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Short Intro

The End-of-Chapter Exercises of Chapter 2 “Introduction to Linear Polynomials” help students revise important concepts like linear polynomials, polynomial evaluation, linear equations, graph plotting, linear growth, linear decay, and linear relationships. These detailed solutions are written in simple step-by-step format for better understanding and exam preparation.

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Quick Information Box

Topic	Details
Chapter	Introduction to Linear Polynomials
Exercise	End-of-Chapter Exercises
Subject	Mathematics
Class	Grade 9
Main Concepts	Polynomials & Linear Relationships
Difficulty Level	Moderate to Advanced
Useful For	School Exams & Olympiad Preparation

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Concepts Used (Topics Covered)

• Linear Polynomials
• Degree of Polynomial
• Polynomial Evaluation
• Linear Equations
• Linear Growth & Decay
• Graphs of Linear Equations
• Slope & y-intercept
• Coordinate Geometry
• Linear Relationships
• Pattern Formation

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Important Formulas

General Linear Equation

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Quadratic Polynomial

$ax^2+bx+c$

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Work Done Formula

W=Fd

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Temperature Conversion Formula

Questions & Step-by-Step Solutions

Question 1

Write a polynomial of degree 3 in variable x where coefficient of $x^2$ is −7.

Solution

One such polynomial is:$$x^3 – 7x^2 + 5x + 2$$

Final Answer

✅ Polynomial:$$x^3 – 7x^2 + 5x + 2$$

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Question 2

Find the value of the following polynomials.

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(i) $5x^2 – 3x + 7$ when $x=1$

Solution

$$5(1)^2 – 3(1) + 7$$$$= 5 – 3 + 7$$$$= 9$$

Final Answer

✅ Value = 9

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(ii) $4t^3 – t^2 + 6$ when $t=a$

Solution

$$4a^3 – a^2 + 6$$

Final Answer

✅ Value = 4a³−a²+6

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Question 3

If a number is multiplied by $\frac{5}{2}$​ and $\frac{2}{3}$​ is added, the result becomes $-\frac{7}{12}$​. Find the number.

Solution

Let number = $x$$$\frac{5}{2}x + \frac{2}{3} = -\frac{7}{12}$$

Multiply by 12:$$30x + 8 = -7$$$$30x = -15$$$$x = -\frac{1}{2}$$​

Final Answer

✅ Number = −1/2

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Question 4

A positive number is 5 times another number. If 21 is added to both numbers, one becomes twice the other. Find the numbers.

Solution

Let smaller number = $x$

Larger number = $5x$

After adding 21:$$5x+21 = 2(x+21)$$$$5x+21 = 2x+42$$$$3x = 21$$$$x = 7$$

Larger number:$$5(7)=35$$

Final Answer

✅ Numbers are 7 and 35

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Question 5

You have ₹800 and save ₹250 every month. Find amount after:

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(i) 6 months

Solution

$$800 + 250(6)$$$$= 800 + 1500$$$$= 2300$$

Final Answer

✅ Amount after 6 months = ₹2300

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(ii) 2 years

2 years = 24 months$$800 + 250(24)$$$$= 800 + 6000$$$$= 6800$$

Final Answer

✅ Amount after 2 years = ₹6800

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Linear Pattern

$$800 + 250n$$

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Question 6

The digits of a two-digit number differ by 3. If digits are interchanged and both numbers are added, sum becomes 143. Find numbers.

Solution

Let tens digit = $x$

Units digit = $x+3$

Original number:$$10x + (x+3)=11x+3$$

Interchanged number:$$10(x+3)+x=11x+30$$

Their sum:$$11x+3+11x+30=143$$$$22x+33=143$$$$22x=110$$$$x=5$$

Digits are 5 and 8.

Final Answer

✅ Numbers are 58 and 85

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Question 7

Find slope and y-intercept of given equations.

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(i) $y=-3x+4$

Slope = −3

y-intercept = 4

Cuts y-axis at:$$(0,4)$$

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(ii) $2y=4x+7$2y=4x+7

$$y=2x+\frac{7}{2}$$

Slope = 2

y-intercept = $7/2$

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(iii) $5y=6x-10$

$$y=\frac{6}{5}x-2$$

Slope = $6/5$

y-intercept = −2

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(iv) $3y=6x-11$

$$y=2x-\frac{11}{3}$$

Slope = 2

y-intercept = $-11/3$

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Parallel Lines

Equations (ii) and (iv) are parallel because slopes are same.

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Question 8

Temperature relation:

$y=\frac{9}{5}(x-273)+32$

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(i) Find Fahrenheit when $x=313K$

Solution

$$y=\frac{9}{5}(313-273)+32$$$$=\frac{9}{5}(40)+32$$$$=72+32$$$$=104$$

Final Answer

✅ Temperature = 104°F

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(ii) Find Kelvin when $y=158°F$

Solution

$$158=\frac{9}{5}(x-273)+32$$

$$126=\frac{9}{5}(x-273)$$$$70=x-273$$$$x=343$$

Final Answer

✅ Temperature = 343 K

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Question 9

Express work done relation when force = 3 units.

Solution

Using:$$W=Fd$$

Force = 3$$W=3d$$

When distance = 2:$$W=3(2)=6$$

Final Answer

✅ Equation:$$W=3d$$

✅ Work done = 6 units

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Question 10

Graph passes through (1,5) and (3,11). Find polynomial.

Solution

Slope:$$m=\frac{11-5}{3-1}$$$$=\frac{6}{2}=3$$

Equation:$$y=3x+b$$

Substitute point (1,5):$$5=3(1)+b$$$$b=2$$

Final Answer

✅ Polynomial:$$p(x)=3x+2$$

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Common Mistakes

❌ Sign errors while solving equations
❌ Wrong substitution in polynomial evaluation
❌ Confusing slope with intercept
❌ Incorrect graph plotting

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Exam Tips

✔ Always simplify equations step-by-step
✔ Remember slope formula carefully
✔ Verify answers after solving
✔ Practice graph questions regularly

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Practice MCQs

1. Degree of polynomial $x^3+2x^2+1$ is:

A) 1
B) 2
C) 3
D) 4

Answer

✅ C) 3

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2. Which equation represents a straight line?

A) $y=x^2$
B) $y=2x+1$
C) $y=x^3$
D) $y=2^x$

Answer

✅ B) $y=2x+1$

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3. In equation $y=ax+b$, $a$ represents:

A) intercept
B) constant
C) slope
D) variable

Answer

✅ C) slope

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4. Parallel lines have:

A) Same intercept
B) Same slope
C) Same coordinates
D) Same graph

Answer

✅ B) Same slope

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FAQ Section

Q1. What is a linear polynomial?

A polynomial having degree 1 is called a linear polynomial.

Q2. What is slope?

Slope measures the steepness of a straight line.

Q3. What is y-intercept?

It is the point where graph cuts the y-axis.

Q4. Why are linear relationships important?

They help represent real-life situations mathematically.

Q5. How can graph questions be solved easily?

By plotting accurate points and joining them carefully.

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