Exercise 2.3 Practice
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Overview
This page provides comprehensive Ch 2: Polynomials - Exercise 2.3 Practice. Practice expressing practical word problems as linear expressions, identifying linear patterns, and evaluating sequences. Free step-by-step NCERT solutions and explanations.
Formulating and Representing Linear Patterns
Q1: Pocket Money Linear Expression
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 $n$-th month.
Initially (at Month 0), the student has:
$$A_0 = \text{₹}500$$
She receives ₹150 every month. Let's calculate the amount at the end of each month:
- Month 1: $500 + 150(1) = \text{₹}650$
- Month 2: $500 + 150(2) = \text{₹}800$
- Month 3: $500 + 150(3) = \text{₹}950$
- Month 4: $500 + 150(4) = \text{₹}1100$
Therefore, the values from the second month onwards are ₹800, ₹950, ₹1100, and so on, increasing by ₹150 each month.
For the $n$-th month, the amount $A(n)$ she will have is the initial amount plus ₹150 times $n$:
$$A(n) = 500 + 150n$$
Sequence: ₹800, ₹950, ₹1100, ...; Linear Expression: $150n + 500$
Q2: Rally Members Dropout
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 $n$-th hour.
Initial number of members is 120.
Let's calculate the remaining members after the first few hours by subtracting 9 members for each hour:
- After 1 hour: $120 - 9(1) = 111$ members
- After 2 hours: $120 - 9(2) = 102$ members
- After 3 hours: $120 - 9(3) = 93$ members
- After 4 hours: $120 - 9(4) = 84$ members
For the $n$-th hour, the remaining members $M(n)$ is represented by the initial members minus $9$ times the number of hours $n$:
$$M(n) = 120 - 9n$$
Sequence: 111, 102, 93, ...; Linear Expression: $120 - 9n$
Q3: Rectangle Area Pattern
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.
The area $A$ of a rectangle with length $l$ and breadth $b$ is given by:
$$A = l \times b$$
Given length $l = 13$ cm.
Let's find the area for each given breadth:
(i) Breadth = 12 cm: $$A = 13 \times 12 = 156\text{ cm}^2$$
(ii) Breadth = 10 cm: $$A = 13 \times 10 = 130\text{ cm}^2$$
(iii) Breadth = 8 cm: $$A = 13 \times 8 = 104\text{ cm}^2$$
(i) Breadth = 12 cm: $$A = 13 \times 12 = 156\text{ cm}^2$$
(ii) Breadth = 10 cm: $$A = 13 \times 10 = 130\text{ cm}^2$$
(iii) Breadth = 8 cm: $$A = 13 \times 8 = 104\text{ cm}^2$$
Linear Pattern:
If the breadth is represented by the variable $x$ (in cm), the area $A(x)$ is represented by: $$A(x) = 13x\text{ cm}^2$$
If the breadth is represented by the variable $x$ (in cm), the area $A(x)$ is represented by: $$A(x) = 13x\text{ cm}^2$$
(i) $156\text{ cm}^2$, (ii) $130\text{ cm}^2$, (iii) $104\text{ cm}^2$; Pattern: $13x$
Q4: Box Volume Pattern
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.
The volume $V$ of a rectangular box with length $l$, breadth $b$, and height $h$ is given by:
$$V = l \times b \times h$$
Given: length $l = 7$ cm and breadth $b = 11$ cm.
The base area is $7 \times 11 = 77\text{ cm}^2$.
The base area is $7 \times 11 = 77\text{ cm}^2$.
Let's find the volume for each given height:
(i) Height = 5 cm: $$V = 77 \times 5 = 385\text{ cm}^3$$
(ii) Height = 9 cm: $$V = 77 \times 9 = 693\text{ cm}^3$$
(iii) Height = 13 cm: $$V = 77 \times 13 = 1001\text{ cm}^3$$
(i) Height = 5 cm: $$V = 77 \times 5 = 385\text{ cm}^3$$
(ii) Height = 9 cm: $$V = 77 \times 9 = 693\text{ cm}^3$$
(iii) Height = 13 cm: $$V = 77 \times 13 = 1001\text{ cm}^3$$
Linear Pattern:
If the height is represented by the variable $h$ (in cm), the volume $V(h)$ is represented by: $$V(h) = 77h\text{ cm}^3$$
If the height is represented by the variable $h$ (in cm), the volume $V(h)$ is represented by: $$V(h) = 77h\text{ cm}^3$$
(i) $385\text{ cm}^3$, (ii) $693\text{ cm}^3$, (iii) $1001\text{ cm}^3$; Pattern: $77h$
Q5: Book Reading Progress
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.
Total pages in the book = 500.
Pages read per day = 20.
Pages read per day = 20.
Let's find the pages read after 15 days:
$$\text{Pages read} = 20 \times 15 = 300\text{ pages}$$
Now, calculate pages left after 15 days:
$$\text{Pages left} = 500 - 300 = 200\text{ pages}$$
Linear Pattern:
If the number of days is represented by the variable $d$, the pages left $P(d)$ is represented by: $$P(d) = 500 - 20d$$
If the number of days is represented by the variable $d$, the pages left $P(d)$ is represented by: $$P(d) = 500 - 20d$$
Pages left after 15 days = 200; Pattern: $500 - 20d$