nahi

1, 4, 9, 16, … (jumps 3, 5, 7)
3, 6, 12, 24, … (har term double hota hai)
60, 60, 60, 85, … (jumps 0, 0, 25)

Jo rules humne nikaale unhe phir se dekho: $4 n + 1$ , $250 - 30 n$ aur $25 n - 15$ . Har ek $n$ mein degree 1 ka polynomial hai, isliye upar wali property pakka karti hai ki unke jumps barabar hain. Yeh link ulti taraf bhi kaam karta hai. Jo list $f$ se shuru ho aur har baar ek fixed $d$ se badle, uska $n$ th term $f + d (n - 1)$ hota hai, aur yeh $d \neq 0$ hone par $n$ mein linear hai. Rangoli ke liye, $5 + 4 (n - 1) = 4 n + 1$ .

Quick check

4 mein se 1

Inmein se kaun si lists linear patterns hain?

7, 11, 15, 19

2, 4, 8, 16

40, 34, 28, 22

1, 3, 6, 10

Saare sahi options chuno

Answer dekho

Answer

(A) $7, 11, 15, 19$
(C) $40, 34, 28, 22$

7, 11, 15, 19

har baar 4 jodta hai aur

40, 34, 28, 22

har baar

- 6

jodta hai. Doubling (2, 4, 8, 16) se jumps 2, 4, 8 milte hain, aur 1, 3, 6, 10 ke jumps 2, 3, 4 hain; dono mein se koi bhi constant nahi.

Practice · 5 questions

Practice 2.3: answers ke saath

1
Short answerEasyNCERT Exercise Set 2.3, Q1
Ek school library saal ki shuruaat 320 books se karti hai aur har mahine 45 nayi books jodti hai.
1. 1, 2, 3 aur 4 mahine baad kitni books hain?
  Solution dikhao
  Answer365, 410, 455 aur 500.
  Har baar 45 jodo: $320 + 45 = 365$ , phir 410, 455 aur 500.
2. $n$ mahine baad books ki ginti ka expression likho.
  Solution dikhao
  Answer $320 + 45 n$
  $n$ mahine baad 320 mein 45 ke $n$ set jud chuke hain, isliye $320 + 45 n$ books hain.
3. Poora ek saal hone par kitni books hongi?
  books
  Answer dekho
  Answer 860 books
  $320 + 45 \times 12 = 320 + 540 = 860$ .
2
Short answerEasyNCERT Exercise Set 2.3, Q2
Ek school bus apne pehle stop se 56 students ke saath chalti hai. Har agle stop par 4 students utarte hain aur koi nahi chadhta.
1. 1, 2 aur 3 aur stops ke baad bus mein kitne students hain?
  Solution dikhao
  Answer52, 48 aur 44.
  Har baar 4 ghatao: 52, 48, 44.
2. $n$ stops ke baad students ki ginti ka expression likho.
  Solution dikhao
  Answer $56 - 4 n$
  $n$ stops par $4 n$ students utar jaate hain, aur $56 - 4 n$ bachte hain.
3. Kitne stops ke baad bus khaali ho jaati hai?
  stops
  Answer dekho
  Answer 14 stops
  $56 - 4 n = 0$ se $4 n = 56$ milta hai, isliye $n = 14$ .
3
Short answerEasyNCERT Exercise Set 2.3, Q3
Ek triangle ka base 8 cm hai. Uski height $h$ cm badal sakti hai.
1. Height 2 cm, 3 cm aur 4 cm hone par area nikalo. $h$ aur area ko kaun sa linear pattern jodta hai?
  Solution dikhao
  Answer8, 12 aur 16 square cm; area $= 4 h$ .
  Area $= \frac{1}{2} \times 8 \times h = 4 h$ square centimetres. $h = 2, 3, 4$ ke liye isse 8, 12 aur 16 milte hain. Height ka har extra centimetre 4 square centimetres jodta hai, isliye areas rule $4 h$ wala linear pattern banate hain.
2. Kaun si height se area 36 square centimetres hota hai?
  cm
  Answer dekho
  Answer 9 cm
  $4 h = 36$ , isliye $h = 9$ cm.
4
Short answerMediumNCERT Exercise Set 2.3, Q4
Ek glass fish tank ka rectangular base 40 cm lamba aur 25 cm chauda hai. Usmein dheere-dheere paani daala jaata hai, isliye paani ki depth $d$ cm badhti hai.
1. $d$ cm depth ke liye paani ka volume cubic centimetres mein likho.
  Solution dikhao
  Answer $V = 1000 d$
  Base ka area $40 \times 25 = 1000$ square centimetres hai, isliye $V = 1000 d$ cubic centimetres: 10 cm ke liye 10000, 15 cm ke liye 15000, aur aise hi aage, ek linear pattern.
2. Depth ka har extra centimetre kitne litres jodta hai?
  L
  Answer dekho
  Answer 1 L
  Har centimetre 1000 cubic centimetres jodta hai, jo 1 litre hai.
3. Jab tank mein 18 litres hon, to paani kitna gehra hai?
  cm
  Answer dekho
  Answer 18 cm
  18 litres yaani 18000 cubic centimetres, aur $1000 d = 18000$ se $d = 18$ cm milta hai.
5
Short answerEasyNCERT Exercise Set 2.3, Q5
Ek teacher ke paas check karne ke liye 180 answer sheets ka dher hai aur woh har ghante 12 sheets check karti hain.
1. 7 ghante baad kitni sheets abhi bhi bina check ki hain?
  sheets
  Answer dekho
  Answer 96 sheets
  $180 - 12 \times 7 = 180 - 84 = 96$ .
2. $n$ ghante baad bachi sheets ka linear pattern likho, aur pata karo ki woh kab khatam karengi.
  Solution dikhao
  Answer $180 - 12 n$ ; 15 ghante baad.
  Bachi sheets $= 180 - 12 n$ : 168, 156, 144, … Woh tab khatam karti hain jab $180 - 12 n = 0$ , yaani $n = 15$ ghante baad.

2.4 Linear growth aur linear decay

Bahut si quantities time ke saath steady tarah se badalti hain: tank mein bharta paani, phone battery ka charge, banti hui sadak ki length. Jab time ke har barabar hisse mein change same ho, to time mein ek linear polynomial achha model hota hai.

Example 9NCERT Example 9 · p. 24

Overhead tank bharna

Ek pump ek overhead tank bharta hai jismein pehle se 200 litres hain. Woh har minute 40 litres jodta hai, isliye $t$ minutes baad volume $V (t) = 200 + 40 t$ litres hai. $t = 0$ se 5 tak ki table banao. 12 minutes baad kitna paani hai? Tank 1000 litres par full hota hai. Woh kab bharega?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 10NCERT Example 10 · p. 24

Phone par streaming

Jab Kiran videos dekhna shuru karti hai, to uski phone battery 92% dikhati hai, aur har ghante 8 percentage points girti hai. $t$ ghante baad charge $B (t) = 92 - 8 t$ per cent hai. $t = 0$ se 4 tak ki table banao. 6 ghante baad charge kitna hai? Phone 20% par warning deta hai. Warning kab aayegi?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Fig. 2.6 dono situations ko aamne-saamne dikhata hai, har minute ya ghante ke liye ek bar. Tank ke bars har baar same amount se upar chadhte hain: yeh linear growth (रैखिक वृद्धि) hai. Battery ke bars har baar same amount se chhote hote hain: yeh linear decay (रैखिक ह्रास) hai. Bars ke tops ko jodo to seedhi lines milti hain, ek upar jaati hui aur ek neeche aati hui.

6/6Bars ke tops seedhi lines par hain: ek upar jaati, ek neeche aati.

Fig. 2.6Barabar time mein barabar change: left mein linear growth, right mein linear decay.

Do bar charts aamne-saamne hain: left mein tank ka paani litres mein, time minutes mein, aur right mein phone battery per cent mein, time ghanton mein. Step 1 pehle bars dikhata hai, 200 litres aur 92 per cent. Steps 2 se 5 har baar ek bar jodte hain: tank ke bars 240, 280, 320 aur 360 padhte hain, har ek 40 zyaada, aur battery ke bars 84, 76, 68 aur 60 padhte hain, har ek 8 kam. Step 6 har chart mein bars ke tops ke saath ek dashed seedhi line banata hai, tank ke liye upar jaati hui aur battery ke liye neeche aati hui.

Definition

Linear growth

Barabar jodne se growth: time ke har barabar hisse mein quantity mein wahi fixed amount judta hai.

ExampleTank mein har minute 40 litres judte hain.

Definition

Linear decay

Barabar ghatane se kami: time ke har barabar hisse mein quantity wahi fixed amount khoti hai.

ExampleBattery har ghante 8 percentage points khoti hai.

starting value + rate × time wale rule mein rate ka sign batata hai ki change kis tarah ka hai. $+ 40$ litres per minute ka rate matlab growth, aur $- 8$ per hour ka rate matlab decay. Har model ki limits bhi hoti hain. Battery ka rule $B (12) = 92 - 96 = - 4$ deta hai, jo impossible hai. Rule tabhi tak sense banata hai jab tak battery chale, $t = 0$ se $t = 11.5$ ghante tak, jab $B$ 0 ho jaata hai.

