NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Exercise 4.1

NCERT Solutions for Class 11 Maths Chapter 4 Exercise 4.1 Complex Numbers and Quadratic Equations help students understand the basic concepts of complex numbers and their properties. This exercise introduces students to the idea of numbers that include an imaginary component, which is essential for solving equations that do not have real solutions.

Prepared according to the latest CBSE Class 11 Maths syllabus, Exercise 4.1 focuses on the definition of complex numbers, imaginary numbers, and operations on complex numbers such as addition, subtraction, multiplication, and division. Practicing this exercise helps students develop a strong foundation for solving problems involving complex numbers.

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Exercise 4.1

Complex Numbers and Quadratic Equations

Medium

Q.

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

$\begin{array}{l}1.\left(5\text{i}\right)\left(-\frac{3}{5}\text{i}\right)\\ \text{2.\hspace{0.33em}}{\mathrm{i}}^9+{\mathrm{i}}^{19}\\ \text{3.\hspace{0.33em}}{\mathrm{i}}^{-39}\\ \text{4.\hspace{0.33em}}3(7+\mathrm{i}7)+\mathrm{i}(7+\mathrm{i}7)\\ \text{5.\hspace{0.33em}}(1-\mathrm{i})-(-1+\mathrm{i}6)\\ \text{6.\hspace{0.33em}}\left(\frac{1}{5}+\mathrm{i}\frac{2}{5}\right)-\left(4+\mathrm{i}\frac{5}{2}\right)\\ \text{7.\hspace{0.33em}}\left[\left(\frac{1}{3}+\mathrm{i}\frac{7}{3}\right)+\left(4+\mathrm{i}\frac{1}{3}\right)\right]-\left(-\frac{4}{3}+\mathrm{i}\right)\\ \text{8.\hspace{0.33em}}{(1-\mathrm{i})}^4\\ 9.{\left(\frac{1}{3}+3\text{i}\right)}^3\\ 10.{\left(-2-\frac{1}{3}\text{i}\right)}^3\end{array}$

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Complex Numbers and Quadratic Equations

Difficult

Q.

$\text{If a + ib = }\frac{{\left(\text{x + iy}\right)}^{\text{2}}}{{\text{2x}}^{\text{2}}\text{+1}}{\text{, prove that a}}^{\text{2}}{\text{+b}}^{\text{2}}\text{ = }\frac{{\left({\text{x}}^{\text{2}}\text{+1}\right)}^{\text{2}}}{{\left({\text{2x}}^{\text{2}}\text{+1}\right)}^{\text{2}}}\text{.}$

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Complex Numbers and Quadratic Equations

Medium

Q.

For any two complex numbers, z₁ and z₂, prove that
Re (z₁ z₂) = Re z₁ Re z₂ – Imz₁ Imz₂.

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Complex Numbers and Quadratic Equations

Medium

Q.

$\text{Reduce\hspace{0.17em}\hspace{0.17em}}\left(\frac{\text{1}}{\text{1 - 4i}}\text{ - }\frac{\text{2}}{\text{1 + i}}\right)\left(\frac{\text{3 - 4i}}{\text{5 + i}}\right)\text{ to the standard form.}$

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Complex Numbers and Quadratic Equations

Difficult

Q.

$\mathrm{If} \mathrm{x}-\mathrm{iy}=\sqrt{\frac{\mathrm{a}-\mathrm{ib}}{\mathrm{c}-\mathrm{id}}}\mathrm{prove} \mathrm{that} {\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}^2=\frac{{\mathrm{a}}^2+{\mathrm{b}}^2}{{\mathrm{c}}^2+{\mathrm{d}}^2}.$

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Complex Numbers and Quadratic Equations

Medium

Q.

$\begin{array}{l}\text{Convert the following in the polar form:}\\ \left(\text{i}\right)\frac{\text{1+7i}}{{\left(\text{2}-\text{i}\right)}^{\text{2}}}\\ \left(\text{ii}\right)\frac{\text{1+3i}}{\text{1}-\text{2i}}\end{array}$

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Complex Numbers and Quadratic Equations

Medium

Q.

