Chapter 2 Important Questions: Introduction to Linear Polynomials

The degree of the polynomial 5y³ + y² + 2y - 1 is 3, as the highest power of the variable y is 3. Such polynomials are called cubic polynomials.

(B) 3z + 7 is a linear polynomial because it has degree 1 (the highest power of the variable z is 1).

The y-intercept is -5. In the equation y = ax + b, the constant term b represents the y-intercept, which is the point where the line cuts the y-axis at (0, b). Here, the line cuts the y-axis at (0, -5).

The slope is 3. In the equation y = ax + b, the coefficient a represents the slope of the line. Here, a = 3, so the slope is 3.

(i) When x = 0: 5(0) - 3 = 0 - 3 = -3

(ii) When x = -1: 5(-1) - 3 = -5 - 3 = -8

(iii) When x = 2: 5(2) - 3 = 10 - 3 = 7

The linear polynomial for total cost is: 200 + 50m

For 8 matches: Total cost = 200 + 50(8) = 200 + 400 = ₹600

This is a linear polynomial in variable m with degree 1.

Terms: 4x, 5y, and 3

Variables: x and y

Coefficients: 4 (coefficient of x) and 5 (coefficient of y)

Constant term: 3

This expression involves two variables, so it is a two-variable polynomial.

This represents linear decay because as the value of t increases, the value of h decreases by a constant amount (0.5 m per month). The negative coefficient of t (-0.5) indicates that the height is decreasing. A linear function with a negative slope (negative coefficient of the variable) represents linear decay.

Coefficient of x³ = -3 (the number multiplying x³)

Coefficient of x² = 6 (the number multiplying x²)

Note: The coefficient of x⁴ is 1 (implied) and the constant term is 7.

Yes, these lines are parallel. Both lines have the same slope (a = 2), but different y-intercepts (b = 3 and b = -5 respectively). According to the property of linear equations, lines with equal slopes but different y-intercepts are parallel to each other.

Given: When x = 10, y = 350 and when x = 20, y = 550

Substituting in y = ax + b:

350 = 10a + b ... (1)

550 = 20a + b ... (2)

Subtracting (1) from (2):

550 - 350 = 20a - 10a

200 = 10a

a = 20

Substituting a = 20 in equation (1):

350 = 10(20) + b

350 = 200 + b

b = 150

Therefore, the linear relationship is y = 20x + 150

The coefficient 20 represents the cost per GB, and 150 represents the fixed monthly fee.

Observing the pattern: 1, 3, 5, 7, ...

The difference between consecutive terms is constant = 2

Number of tiles at Stage n = 2n - 1

Verification:

• Stage 1: 2(1) - 1 = 1 ✓

• Stage 2: 2(2) - 1 = 3 ✓

• Stage 3: 2(3) - 1 = 5 ✓

For Stage 15:

Number of tiles = 2(15) - 1 = 30 - 1 = 29 tiles

This is a linear polynomial with degree 1 showing linear growth pattern.

Linear expression for amount left after n days = 100 - 5n

(i) Amount left after 12 days:

Amount = 100 - 5(12) = 100 - 60 = ₹40

(ii) To find days when amount is ₹40:

40 = 100 - 5n

5n = 100 - 40

5n = 60

n = 12 days

Therefore, Bela will have ₹40 left after 12 days.

Let the smaller number be x.

Then the larger number = x + 10

Given: Sum of two numbers = 64

Linear equation: x + (x + 10) = 64

Simplifying:

2x + 10 = 64

2x = 64 - 10

2x = 54

x = 27

Therefore:

Smaller number = 27

Larger number = 27 + 10 = 37

Verification: 27 + 37 = 64 ✓

Given: When x = 10, y = 400 and when x = 14, y = 500

Substituting in y = ax + b:

400 = 10a + b ... (1)

500 = 14a + b ... (2)

Subtracting (1) from (2):

500 - 400 = 14a - 10a

100 = 4a

a = 25

Substituting a = 25 in equation (1):

400 = 10(25) + b

400 = 250 + b

b = 150

Therefore: a = 25 and b = 150

The linear relationship is y = 25x + 150

This means the cost per module is ₹25 and the fixed monthly fee is ₹150.