Quick check

3 mein se 1

Kaun si situation linear decay dikhati hai?

Ek bus pass jo har saal ₹5 mehenga hota haiBacteria jo har ghante double hote hainEk savings jar jismein har hafte ₹10 daale jaate hainTel ka 20 L ka dabba jisse har din 1.5 L use hota hai

Answer dekho

Answer (D) Tel ka 20 L ka dabba jisse har din 1.5 L use hota hai

Tel har din wahi 1.5 L kam hota hai, jo linear decay hai. Bus pass aur jar linear growth dikhate hain, aur doubling to linear hai hi nahi.

Practice · 4 questions

Practice 2.4: answers ke saath

1
Short answerEasyNCERT Exercise Set 2.4, Q1
Ek electric kettle 1 litre paani garam karti hai jo 24 °C se shuru hota hai. Garam hote samay temperature har minute 6 °C badhta hai.
1. $t = 0$ se $t = 12$ tak har 2 minutes par temperature bharo.
  Solution dikhao
  Answer24, 36, 48, 60, 72, 84, 96
  $t$ (min) 0 2 4 6 8 10 12
  $T$ (°C) 24 36 48 60 72 84 96
2. $t$ minutes baad temperature $T$ °C ka rule likho. Tumhari table ki kaun si baat dikhati hai ki yeh linear growth hai?
  Solution dikhao
  Answer $T = 24 + 6 t$
  $T = 24 + 6 t$ . Table mein har 2 minutes mein wahi 12 °C judta hai, yaani har minute mein 6 °C. Barabar time mein barabar badhna ise linear growth banata hai.
3. Kitne minutes baad paani 90 °C tak pahunchta hai?
  min
  Answer dekho
  Answer 11 min
  $24 + 6 t = 90$ se $6 t = 66$ milta hai, isliye $t = 11$ . Rule 100 °C par kaam karna band kar deta hai, jab paani ubalne lagta hai, yaani $t = 13$ se thoda pehle.
2
Short answerEasyNCERT Exercise Set 2.4, Q2
Ek family ke cooking-gas cylinder mein 14.2 kg gas hai. Family roz lagbhag 0.4 kg use karti hai.
1. $t$ din baad bachi gas, $g$ kg, ka rule likho, aur usse $t = 0, 5, 10, \dots, 35$ ke liye table bharo.
  Solution dikhao
  Answer $g = 14.2 - 0.4 t$ : 14.2, 12.2, 10.2, 8.2, 6.2, 4.2, 2.2, 0.2
  $g = 14.2 - 0.4 t$ . Har 5 din mein $5 \times 0.4 = 2$ kg use hoti hai, isliye har entry pichhli se 2 kam hai.
  $t$ (din) 0 5 10 15 20 25 30 35
  $g$ (kg) 14.2 12.2 10.2 8.2 6.2 4.2 2.2 0.2
2. Jaise hi 2 kg se kam gas bachti hai, family refill book karti hai. Kitne din baad aisa hota hai?
  din
  Answer dekho
  Answer 31 din
  30 din baad $g = 14.2 - 12 = 2.2$ kg, jo abhi 2 se kam nahi hai. 31 din baad $g = 14.2 - 12.4 = 1.8$ kg, isliye woh 31 din baad book karte hain.
3. Yeh linear decay kyun hai? Agar woh kabhi refill na karein, to gas kab khatam hogi?
  Solution dikhao
  AnswerHar din wahi 0.4 kg use hoti hai; gas 35.5 din baad khatam hogi.
  Gas har din wahi 0.4 kg kam hoti hai, isliye yeh linear decay hai. Yeh tab khatam hoti hai jab $14.2 - 0.4 t = 0$ , yaani $t = 35.5$ din.
3
Short answerEasyNCERT Exercise Set 2.4, Q3
Ek nayi metro line ka 18 km track pehle se bichh chuka hai. Builders har mahine 2.5 km track jodte hain.
1. Kaun sa rule $t$ mahine baad track ki length, $L$ km, deta hai?
  $L = 2.5 + 18 t$ $L = 20.5 t$ $L = 18 + 2.5 t$ $L = 18 - 2.5 t$
  Answer dekho
  Answer (C) $L = 18 + 2.5 t$
  Shuru mein ( $t = 0$ ) 18 km hai, aur har mahina 2.5 km jodta hai, isliye $L = 18 + 2.5 t$ . Rule $2.5 + 18 t$ start aur rate ki jagah badal deta hai, $20.5 t$ shuru mein 0 km deta hai, aur $18 - 2.5 t$ track ko chhota karta hai.
2. 8 mahine baad kitna track hoga?
  km
  Answer dekho
  Answer 38 km
  $18 + 2.5 \times 8 = 18 + 20 = 38$ km.
3. $t = 12$ tak har doosre mahine ke liye $L$ ki table banao. Poori line 55.5 km lambi hogi. Woh kab poori hogi, aur yeh linear growth kyun hai?
  Solution dikhao
  Answer18, 23, 28, 33, 38, 43, 48; 15 mahine baad poori.
  $t$ (mahine) 0 2 4 6 8 10 12
  $L$ (km) 18 23 28 33 38 43 48
  Har mahine wahi 2.5 km judta hai, isliye length linearly badhti hai. $18 + 2.5 t = 55.5$ se $2.5 t = 37.5$ milta hai, isliye $t = 15$ mahine.
4
Short answerEasyNCERT Exercise Set 2.4, Q4
Ek school office ke paas printer paper ki 2400 sheets hain aur woh roz 150 sheets use karta hai.
1. Maan lo $P (d)$ woh sheets hain jo $d$ din baad bachti hain. $P (d)$ ka rule likho. Rule kaise dikhata hai ki yeh growth nahi, decay hai?
  Solution dikhao
  Answer $P (d) = 2400 - 150 d$
  $P (d) = 2400 - 150 d$ . $d$ se multiply hone wala number negative hai: har din wahi 150 sheets kam hoti hain, isliye yeh linear decay hai.
2. Jab 600 sheets bachti hain, office paper dobara mangata hai. Kitne din baad woh order karta hai?
  din
  Answer dekho
  Answer 12 din
  $2400 - 150 d = 600$ se $150 d = 1800$ milta hai, isliye $d = 12$ .
3. $d = 0, 4, 8, 12, 16$ ke liye $P (d)$ ki table banao. Paper kab khatam hota hai?
  Solution dikhao
  Answer2400, 1800, 1200, 600, 0; 16 din baad.
  $d$ (din) 0 4 8 12 16
  $P (d)$ 2400 1800 1200 600 0
  Har 4 din mein $4 \times 150 = 600$ sheets use hoti hain. Stock $d = 16$ par 0 ho jaata hai, kyunki $2400 \div 150 = 16$ .

$t$ (min)	0	2	4	6	8	10	12
$T$ (°C)	24	36	48	60	72	84	96

$t$ (din)	0	5	10	15	20	25	30	35
$g$ (kg)	14.2	12.2	10.2	8.2	6.2	4.2	2.2	0.2

$t$ (mahine)	0	2	4	6	8	10	12
$L$ (km)	18	23	28	33	38	43	48

$d$ (din)	0	4	8	12	16
$P (d)$	2400	1800	1200	600	0

2.5 Linear Relationships

Ab tak har rule mein ek variable tha. Lekin aksar hum do aisi quantities ko jodna chahte hain jo saath-saath badalti hain: stage number aur tiles ki ginti, minutes aur litres, distance aur fare. Jab ek quantity $y$ kisi quantity $x$ par rule $y = a x + b$ se depend karti hai, to hum kehte hain ki $x$ aur $y$ ek linear relationship (रैखिक संबंध) mein hain.

Definition

Linear relationship

Do badalti quantities ke beech aisa link jismein ek quantity doosri ka fixed multiple plus ek fixed number hoti hai: $y = a x + b$ .

ExampleRangoli ke liye $y = 4 x + 1$ , jahan $x$ stage hai aur $y$ tiles ki ginti.

Pichhle sections ke do rules pehle se is form mein hain. Example 9 wala tank $V = 40 t + 200$ follow karta hai: time $t$ $x$ ka role nibhata hai, volume $V$ $y$ ka role, $a = 40$ aur $b = 200$ . Section 2.3 wali rangoli $y = 4 x + 1$ deti hai jab $x$ stage ho aur $y$ tiles ki ginti. Lekin aksar humein rule nahi, sirf kuch readings di jaati hain. Agar hum jaante hain ki relationship linear hai, to $a$ aur $b$ wapas nikaalne ke liye do readings kaafi hain.

