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}+ifrac{2}{5})-(4+ifrac{5}{2}))
7. ([(frac{1}{3}+ifrac{7}{3})+(4+ifrac{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+isqrt5)(3-isqrt5)}{(sqrt3+isqrt2)-(sqrt3-isqrt2)}).
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.