Solve each of the equation in Exercises 6 to 9.

$\begin{array}{l}6.3{\text{x}}^2-4\text{x}+\frac{20}{3}=0\\ 7.{\text{\hspace{0.33em}x}}^2-2\text{x}+\frac{3}{2}=0\\ \text{8.\hspace{0.33em}}27{\mathrm{x}}^2-10\mathrm{x}+1=0\\ \text{9.\hspace{0.33em}}21{\mathrm{x}}^2-28\mathrm{x}+10=0\end{array}$

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Complex Numbers and Quadratic Equations

Medium

Q.

$\mathrm{If}{\mathrm{z}}_1=2-\mathrm{i},{\mathrm{z}}_2=1+\mathrm{i}, \mathrm{find} \left|\frac{{\mathrm{z}}_1+{\mathrm{z}}_2+1}{{\mathrm{z}}_1-{\mathrm{z}}_2+1}\right|.$

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Complex Numbers and Quadratic Equations

Medium

Q.

$\begin{array}{l}{\text{Let z}}_1{\text{= 2 - i,\hspace{0.17em}\hspace{0.17em}z}}_{\text{2}}\text{ = -2 + i. Find}\\ \left(\text{i}\right) \text{Re}\left(\frac{{\text{z}}_{\text{1}}{\text{z}}_{\text{2}}}{{\overline{\text{z}}}_{\text{1}}}\right)\text{, }\left(\text{ii}\right)\text{ Im }\left(\frac{\text{1}}{{\text{z}}_{\text{1}}{\overline{\text{z}}}_{\text{1}}}\right)\end{array}$

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Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: 1 + i

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Complex Numbers and Quadratic Equations

Difficult

Q.

$\mathbf{\text{Find the modulus and argument of the complex number}}\frac{\mathbf{\text{1+2i}}}{\mathbf{\text{1-3i}}}\mathbf{\text{.}}$

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Complex Numbers and Quadratic Equations

Difficult

Q.

Find the real numbers x and y if
(x – iy) (3 + 5i) is the conjugate of –6 – 24i.

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Complex Numbers and Quadratic Equations

Medium

Q.

$\text{Find the modulus of\hspace{0.17em}\hspace{0.17em}}\frac{\text{1 + i}}{\text{1 – i}}–\frac{\text{1 }–\text{ i}}{\text{1 + i}}\text{.}$

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Complex Numbers and Quadratic Equations

Medium

Q.

${\text{If (x + iy)}}^{\text{3}}\text{ = u + iv, then show that\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{\text{u}}{\text{x}}\text{+}\frac{\text{v}}{\text{y}}\text{ = 4}\left({\text{x}}^{\text{2}}{\text{ – y}}^{\text{2}}\right)\text{.}$

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Complex Numbers and Quadratic Equations

Difficult

Q.

$\text{If α and β are different complex numbers with }\left|\text{β}\right|\text{ =1, then find }\left|\frac{\text{β-α}}{\text{1-}\overline{\text{α}}\text{\hspace{0.17em}β}}\right|\text{.}$

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Complex Numbers and Quadratic Equations

Medium

Q.

$\begin{array}{l}\text{Find the number of non-zero integral solutions of the}\\ \text{equation }{\left|\text{1 - i}\right|}^{\text{x}}{\text{= 2}}^{\text{x}}\text{.}\end{array}$

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Complex Numbers and Quadratic Equations

Medium

Q.

If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that
(a² + b²) (c² + d²) (e² + f²) (g² + h²) = A² + B²

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Complex Numbers and Quadratic Equations

Medium

Q.

$\mathrm{Evaluate}:{[{\mathrm{i}}^{18}+{\left(\frac{1}{\mathrm{i}}\right)}^{25}]}^3$

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Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: i

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Complex Numbers and Quadratic Equations

Medium

Q.

Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.

11. 4 – 3i
12. $\sqrt{5}+3\mathrm{i}$
13. – i

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Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
          x² + 3x + 5 = 0

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Complex Numbers and Quadratic Equations

Medium

Q.

$\begin{array}{l}\text{Express the following expression in the form of a+ib:}\\ \frac{\left(\text{3+i}\sqrt{\text{5}}\right)\left(\text{3-i}\sqrt{\text{5}}\right)}{\left(\sqrt{\text{3}}\text{+}\sqrt{\text{2}}\text{i}\right)\text{-}\left(\sqrt{\text{3}}\text{-}\sqrt{\text{2}}\text{i}\right)}\end{array}$

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Complex Numbers and Quadratic Equations

Medium

Q.

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

$\begin{array}{l}1.\mathrm{z}=-1-\mathrm{i}\sqrt{3}\\ 2.\text{\hspace{0.33em}z}=-\sqrt{3}+\text{i}\end{array}$

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Complex Numbers and Quadratic Equations

Medium

Q.

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

3. 1 – i
4. – 1 + i
5. – 1 – i
6. – 3
7. $\sqrt{3}+\mathrm{i}$
8. i

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Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
         x² + 3 = 0

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Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
        2x² + x + 1 = 0

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Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
         x² + 3x + 9 = 0

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Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
         – x² + x – 2 = 0

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Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
          x² – x + 2 = 0

 

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Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: –i

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Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:

$\sqrt{2}{x}^2+x+\sqrt{2}=0$

 

 

 

 

 

 

 

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Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:

$\sqrt{3}{x}^2-\sqrt{2}x+3\sqrt{3}=0$

 

 

 

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Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:

$x^2+x+\frac{1}{\sqrt{2}}=0$

 

 

 

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Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:
${\mathrm{x}}^2+\frac{\mathrm{x}}{\sqrt{2}}+1=0$

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Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: – 15 – 8i

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Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: – 8 – 6i.

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Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: 1 – i

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Complex Numbers and Quadratic Equations

Medium

Q.

$\text{If }{\left(\frac{\text{1 + i}}{\text{1 - i}}\right)}^{\text{m}}\text{=1, then find the least positive integral\hspace{0.33em}value of m.}$

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The solutions are explained in a clear and step-by-step format so students can easily understand the concept and apply the methods correctly in examinations.

1. Solve $24x<100$ when

(i) x is a natural number
$24x<100$
$\Rightarrow x<\frac{100}{24}=\frac{25}{6}$
Natural numbers less than $\frac{25}{6}$ are 1, 2, 3, 4
Solution set: $\{1,2,3,4\}$

(ii) x is an integer
$x<\frac{25}{6}$
Integers less than $\frac{25}{6}$ are ..., -3, -2, -1, 0, 1, 2, 3, 4
Solution set: $\{\mathrm{.}\mathrm{.}\mathrm{.},-3,-2,-1,0,1,2,3,4\}$

2. Solve $-12x>30$ when

(i) x is a natural number
$-12x>30\Rightarrow -x>\frac{5}{2}\Rightarrow x<-\frac{5}{2}$
No natural number is less than $-\frac{5}{2}$
Solution set: No solution

(ii) x is an integer
$-12x>30\Rightarrow x<-\frac{5}{2}$
Integers less than $-\frac{5}{2}$ are -5, -4, -3
Solution set: $\{-5,-4,-3\}$

3. Solve $5x-3<7$ when

(i) x is an integer
$5x-3<7\Rightarrow 5x<10\Rightarrow x<2$
Integers less than 2 are ..., -4, -3, -2, -1, 0, 1
Solution set: $\{\mathrm{.}\mathrm{.}\mathrm{.},-4,-3,-2,-1,0,1\}$