Example 11NCERT Example 11 · p. 26

Taxi app ke charges nikaalna

Ek taxi app ek fixed booking fee aur har kilometre ki fixed amount leta hai. 4 km ki ride ₹92 ki padi aur 10 km ki ride ₹170 ki. Agar $x$ km ka fare $y$ rupees, $y = a x + b$ hai, to $a$ aur $b$ nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Dono numbers ka saaf matlab hai. Extra 6 km ke ₹78 zyaada lage, isliye har kilometre $78 \div 6 = 13$ rupees ka hai: $a$ rate per kilometre hai. 0 km par rule ₹40 deta hai: $b$ booking fee hai, jo ride kitni bhi chhoti ho, deni padti hai. Fig. 2.7 yahi soch ek graph par dikhata hai.

4/4Line ko 0 km tak badhao: fare ₹40 hai, isliye $b = 40$ .

Fig. 2.7Rise ko run se divide karne par rate milta hai; 0 km tak wapas jaane par booking fee milti hai.

Axes par aadi taraf 0 se 12 tak distance kilometres mein aur upar 0 se 200 tak fare rupees mein hai. Step 1 do points plot karta hai, (4, 92) aur (10, 170). Step 2 pehle point se doosre tak ek dashed right-angled raasta banata hai, 6 km aadi taraf aur 78 rupees upar. Step 3 ek card jodta hai jispar likha hai 78 divided by 6 barabar 13 per km. Step 4 points ko ek line se jodta hai aur use dashed karke wapas fare axis tak badhata hai, jahan ek ring 0 km par 40 rupees mark karti hai, yaani booking fee.

Graph ek shortcut sujhata hai jo tum tab use kar sakte ho jab bhi relationship linear ho: $a = \frac{change in y}{change in x} = \frac{170 - 92}{10 - 4} = 13$ . $a$ pata chalne ke baad kisi bhi measurement se wapas chalkar $b$ nikalo. Neeche wali machine mein ek linear rule chhupa hai; use todne ke liye yahi idea use karo.

InteractiveMachine ka rule todo

Ek machine jismein y barabar a x plus b form ka ek chhupa rule hai. Tum input mein koi bhi number type karte ho aur Feed it dabate ho; output dikhta hai aur har input-output pair ek table mein jud jaata hai jo input ke hisaab se sorted hai, upar wali row se change ke saath. Tum a aur b ke guesses daalte ho aur Check dabate ho; galat guess ko woh input dikhaya jaata hai jahan woh machine se match nahi karta. Buttons hints dete hain, rule dikhate hain aur nayi machine shuru karte hain, aur ek menu teen levels mein se ek chunta hai: positive whole numbers, negatives allowed, ya aisa slope jo half par khatam hota hai, jaise 1.5 ya minus 0.5.

Fig. 2.8Machine mein numbers daalo, uske outputs dekho, phir uska rule

y = a x + b

guess karo.

Quick check

3 mein se 1

Ek plumber

x

ghante kaam ke liye

y = 150 x + 200

rupees leta hai. 200 kis cheez ke liye hai?

Ek fixed visiting chargeHar ghante ka chargeKaam ke ghanton ki ginti2 ghante ka bill

Answer dekho

Answer (A) Ek fixed visiting charge

x = 0

ghante par bhi charge 200 hai, isliye yeh visit ka fixed charge hai. Har ghante ka charge 150 hai, jo

x

ka coefficient hai.

Practice · 3 questions

Practice 2.5: answers ke saath

1
Short answerMediumNCERT Exercise Set 2.5, Q1
Ek electrician ek fixed visiting fee aur ek hourly rate leta hai. 2 ghante ka kaam ₹650 ka pada aur 5 ghante ka kaam ₹1,250 ka. $x$ ghante ka charge $y = a x + b$ rupees hai. $a$ aur $b$ nikalo.
1. $a$ , hourly rate
  rupees per ghanta
  Answer dekho
  Answer 200 rupees per ghanta
  $a = \frac{1250 - 650}{5 - 2} = \frac{600}{3} = 200$ .
2. $b$ , visiting fee
  rupees
  Answer dekho
  Answer 250 rupees
  $650 = 2 \times 200 + b$ , isliye $b = 250$ . Check: $5 \times 200 + 250 = 1250$ .
2
Short answerMediumNCERT Exercise Set 2.5, Q2
Ek spring hook se latki hai. 200 g mass ke saath woh 17 cm lambi hai, aur 500 g mass ke saath 23 cm lambi. Uski length $L$ cm aur mass $m$ g, $L = a m + b$ follow karte hain.
1. $a$ nikalo.
  cm per g
  Answer dekho
  Answer 0.02 cm per g
  $a = \frac{23 - 17}{500 - 200} = \frac{6}{300} = 0.02$ cm har gram ke liye.
2. $b$ nikalo. Yeh spring ke baare mein kya batata hai?
  Solution dikhao
  Answer $b = 13$ : bina kisi mass ke spring 13 cm lambi hai.
  $17 = 0.02 \times 200 + b = 4 + b$ , isliye $b = 13$ . $m = 0$ rakhne par $L = 13$ milta hai, isliye $b$ spring ki woh length hai jab usse kuch bhi latka na ho.
3
Short answerHardNCERT Exercise Set 2.5, Q3
Ek candle steady tarah se jalkar chhoti hoti hai. 2 ghante jalne ke baad woh 21 cm oonchi hai, aur 5 ghante baad 12 cm oonchi. $t$ ghante baad uski height $h$ cm, $h = a t + b$ follow karti hai.
1. $a$ nikalo.
  cm per ghanta
  Answer dekho
  Answer -3 cm per ghanta
  $a = \frac{12 - 21}{5 - 2} = \frac{- 9}{3} = - 3$ . Yeh negative hai kyunki height kam hoti hai.
2. $b$ nikalo. $a$ aur $b$ candle ke baare mein kya batate hain?
  Solution dikhao
  Answer $b = 27$ : jalaate samay candle 27 cm oonchi thi, aur woh har ghante 3 cm kam hoti hai.
  $21 = - 3 \times 2 + b$ , isliye $b = 27$ . Check: $- 3 \times 5 + 27 = 12$ . $t = 0$ par height $b = 27$ cm hai, yaani nayi candle ki height. Slope $a = - 3$ batata hai ki woh har ghante 3 cm kam hoti hai: linear decay.
3. Kitne ghante jalne ke baad candle khatam ho jaati hai?
  ghante
  Answer dekho
  Answer 9 ghante
  $27 - 3 t = 0$ se $t = 9$ ghante milte hain.

2.6 Linear relationships ko dekhna

Example 11 mein taxi fares ke graph ne rate aur booking fee ko dekhna aasaan bana diya. Ab hum kisi bhi relationship $y = a x + b$ ke liye aise graphs banayenge. Mukhya baat simple hai: linear relationship mein fit hone wale har pair $(x, y)$ ko plot karo, to saare points ek hi seedhi line par aate hain. Isi liye in polynomials ko linear kehte hain.

Do points se line tay hoti hai

Do points se theek ek hi seedhi line ban sakti hai. Isliye $y = a x + b$ ka graph banane ke liye do points nikalo, unhe plot karo, unse hokar ek line kheencho aur use dono points ke aage tak le jao. Teesra point achha check hota hai. Fig. 2.9 yahi $y = 2 x - 3$ ke liye karta hai: $x = 0$ se $y = - 3$ milta hai, aur $x = 2$ se $y = 1$ .

4/4 $C (4, 5)$ equation mein fit hota hai aur line par hai; $D (5, 4)$ nahi.

Fig. 2.9Do points line tay karte hain; teesra point use check karta hai.NCERT Fig. 2.5 · p. 28

Left mein ek coordinate grid aur right mein x aur y values ki chhoti table hai. Step 1 do rows wali table dikhata hai: x barabar 0 se y barabar minus 3 milta hai, jispar A likha hai, aur x barabar 2 se y barabar 1, jispar B likha hai. Step 2 A ko (0, minus 3) par aur B ko (2, 1) par plot karta hai. Step 3 unhe ek seedhi line se jodta hai jo poore grid par badhayi gayi hai aur jispar y barabar 2x minus 3 likha hai. Step 4 C ko (4, 5) par jodta hai, jo line par hai aur tick laga hai, aur D ko (5, 4) par, jo line ke neeche hai aur cross laga hai; table dikhati hai ki x barabar 5 ke liye y barabar 7 chahiye, 4 nahi.

Kisi bhi point ka test yeh hai: uska $x$ -coordinate rule mein rakho. Agar rule point ka $y$ -coordinate deta hai, to point line par hai; warna nahi. $C (4, 5)$ pass hota hai, kyunki $2 \times 4 - 3 = 5$ . $D (5, 4)$ fail hota hai, kyunki $2 \times 5 - 3 = 7$ , 4 nahi, isliye $D$ line ke neeche hai.

Think and reflectNCERT Think and Reflect · p. 28

Usi line par aur points

$y = 2 x - 3$ ke liye table poori karo. Phir points ko graph paper par plot karo. Kya woh sab $A$ aur $B$ se jaati line par aate hain?

