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Short Intro

Exercise 2.2 focuses on evaluating linear and quadratic polynomials and solving real-life word problems using linear equations. These detailed step-by-step solutions help students understand algebra concepts easily and improve their problem-solving skills for school exams and competitive preparation.

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Quick Information Box

Topic	Details
Chapter	Introduction to Linear Polynomials
Exercise	Exercise 2.2
Subject	Mathematics
Class	Grade 9
Main Topics	Polynomial Evaluation & Linear Equations
Difficulty Level	Moderate
Useful For	School Exams & Olympiad Foundation

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Concepts Used (Topics Covered)

• Linear Polynomials
• Quadratic Polynomials
• Polynomial Evaluation
• Substitution Method
• Linear Equations
• Ratio Problems
• Age Problems
• Coin Problems
• Rectangle Problems

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Important Formulas

Linear Polynomial

$ax+b$

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Quadratic Polynomial

$ax^2+bx+c$ax2+bx+c

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Perimeter of Rectangle

$P=2(l+b)$

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Value of Polynomial

Substitute the given value of the variable into the expression.

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Questions & Step-by-Step Solutions

Question 1

Find the value of the linear polynomial $5x – 3$

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(i) $x = 0$

Solution

Given polynomial:

$5x-3$

Substitute $x = 0$

$5(0) – 3 = -3$

Final Answer

✅ Value = −3

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(ii) $x = -1$

Solution

$5(-1) – 3 = -5 – 3$

$= -8$

Final Answer

✅ Value = −8

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(iii) $x = 2$

Solution

$5(2) – 3 = 10 – 3$

$= 7$

Final Answer

✅ Value = 7

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Question 2

Find the value of the quadratic polynomial $7s^2 – 4s + 6$ if:

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(i) $s = 0$

Solution

$$7(0)^2 – 4(0) + 6$$$$= 6$$

Final Answer

✅ Value = 6

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(ii) $s = -3$

Solution

$$7(-3)^2 – 4(-3) + 6$$$$= 7(9) + 12 + 6$$$$= 63 + 18$$$$= 81$$

Final Answer

✅ Value = 81

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(iii) $s = 4$

Solution

$$7(4)^2 – 4(4) + 6$$$$= 7(16) – 16 + 6$$$$= 112 – 10$$$$= 102$$

Final Answer

✅ Value = 102

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Question 3

The present age of Salil’s mother is three times Salil’s present age. After 5 years, their ages will add up to 70 years. Find their present ages.

Solution

Let Salil’s present age = $x$ years

Mother’s present age = $3x$

After 5 years:

Salil’s age = $x + 5$

Mother’s age = $3x + 5$

According to question:$$x + 5 + 3x + 5 = 70$$$$4x + 10 = 70$$$$4x = 60$$$$x = 15$$

Mother’s age:$$3 \times 15 = 45$$

Final Answer

✅ Salil’s age = 15 years
✅ Mother’s age = 45 years

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Question 4

The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.

Solution

Let integers be $2x$ and $5x$

Difference = 63$$5x – 2x = 63$$$$3x = 63$$$$x = 21$$

Numbers are:$$2x = 42$$$$5x = 105$$

Final Answer

✅ Integers are 42 and 105

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Question 5

Ruby has 3 times as many two-rupee coins as she has five-rupee coins. If she has a total ₹88, how many coins does she have of each type?

Solution

Let number of five-rupee coins = $x$

Number of two-rupee coins = $3x$

Total value:$$5x + 2(3x) = 88$$$$5x + 6x = 88$$$$11x = 88$$$$x = 8$$

Two-rupee coins:$$3x = 24$$

Final Answer

✅ Five-rupee coins = 8
✅ Two-rupee coins = 24

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Question 6

A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?

Solution

Let shorter piece = $x$ feet

Longer piece = $4x$ feet

Total length:$$x + 4x = 300$$$$5x = 300$$$$x = 60$$

Longer piece:$$4x = 240$$

Final Answer

✅ Shorter piece = 60 feet
✅ Longer piece = 240 feet

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Question 7

If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?

Solution

Let width = $x$ cm

Length = $2x + 3$ cm

Perimeter formula:$$2(l+b)=24$$

Substitute values:$$2[(2x+3)+x]=24$$$$2(3x+3)=24$$$$6x+6=24$$$$6x=18$$$$x=3$$

Width = 3 cm

Length:$$2(3)+3=9$$

Final Answer

✅ Width = 3 cm
✅ Length = 9 cm

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Common Mistakes

❌ Forgetting brackets during substitution
❌ Wrong sign handling for negative numbers
❌ Using incorrect perimeter formula
❌ Confusing ratio with difference

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Exam Tips

✔ Write variables clearly before solving
✔ Substitute values carefully
✔ Always verify final answers
✔ Practice word problems regularly

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Practice MCQs

1. Find value of $3x + 2$ when $x = 4$

A) 10
B) 12
C) 14
D) 16

Answer

✅ C) 14

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2. Which is a quadratic polynomial?

A) $2x+1$
B) $x^2+5x+1$
C) $7$7
D) $5x^3$

Answer

✅ B) $x^2+5x+1$

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3. Perimeter of rectangle formula is:

A) $l+b$
B) $2l+b$
C) $2(l+b)$
D) $lb$

Answer

✅ C) $2(l+b)$

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4. If ratio is 2:5, numbers can be written as:

A) $2+x$ and $5+x$
B) $2x$ and $5x$
C) $x/2$ and $x/5$
D) None

Answer

✅ B) $2x$ and $5x$

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FAQ Section

Q1. What is polynomial evaluation?

It means finding the value of a polynomial after substituting a variable value.

Q2. What is a linear polynomial?

A polynomial having degree 1 is called a linear polynomial.

Q3. What is a quadratic polynomial?

A polynomial having degree 2 is called a quadratic polynomial.

Q4. Why are word problems important?

They improve logical thinking and real-life mathematical understanding.

Q5. Can negative numbers be substituted in polynomials?

Yes, negative values can also be substituted.

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