(ii) x is a real number
$x<2$
Solution set: $(-\mathrm{\infty },2)$

4. Solve $3x+8>2$ when

(i) x is an integer
$3x+8>2\Rightarrow 3x>-6\Rightarrow x>-2$
Integers greater than -2 are -1, 0, 1, 2, 3, ...
Solution set: $\{-1,0,1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}\}$

(ii) x is a real number
$x>-2$
Solution set: $(-2,\mathrm{\infty })$

5. Solve for real x: $4x+3<5x+7$

$4x+3<5x+7\Rightarrow 4x-5x<7-3\Rightarrow -x<4\Rightarrow x>-4$
Solution set: $(-4,\mathrm{\infty })$

6. Solve for real x: $3x-7>5x-1$

$3x-7>5x-1\Rightarrow 3x-5x>-1+7\Rightarrow -2x>6\Rightarrow x<-3$
Solution set: $(-\mathrm{\infty },-3)$

7. Solve for real x: $3(x-1)\le 2(x-3)$

$3x-3\le 2x-6\Rightarrow 3x-2x\le -6+3\Rightarrow x\le -3$
Solution set: $(-\mathrm{\infty },-3]$

8. Solve for real x: $3(2-x)\ge 2(1-x)$

$6-3x\ge 2-2x\Rightarrow -3x+2x\ge 2-6\Rightarrow -x\ge -4\Rightarrow x\le 4$
Solution set: $(-\mathrm{\infty },4]$

9. Solve for real x: $x+\frac{x}{2}+\frac{x}{3}<11$

$x(1+\frac{1}{2}+\frac{1}{3})<11\Rightarrow x\cdot \frac{11}{6}<11\Rightarrow x<6$
Solution set: $(-\mathrm{\infty },6)$

10. Solve for real x: $\frac{x}{3}>\frac{x}{2}+1$

$\frac{x}{3}-\frac{x}{2}>1\Rightarrow x(\frac{1}{3}-\frac{1}{2})>1\Rightarrow x(-\frac{1}{6})>1\Rightarrow x<-6$
Solution set: $(-\mathrm{\infty },-6)$

11. Solve for real x: $\frac{3(x-2)}{5}\le \frac{5(2-x)}{3}$

$9(x-2)\le 25(2-x)\Rightarrow 9x-18\le 50-25x\Rightarrow 34x\le 68\Rightarrow x\le 2$
Solution set: $(-\mathrm{\infty },2]$

12. Solve for real x: $\frac{1}{2}(\frac{3x}{5}+4)\ge \frac{1}{3}x-2$

$\frac{3x}{10}+2\ge \frac{x}{3}-2\Rightarrow \frac{3x}{10}-\frac{x}{3}\ge -4\Rightarrow \frac{9x-10x}{30}\ge -4\Rightarrow -\frac{x}{30}\ge -4\Rightarrow x\le 120$
Solution set: $(-\mathrm{\infty },120]$

13. Solve for real x: $2(2x+3)-10<6(x-2)$

$4x+6-10<6x-12\Rightarrow 4x-4<6x-12\Rightarrow -2x<-8\Rightarrow x>4$
Solution set: $(4,\mathrm{\infty })$

14. Solve for real x: $37-(3x+5)\ge 9x-8(x-3)$

$37-3x-5\ge 9x-8x+24\Rightarrow -3x+32\ge x+24\Rightarrow -4x\ge -8\Rightarrow x\le 2$
Solution set: $(-\mathrm{\infty },2]$

15. Solve for real x: $\frac{x}{4}<\frac{(5x-2)}{3}-\frac{(7x-3)}{5}$

$\frac{x}{4}<\frac{25x-10-21x+9}{15}\Rightarrow \frac{x}{4}<\frac{4x-1}{15}\Rightarrow 15x<16x-4\Rightarrow -x<-4\Rightarrow x>4$
Solution set: $(4,\mathrm{\infty })$

16. Solve for real x: $\frac{(2x-1)}{3}\ge \frac{(3x-2)}{4}-\frac{(2-x)}{5}$

$\frac{2x-1}{3}\ge \frac{15x-10-8+4x}{20}\Rightarrow \frac{2x-1}{3}\ge \frac{19x-18}{20}\Rightarrow 40x-20\ge 57x-54\Rightarrow 34\ge 17x\Rightarrow x\le 2$
Solution set: $(-\mathrm{\infty },2]$

17. Solve and graph on number line: $3x-2<2x+1$

$3x-2x<1+2\Rightarrow x<3$
Graph: Open circle at 3, line to left.