$x$	−1	1	3	4	6	10
$y$	−5			5

Tumhari entries apne aap save hoti hain.

Quick check

3 mein se 1

Line

y = 2 x - 3

par,

x = 10

hone par

y

kya hai?

Answer dekho

Answer 17

2 \times 10 - 3 = 17

, isliye

(10, 17)

line par hai.

Hum ulti taraf bhi ja sakte hain: plot kiye points se shuru karke rule nikaal sakte hain. Fig. 2.10 try karne ke liye points ke do sets dikhata hai.

4/4Har $y$ apne $x$ ka $- 1.5$ guna hai: $y = - 1.5 x$ .

Fig. 2.10Jo points ek line par hain, woh ek hi linear rule follow karte hain.NCERT Figs 2.6–2.7 · p. 29

Do chhote grids, har ek dono axes par minus 4 se 4 tak. Step 1 Set A dikhata hai: paanch gol points (minus 4, minus 2), (minus 2, minus 1), (0, 0), (2, 1) aur (4, 2) par. Step 2 unse hokar ek seedhi line banata hai jispar y barabar 0.5x likha hai. Step 3 doosre grid par Set B dikhata hai: paanch square points (minus 2, 3), (minus 1, 1.5), (0, 0), (1, minus 1.5) aur (2, minus 3) par. Step 4 unse hokar ek dashed neeche aati line banata hai jispar y barabar minus 1.5x likha hai.

Example 12NCERT Example 12 · p. 28

Points se rule padhna

Points $(- 4, - 2)$ , $(- 2, - 1)$ , $(0, 0)$ , $(2, 1)$ aur $(4, 2)$ Fig. 2.10 mein Set A banate hain. Check karo ki woh ek seedhi line par hain aur uski equation nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 13NCERT Example 13 · p. 29

Neeche aati line

Fig. 2.10 ke Set B mein $(- 2, 3)$ , $(- 1, 1.5)$ , $(0, 0)$ , $(1, - 1.5)$ aur $(2, - 3)$ hain. Inse guzarne wali line ki equation nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Origin se guzarti lines: slope

Example 14NCERT Example 14 · p. 29

Steep aur gentle lines

Same axes par $y = x$ , $y = 2.5 x$ aur $y = 0.4 x$ banao. Teeno lines mein kya fark hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

4/4 $a < 0$ hone par lines left se right neeche aati hain.

Fig. 2.11Har line

y = a x

origin se guzarti hai;

a

uski steepness aur direction tay karta hai.NCERT Figs 2.8–2.11 · p. 30

Dono axes par minus 6 se 6 tak ek grid, aur right mein ek legend. Step 1 dotted line y barabar x dikhata hai. Step 2 zyaada steep solid line y barabar 2.5x jodta hai, jis par point (2, 5) mark hai. Step 3 halki dashed line y barabar 0.4x jodta hai, jis par point (5, 2) mark hai, jo line y barabar x mein (2, 5) ki mirror image hai. Step 4 teen neeche aati lines jodta hai, y barabar minus x, y barabar minus 2.5x aur y barabar minus 0.4x, jo falling heading ke neeche likhi hain.

Teeno lines mein fark sirf itna hai ki right ki taraf 1 ke har step par woh kitna upar chadhti hain: $y = 2.5 x$ ke liye 2.5, $y = x$ ke liye 1 aur $y = 0.4 x$ ke liye 0.4. Yeh chadhai khud number $a$ hai, aur ise line ka slope (ढाल) kehte hain. 1 se bada slope $y = x$ se zyaada steep line deta hai, aur 0 aur 1 ke beech ka slope halki line deta hai. Lines $y = 2.5 x$ aur $y = 0.4 x$ line $y = x$ mein ek doosre ki mirror images bhi hain. Ek par point $(2, 5)$ doosri par $(5, 2)$ se match karta hai, aur unke slopes $\frac{5}{2}$ aur $\frac{2}{5}$ reciprocals hain.

Definition

Slope

$y = a x + b$ mein number $a$ . Yeh batata hai ki $x$ ke 1 badhne par $y$ kitna badalta hai: $a > 0$ hone par $y$ upar jaata hai aur $a < 0$ hone par neeche.

Example $y = 2.5 x$ ka slope 2.5 hai; $y = - 0.4 x$ ka slope $- 0.4$ hai.

Example 15NCERT Example 15 · p. 31

Neeche aati lines

Same axes par $y = - x$ , $y = - 2.5 x$ aur $y = - 0.4 x$ banao aur unhe Example 14 ki lines se compare karo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

To slope ka sign direction batata hai aur uska size steepness. Yeh Section 2.4 se judta hai. Tank, jo linearly badhta hai, ka graph upar jaata hai aur slope positive hai. Battery, jo linearly ghatti hai, ka graph neeche aata hai aur slope negative hai.

Slope linear pattern ke constant difference se bhi match karta hai. Fig. 2.12 Section 2.3 ki rangoli wali ginti plot karta hai, aadi taraf stage number aur upar tiles ki ginti. Ek stage se agle tak points 1 right aur 4 upar jaate hain. To 4, list 5, 9, 13, 17 ka jump bhi hai aur $y = 4 x + 1$ ka slope bhi. Yahan dono axes ke scales alag hain, isliye steepness ko sirf aankh se mat aanko.

3/3Saare points $y = 4 x + 1$ par hain, jiska slope 4 hai.

Fig. 2.121 aadi taraf ke har step par 4 upar: slope common jump ke barabar hai.

Axes par aadi taraf 0 se 4 tak stage number aur upar 0 se 20 tak tiles ki ginti hai. Step 1 heights 5, 9, 13 aur 17 par chaar points plot karta hai, aur unse stage axis tak dotted lines. Step 2 unke beech ek seedhi (staircase) banata hai: har pair ke beech plus 1 ka blue step aadi taraf aur plus 4 ka green step upar. Step 3 chaaron points se hokar seedhi line y barabar 4x plus 1 banata hai aur use wapas tile axis tak badhata hai, jahan ek ring (0, 1) mark karti hai.

SimulationLine ke saath barabar steps

Ek coordinate plane jo aadi taraf minus 2 se 8 aur upar minus 3 se 10 tak hai, chaar draggable points ke saath, A (0, minus 1), B (2, 2), C (4, 5) aur D (6, 8), jo sab line y barabar 1.5x minus 1 par hain. Segments A ko B se, B ko C se aur C ko D se jodte hain, aur har ek naapa jaata hai; teeno root 13 se shuru hote hain. Kisi point ko press karke pakadne, drag karne ya keyboard se focus karne par us par milne wale segments ke aadi aur upar wale legs, delta x aur delta y, dikhte hain. Points half units par snap hote hain, aur drag karte samay lengths update hoti hain.

Fig. 2.13Points

A

B

C

aur

D

line

y = 1.5 x - 1

par shuru hote hain, segments se jude hue.

Try karo

$D$ ko press karke pakdo (ya Tab key se select karo) taaki $C D$ ke aadi aur upar wale legs dikhein. $Δ x$ aur $Δ y$ padho: $Δ y \div Δ x$ kya hai? $A B$ ke liye $A$ ke saath yahi karo.
$D$ ko upar $(6, 10)$ tak drag karo. Ab raasta kahan mudta hai, aur $C D$ ki length ka kya hota hai? Ise phir se seedha karne ke liye $D$ ko kahan jaana hoga?
$D$ ko wapas $(6, 8)$ par rakho aur $C$ ko neeche $(3, 3.5)$ tak drag karo. Ab steps alag-alag size ke hain. $C$ ko press karke pakdo: kya $B C$ aur $C D$ ke liye $Δ y \div Δ x$ abhi bhi 1.5 hai?

$b$ badalna: parallel lines

Example 16NCERT Example 16 · p. 33

Same slope wali teen lines

Same axes par $y = - x - 2$ , $y = - x + 1$ aur $y = - x + 4$ banao. Tumhe kya dikhta hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

4/4Gaps har jagah 3 rehte hain, isliye lines parallel hain.

Fig. 2.14Same slope, alag

b

: line bina ghoome upar khisakti hai.NCERT Figs 2.12–2.13 · p. 33

Aadi taraf minus 5 se 6 aur upar minus 6 se 7 tak ek grid, right mein ek legend. Step 1 solid line y barabar minus x minus 2 dikhata hai. Step 2 uski ek copy ko 3 units upar khiskakar dashed line y barabar minus x plus 1 banata hai. Step 3 ek aur copy ko 3 units aur upar khiskakar dotted line y barabar minus x plus 4 banata hai. Step 4 woh points mark karta hai jahan lines y-axis ko cross karti hain, (0, minus 2), (0, 1) aur (0, 4), aur x barabar minus 2 par arrows banata hai jo paas-paas ki lines ke beech 3 ke barabar vertical gaps dikhate hain.

