The perimeter of a square with side 'x' is given by 4x. Is 4x a linear polynomial?

Yes, because the highest power of x is 1.

No, because it only has one term.

Yes, because it contains a number.

No, because it is a geometric formula.

Which of the following represents linear growth?

A plant growing 2 cm taller each week.

A population doubling every year.

A plant growing 2 cm taller each week.

The area of a square increasing with its side squared.

The intensity of light decreasing with distance squared.

Assertion (A): The expression 2x + 5 is a linear polynomial. Reason (R): A linear polynomial is an algebraic expression with the highest power of the variable as 1.

A and R both correct, R explains A

The graph shows the cost of a service based on hours used. What is the fixed charge (y-intercept) and the hourly rate (slope) from the graph?

Fixed charge = 10, Hourly rate = 5

Fixed charge = 5, Hourly rate = 10

Fixed charge = 0, Hourly rate = 10

Fixed charge = 10, Hourly rate = 10

Is this CBSE Class 9 Maths Chapter 2 quiz free?

Yes, absolutely! All quizzes on Shrutam, including this one for Introduction to Linear Polynomials, are completely free to use without any signup or hidden charges.

Is this quiz aligned with the NCERT syllabus?

Yes, this quiz is meticulously designed based on the latest NCERT Class 9 Mathematics textbook, specifically Chapter 2: Introduction to Linear Polynomials, ensuring full syllabus coverage.

How many questions are in the game mode for this chapter?

The game mode for 'Introduction to Linear Polynomials' features a total of 40 questions, carefully balanced across easy, medium, and hard difficulty levels to provide a comprehensive assessment.

What topics does this quiz cover?

This quiz covers all major topics from Chapter 2, including identifying linear polynomials, understanding linear patterns, linear growth and decay, and visualising linear relationships through equations like y = ax + b.

Can I access explanations for the answers?

Yes, for every question, you will receive an instant short explanation after answering, and a detailed step-by-step explanation is available on tapping 'Why?' to help you understand the concepts better.

Are there questions from the NCERT textbook exercises?

Yes, our 'Book Questions' mode provides solved versions of questions directly from the NCERT textbook exercises, helping you practice and understand the solutions.

Home › CBSE › Class 9 › Maths › Introduction to Linear Polynomials

CBSE · Class 9 · 🧮 Maths · Chapter 2

Introduction to Linear Polynomials

Algebraic ExpressionsLinear PolynomialsLinear PatternsLinear Growth and DecayLinear RelationshipsVisualising Linear Relationships

Chapter 2, 'Introduction to Linear Polynomials', lays the foundation for understanding algebraic expressions with a special focus on linear polynomials. Students learn to identify terms, coefficients, and variables, and explore how linear patterns manifest in real-world scenarios. The chapter delves into concepts of linear growth and decay, and introduces the representation of linear relationships as equations of the form y = ax + b, which can be visualised as straight lines on a graph. This chapter is crucial for developing a strong base in algebra.

Algebraic Expressions ka Revision

Pehle ki classes mein humne algebraic expressions ke baare mein padha hai. Ek algebraic expression mein variables, constants aur mathematical operations (addition, subtraction, multiplication, division) hote hain.

Variables: Letters jaise $x, y, z, p, q$ jo different values le sakte hain. Inhe literal numbers bhi kehte hain.
Constants: Fixed numerical values jaise $2, -5, \frac{1}{3}, \pi$.
Terms: Expression ke parts jo addition ya subtraction se separate hote hain. Jaise $2x^2 - 5x + 3$ mein terms hain $2x^2, -5x, 3$.
Coefficients: Variable ke saath multiply hone wala numerical part. Jaise $2x^2$ mein $2$ coefficient hai, $-5x$ mein $-5$ coefficient hai.
Polynomial: Ek special type ka algebraic expression jismein variables ki powers non-negative integers (whole numbers) hoti hain. Jaise $x^2 + 2x - 3$ ek polynomial hai, lekin $x + \frac{1}{x}$ ya $\sqrt{x} + 1$ polynomial nahi hain (kyunki $\frac{1}{x} = x^{-1}$ aur $\sqrt{x} = x^{1/2}$ mein powers negative ya fractional hain).
Degree of a Polynomial: Polynomial mein variable ki highest power. Jaise $2x^3 + 4x^2 - 7$ ki degree $3$ hai.