18. Solve and graph: $5x-3\ge 3x-5$

$5x-3x\ge -5+3\Rightarrow 2x\ge -2\Rightarrow x\ge -1$
Graph: Closed circle at -1, line to right.

19. Solve and graph: $3(1-x)<2(x+4)$

$3-3x<2x+8\Rightarrow -3x-2x<8-3\Rightarrow -5x<5\Rightarrow x>-1$
Graph: Open circle at -1, line to right.

20. Solve and graph: $\frac{x}{2}\ge \frac{(5x-2)}{3}-\frac{(7x-3)}{5}$

$\frac{x}{2}\ge \frac{4x-1}{15}\Rightarrow 15x\ge 8x-2\Rightarrow 7x\ge -2\Rightarrow x\ge -\frac{2}{7}$
Graph: Closed circle at -2/7, line to right.

21. Ravi’s marks average problem

Let third test marks = $x$
$\frac{70+75+x}{3}\ge 60\Rightarrow 145+x\ge 180\Rightarrow x\ge 35$
Minimum marks: 35

22. Sunita’s grade problem

Let fifth exam marks = $x$
$\frac{87+92+94+95+x}{5}\ge 90\Rightarrow 368+x\ge 450\Rightarrow x\ge 82$
Minimum marks: 82

23. Consecutive odd positive integers

Let integers be $x$ and $x+2$
$x+2<10\Rightarrow x<8$
$x+(x+2)>11\Rightarrow 2x>9\Rightarrow x>4.5$
So $x=5,7$
Pairs: (5, 7), (7, 9)

24. Consecutive even positive integers

Let integers be $x$ and $x+2$
$x>5$ and $x+(x+2)<23\Rightarrow 2x<21\Rightarrow x<10.5$
So $x=6,8,10$
Pairs: (6, 8), (8, 10), (10, 12)

25. Triangle sides problem

Shortest side = $x$
Longest side = $3x$, Third side = $3x-2$
Perimeter $\ge 61$
$x+3x+(3x-2)\ge 61\Rightarrow 7x\ge 63\Rightarrow x\ge 9$
Minimum length: 9 cm

26. Board cutting problem

Shortest piece = $x$ cm
Second = $x+3$ cm, Third = $2x$ cm
Total length $\le 91$:
$x+(x+3)+2x\le 91\Rightarrow 4x+3\le 91\Rightarrow 4x\le 88\Rightarrow x\le 22$
Also, third $\ge$ second + 5:
$2x\ge (x+3)+5\Rightarrow 2x\ge x+8\Rightarrow x\ge 8$
Possible lengths: $8\le x\le 22$ cm

FAQs – Class 11 Maths Chapter 4 Exercise 4.1 Complex Numbers and Quadratic Equations

Q1. What is the focus of Exercise 4.1?
Exercise 4.1 focuses on understanding complex numbers, imaginary numbers, and performing basic operations on them.

Q2. What is a complex number?
A complex number is a number of the form a + ib, where a is the real part and b is the imaginary part, and i = √(-1).

Q3. Why is Exercise 4.1 important for exams?
This exercise builds the foundation for complex numbers and quadratic equations, which are important topics in higher mathematics.

Q4. How can students prepare effectively for Exercise 4.1?
Students should understand the definition of complex numbers clearly, practice basic operations on complex numbers, and solve different numerical problems for better understanding.