Tiêu đề CH 2. Complex Number (Math +1).pdf ✅

 Theory ............................................................................................................................................... 2  Solved examples ............................................................................................................................... 8  Exercise - 1 : Basic Objective Questions ......................................................................................... 17  Exercise - 2 : Previous Year JEE Mains Questions .......................................................................... 21  Exercise - 3 : Advanced Objective Questions ................................................................................. 24  Exercise - 4 : Previous Year JEE Advanced Questions .................................................................... 31  Answer Key ....................................................................................................................................... 38 Complex Number Table of Contents


4 COMPLEX NUMBER 4. POLAR FORM P (a, b) r  r sin  r cos  a = r cos  & b = r sin ; where r = |z| and  = arg(z)  z = a + ib = r (cos  + isin ) Z = rei is known as Euler’s form; where r =|Z| &  = arg(Z) 5. SOME IMPORTANT PROPERTIES 1. (z) z  2. z z 2Re z     3. z z 2i Im(z)   4. 1 2 1 2 z z z z    5. 1 2 1 2 z z z z  6. | z | = 0 z = 0 7. 2 zz | z |  8. 1 1 1 2 1 2 2 2 z z | z z | | z | | z | ; z z   9. | z | | z | | z |    10. 2 2 2 1 2 1 2 1 2 | z z | | z | | z | 2 Re (z z )     11. | z1 + z2 |  | z1 |+ | z2 | (Triangle Inequality) 12. | z1 – z2 | | |z1 | – | z2 || 13. | az1 – bz2 |2 + | bz1 + az2 | 2 = (a2 + b2 ) (| z1 | 2 + | z2 | 2 ) 14. amp (z1 . z2) = amp z1 + amp z2 + 2 k; k  I 15. amp z z 1 2       = amp z1  amp z2 + 2 k; k  I 16. amp(zn ) = n amp(z) + 2k ; k  I 6. DE-MOIVRE’S THEOREM Statement : cos n  + i sin n  is the value or one of the values of (cos + i sin )n according as if ‘n’ is integer or a rational number. The theorem is very useful in determining the roots of any complex quantity 7. CUBE ROOT OF UNITY Roots of the equation x3 = 1 are called cube roots of unity. x 3 – 1 = 0 (x – 1) (x2 + x + 1) = 0 x = 1 or x 2 + x + 1 = 0 i.e w 2 w 1 3i 1 3i x or x 2 2         (i) The cube roots of unity are 1 , 1 i 3 2   , 1 i 3 2   . (ii) W3 = 1 (iii) If w is one of the imaginary cube roots of unity then 1 + w + w2 = 0. (iv) In general 1 + wr + w2r = 0 ; where r  I but is not the multiple of 3. (v) In polar form the cube roots of unity are : cos 0 + i sin 0 ; cos 2 3  + i sin 2 3  , cos 4 3  + i sin 4 3  (vi) The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral triangle. (vii) The following factorisation should be remembered : a3  b3 = (a  b) (a  b) (a  2b) ; x2 + x + 1 = (x  ) (x  2) ; a3 + b3 = (a + b) (a + b) (a + 2b); a3 + b3 + c3  3abc = (a + b + c) (a + b + 2c) (a + 2b + c)