Example: Expression $5x^2 - 3x + 7$

Variables: $x$
Constants: $7$
Terms: $5x^2, -3x, 7$
Coefficients: $5$ (for $x^2$), $-3$ (for $x$)
Degree: $2$

❗Important

Har polynomial ek algebraic expression hota hai, par har algebraic expression polynomial nahi hota. Polynomial hone ke liye variable ki power whole number honi chahiye.

Linear Polynomials: Definition aur Types

Linear Polynomials aise polynomials hote hain jinki degree 1 hoti hai. Matlab, variable ki highest power $1$ hoti hai.

General Form: Ek variable mein linear polynomial ka general form hai $ax + b$, jahan $a$ aur $b$ real numbers hain aur $a \neq 0$. Agar $a = 0$ ho gaya, toh $ax + b$ sirf $b$ ban jayega, jo ki ek constant polynomial hai (degree 0).
Examples:
$2x + 5$ (yahan $a=2, b=5$)
$3y - 7$ (yahan $a=3, b=-7$)
$z$ (yahan $a=1, b=0$)
$\frac{1}{2}p + \frac{3}{4}$
Non-examples (kyunki degree 1 nahi hai):
$x^2 + 3x - 1$ (Degree 2, quadratic polynomial)
$5$ (Degree 0, constant polynomial)
$x^3 - 2$ (Degree 3, cubic polynomial)

Linear Equation: Jab ek linear polynomial ko kisi constant ke equal rakha jata hai, toh woh linear equation ban jaati hai. Jaise $ax + b = 0$ ya $ax + b = c$.

One Variable: $2x + 5 = 0$ (ek variable $x$ mein linear equation)
Two Variables: $2x + 3y = 7$ (do variables $x, y$ mein linear equation)

Is chapter mein hum mainly ek variable wale linear polynomials aur unke applications par focus karenge.

Function: Jab hum kisi expression ko ek variable ke function ke roop mein dekhte hain, toh usse $f(x)$ ya $P(x)$ se denote karte hain. Jaise $P(x) = 2x + 5$. Yahan $x$ input hai aur $P(x)$ output.

📖Definition

Linear Polynomial: Ek polynomial jiski degree $1$ hoti hai. Iska general form $ax + b$ hai, jahan $a \neq 0$.

💡Tip

Degree identify karte waqt variable ki highest power dekhna. Agar variable denominator mein hai ya square root mein hai, toh woh polynomial nahi hoga.

Linear Patterns: Pehchan aur Generalization

Linear patterns aise sequences hote hain jahan consecutive terms ke beech ka difference constant hota hai. Isse arithmetic progression (AP) bhi kehte hain.

Pehchan: Jab values ek fixed amount se badhti (linear growth) ya gharti (linear decay) hain har step par.
Generalization: Agar first term $a$ hai aur common difference $d$ hai, toh $n^{th}$ term ka formula $a_n = a + (n-1)d$ hota hai. Yeh ek linear expression hai $n$ mein.

Example 1: Square Tiles Pattern (NCERT Fig. 2.4)

Stage 1: 1 tile
Stage 2: 3 tiles
Stage 3: 5 tiles
Stage 4: 7 tiles

| Stage (n) | Number of Tiles (T) | |---|---| | 1 | 1 | | 2 | 3 | | 3 | 5 | | 4 | 7 |

Observations:

Consecutive terms ka difference: $3-1=2$, $5-3=2$, $7-5=2$. Difference constant hai (2).
Iska matlab yeh ek linear pattern hai.
General rule ($n^{th}$ term): $T_n = 1 + (n-1)2 = 1 + 2n - 2 = 2n - 1$.

