Ncert Solutions for Class 11
Complex Numbers and Quadratic Equations
Complex Numbers and Quadratic Equations
Exercise 4.1
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.
6. \((\frac{1}{5}+i\frac{2}{5})-(4+i\frac{5}{2})\)
7. \([(\frac{1}{3}+i\frac{7}{3})+(4+i\frac{1}{3})]-(-\frac{4}{3}+i)\)
Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.
14. Express the following expression in the form of a + ib:
\(\frac{(3+i\sqrt5)(3-i\sqrt5)}{(\sqrt3+i\sqrt2)-(\sqrt3-i\sqrt2)}\).
Complex Numbers and Quadratic Equations
Miscellaneous Exercise
1. Evalaute: \([i^{18}+(\frac{1}{i})^{25}]^3\).
3. Reduce \((\frac{1}{1-4i}-\frac{2}{1+i})(\frac{3-4i}{5+i})\) to the standard form.
4. If \(x-iy=\sqrt{\frac{a-ib}{c-id}}\), prove that \((x^2+y^2)^2=\frac{a^2+b^2}{c^2+d^2}\).
5. If \(z_1=2-i, z_2=1+i\), find \(|\frac{z_1+z_2+1}{z_1-z_2+1}|\).
6. If \(a+ib=\frac{(x+i)^2}{2x^2+1}\), prove that \(a^2+b^2=\frac{(x^2+1)^2}{(2x^2+1)^2}\).
7. Let \(z_1=2-i, z_2=-2+i\). Find
(i) \(Re(\frac{z_1 z_2}{\bar{z_1}})\)
(ii) \(Im(\frac{1}{z_1 \bar{z_1}})\)
8. Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.
9. Find the modulus of \(\frac{1+i}{1-i}-\frac{1-i}{1+i}\).
10. If \((x+iy)^3=u+iv\), then show that \(\frac{u}{x}+\frac{v}{y}=4(x^2-y^2)\).
11. If α and β different complex numbers with |β|=1, then find \(|\frac{β-α}{1-\bar{α}β}|\)
12. Find the number of non-zero integral solutions of the equation \(|1-i|^x=2^x\).
14. If \((\frac{1+i}{1-i})^m=1\), then find the least positive integral value of m.