# NCERT Solutions for Class 11 Maths

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# Complex Numbers and Quadratic Equations

Exercise 1

1.

2.     i9 + i19

3.      i-39

4.     3(7 + i7) + i(7 + i7)

5.     (1 – i) – (-1 + i6)

6.

7.

8.    (1 – i)4

9.

10.

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

11.   4 – 3i

12.   √5 + 3i

13.   -i

14.   Express the following expression in the form of

Exercise 2

Find the modulus and the argument of each of the complex numbers in exercise 1 to 2.

15.   z = -1 – i√3

16.   z = -√3 + i

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

17.   1 – i

18.   -1 + i

19.   -1 -i

20.   -3

21.   √3 + i

22.   i

Exercise 3

Solve each of the following equations:

23.   x2 + 3 = 0

24.   2x2 + x + 1 = 0

25.   x2 + 3x + 9 = 0

26.   -x2 + x – 2 = 0

27.   x2 + 3x + 5 = 0

28.   x2 – x + 2 = 0

29.   √2x2 + x + √2 = 0

30.   √3x2 – √2x + 3√3 = 0

31.

32.

Exercise 4

33.

34.   For any two complex numbers z1 and z2 prove that Re(z1z2) = Rez1Rez2 – Im z1 Im z2

35.   Reduce  to the standard form.

36.   If   prove that

37.   Convert the following in the polar form:

Solve each of the equations in exercises 6 to 9:

38.

39.

40.   27x2 – 10x + 1 = 0

41.   21x2 – 28x + 10 = 0

42.

43.

44.   Let z1 = 2 – i, z2 = -2 + i find:

45.   Find the modulus and argument of the complex number

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

47.   Find the modulus of

48.   If (x + iy)3 = u + iv, then show that

49.   If α and β are different complex numbers with |β| = 1 then find

50.   Find the number of non-zero integral solutions of the equation |1 – i|x = 2x.

51.   If (a + ib) (c + id) (e + if) (g + ih) = A + iB then show that:

(a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.

52.   If  then find the least positive integral value of m.

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