Check: $T_1 = 2(1)-1 = 1$, $T_2 = 2(2)-1 = 3$, etc. Correct.

Example 2: Amount Paid by a Player (NCERT Table 1)

Matches played (m): 1, 2, 3, 4, 5, ...
Amount paid ($\text{`}$): 250, 300, 350, 400, 450, ...

Observations:

Consecutive terms ka difference: $300-250=50$, $350-300=50$. Difference constant hai (50).
Yeh bhi ek linear pattern hai.
General rule ($m^{th}$ match ke liye): First term $a=250$, common difference $d=50$.

$A_m = a + (m-1)d = 250 + (m-1)50 = 250 + 50m - 50 = 200 + 50m$.

Linear patterns ko identify karna aur unka general rule (linear expression) nikalna important hai.

⭐Remember

Linear pattern ka matlab hai constant rate of change. Har step par value ek fixed amount se badhti ya gharti hai.

Linear Growth aur Linear Decay

Linear patterns ko hum linear growth ya linear decay ke roop mein classify kar sakte hain.

Linear Growth: Jab ek quantity fixed amount se badhti hai equal time intervals mein. Iska graph ek upward-sloping straight line hota hai.
Example: Har mahine bank account mein $₹500$ jama karna. Balance linear growth show karega.
Expression: $P(t) = P_0 + rt$, jahan $P_0$ initial amount hai, $r$ growth rate hai per interval, aur $t$ number of intervals.

Linear Decay: Jab ek quantity fixed amount se gharti hai equal time intervals mein. Iska graph ek downward-sloping straight line hota hai.
Example: Har din ek tank se $10$ litres paani nikalna. Tank mein paani ka level linear decay show karega.
Expression: $P(t) = P_0 - rt$, jahan $P_0$ initial amount hai, $r$ decay rate hai per interval, aur $t$ number of intervals.

Key Difference:

Growth mein positive slope (value badhti hai).
Decay mein negative slope (value gharti hai).

Both linear growth aur decay ko linear polynomials ya linear equations se represent kiya ja sakta hai.

📖Definition

Linear Growth: Quantity ka fixed amount se badhna over equal intervals. Linear Decay: Quantity ka fixed amount se ghatna over equal intervals.

Linear Relationships: Equation $y = ax + b$

Do variables, $x$ aur $y$, ke beech ke linear relationship ko ek equation se represent kiya ja sakta hai jiska general form hai:

$$y = ax + b$$

Jahan:

$x$ aur $y$ variables hain.
$a$ aur $b$ constants hain (real numbers).
$a$ ko slope kehte hain. Yeh batata hai ki $y$ kitni tezi se badal raha hai $x$ ke respect mein. Agar $a > 0$, toh line upward slope karegi (growth). Agar $a < 0$, toh line downward slope karegi (decay).
$b$ ko y-intercept kehte hain. Yeh woh point hai jahan line y-axis ko cut karti hai. Matlab, jab $x = 0$ hota hai, toh $y = b$ hota hai.

Understanding $a$ (Slope):

Slope $a = \frac{\text{change in y}}{\text{change in x}}$. Yeh rate of change ko represent karta hai.
Agar $a = 2$, toh iska matlab hai ki $x$ mein har $1$ unit change ke liye, $y$ mein $2$ units ka change hoga.

Understanding $b$ (Y-intercept):

Y-intercept $b$ us initial value ko represent karta hai jab $x$ zero hota hai.
Example: Agar ek taxi ka fare $₹50$ fixed charge hai aur $₹10$ per km hai, toh equation hogi $C = 10k + 50$. Yahan $C$ total cost hai aur $k$ kilometers hain.
$a = 10$ (slope): Har kilometer ke liye $₹10$ badhte hain.
$b = 50$ (y-intercept): $0$ km ke liye bhi $₹50$ dene padenge (fixed charge).

Input-Output Process (NCERT Fig. 2.3): Ek linear expression ko ek machine ki tarah dekha ja sakta hai. Jab hum $x$ ki value input karte hain, toh machine $y = ax + b$ formula use karke ek output $y$ deti hai.