Example 16 ki table ka koi bhi column neeche tak padho: $y$ ki teeno values mein hamesha 3 ka fark hai, wahi jo $b$ ki values ke beech hai. Rule mein ek number jodne se line ka har point utna upar uth jaata hai, isliye poori line bina tilt hue upar khisak jaati hai. Isliye same slope wali lines ke beech fixed gap rehta hai aur woh kabhi cross nahi karti.

Line $y$ -axis se kahan milti hai

$y$ -axis ke har point par $x = 0$ hota hai. $y = a x + b$ mein $x = 0$ rakhne par $y = a \times 0 + b = b$ milta hai. Isliye line $y = a x + b$ hamesha $y$ -axis se $(0, b)$ par milti hai, aur number $b$ ko uska y-intercept (y-अंतःखंड) kehte hain. Fig. 2.15 $b$ ki alag values wali teen lines dikhata hai.

4/4 $y$ -axis par $x = 0$ , isliye $y = b$ .

Fig. 2.15Line

y = a x + b

y

-axis ko

(0, b)

par cross karti hai.NCERT Fig. 2.14 · p. 35

Aadi taraf minus 5 se 5 aur upar minus 5 se 6 tak ek grid, right mein ek legend. Step 1 solid line y barabar 2x plus 4 banata hai aur y-axis ke saath uske crossing (0, 4) par ring lagata hai. Step 2 dashed line y barabar minus 0.5x plus 1 jodta hai, jo (0, 1) par cross karti hai. Step 3 dotted line y barabar x minus 3 jodta hai, jo (0, minus 3) par cross karti hai. Step 4 poore y-axis ko highlight karta hai aur likhta hai ki y-axis par x barabar 0, isliye y barabar b.

Definition

y-intercept

$y = a x + b$ mein number $b$ . Yeh batata hai ki line $y$ -axis se kahan milti hai: $(0, b)$ par.

Example $y = x - 3$ ka $y$ -intercept $- 3$ hai: yeh $y$ -axis ko origin se 3 units neeche cross karti hai.

$b$ ka sign batata hai ki crossing origin ke kis taraf hai. Line $y = 2 x + 4$ $O$ se 4 units upar cross karti hai, jabki $y = x - 3$ usse 3 units neeche, $(0, - 3)$ par. Isliye $y$ -intercept ek signed number hai, sirf distance nahi. Jab $b = 0$ , to crossing point khud origin hota hai, jaise Origin se guzarti lines: slope ki lines ke liye.

PropertyEquation se line padhna

$y = a x + b$ form ki lines lo. (i) Har ek $y$ -axis ko $(0, b)$ par cross karti hai, aur right ki taraf 1 ka har step $y$ ko $a$ se badalta hai. (ii) Same $a$ lekin alag $b$ wali lines parallel hoti hain: har ek baaki lines ki copy hai, upar ya neeche khiskayi hui. (iii) Same $b$ wali saari lines $(0, b)$ se guzarti hain, aur $a$ badalne se line us point ke charon taraf ghoomti hai.

(i) $x = 0$ par $y = a \times 0 + b = b$ , isliye line $y$ -axis ko $(0, b)$ par cross karti hai. $x$ se $x + 1$ par jaane se $y$ mein $a (x + 1) + b - (a x + b) = a$ ka change hota hai, jo theek wahi hai jo slope naapta hai.

(ii) $b_{1} \neq b_{2}$ ke saath $y = a x + b_{1}$ aur $y = a x + b_{2}$ lo. Har $x$ par doosri line pehli se $b_{2} - b_{1}$ upar hai: har jagah same gap. Agar lines kisi $x$ par milti, to $a x + b_{1} = a x + b_{2}$ hota, jisse $b_{1} = b_{2}$ milta. Yeh galat hai, isliye lines kabhi nahi milti: woh parallel hain.

(iii) Same $b$ wali har line $y = a x + b$ , $(0, b)$ se guzarti hai, $a$ ki value chahe kuch bhi ho. Sirf uski steepness aur direction badalti hai, isliye lines us point ke charon taraf ghoomti hain.

Line lab yeh sab tumhare haath mein de deta hai. $a$ aur $b$ ke sliders chalao, compare karne ke liye lines ki copies rakho, aur phir Match the line khelo.

InteractiveLine lab: slope aur intercept

Minus 6 se 6 tak ke grid par y barabar a x plus b ka ek interactive graph. Sliders aur plus aur minus buttons slope a ko minus 4 se 4 tak aur y-intercept b ko minus 6 se 6 tak badalte hain. Line ke crossing point (0, b) par ring hoti hai aur ek chhota step dikhta hai, 1 aadi taraf aur a upar. Ek table x minus 2 se 3 tak ke liye y aur paas-paas ki values ke beech constant change a dikhati hai, aur ek status box batata hai ki line upar jaati hai ya neeche aati hai, kya woh y barabar x se zyaada steep hai, aur kya woh kisi rakhi hui line ke parallel hai ya usse y-axis par milti hai. Tum line y barabar x dikha sakte ho, compare karne ke liye lines ki chaar copies tak rakh sakte ho, aur Match the line game par switch kar sakte ho jo match karne ke liye ek dashed target line banata hai, hints ke saath.

Fig. 2.16

y = a x + b

mein

a

aur

b

badlo aur line, uska

y

-intercept, uska 1-across step aur uski values ki table dekho.

Try karke dekho

Line lab ke saath explore karo

$b = 0$ set karo aur Show the line y = x par tick karo. $a$ ko 0.4 se 4 tak slide karo. Kaun sa point kabhi nahi hilta, aur $a$ ki kin values ke liye tumhari line $y = x$ se zyaada steep hai?
Keep this line dabao, phir sirf $b$ badlo. Status box dono lines ke baare mein kya kehta hai? Ab rakhi hui lines clear karo, $a = 1.5$ , $b = 2$ set karo, use rakho, aur same $b$ ke saath $a = - 1.5$ try karo. Ab status box kya kehta hai?
Match the line par switch karo aur teen dashed lines match karo. Tum pehle kya dhoondhte ho, $a$ ya $b$ , aur kaise?

Recap

Summary aur key points

Ek variable wala polynomial terms ka sum hota hai, jismein har term ek number guna variable ki whole-number power hai. Uski degree usmein maujood sabse badi power hai (Ek variable wale polynomials).
Degree 0, 1, 2 aur 3 wale polynomials ko constant, linear, quadratic aur cubic kehte hain. Jo power nahi hai uska coefficient 0 hota hai (Table 2.2).
Linear polynomial ko $a \neq 0$ ke saath $a x + b$ likha ja sakta hai. Ise kisi number ke barabar rakhne se linear equation banti hai (Section 2.2).
Polynomial ko input–output rule, yaani function, ki tarah padh sakte hain, jaise $p (x) = 3 x - 4$ jismein $p (6) = 14$ (Polynomials input–output rules ki tarah).
Linear polynomial ke input mein barabar steps se output mein barabar steps milte hain: $p (n + 1) - p (n) = a$ (Property).
Linear pattern har baar same number jodta hai, aur uska $n$ th term $n$ mein linear hota hai, jaise rangoli ke liye $4 n + 1$ (Section 2.3).
Linear growth har barabar time interval mein same amount jodti hai aur linear decay same amount ghatati hai. Har model ki limits hoti hain (Section 2.4).
Do variables linear relationship mein hote hain jab $y = a x + b$ . $a$ aur $b$ nikaalne ke liye do measurements kaafi hain (Section 2.5).
$y = a x + b$ ka graph ek seedhi line hai. Do points use tay karte hain, aur koi point us par theek tab hai jab uske coordinates equation mein fit hon (Do points se line tay hoti hai).
Slope $a$ , $x$ mein 1 ke har step par $y$ ka change hai. Positive slopes upar jaate hain, negative slopes neeche aate hain, aur bada size matlab zyaada steep line (Origin se guzarti lines: slope).
Line $y = a x + b$ $y$ -axis ko $(0, b)$ par cross karti hai, aur $b$ uska y-intercept hai: origin ke upar positive, neeche negative, aur origin se guzarti line ke liye 0 (Line $y$-axis se kahan milti hai).
Same slope, alag $y$ -intercept: dono lines parallel hain aur kabhi nahi milti ($b$ badalna: parallel lines).

Flash cards · 8

Key terms

1 / 8

Fig. 2.17 wala steel truss neeche exercises ke question 12 mein use hota hai.

4/4Stage 4: 9 rods.

Fig. 2.17Har naya triangle pichhle triangle ke saath ek rod share karta hai aur do nayi rods jodta hai.NCERT End-of-chapter Exercises, Q12 · p. 38

Joints par bolt ki gayi barabar steel rods se bane footbridge truss ke chaar stages, aamne-saamne bane hue. Step 1 stage 1 dikhata hai, 3 rods ka ek triangle. Step 2 stage 2 jodta hai, ek rod share karte do triangles, jiske right-hand end ki 2 nayi rods blue mein hain aur plus 2 mark hai, kul 5 rods. Step 3 stage 3 jodta hai, upar, neeche aur upar point karte teen triangles, 7 rods. Step 4 stage 4 jodta hai, chaar triangles, 9 rods. Har label stage number aur rods ki ginti batata hai.

