SUPPLEMENTARY PROBLEMS

(1) Simplify the following

(i) (5 - 9i) ( 7/2 - 5/2i) (ii) 4 - Ö - 9 ) ( - 2 + Ö 27 )

(iii) (2 + 5i)³

Ans :- (i) 1/2 - 9/2i (ii) ( - 8 + 9 Ö 3 ) + (6 + 2 Ö 3 ) i (iii) - 142 - 65i (iv) (24 / 13) i (v) 1 / 2 .

(2) Express the following in the form a + i b, a, b Î R and i = Ö-1

(3) (a) If show that

(b) Show that is a cube root of unity.

(vii) Solve 3x² + 2x + 1 = 0

Ans :-

(viii) Solve 2x² + 3ix + 2 = 0

Ans :- x = -2i, (1/2) i

(ix) Solve ( 3 c is 41⁰) (2 c is197) = 6 c is 238⁰ where c is q = cosq + sin (Sketch it graphically).

(x) Sketch

(xi) Compute directly. Change into trigonometric form. and then recompute .

(xii) Find all solutions to x⁶ = Ö 3 - i

Ans :- 1.11819 - 0.0078290i, 0.643818 + 0.96946i, -0.474373 + 1.01730 i,

1.11819 + 0.0978290i, -0.643818 -0.919467i and 0.474373 - 1.01730i

5. Express in the Polar form :-

(i) 3 + 3i

Ans :- 3 Ö 3 (cos 45⁰ + sin 45⁰ )

(ii) - 4 - 3i

Ans :- 5 (cos 216⁰ 52' + i sin 216⁰ 52')

(iii) - 12 + 5i = 13 (cos 157⁰ 23' + i sin 157⁰ 23')

(iv) - 2i = 2 (cos 270⁰ + i sin 270⁰)

6. Use DeMoivre's Theorem to evaluate :-

(i) ( Ö 3 / 2 i / 2 ) ⁹ Ans : - i

(ii) (1 + i)⁸ Ans :- 16

(iii) (3 + 4i)⁴ Ans : - 526.9 - 336. 1 i

(iv) (1/ 2 - i Ö 3 / 2) ²⁰ Ans :- - 1/2 - i Ö 3 / 2

Ans :- 1 - i

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Index

8.1 Geometry of complex numbers
8.2 De - Moivres's theorem
8.3 Roots of complex numbers
8.4 Cirsular functions of complex angles & hyperbolic function
Supplementary Problems

Chapter 9