Example: $y = 2x + 3$
Agar $x = 4$ (input), toh $y = 2(4) + 3 = 8 + 3 = 11$ (output).
Agar $x = -1$ (input), toh $y = 2(-1) + 3 = -2 + 3 = 1$ (output).

🧮Formula

Linear Relationship Equation: $y = ax + b$

$a$ = Slope (rate of change)
$b$ = Y-intercept (value of $y$ when $x=0$)

Linear Relationships ka Visualisation (Graphing)

Ek linear relationship $y = ax + b$ ko Cartesian plane par ek straight line se represent kiya ja sakta hai. Har point $(x, y)$ jo equation ko satisfy karta hai, woh line par lie karta hai.

Graph Plot Karne ke Steps:

Table of Values Banayein: $x$ ki kam se kam do (preferably teen) convenient values choose karein aur corresponding $y$ values calculate karein equation $y = ax + b$ use karke.

Tip: $x=0$ choose karne se y-intercept $(0, b)$ milta hai. $y=0$ choose karne se x-intercept $(-\frac{b}{a}, 0)$ milta hai.

Points Plot Karein: In $(x, y)$ ordered pairs ko Cartesian plane par plot karein.
Line Draw Karein: Plot kiye gaye points ko ek straight line se join karein. Line ko dono directions mein extend karein aur arrows lagayein, indicate karne ke liye ki line infinite hai.

Example: Graph $y = 2x + 1$ (NCERT Fig. 2.5)

Table of Values:

| $x$ | $y = 2x + 1$ | Point $(x, y)$ | |---|---|---| | 0 | $2(0) + 1 = 1$ | $(0, 1)$ | | 1 | $2(1) + 1 = 3$ | $(1, 3)$ | | 2 | $2(2) + 1 = 5$ | $(2, 5)$ | | -1 | $2(-1) + 1 = -1$ | $(-1, -1)$ |

Points Plot Karein: $(0, 1)$, $(1, 3)$, $(2, 5)$, $(-1, -1)$ ko graph paper par plot karein.

Line Draw Karein: In points ko join karne par ek straight line banegi.

Interpretation of Graph:

Slope ($a$): $y = 2x + 1$ mein slope $a = 2$ hai. Iska matlab hai ki jab $x$ $1$ unit badhta hai, toh $y$ $2$ units badhta hai. Line upward slope kar rahi hai.
Y-intercept ($b$): $y = 2x + 1$ mein y-intercept $b = 1$ hai. Line y-axis ko point $(0, 1)$ par cut karti hai.

Har linear equation ka graph ek straight line hota hai, aur har straight line ek linear equation ko represent karti hai.

💡Tip

Graph plot karte waqt scale choose karna bahut important hai. Aur kam se kam do points (teesra check ke liye) plot karna zaruri hai. Line ko extend karna aur arrows lagana na bhulein.

Easy (15)

