---
title: "NCERT Solutions for Class 9 Maths Chapter 4 Exercise 4.1"
url: https://www.swavid.com/maths/class/9/chapter/exploring-algebraic-identities/ncert-solutions/exercise-4-1
dateModified: 2026-10-07T15:43:39+00:00
---

# NCERT Solutions for Class 9 Maths Chapter 4 Exercise 4.1

Chapter 4: Exploring Algebraic Identities. Every question from Exercise 4.1, with full working and the final answer.

Free PDF (27 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-9/swavid-ncert-solutions-class-9-maths-chapter-4-exploring-algebraic-identities-4d2a728afe.pdf

## Exercise Set 4.1

### Question 1

*3 marks · Short answer*

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, expand the following:
(i) $(7x + 4y)^2$
(ii) $\left(\frac{7}{5}x + \frac{3}{2}y\right)^2$
(iii) $(2.5p + 1.5q)^2$
(iv) $\left(\frac{3}{4}s + 8t\right)^2$
(v) $\left(x + \frac{1}{2y}\right)^2$
(vi) $\left(\frac{1}{x} + \frac{1}{y}\right)^2$

**Part (i)**

1. Given expression is $(7x + 4y)^2$.
2. Substitute $a = 7x$ and $b = 4y$ into $(a+b)^2 = a^2 + 2ab + b^2$.
3. $(7x + 4y)^2 = (7x)^2 + 2(7x)(4y) + (4y)^2$.
4. $= 49x^2 + 56xy + 16y^2$.

Answer (i): $49x^2 + 56xy + 16y^2$

**Part (ii)**

1. Given expression is $\left(\frac{7}{5}x + \frac{3}{2}y\right)^2$.
2. Substitute $a = \frac{7}{5}x$ and $b = \frac{3}{2}y$ into $(a+b)^2 = a^2 + 2ab + b^2$.
3. $= \left(\frac{7}{5}x\right)^2 + 2\left(\frac{7}{5}x\right)\left(\frac{3}{2}y\right) + \left(\frac{3}{2}y\right)^2$.
4. $= \frac{49}{25}x^2 + \frac{21}{5}xy + \frac{9}{4}y^2$.

Answer (ii): $\frac{49}{25}x^2 + \frac{21}{5}xy + \frac{9}{4}y^2$

**Part (iii)**

1. Given expression is $(2.5p + 1.5q)^2$.
2. Substitute $a = 2.5p$ and $b = 1.5q$ into $(a+b)^2 = a^2 + 2ab + b^2$.
3. $= (2.5p)^2 + 2(2.5p)(1.5q) + (1.5q)^2$.
4. $= 6.25p^2 + 7.5pq + 2.25q^2$.

Answer (iii): $6.25p^2 + 7.5pq + 2.25q^2$

**Part (iv)**

1. Given expression is $\left(\frac{3}{4}s + 8t\right)^2$.
2. Substitute $a = \frac{3}{4}s$ and $b = 8t$ into $(a+b)^2 = a^2 + 2ab + b^2$.
3. $= \left(\frac{3}{4}s\right)^2 + 2\left(\frac{3}{4}s\right)(8t) + (8t)^2$.
4. $= \frac{9}{16}s^2 + 12st + 64t^2$.

Answer (iv): $\frac{9}{16}s^2 + 12st + 64t^2$

**Part (v)**

1. Given expression is $\left(x + \frac{1}{2y}\right)^2$.
2. Substitute $a = x$ and $b = \frac{1}{2y}$ into $(a+b)^2 = a^2 + 2ab + b^2$.
3. $= (x)^2 + 2(x)\left(\frac{1}{2y}\right) + \left(\frac{1}{2y}\right)^2$.
4. $= x^2 + \frac{x}{y} + \frac{1}{4y^2}$.

Answer (v): $x^2 + \frac{x}{y} + \frac{1}{4y^2}$

**Part (vi)**

1. Given expression is $\left(\frac{1}{x} + \frac{1}{y}\right)^2$.
2. Substitute $a = \frac{1}{x}$ and $b = \frac{1}{y}$ into $(a+b)^2 = a^2 + 2ab + b^2$.
3. $= \left(\frac{1}{x}\right)^2 + 2\left(\frac{1}{x}\right)\left(\frac{1}{y}\right) + \left(\frac{1}{y}\right)^2$.
4. $= \frac{1}{x^2} + \frac{2}{xy} + \frac{1}{y^2}$.

Answer (vi): $\frac{1}{x^2} + \frac{2}{xy} + \frac{1}{y^2}$

**Answer:** Expanded forms of the given binomials using $(a+b)^2 = a^2 + 2ab + b^2$.

> Common mistake: Forgetting to multiply the middle term $2ab$ by 2.

### Question 2

*3 marks · Short answer*

Using the same identity, find the values of the following:
(i) $(64)^2$
(ii) $(105)^2$
(iii) $(205)^2$

**Part (i)**

1. Express $64$ as $(60 + 4)^2$.
2. Apply $(a+b)^2 = a^2 + 2ab + b^2$ with $a = 60$ and $b = 4$.
3. $= 60^2 + 2(60)(4) + 4^2$.
4. $= 3600 + 480 + 16 = 4096$.

Answer (i): $4096$

**Part (ii)**

1. Express $105$ as $(100 + 5)^2$.
2. Apply $(a+b)^2 = a^2 + 2ab + b^2$ with $a = 100$ and $b = 5$.
3. $= 100^2 + 2(100)(5) + 5^2$.
4. $= 10000 + 1000 + 25 = 11025$.

Answer (ii): $11025$

**Part (iii)**

1. Express $205$ as $(200 + 5)^2$.
2. Apply $(a+b)^2 = a^2 + 2ab + b^2$ with $a = 200$ and $b = 5$.
3. $= 200^2 + 2(200)(5) + 5^2$.
4. $= 40000 + 2000 + 25 = 42025$.

Answer (iii): $42025$

**Answer:** Values computed using $(a+b)^2 = a^2 + 2ab + b^2$.

> Common mistake: Incorrect splitting of the number or wrong multiplication in the middle term.

## Related pages

- [All Chapter 4 NCERT solutions](https://www.swavid.com/maths/class/9/chapter/exploring-algebraic-identities/ncert-solutions)
- [Exercise 4.2](https://www.swavid.com/maths/class/9/chapter/exploring-algebraic-identities/ncert-solutions/exercise-4-2)
- [Exercise 4.3](https://www.swavid.com/maths/class/9/chapter/exploring-algebraic-identities/ncert-solutions/exercise-4-3)
- [Exercise 4.4](https://www.swavid.com/maths/class/9/chapter/exploring-algebraic-identities/ncert-solutions/exercise-4-4)
- [Exercise 4.5](https://www.swavid.com/maths/class/9/chapter/exploring-algebraic-identities/ncert-solutions/exercise-4-5)

Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student