Exercise 5.1 class 11 maths solutions | class 11 ch 5 exercise 5.1 solutions | class 11 chapter 5 exercise 5.1 solution | class 11 maths ncert solutions chapter 5 | ncert solutions for class 11 maths chapter 5 | ncert exemplar class 11 maths | class 11 complex numbers and quadratic equations
Looking for Exercise 5.1 Class 11 Maths solutions? You’re in the right place! This section offers detailed and easy-to-understand solutions for all questions from Exercise 5.1 of Chapter 5 – Complex Numbers and Quadratic Equations. Based on the latest NCERT curriculum, these solutions help you grasp the fundamental concepts of complex numbers, including imaginary units, algebra of complex numbers, and their basic properties. Whether you’re working on Class 11 Ch 5 Exercise 5.1 solutions or referring to the NCERT Exemplar Class 11 Maths, these step-by-step answers are perfect for building a strong foundation. Download or view the Class 11 Maths NCERT Solutions Chapter 5 now and take your understanding of complex numbers and quadratic equations to the next level!
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Exercise 5.1
1. Express each of the complex number given in the in the form \( \mathrm{a}+\mathrm{ib} \).
Answer
\( (5i) \)\(\left(\frac{-3}{5} i\right)=-5 \times \frac{3}{5} \times i \times i\)
\( =-3 \mathrm{i}^{2}\)
\( =-3(-1)\)
\( =3\)
2. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\(\mathrm{i}^{9}+\mathrm{i}^{19}\)
\(\mathrm{i}^{9}+\mathrm{i}^{19}\)
Answer
\(i^{9}+\mathrm{i}^19 =\mathrm{i} 4 x-1+\mathrm{i} 4 x+1\)
\( =\left(\mathrm{i}^{4}\right)^{2} \cdot \mathrm{j}+\left(\mathrm{j}^{4}\right)^{4} \cdot \mathrm{i}^{3}\)
\( =1 \times i+1 \times(-\mathrm{j})\)
\( =i+(-i)\)
\( =0\)
3. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\(i^{-39}\)
\(i^{-39}\)
Answer
\(i^{-14}=f^{1 \times 4-3}=\left(i^{4}\right)^{-9} \cdot l^{-3}\)
\(=(1)^{-9}-F^{-3}\)
\(=\frac{1}{p^{3}}=\frac{1}{-i}\)
\(=\frac{-1}{i} \times \frac{j}{i}\)
\(=\frac{-i}{i^{2}}=\frac{-i}{-1}=i\)
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4. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\(3(7+\mathrm{i} 7)+i(7+\mathrm{i} 7)\)
\(3(7+\mathrm{i} 7)+i(7+\mathrm{i} 7)\)
Answer
\(3(7+\mathrm{i} 7)+i(7+\mathrm{i} 7)=21+21 \mathrm{i}+7 \mathrm{i}+7 \mathrm{i}^{2}\)
\(=21+28 \mathrm{i}+7 \times(-1)\)
\(=14+28 \mathrm{i}\)
5. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\((1-i)-(-1+\mathrm{i} 6)\)
\((1-i)-(-1+\mathrm{i} 6)\)
Answer
\((1-\mathrm{i})-(-1+\mathrm{i} 6) \Rightarrow 1-\mathrm{i}+1-\mathrm{i} 6\)
\(\Rightarrow 2-7 \mathrm{i}\)
6. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\(\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)\)
\(\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)\)
Answer
\(
\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)\)
\(\quad=\frac{1}{5}+\frac{2}{5} i-4-\frac{5}{2} l\)
\(=\left(\frac{1}{5}-4\right)+i\left(\frac{2}{5}-\frac{5}{2}\right)\)
\(=\frac{-19}{5}+i \frac{-21}{10}\)
\(=\frac{-19}{5}-\frac{21}{10} i\)
7. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
Answer
\({\left[\left(\frac{1}{3}+i \frac{7}{3}\right)+\left(4+i \frac{1}{3}\right)\right]-\left(\frac{-4}{3}+i\right)}\)
\(=\frac{1}{3}+\frac{7}{3} i+4+\frac{1}{3} i+\frac{4}{3}-i\)
\(=\left(\frac{1}{3}+4+\frac{4}{3}\right)+\left(\frac{7}{3} i+\frac{4}{3}-i\right)\)
\(=\frac{17}{3}+i \frac{5}{3}\)
8. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\((1-i)^{4}\)
\((1-i)^{4}\)
Answer
\((1-i)^{t}=\left[\left(1-i^{2}\right)\right]^{2}\)
\(=\left[1^{2}+i^{2}-2 i\right]^{2}\)
\(=[1-1-2 i]^{2}\)
\(=(-2 i)^{2}\)
\(=(-2 i) \times(-2 i)\)
\(=4 i^{2}\)
\( \Rightarrow-4\)
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9. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\(\left(\frac{1}{3}+3 i\right)^{3}\)
\(\left(\frac{1}{3}+3 i\right)^{3}\)
Answer
\(\left(\frac{1}{3}+3 i\right)^{3}=\left(\frac{1}{3}\right)^{3}+(3 i)^{3}+3 \times \frac{1}{3}(3 i)\left(\frac{1}{3}+3 i\right)\)
\(=\frac{1}{27}+27 i^{3}+3 i\left(\frac{1}{3}+3 i\right)\)
\(=\frac{1}{27}+27(-i)+i+9 i^{2}\)
\(=\frac{1}{27}-27 i+i-9\)
\(=\left(\frac{1}{27}-9\right)+i(-27+1)\)
\(=\frac{-242}{27}-26 i\)