Which of the following is an example of an algebraic expression?
- 5
- x + 7 (correct)
- True
- Red
Why: An algebraic expression combines numbers, variables, and operation symbols.
What is the degree of a linear polynomial?
- 0
- 1 (correct)
- 2
- Undefined
Why: A linear polynomial is defined as a polynomial of degree one.
In the expression 3x + 5, what is the coefficient of x?
- x
- 5
- 3 (correct)
- 3x
Why: The coefficient is the numerical part multiplying the variable.
Which term refers to a fixed number in an algebraic expression?
- Variable
- Coefficient
- Constant (correct)
- Term
Why: A constant is a fixed numerical value in an expression.
The perimeter of a square with side 'x' is given by 4x. Is 4x a linear polynomial?
- Yes, because the highest power of x is 1. (correct)
- No, because it only has one term.
- Yes, because it contains a number.
- No, because it is a geometric formula.
Why: 4x is a linear polynomial because the variable x has a degree of 1.
Which of the following represents linear growth?
- A population doubling every year.
- A plant growing 2 cm taller each week. (correct)
- The area of a square increasing with its side squared.
- The intensity of light decreasing with distance squared.
Why: Linear growth means a quantity increases by a fixed amount over equal intervals.
In the linear relationship y = ax + b, what does 'a' represent when visualising the line?
- Y-intercept
- X-intercept
- Slope (correct)
- Constant term
Why: In y = ax + b, 'a' is the slope, indicating the steepness of the line.
Which of these is NOT a linear polynomial?
- x - 5
- 7y
- z^2 + 2z (correct)
- 10 - 3p
Why: A linear polynomial has a degree of 1; z^2 + 2z has a degree of 2.
The expression 5x + 2 is a univariate polynomial.
Why: A univariate polynomial has only one variable, which is 'x' in this case.
Linear decay means a quantity increases by a fixed amount over equal intervals.
Why: Linear decay means a quantity decreases, not increases, by a fixed amount.
The relationship y = x^2 + 1 is an example of a linear relationship.
Why: A linear relationship is of the form y = ax + b, not involving squared terms.
The constant 'b' in the equation y = ax + b represents the y-intercept.
Why: The y-intercept is the point where the line crosses the y-axis, given by 'b' when x=0.
The parts of an algebraic expression separated by addition or subtraction signs are called _________.
Why: Individual components of an expression separated by operations are called terms.
A sequence of numbers where the difference between two consecutive terms is constant is called a _________ pattern.
Why: A constant difference between consecutive terms defines a linear pattern.
The process where an expression is a function of a variable is called a _________.
Why: When an expression's output depends on a variable's input, it's a function.

Medium (15)

If the cost of a pen is ₹10 and a notebook is ₹20, what is the total cost of 'p' pens and 'n' notebooks?
- 10p + 20n (correct)
- 30pn
- 10p - 20n
- 20p + 10n
Why: The total cost is the sum of the cost of pens (10p) and notebooks (20n).
A taxi charges a base fare of ₹50 and ₹15 per kilometer. If 'd' is the distance in kilometers, which expression represents the total fare?
- 50d + 15
- 15d + 50 (correct)
- 65d
- 50 - 15d
Why: The fare is 15 times distance plus the base fare of 50.
A water tank is being filled at a constant rate. It initially contains 100 liters and gains 5 liters per minute. If 't' is the time in minutes, what is the linear polynomial representing the volume of water?
- 100t + 5
- 5t + 100 (correct)
- 105t
- 100 - 5t
Why: The volume is the initial 100 liters plus 5 liters per minute (5t).
Consider the pattern: 2, 5, 8, 11, ... What is the next term in this linear pattern?
- 12
- 13
- 14 (correct)
- 15
Why: The common difference is 3, so add 3 to the last term.
If y = 3x - 2, what is the value of y when x = 4?
- 8
- 10 (correct)
- 12
- 14
Why: Substitute x=4 into the equation: y = 3(4) - 2 = 12 - 2 = 10.
Match the following terms with their definitions:
Why: This matches key definitions from the chapter.
Match the type of change with its description:
Why: This matches the definitions of linear growth, decay, pattern, and relationship.
Match the following algebraic terms:
Why: This matches examples and characteristics to their correct algebraic terms.
What is the slope of the line represented by the equation y = 5x - 3?
Why: In y = ax + b, 'a' is the slope.
What is the y-intercept of the line represented by the equation y = 2x + 7?
Why: In y = ax + b, 'b' is the y-intercept.
Arrange the following steps to evaluate the expression 2x + 3 when x = 4:
Why: Follow the order of operations: substitute, multiply, then add.
Arrange the terms to form a linear polynomial:
Why: The standard form for a linear polynomial is ax + b.
Observe the given graph of a linear equation. What is the y-intercept?
- 1 (correct)
- 2
- 3
- 7
Why: The y-intercept is the point where the line crosses the y-axis (when x=0).
Consider the pattern of square tiles shown in Fig. 2.4. If Stage 1 has 1 tile, Stage 2 has 3 tiles, and Stage 3 has 5 tiles, how many tiles would Stage 5 have?
- 7
- 9 (correct)
- 11
- 13
Why: The pattern adds 2 tiles per stage. Stage 4 has 7, Stage 5 has 9.
The diagram shows an input-output process for y = 2x + 3. If the input x is 5, what would be the output y?
- 10
- 11
- 13 (correct)
- 15
Why: Substitute x=5 into y = 2x + 3: y = 2(5) + 3 = 10 + 3 = 13.