Practice · 18 questions

Practice questions answers ke saath (NCERT “End-of-chapter exercises”)

Short answerEasyNCERT End-of-chapter Exercises, Q1
$y$ mein degree-4 ka ek polynomial banao jiska constant term 3 ho, $y$ ka coefficient $- 2$ ho, aur koi $y^{3}$ term na ho.
Solution dikhao
AnswerJaise, $y^{4} + 5 y^{2} - 2 y + 3$ .
$y^{4}$ term degree ko 4 par fix karta hai. $y^{3}$ chhod do, $- 2 y$ aur constant 3 shaamil karo. Koi bhi $y^{2}$ term joda ja sakta hai, isliye $y^{4} + 5 y^{2} - 2 y + 3$ , $2 y^{4} - 2 y + 3$ aur kai aur sahi hain.
2
Short answerEasyNCERT End-of-chapter Exercises, Q2
Har polynomial ki value nikalo.
1. $3 m^{2} + 2 m - 8$ jab $m = - 2$
  Answer dekho
  Answer 0
  $3 (- 2)^{2} + 2 (- 2) - 8 = 12 - 4 - 8 = 0$ .
2. $2 r^{3} - 5 r + 1$ jab $r = k$
  Solution dikhao
  Answer $2 k^{3} - 5 k + 1$
  Har $r$ ki jagah $k$ rakho: $2 k^{3} - 5 k + 1$ . Letter rakhne se expression milta hai, number nahi.
Short answerMediumNCERT End-of-chapter Exercises, Q3
Ek number ka teen-chauthai, minus aadha, paanch-aathvaan ke barabar hai. Number nikalo.
Answer dekho
Answer 1.5
$\frac{3}{4} x = \frac{5}{8} + \frac{1}{2} = \frac{9}{8}$ , isliye $x = \frac{9}{8} \times \frac{4}{3} = \frac{3}{2} = 1.5$ . Check: $\frac{3}{4} \times \frac{3}{2} - \frac{1}{2} = \frac{9}{8} - \frac{4}{8} = \frac{5}{8}$ .
4
Short answerMediumNCERT End-of-chapter Exercises, Q4
Aditi ke paas Irfan se teen guna stickers hain. Apne 10 stickers Irfan ko dene ke baad, uske paas Irfan ke theek double stickers hain. Shuru mein dono ke paas kitne stickers the?
1. Shuru mein Irfan
  stickers
  Answer dekho
  Answer 30 stickers
  $3 x - 10 = 2 (x + 10)$ se $3 x - 10 = 2 x + 20$ milta hai, isliye $x = 30$ .
2. Shuru mein Aditi
  stickers
  Answer dekho
  Answer 90 stickers
  $3 \times 30 = 90$ . Check: dene ke baad Aditi ke paas 80 hain aur Irfan ke paas 40, aur $80 = 2 \times 40$ .
5
Short answerEasyNCERT End-of-chapter Exercises, Q5
Highway ke kinaare ped lagaane ki ek muhim mein pehle se 1,200 paudhe lag chuke hain, aur har mahine 175 aur lagte hain.
1. 8 mahine baad kul kitne paudhe lag chuke honge?
  paudhe
  Answer dekho
  Answer 2600 paudhe
  $1200 + 175 \times 8 = 1200 + 1400 = 2600$ .
2. Kitne mahine baad total pehli baar 5,000 se zyaada hoga?
  mahine
  Answer dekho
  Answer 22 mahine
  21 mahine baad total $1200 + 175 \times 21 = 4875$ hai, jo kaafi nahi. 22 mahine baad yeh $1200 + 175 \times 22 = 5050$ hai.
3. $n$ mahine baad lage paudhon ki ginti ka linear pattern likho.
  Solution dikhao
  Answer $1200 + 175 n$
  Har mahina 175 jodta hai, isliye $n$ mahine baad total $1200 + 175 n$ hai: 1375, 1550, 1725, … 175 ke jump wala linear pattern.
Short answerHardNCERT End-of-chapter Exercises, Q6
Ek two-digit number mein tens digit units digit ka double hai. Dono digits ki jagah badalne se jo number banta hai woh original se 36 kam hai. Number nikalo.
Answer dekho
Answer 84
Number $21 u$ hai aur digits badalne par number $10 u + 2 u = 12 u$ hai. Isliye $21 u - 12 u = 36$ , jisse $9 u = 36$ aur $u = 4$ milta hai. Number 84 hai; digits badalne par 48 milta hai, aur $84 - 48 = 36$ .
7
DiagramMediumNCERT End-of-chapter Exercises, Q7
Har line ke liye slope aur $y$ -intercept nikalo, aur woh point jahan woh $y$ -axis ko cross karti hai. Phir lines banao.
1. $2 y = 3 x - 8$
  Solution dikhao
  AnswerSlope 1.5, $y$ -intercept $- 4$ , aur crossing point $(0, - 4)$ hai.
  2 se divide karo: $y = 1.5 x - 4$ . Doosra point $(2, - 1)$ hai.
2. $y = 5 - x$
  Solution dikhao
  AnswerSlope $- 1$ , $y$ -intercept 5, aur crossing point $(0, 5)$ hai.
  Ise $y = - x + 5$ likho. Doosra point $(5, 0)$ hai.
3. $4 y = 6 x + 2$
  Solution dikhao
  AnswerSlope 1.5, $y$ -intercept 0.5, aur crossing point $(0, 0.5)$ hai.
  4 se divide karo: $y = 1.5 x + 0.5$ . Doosra point $(1, 2)$ hai.
4. $3 y + 6 x = 9$
  Solution dikhao
  AnswerSlope $- 2$ , $y$ -intercept 3, aur crossing point $(0, 3)$ hai.
  Dono sides se $6 x$ ghatao: $3 y = - 6 x + 9$ . 3 se divide karo: $y = - 2 x + 3$ . Doosra point $(1, 1)$ hai.
5. Inmein se kaun si do lines parallel hain?
  (b) aur (d)(a) aur (c)(a) aur (d)(b) aur (c)
  Answer dekho
  Answer (B) (a) aur (c)
  Lines (a) aur (c) dono ka slope 1.5 hai aur $y$ -intercepts alag hain, isliye woh parallel hain. Lines (b) aur (d) dono neeche aati hain, lekin unke slopes, $- 1$ aur $- 2$ , alag hain.
8
Short answerMediumNCERT End-of-chapter Exercises, Q8
Saaf din par, ek shehar ke upar $h$ km ki height par hawa ka temperature $T$ (°C) lagbhag $T = 30 - 6.5 h$ hota hai.
1. 2 km ki height par temperature nikalo.
  °C
  Answer dekho
  Answer 17 °C
  $30 - 6.5 \times 2 = 30 - 13 = 17$ °C.
2. Kis height par hawa $- 9$ °C par hai?
  km
  Answer dekho
  Answer 6 km
  $30 - 6.5 h = - 9$ se $6.5 h = 39$ milta hai, isliye $h = 6$ km.
3. Height badhne par temperature dikhata hai
  linear growthlinear decaykoi steady change nahi
  Answer dekho
  Answer (B) linear decay
  Har kilometre upar chadhne par yeh wahi 6.5 °C girta hai, jo linear decay hai.
9
DiagramMediumNCERT End-of-chapter Exercises, Q9
Ek cyclist steady 5 m/s se chalata hai. $t$ seconds mein tay ki gayi distance $d$ metres, $d = 5 t$ hai. $0 \leq t \leq 10$ ke liye $d$ aur $t$ ka graph banao.
1. 3 seconds mein kitni distance tay hoti hai? Apne graph par check karo.
  m
  Answer dekho
  Answer 15 m
  $d = 5 \times 3 = 15$ m. Graph par $t = 3$ se upar line tak jao aur aadi taraf 15 tak.
2. 40 m tay karne mein kitna time lagta hai?
  s
  Answer dekho
  Answer 8 s
  $5 t = 40$ se $t = 8$ seconds milte hain.
3. Graph origin se kyun guzarta hai?
  Solution dikhao
  AnswerKyunki $t = 0$ par $d = 0$ hai.
  Rule mein koi constant term nahi hai ( $b = 0$ ): time 0 par koi distance tay nahi hui, isliye line $(0, 0)$ se shuru hoti hai. Iska slope, 5, speed hai.
10
Short answerHardNCERT End-of-chapter Exercises, Q10
Ek linear polynomial $p (x)$ ke liye $p (2) = 1$ aur $p (5) = 10$ hai.
1. $p (x)$ nikalo.
  $3 x - 5$ $3 x + 1$ $9 x - 17$ $\frac{1}{3} x + \frac{1}{3}$
  Answer dekho
  Answer (A) $3 x - 5$
