1. Find the value of the linear polynomial 5x – 3 if:
(i) x = 0 (ii) x = –1 (iii) x = 2
2. Find the value of the quadratic polynomial 7s2
– 4s + 6 if:
(i) s = 0 (ii) s = –3 (iii) s = 4
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.
4. The difference between two positive integers is 63. The ratio of the
two integers is 2:5. Find the two integers.
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?
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?
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?
2.3 Exploring linear patterns
Observe the following growing pattern of square tiles.
Think and Reflect
Predict the number of squares in the next three stages of the pattern and
write the sequence of numbers up to Stage 7 of the pattern.
Each stage is obtained by adding two more tiles to the previous stage.
The table mentions the number of tiles for the first seven stages.
To generalise this pattern, we observe that the number of squares at
each stage is one less than twice the number of the term. For example,
in Term 2, the number of squares is 2 × 2 – 1 = 3, in Term 5, the number
of squares is 2 × 5 – 1 = 9 and so on.
This leads us to conclude that the number of squares at Stage n is
given by 2n – 1.
The polynomial 2n – 1 has degree 1. Hence, it is an example of a
linear polynomial. Also, the difference between consecutive terms in
the sequence of the number of squares, that is, 1, 3, 5, 7, 9 … is the
constant value 2. Thus, with each stage, the number of squares increases
by 2. The relationship between the number of the stage and the number
of square tiles is a linear relationship.
Think and Reflect
Using the expression 2n – 1, can you find out how many tiles will be there in
the 15th stage and the 26th stage of the pattern? Also, which stage will contain
21 tiles and 47 tiles?
Example 7: Bela has `100 for pocket money. She spends rupee 5 every
day. After how many days will she be left with rupees 40?
Observe that the amount left on the nth day will be `(100 – 5n).
Therefore, on the 12th day the amount left will be `(100 – 12 × 5) = `40.
Think and Reflect
What amount will be left on the 15th day? How many days will it take for
the entire amount to be spent?
Example 8: An auto-rikshaw fare starts at `25 and remains the same
for the initial 2 km. Then it increases by `15 per km. What will be the
fare for a travel of 10 km?
For the initial 2 km, the fare is `25. For every kilometre (km)
thereafter, the fare will increase by `15. Therefore, the total fare for a
travel of 10 km will be `25 + 15 × 8 = `145.
Observe that the total fare for a travel of n km will be
`25 + 15 × (n – 2) = 15n – 5, when n ≥ 2. Here, the fare for a specific
distance covered is a function of the distance n km.
Think and Reflect
For how many km will the fare be `130?
Note that in all the above examples, the nth term is a linear expression
in n. A linear pattern is a sequence of numbers where the difference
between two consecutive terms is constant. We will learn more about
linear patterns in the chapter on Sequences and Progressions.