END OF CHAPTER EXERCISE
1. Write a polynomial of degree 3 in the variable x, in which the
coefficient of the x2
term is –7.
2. Find the values of the following polynomials at the indicated
values of the variables.
(i) 5×2
– 3x + 7 if x = 1
(ii) 4t3
– t2
+ 6 if t = a
3. If we multiply a number by 5
2 and add 2
3 to the product, we get
–7
12. Find the number.
4. 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?
5. 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.
*6. 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.
*7. 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?
*8. 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 =
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.
*9. 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 isthe 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 nth 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?