Introduction to Linear Polynomials

Find the degrees of the following polynomials:

(i) $2x^2 - 5x + 3$

(ii) $y^3 + 2y - 1$

(iii) $-9$

(iv) $4z - 3$

Write polynomials of degrees 1, 2 and 3.

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

What is the coefficient of $z$ in the polynomial $4z^3 + 5z^2 - 11$?

What is the constant term of the polynomial $9x^3 + 5x^2 - 8x - 10$?

Find the value of the linear polynomial $5x - 3$ if:

(i) $x = 0$ (ii) $x = -1$ (iii) $x = 2$

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

(i) $s = 0$ (ii) $s = -3$ (iii) $s = 4$

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.

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

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?

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?

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?

Solve the following:

1. A student has ₹500 in her savings bank account. She gets ₹150 every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the $n^{\text{th}}$ month.

A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, ... hours? Find a linear expression to represent the number of members at the end of the $n^{\text{th}}$ hour.

Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm, (ii) 10 cm, (iii) 8 cm. Find the linear pattern representing the area of the rectangle.

Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm, (ii) 9 cm, (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.

Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.

Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month.

(i) Find the height after 7 months.

(ii) Make a table of values for $t$ varying from 0 to 10 months and show how the height, $h$, increases every month.

(iii) Find an expression that relates $h$ and $t$, and explain why it represents linear growth.

A mobile phone is bought for ₹10,000. Its value decreases by ₹800 every year.

(i) Find the value of the phone after 3 years.

(ii) Make a table of values for $t$ varying from 0 to 8 years and show how the value of the phone, $v$, depreciates with time.

(iii) Find an expression that relates $v$ and $t$, and explain why it represents linear decay.

The initial population of a village is 750. Every year, 50 people move from a nearby city to the village.

(i) Find the population of the village after 6 years.

(ii) Make a table of values for $t$ varying from 0 to 10 years and show how the population, $P$, increases every year.

(iii) Find an expression that relates $P$ and $t$, and explain why it represents linear growth.

A telecom company charges ₹600 for a certain recharge scheme. This prepaid balance is reduced by ₹15 each day after the recharge.

(i) Write an equation that models the remaining balance $b(x)$ after using the scheme for $x$ days. Explain why it represents linear decay.

(ii) After how many days will the balance run out?

(iii) Make a table of values for $x$ varying from 1 to 10 days and show how the balance $b(x)$ reduces with time.

A learning platform charges a fixed monthly fee and an additional cost per digital learning module accessed. A student observes that when she accessed 10 modules, her bill was ₹400. When she accessed 14 modules, her bill was ₹500. If the monthly bill y depends on the number of modules accessed, x, according to the relation y = ax + b, find the values of a and b.

A gym charges a fixed monthly fee and an additional cost per hour for using the badminton court. A student using the gym observed that when she used the badminton court for 10 hours, her bill was ₹800. When she used it for 15 hours, her bill was ₹1100. If the monthly bill y depends on the hours of the use of the badminton court, x, according to the relation y = ax + b, find the values of a and b.

Consider the relationship between temperature measured in degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by °C = a °F + b. Find a and b, given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit. (Hint: When °C = 0, °F = 32 and when °C = 100, °F = 212. Use this information to find a and b, and thus, the linear relationship between °C and °F.)

Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'.

(i) $y = 4x, y = 2x, y = x$

(ii) $y = -6x, y = -3x, y = -x$

(iii) $y = 5x, y = -5x$

(iv) $y = 3x - 1, y = 3x, y = 3x + 1$

(v) $y = -2x - 3, y = -2x, y = 2x + 3$

Write a polynomial of degree 3 in the variable $x$, in which the coefficient of the $x^2$ term is $-7$.

Find the values of the following polynomials at the indicated values of the variables.

(i) $5x^2 - 3x + 7$ if $x = 1$

(ii) $4t^3 - t^2 + 6$ if $t = a$

If we multiply a number by $\frac{5}{2}$ and add $\frac{2}{3}$ to the product, we get $\frac{-7}{12}$. Find the number.

A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?

If you have ₹800 and you save ₹250 every month, find the amount you have after (i) 6 months (ii) 2 years. Express this as a linear pattern.

The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. Find both the numbers.

Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis.

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

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

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

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

Are any of the lines parallel?

If the temperature of a liquid can be measured in Kelvin units as $x$ K and in Fahrenheit units as $y$ °F, the relation between the two systems of measurement of temperature is given by the linear equation $y = \frac{9}{5}(x - 273) + 32$.

(i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313 K.

(ii) If the temperature is 158 °F, then find the temperature in Kelvin.

The work done by a body on the application of a constant force is the product of the constant force and the distance travelled by the body in the direction of the force. Express this in the form of a linear equation in two variables (work $w$ and distance $d$), and draw its graph by taking the constant force as 3 units. What is the work done when the distance travelled is 2 units? Verify it by plotting it on the graph.

*10. The graph of a linear polynomial $p(x)$ passes through the points $(1, 5)$ and $(3, 11)$.

(i) Find the polynomial $p(x)$.

(ii) Find the coordinates where the graph of $p(x)$ cuts the axes.

(iii) Draw the graph of $p(x)$ and verify your answers.

*11. Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(i) $p(0) = 5$.

(ii) The polynomial $p(x) - q(x)$ cuts the x-axis at $(3, 0)$.

(iii) The sum $p(x) + q(x)$ is equal to $6x + 4$ for all real $x$.

Find the polynomials $p(x)$ and $q(x)$.

*12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage.

(i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

(ii) Complete the following table.

(iii) Find a rule to determine the number of matchsticks required for the $n^{\text{th}}$ stage.

(iv) How many matchsticks will be required for the 15th stage of the pattern?

(v) Can 200 matchsticks form a stage in this pattern? Justify your answer.

*13. Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(i) The graph of $p(x)$ passes through the points $(2, 3)$ and $(6, 11)$.

(ii) The graph of $q(x)$ passes through the point $(4, -1)$.

(iii) The graph of $q(x)$ is parallel to the graph of $p(x)$.

Find the polynomials $p(x)$ and $q(x)$. Also, find the coordinates of the point where these lines meet the x-axis.

*14. What do all linear functions of the form $f(x) = ax + a, a > 0$, have in common?