  Slope $= \frac{10 - 1}{5 - 2} = 3$ . Phir $1 = 3 \times 2 + b$ se $b = - 5$ milta hai, isliye $p (x) = 3 x - 5$ . Check: $3 \times 5 - 5 = 10$ .
2. $p (0)$ nikalo. Yeh graph ke baare mein kya batata hai?
  Answer dekho
  Answer -5
  $p (0) = 3 \times 0 - 5 = - 5$ , isliye graph $y$ -axis ko $(0, - 5)$ par cross karta hai.
3. $p (x) = 0$ solve karo. Isse graph ka kaun sa point milta hai?
  Answer dekho
  Answer 1.6667
  $3 x - 5 = 0$ se $x = \frac{5}{3} \approx 1.67$ milta hai: graph $x$ -axis ko $(\frac{5}{3}, 0)$ par cross karta hai.
4. $(2, 1)$ aur $(5, 10)$ plot karo, unse hokar line banao, aur usse (b) aur (c) ke apne answers check karo.
  Solution dikhao
  Answer $(0, - 5)$ , $(\frac{5}{3}, 0)$ , $(2, 1)$ aur $(5, 10)$ se guzarti ek upar jaati line.
  $(2, 1)$ aur $(5, 10)$ plot karo aur unhe jodo. Aage badhane par line $y$ -axis se $(0, - 5)$ par milti hai aur $x$ -axis se 1 aur 2 ke beech, $x = \frac{5}{3}$ par.
11
Long answerHardNCERT End-of-chapter Exercises, Q11
Do linear polynomials $f (x) = a x + b$ aur $g (x) = c x + d$ do tarah se jude hain: $g$ ka graph $(0, 1)$ aur $(2, 7)$ se guzarta hai, aur $x$ ki har value ke liye $f (x) - g (x) = x - 5$ hai.
1. $g (x)$ aur $f (x)$ nikalo.
  Solution dikhao
  Answer $g (x) = 3 x + 1$ aur $f (x) = 4 x - 4$ .
  $g$ ka graph $(0, 1)$ se guzarta hai, isliye $d = 1$ . $(0, 1)$ se $(2, 7)$ tak yeh 2 aadi taraf ke liye 6 upar jaata hai, isliye $c = 6 \div 2 = 3$ . Isliye $g (x) = 3 x + 1$ .
  Phir $f (x) = g (x) + (x - 5) = 3 x + 1 + x - 5 = 4 x - 4$ . Check: $(4 x - 4) - (3 x + 1) = x - 5$ .
2. $x$ ki kis value par $f$ aur $g$ ke graphs milte hain?
  Answer dekho
  Answer 5
  Graphs wahan milte hain jahan $f (x) = g (x)$ , yaani jahan $f (x) - g (x) = x - 5 = 0$ . Isliye $x = 5$ , aur dono $y = 16$ dete hain: graphs $(5, 16)$ par milte hain.
12
Short answerMediumNCERT End-of-chapter Exercises, Q12
Fig. 2.17 dikhata hai ki footbridge truss barabar steel rods se kaise banta hai. Har naya triangle pichhle triangle ke saath ek rod share karta hai aur use do nayi rods chahiye.
1. Har stage ko pichhle stage se theek 2 zyaada rods kyun chahiye?
  Solution dikhao
  AnswerNaye triangle ki 3 sides hain, lekin unmein se ek pichhle triangle ki rod hai.
  Triangle ko 3 rods chahiye. Naya triangle pichhle stage ke aakhri triangle ke saath ek side share karta hai, isliye sirf uski baaki 2 sides nayi rods hain.
2. Stages 1 se 5 ke liye table poori karo, aur stage $n$ mein rods ki ginti ka rule nikalo.
  Solution dikhao
  Answer3, 5, 7, 9, 11, …; $2 n + 1$
  Stage 1 2 3 4 5 … $n$
  Rods 3 5 7 9 11 … $2 n + 1$
  3 se shuru karo aur har agle stage ke liye 2 jodo: $3 + 2 (n - 1) = 2 n + 1$ . Ya direction ke hisaab se gino: $n$ triangles mein $n + 1$ tirchhi rods aur $n$ seedhi (level) rods hain, aur $(n + 1) + n = 2 n + 1$ .
3. 40 triangles wale truss ko kitni rods chahiye?
  rods
  Answer dekho
  Answer 81 rods
  $2 \times 40 + 1 = 81$ .
4. Ek workshop ke paas 100 rods hain. Woh sabse bada kaun sa stage bana sakti hai?
  Answer dekho
  Answer 49
  Stage 49 ko $2 \times 49 + 1 = 99$ rods chahiye, aur 1 bachti hai, aur stage 50 ko 101 chahiye. Har stage mein odd number of rods lagti hain, isliye koi bhi stage theek 100 rods use nahi karta.
13
Long answerHardNCERT End-of-chapter Exercises, Q13
Line $L_{1}$ $(1, - 1)$ aur $(4, 5)$ se guzarti hai. Line $L_{2}$ , $L_{1}$ ke parallel hai aur $x$ -axis ko $(- 4, 0)$ par cross karti hai.
1. $L_{1}$ ki equation $y = a x + b$ form mein nikalo.
  Solution dikhao
  Answer $y = 2 x - 3$
  Slope $= \frac{5 - (- 1)}{4 - 1} = \frac{6}{3} = 2$ . Phir $- 1 = 2 \times 1 + b$ se $b = - 3$ milta hai.
2. $L_{2}$ ki equation nikalo.
  Solution dikhao
  Answer $y = 2 x + 8$
  Parallel hai, isliye iska slope bhi 2 hai. Yeh $(- 4, 0)$ se guzarti hai: $0 = 2 \times (- 4) + b$ se $b = 8$ milta hai.
3. $x$ ki kis value par $L_{1}$ $x$ -axis ko cross karti hai?
  Answer dekho
  Answer 1.5
  $2 x - 3 = 0$ se $x = 1.5$ milta hai, isliye $L_{1}$ $(1.5, 0)$ par cross karti hai.
4. Seedha upar naapne par $L_{2}$ , $L_{1}$ se kitni upar hai?
  units
  Answer dekho
  Answer 11 units
  Har $x$ par gap $(2 x + 8) - (2 x - 3) = 11$ hai. Jaise, $x = 0$ par lines 8 aur $- 3$ par hain.
14
Short answerHardNCERT End-of-chapter Exercises, Q14
Lines $y = x - 1$ , $y = 3 x - 5$ aur $y = - 0.5 x + 2$ sab ek hi point se guzarti hain.
1. Woh kaun sa point hai?
  $(0, 2)$ $(4, 3)$ $(2, 1)$ $(0, - 1)$
  Answer dekho
  Answer (C) $(2, 1)$
  $x = 2$ par: $2 - 1 = 1$ , $3 \times 2 - 5 = 1$ aur $- 0.5 \times 2 + 2 = 1$ . Points $(4, 3)$ aur $(0, - 1)$ sirf $y = x - 1$ mein fit hote hain, aur $(0, 2)$ sirf $y = - 0.5 x + 2$ mein.
2. Samjhao ki $a \neq 0$ ke saath har line $y = a x + 1 - 2 a$ is point se kyun guzarti hai. In lines mein ek se doosri tak kaun si cheez badalti hai?
  Solution dikhao
  AnswerSab $(2, 1)$ se guzarti hain; sirf slope $a$ badalta hai.
  $y = a x + 1 - 2 a = a (x - 2) + 1$ . $x = 2$ par yeh $a \times 0 + 1 = 1$ hai, $a$ chahe kuch bhi ho, isliye aisi har line $(2, 1)$ se guzarti hai. Upar ki teen lines cases $a = 1$ , $3$ aur $- 0.5$ hain. Sirf slope badalta hai, isliye lines $(2, 1)$ ke charon taraf ghoomti hain, bilkul waise jaise same $b$ wali lines $(0, b)$ ke charon taraf ghoomti hain.
Quick answerEasy
Ek linear pattern 12, 9, 6, … se shuru hota hai. Uska 10th term kya hai?
$- 15$ $- 18$ $- 3$ $- 12$
Answer dekho
Answer (A) $- 15$
Jump $- 3$ hai, isliye $n$ th term $15 - 3 n$ hai, aur 10th term $15 - 30 = - 15$ hai.
Quick answerEasy
$y = a x$ form ki har line origin se guzarti hai.
SahiGalat
Answer dekho
Answer Sahi
$x = 0$ rakhne par $y = 0$ milta hai, $a$ chahe kuch bhi ho.
Quick answerMedium
Line $y = a x + 4$ point $(2, 10)$ se guzarti hai. $a$ nikalo.
Answer dekho
Answer 3
$10 = 2 a + 4$ se $2 a = 6$ milta hai, isliye $a = 3$ .
18
Quick answerMedium
In lines ko sabse kam steep se sabse steep ke order mein lagao.
1. $y = x - 7$
2. $y = - 3 x$
3. $y = - 0.5 x + 1$
4. $y = 0.2 x$
Answer dekho
Answer
$y = 0.2 x$
$y = - 0.5 x + 1$
$y = x - 7$
$y = - 3 x$
Steepness slope ke size par depend karti hai, uske sign ya $b$ par nahi: 0.2, 0.5, 1 aur 3.

