Class 9 Maths (Ganita Manjari) Chapter 2: Introduction to Linear Polynomials solutions

 complete solutions for Class 9 Maths (Ganita Manjari) Chapter 2: Introduction to Linear Polynomials, covering all in-text questions, examples, and end-of-chapter exercises with full explanations .

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Basic Concepts and Definitions 

Before starting the exercises, let's understand the key terms:

• Polynomial: An algebraic expression with variables, constants, and powers (using addition/subtraction).

• Degree: The highest power of the variable in a polynomial.

• Linear Polynomial: A polynomial with degree 1 (General form: $ax+b$, where $a\mathrm{\ne }0$).

• Constant Polynomial: A polynomial with no variable (Degree = 0).

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Exercise Set 2.1 (Basic Concepts)

Focus: Finding degrees, coefficients, and constants.

1. Find the degrees of the following polynomials:

(i) $2x^2-5x+3$
Highest power of $x$ = 2
Degree = 2 (Quadratic)

(ii) $y^3+2y-1$
Highest power of $y$ = 3
Degree = 3 (Cubic)

(iii) $-9$
This is a constant polynomial (no variable)
Degree = 0

(iv) $4z-3$
Highest power of $z$ = 1
Degree = 1 (Linear Polynomial)

2. Write polynomials of degrees 1, 2 and 3.

• Degree 1 (Linear): $2x+5$

• Degree 2 (Quadratic): $x^2+3x+1$

• Degree 3 (Cubic): $x^3-2x^2+x+4$

3. What are the coefficients of $x^2$ and $x^3$ in $x^4-3x^3+6x^2-2x+7$?

• Coefficient of $x^3$ = $-3$

• Coefficient of $x^2$ = $6$

4. What is the coefficient of $z$ in $4z^3+5z^2-11$?
There is no term with $z^1$ (just $z$)
Hence, coefficient of $z$ = 0

5. What is the constant term of $9x^3+5x^2-8x-10$?
The constant term is the term without any variable = $-10$

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Exercise Set 2.2 (Finding Values and Word Problems)

Focus: Substituting values and solving real-life linear equations.

1. Find the value of $5x-3$ for different x values.

x value	Calculation	Result
$x=0$	$5(0)-3$	$-3$
$x=-1$	$5(-1)-3=-5-3$	$-8$
$x=2$	$5(2)-3=10-3$	$7$

2. Find the value of $7s^2-4s+6$ (This is a Quadratic Polynomial).

(i) $s=0$: $7(0{)}^2-4(0)+6=6$
(ii) $s=-3$: $7(9)+12+6=63+12+6=81$
(iii) $s=4$: $7(16)-16+6=112-16+6=102$

3. Age Problem (Linear Application)

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.

Step 1: Let Salil's age = $x$
Mother's age = $3x$

Step 2: After 5 years:
Salil = $x+5$
Mother = $3x+5$

Step 3: Equation: $(x+5)+(3x+5)=70$
$4x+10=70$
$4x=60$
$x=15$

Solution: Salil is 15 years, Mother is 45 years.

4. Ratio Problem

Difference of two numbers is 63, ratio is 2:5.

Let numbers be $2x$ and $5x$.
Difference: $5x-2x=63$
$3x=63\Rightarrow x=21$
Numbers: $2x=42$, $5x=105$.

5. Coin Problem

*Ruby has 3 times as many ₹2 coins as ₹5 coins. Total money = ₹88.*

Let number of ₹5 coins = $x$
Number of ₹2 coins = $3x$

Equation: $5(x)+2(3x)=88$
$5x+6x=88$
$11x=88\Rightarrow x=8$

Solution: ₹5 coins = 8, ₹2 coins = 24.

6. Fence Problem

300 ft fence cut into two pieces. Longer is 4 times the shorter.

Shorter = $x$, Longer = $4x$
$x+4x=300\Rightarrow 5x=300\Rightarrow x=60$
Shorter = 60 ft, Longer = 240 ft.

7. Rectangle Perimeter

*Length is three more than twice width. Perimeter = 24 cm.*

Width = $x$
Length = $2x+3$

Perimeter formula: $2(\text{Length}+\text{Width})=24$
$2[(2x+3)+x]=24$
$2(3x+3)=24$
$6x+6=24$
$6x=18\Rightarrow x=3$

Solution: Width = 3 cm, Length = 9 cm.

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Exercise Set 2.3 (Linear Patterns and Growth/Decay)

Focus: Recognizing the pattern $y=mx+c$.

1. Savings Growth (Linear Growth)

Initial ₹500, receives ₹150 monthly.

• 2nd month: $500+2(150)=\text{₹}800$

• 3rd month: $500+3(150)=\text{₹}950$

• nth month: $A_n=150n+500$

2. Rally Decay (Linear Decay)

Starts with 120 members, 9 leave every hour.

• 1 hour: $120-9=111$

• 2 hours: $120-18=102$

• nth hour: $M_n=120-9n$

3. Rectangle Area (Direct Proportion)

*Length = 13 cm, Breadth = x cm.*

Area = $13\times x$
Pattern: $A=13x$
*(If breadth = 12: $156{\text{ cm}}^2$; if 10: $130{\text{ cm}}^2$)*

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Summary: Key Formulas and Terms

Concept	Explanation	Example
Degree	Highest power of variable	$5x-3$ → Degree 1
Linear Form	$ax+b,a\mathrm{\ne }0$	$2x+5$
Finding Value	Substitute the variable	Put $x=2$ in $2x=4$
Zero of Polynomial	Value making polynomial = 0	$2x-4=0\Rightarrow x=2$
Linear Pattern	Constant rate of change	$A=150n+500$