Complex Number and Quadratic Exercise - I

Page # 9 COMPLEX NUMBER AND QUADRATIC EQUATION EXERCISE Q.1 Q.18 Find the real values of x and y for which : (ii) i51 (iii) i342 (ii) i38 (iii) i130  i) x + (1 + i) y = 1  3i (ii) (x + iy) (2  3i) = (4 + i) Q.19 Find the conjugate of each of the following Evaluate : (i) (i + i ) 9 Q.4 (i) (1 Evaluate : (i) i63 Q.3 UNSOLVED PROBLEMS Evaluate : (i) i9 Q.2 -I (ii) (i 19 131 Evaluate: (i) +i ) 49 (i) (5  29 1   37 1   i  29  (ii)  i  67  i  i     2i) Q.20 Separate Show that : 1 + i2 + i4 + i6 = 0 Q.6 1 + i10 + i100 Q.7 i104 + i109 + i114 + i119 = 0 Q.8 6. i54 + 5. i37 – 2. i11 + 6. i68 = 7. i Q.9 1 1 1 1    0 i i2 i3 i4 3  1 into real and imaginary parts, 2  1  i1000 = 0 Q.21 If z is a complex number such that |z| = 1, prove that x y   =(4a2–b2). a b 81 (ii) Q.24 If (x + iy) = 25  81 4 4  5 9  3 16 Q.13 Simplify and express the result in the form (3  16)  (4  9) (iii) (5 + (ii) (7 – 5i) (3 + i) 7 ) (5  7 ) (iv) (4 + i)2 Q.14 Write each of the following in the form (a + ib): (i) (3  2i)2 (ii) (2 + (iv) (1  4i)3 3 )2 (iii) (2 + 5i)3 (v) (3  2i)1 Q.15 Express each of the following in the form (a+ib): (a  ib) , prove that (x2 + y2)= 1. (a  ib) Q.25 If z = (x + iy) and w = Represent each of the following in the polar form (Question No. 26 to Q. 30) Q.26 1+i Q.27 4+4 3 i Q.28 Q.29 1 3i Q.30 (i) 1 (3  4i ) Q.16 Show that (ii) 1 i 1 i (iii) (2  3i ) (2  i)  ( 7  i 3) ( 7  i 3)     is real.  ( 7  i 3) ( 7  i 3)   3  2i sin   Q.17 Find real values of  for which   is  1  2i sin   purely real. 2  6 3i 5  3i (5  i ) (2  3i ) Solve : (Question No. 31 to Q. 45) Q.31 x2 + 5 = 0 2 1  iz such that |w| = 1, z i then show that z is purely real. (a + ib) : (i)  z 1    , where z  -1.  z 1  Q.23 If (x+iy)1/3=(a+ib), prove that  Q.10 (1 + i14 + i18 + i22) is a real number. Q.12 Evaluate 1 (1  i)2 (iii) (3  i) (4  3i ) and hence find its modulus Q.5 Q.11 Evaluate: (i) (ii) Q.33 x2 x+2=0 Q.32 2x2 + 1 = 0 Q.34 x2 + 2x + 2 = 0 Q.35 2x2  4x + 3 = 0 Q.36 x2 + 3x + 5 = 0 Q.37 25x2  30x + 11 = 0 Q.38 8x2 + 2x + 1 = 0 Q.39 27x2 + 10x + 1 = 0 Q.40 2x2 Q.41 x2 + 13 = 4x Q.42 3x2 + 5 = 7x Q.43 17x + 1 = 8x Q.44 x2 + 3ix + 10 = 0 2 Q.45 2x2 + 3ix + 2 = 0  3x+1=0 Page # 10 COMPLEX NUMBER AND QUADRATIC EQUATION ANSWER KEY 1. (i) i (ii) i (iii) 1 4. (i) 0 (ii) 2i 11. (i) 9 i (ii) –45 13. (i) (–1 + 7 i) (ii) (26 – 8 i) 14. (i) (5 15. (i) 19. (i) (5 + 2i) 26.  12i) 2. (i) i (ii) 1 (iii) 1 (iii) (ii) (1 + 4 3 i) (iii) 12. 11 i  (iv)  8 i 142  65 i) (iv) 47 + 52i) 4   3   22 19    i  (ii) 0 – i (iii)  5  5 i    25 25  (ii) 20. (1 + i),          30.   i sin    , 2, 3  3   3     2  cos  i sin , 4 4   32.  1 1  i, i  2   2 36.  3 11 3 11  i,  i   2 2 2   2 37. 39.  5  2i 5  2i  ,   27   27 40.  42.  7  11i 7  11i  ,   6   6 43.  1   i, 2i  2  33. 1  7i 1  7i  ,   2   2  3 2    i  13 13  2,  4 34. {1 + i, 1  3  2i 3  2i  ,1    5   5  3  5i 3  5i  ,  4 4   4 i 4 i  ,    17 17  5 14 ,y = 13 13 2     27. 8  cos   i sin   ,8,   2  cos  i sin  , 2,   3 3  3  4 4  4  2 cos  (v) 17.  = n, n  Z 18. (i) x = 2,y 1 (ii) x=  1 3  1 (4 + 3i) (iii)    i  25  5 5  29. 45. 3. (i) 0 (ii) 0     28. 2 cos  i sin , 2, 3 3 3  31. (  i} 38. 5 i  5 i) 35. 1 1   i,1  i 1  2 2    1  7i 1  7i  ,   8 8   41. {2 + 3i, 2 44. {2i,  5i}  3i}