Stage	1	2	3	4	5	…	$n$
Rods	3	5	7	9	11	…	$2 n + 1$

Apna NCERT exercise question dhoondo

Hamare practice questions ncert.ai ke liye likhe gaye hain aur NCERT exercises ko follow karte hain. Apni textbook ke question se match karne wala question dhoondo; uska answer uske saath hai.

Tumhari NCERT textbook mein	Yahan practice question
NCERT Exercise Set 2.1, Q1	Practice 2.1 · Q1
NCERT Exercise Set 2.1, Q2	Practice 2.1 · Q2
NCERT Exercise Set 2.1, Q3	Practice 2.1 · Q3
NCERT Exercise Set 2.1, Q4	Practice 2.1 · Q4
NCERT Exercise Set 2.1, Q5	Practice 2.1 · Q5
NCERT Exercise Set 2.2, Q1	Practice 2.2 · Q1
NCERT Exercise Set 2.2, Q2	Practice 2.2 · Q2
NCERT Exercise Set 2.2, Q3	Practice 2.2 · Q3
NCERT Exercise Set 2.2, Q4	Practice 2.2 · Q4
NCERT Exercise Set 2.2, Q5	Practice 2.2 · Q5
NCERT Exercise Set 2.2, Q6	Practice 2.2 · Q6
NCERT Exercise Set 2.2, Q7	Practice 2.2 · Q7
NCERT Exercise Set 2.3, Q1	Practice 2.3 · Q1
NCERT Exercise Set 2.3, Q2	Practice 2.3 · Q2
NCERT Exercise Set 2.3, Q3	Practice 2.3 · Q3
NCERT Exercise Set 2.3, Q4	Practice 2.3 · Q4
NCERT Exercise Set 2.3, Q5	Practice 2.3 · Q5
NCERT Exercise Set 2.4, Q1	Practice 2.4 · Q1
NCERT Exercise Set 2.4, Q2	Practice 2.4 · Q2
NCERT Exercise Set 2.4, Q3	Practice 2.4 · Q3
NCERT Exercise Set 2.4, Q4	Practice 2.4 · Q4
NCERT Exercise Set 2.5, Q1	Practice 2.5 · Q1
NCERT Exercise Set 2.5, Q2	Practice 2.5 · Q2
NCERT Exercise Set 2.5, Q3	Practice 2.5 · Q3
NCERT Exercise Set 2.6, Q1	Practice 2.6 · Q1
NCERT End-of-chapter Exercises, Q1	End-of-chapter exercises · Q1
NCERT End-of-chapter Exercises, Q2	End-of-chapter exercises · Q2
NCERT End-of-chapter Exercises, Q3	End-of-chapter exercises · Q3
NCERT End-of-chapter Exercises, Q4	End-of-chapter exercises · Q4
NCERT End-of-chapter Exercises, Q5	End-of-chapter exercises · Q5
NCERT End-of-chapter Exercises, Q6	End-of-chapter exercises · Q6
NCERT End-of-chapter Exercises, Q7	End-of-chapter exercises · Q7
NCERT End-of-chapter Exercises, Q8	End-of-chapter exercises · Q8
NCERT End-of-chapter Exercises, Q9	End-of-chapter exercises · Q9
NCERT End-of-chapter Exercises, Q10	End-of-chapter exercises · Q10
NCERT End-of-chapter Exercises, Q11	End-of-chapter exercises · Q11
NCERT End-of-chapter Exercises, Q12	End-of-chapter exercises · Q12
NCERT End-of-chapter Exercises, Q13	End-of-chapter exercises · Q13
NCERT End-of-chapter Exercises, Q14	End-of-chapter exercises · Q14

Quick check: MCQs

2.1 Introduction3 questions
Ek variable wale polynomials4 questions
Polynomials input–output rules ki tarah4 questions
Rangoli check3 questions
2.3 Linear patterns ko explore karna4 questions
2.4 Linear growth aur linear decay3 questions
2.5 Linear Relationships3 questions
Do points se line tay hoti hai3 questions
Line $y$-axis se kahan milti hai4 questions

Glossary · 15

Key terms (English + Hindi)

coefficientगुणांकgunank: Woh number jo kisi variable ya uski power se multiply hota hai. $- x^{2}$ mein coefficient $- 1$ hai.Chapter mein pehli baar yahan aaya →
constant termअचर पदachar pad: Aisa term jismein koi variable nahi, isliye uski value kabhi nahi badalti, jaise $2 x^{3} - x^{2} + 7 x - 5$ mein $- 5$ .Chapter mein pehli baar yahan aaya →

degreeघातghaat: Polynomial mein aane wali variable ki sabse badi power. $2 x^{3} - x^{2} + 7 x - 5$ ki degree 3 hai.Chapter mein pehli baar yahan aaya →

functionफलनphalan: Aisa rule jo har input ke liye theek ek output deta hai, jaise $p (x) = 3 x - 4$ .Chapter mein pehli baar yahan aaya →

linear decayरैखिक ह्रासrekhik hraas: Barabar ghatane se kami: same length ke har time step mein quantity wahi fixed amount khoti hai.Chapter mein pehli baar yahan aaya →
linear equationरैखिक समीकरणrekhik samikaran: Linear polynomial ko kisi number ke barabar rakhne se bani equation, jaise $25 + 2 p = 85$ .Chapter mein pehli baar yahan aaya →
linear growthरैखिक वृद्धिrekhik vriddhi: Barabar jodne se growth: same length ke har time step mein quantity mein wahi fixed amount judta hai.Chapter mein pehli baar yahan aaya →
linear patternरैखिक प्रतिरूपrekhik pratiroop: Numbers ki list jismein agla term paane ke liye har baar same number joda jaata hai.Chapter mein pehli baar yahan aaya →
linear polynomialरैखिक बहुपदrekhik bahupad: Degree 1 ka polynomial, jise $a \neq 0$ ke saath $a x + b$ likh sakte hain.Chapter mein pehli baar yahan aaya →
linear relationshipरैखिक संबंधrekhik sambandh: Do badalti quantities ke beech $y = a x + b$ form ka link; iska graph ek seedhi line hai.Chapter mein pehli baar yahan aaya →

polynomialबहुपदbahupad: Terms ka sum, jismein har term ek number guna variable ki whole-number power ho, jaise $2 x^{3} - x^{2} + 7 x - 5$ .Chapter mein pehli baar yahan aaya →

slopeढालdhaal: $y = a x + b$ mein number $a$ : $x$ ke 1 badhne par $y$ kitna badalta hai.Chapter mein pehli baar yahan aaya →

termपदpad: Expression ke woh parts jo aapas mein jode jaate hain, jaise $4 f + 6 s + 11$ mein $4 f$ , $6 s$ aur $11$ .Chapter mein pehli baar yahan aaya →

variableचरchar: Ek letter jo aise number ki jagah leta hai jo badal sakta hai, jaise $x$ ya $t$ .Chapter mein pehli baar yahan aaya →

y-intercepty-अंतःखंडy-antahkhand: $y = a x + b$ mein number $b$ ; line $y$ -axis se $(0, b)$ par milti hai.Chapter mein pehli baar yahan aaya →

Introduction to Linear Polynomials (Linear Polynomials se Pehchaan): Class 9 Maths Chapter 2 Hinglish mein

Is chapter mein aap kya seekhenge

2.1 Introduction

Ek variable wale polynomials

Practice 2.1: answers ke saath

2.2 Linear Polynomials

Linear equations

Polynomials input–output rules ki tarah

Practice 2.2: answers ke saath

2.3 Linear patterns ko explore karna

Practice 2.3: answers ke saath

2.4 Linear growth aur linear decay

Practice 2.4: answers ke saath

2.5 Linear Relationships

Practice 2.5: answers ke saath

2.6 Linear relationships ko dekhna

Do points se line tay hoti hai

Origin se guzarti lines: slope

$b$ badalna: parallel lines

Line $y$ -axis se kahan milti hai

Practice 2.6: answers ke saath

Summary aur key points

Practice questions answers ke saath (NCERT “End-of-chapter exercises”)

Apna NCERT exercise question dhoondo

Quick check: MCQs

Key terms (English + Hindi)

Tum chapter ke end tak pahunch gaye

2.1 Introduction

Ek variable wale polynomials

2.2 Linear Polynomials

Linear equations

Polynomials input–output rules ki tarah

2.3 Linear patterns ko explore karna

2.4 Linear growth aur linear decay

2.5 Linear Relationships

2.6 Linear relationships ko dekhna

Do points se line tay hoti hai

Origin se guzarti lines: slope

b badalna: parallel lines

Line y-axis se kahan milti hai

Apna NCERT exercise question dhoondo

Quick check: MCQs

Tum chapter ke end tak pahunch gaye

$b$ badalna: parallel lines

Line $y$ -axis se kahan milti hai