10. Express each of the complex number given in the form \( \mathrm{a}+\mathrm{ib} \).
\(\left(-2-\frac{1}{3} i\right)^{3}\)
\(\left(-2-\frac{1}{3} i\right)^{3}\)
Answer
\((-2 \left.-\frac{1}{3} i\right)^{3}=(-1)^{3}\left(2+\frac{1}{3} i\right)^{3}\)
\( =-\left[2^{3}+\left(\frac{i}{3}\right)^{3}+3(2)\left(\frac{i}{3}\right)\left(2+\frac{i}{3}\right)\right]\)
\( =-\left[8+\frac{i^{3}}{27}+2 i\left(2+\frac{i}{3}\right)\right]\)
\( =-\left[8-\frac{i}{27}+4 i+\frac{2 i^{2}}{3}\right]\)
\( =-\left[8-\frac{i}{27}+4 i-\frac{2}{3}\right]\)
\(=-\left[\frac{22}{3}+\frac{107 i}{27}\right]\)
\(=-\frac{22}{3}-\frac{107}{27} i\)
11. Find the multiplicative inverse of the complex number (4-3i)
Answer
Let \( \mathrm{z}=4-3 i \)
Then, \( \bar{z}=4+3 \mathrm{i} \) and \( |z|^{2}=4^{2}+(-3)^{2}=16+9=25 \)
Therefore, the multiplicative inverse of \( 4-3 i \) is given by
\(z^{-1}=\frac{z}{|z|}=\frac{4+3 i}{25}=\frac{4}{25}+\frac{3}{25} i\)
12. Find the multiplicative inverse of \( \sqrt{5}+3 \mathrm{i} \)
Answer
Let \( \mathrm{z}=\sqrt{5}+3 \mathrm{i} \)
Then, \( \bar{z}=\sqrt{5}-3 i \) and \( |z|^{2}=(\sqrt{5})^{2}+3^{2}=5+9=14 \)
Therefore, the multiplicative inverse of \( \sqrt{5}+3 \mathrm{i} \) is given by
\(z^{-1}=\frac{z}{|z|^{2}}=\frac{\sqrt{5}-3 i}{14}=\frac{\sqrt{5}}{14}-\frac{3 i}{14}\)
13. Find the multiplicative inverse of given -i
Answer
Let \( \mathrm{z}=-i \)
Then, \( \bar{z}=\mathrm{I} \) and \( |z|^{2}=1^{2}=1 \)
Therefore, the multiplicative inverse of \(-1\) a given by
\(z^{-1}=\frac{z}{|z|^{2}}=\frac{i}{1}=i\)
14. Express the following expression in the form of \( \mathrm{a}+i \mathrm{~b} \) :
\(\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-\sqrt{2} i)}\)
\(\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-\sqrt{2} i)}\)
Answer
\(\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-\sqrt{2} i)}\)
\(=\frac{(3)^{2}-(i \sqrt{5})^{2}}{\sqrt{3}+\sqrt{2} i-\sqrt{3}+\sqrt{2} i}\)
\(=\frac{9-5 i^{2}}{2 \sqrt{2} i}\)
\(=\frac{9-5(-1)}{2 \sqrt{2} i}\)
\(=\frac{9+5}{2 \sqrt{2} i} \times \frac{i}{i}\)
\(=\frac{14 i}{2 \sqrt{2} i^{2}}\)
\(=\frac{14 i}{2 \sqrt{2}(-1)}\)
\(=\frac{-7 i}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\)
\(= \frac{-7 i \sqrt{2}}{2}\)