Hard (10)

A car starts with 40 liters of fuel and consumes 2 liters per hour. If 't' is the time in hours, and 'F' is the remaining fuel, what is the value of F when t = 15 hours?
Why: Fuel remaining = Initial fuel - (Consumption rate × time) = 40 - (2 × 15) = 10 liters.
The cost of producing 'x' units of a product is given by C = 50x + 200. If the cost is ₹700, how many units were produced?
Why: Set C=700: 700 = 50x + 200. Solve for x: 500 = 50x, so x = 10.
A linear pattern starts with 8 and decreases by 3 for each subsequent term. What is the 7th term in the sequence?
Why: The general term is a_n = 8 + (n-1)(-3). For n=7, a_7 = 8 + 6(-3) = 8 - 18 = -10.
If a line passes through the points (0, 5) and (2, 9), what is its slope?
Why: Slope = (y2 - y1) / (x2 - x1) = (9 - 5) / (2 - 0) = 4 / 2 = 2.
Assertion (A): The expression 2x + 5 is a linear polynomial. Reason (R): A linear polynomial is an algebraic expression with the highest power of the variable as 1.
Why: Both statements are true, and the reason correctly defines why 2x+5 is a linear polynomial.
Assertion (A): If a quantity decreases by a fixed amount over equal intervals, it shows linear growth. Reason (R): Linear growth is a pattern where a quantity increases by a fixed amount over equal intervals.
Why: Assertion incorrectly describes linear growth (it's linear decay). Reason correctly defines linear growth.
A school bus starts its route with 50 students. At the first stop, 5 students get off. At each subsequent stop, 3 students get off. If there are 's' stops after the first one, how many students are left on the bus?
- 50 - 5s
- 45 - 3s (correct)
- 50 - 3s
- 47 - 3s
Why: After the first stop, 45 students remain. Then, 3 students leave for each of the 's' subsequent stops, making it 45 - 3s.
A plant is 10 cm tall and grows 0.5 cm per day. Another plant is 5 cm tall and grows 1 cm per day. After how many days will both plants be the same height?
- 5 days
- 10 days (correct)
- 15 days
- 20 days
Why: Set up equations: H1 = 10 + 0.5d, H2 = 5 + 1d. Equate them: 10 + 0.5d = 5 + d. Solve for d.
The graph shows the cost of a service based on hours used. What is the fixed charge (y-intercept) and the hourly rate (slope) from the graph?
- Fixed charge = 10, Hourly rate = 5 (correct)
- Fixed charge = 5, Hourly rate = 10
- Fixed charge = 0, Hourly rate = 10
- Fixed charge = 10, Hourly rate = 10
Why: The line crosses y-axis at 10 (fixed charge). Slope = (20-10)/(2-0) = 10/2 = 5 (hourly rate).
A rectangular garden has a length 'l' and width 'w'. The perimeter is given by P = 2(l+w). If the perimeter is 30 units and the length is 10 units, what is the width?
- 3 units
- 5 units (correct)
- 7 units
- 10 units
Why: Substitute P=30, l=10 into P=2(l+w): 30 = 2(10+w). Solve for w: 15 = 10+w, so w=5.

Algebraic Expressions ka Revision

Linear Polynomials: Definition aur Types

Linear Patterns: Pehchan aur Generalization

Linear Growth aur Linear Decay

Linear Relationships: Equation \(y = ax + b\)

Linear Relationships ka Visualisation (Graphing)