TN Maths 12th, Chapter 2 Important Questions

                     Welcome To

Class : XII

Subject : Maths

Chapter : 2

Chapter : Complex Numbers

IMPORTANT QUESTIONS

      

2 & 3 Marks Questions

1. Simplify:

i) i¹⁷²⁹

ii) i-¹⁹²⁴ +i²⁰¹⁸

      102

iii) Σ      i^n

      n=1

iv) i i² ……i⁴⁰

v) i¹⁹⁴⁷+ i²⁰⁰⁰

      12

vi) Σ i^n

     n=1

       10

vii) Σ i^n+⁵⁰

      n=1

viii) i¹⁹⁴⁸ – i¹⁸⁶⁹

ix) i⁵⁹+(1/ i⁵⁹)

x) i i² i³…… i²⁰⁰⁰

2. Evaluate the following if Z= 5-2i and W= -1+3i

i) z+w

ii) z-iw

iii) 2z+3w

iv) zw

v) z²+2zw+w²

vi)(z+w)²

3. If Z1 = 1 -3i, Z2 = -4i and Z3 =5 , show that

i)  (z1+z2)+z3 = z1+(z2+z3)

ii) (z1z2) z3 = z1(z2z3)

4. Find z-¹, if z = (2+3i) (1-i)

5.Write the following in the rectangular form

i) (5+9i )+ (2-4i)

ii) (10-5i)/(6+2i)

iii) 3i +(1/2-i)

6. Prove the following properties:

                             _

i) z is real if z = z

                          _                                 _

ii) Re(z) = (z + z)/2 and Im(z) = (z – z)/2i

7. Find the modules of the following complex numbers,

i) 2i

ii) (2 – i)/(1+i) + (1 - 2i)/(1-i)

iii) (1 - i)¹⁰

iv) 2i (3 – 4i) (4 – 3i)

v) |(2+i)/(1+2i)|

         _____

vi) |(1 + i) (2 + 3i) (4i – 3)|

vii)|i (2 + i)³/(1 + i)²|

8. Show that the following equations represent a circle, and find its center and radius 

i) |z – 2 – i|= 3

ii) |2z + 2 – 4i| = 2

iii) |3z – 6 + 12i| = 8

iv) |3z – 5 + i| = 4

9. Find the real values of numbers x and y if the complex numbers

i) (2 + i) x + (1 – i) y + 2i – 3 and x + (-1 + 2i )y + 1 + i

ii) (3 – i) x – (2-i) y + 2i +5 and 2x + (-1 + 2i)y + 3 + 2i are equal.

10. If z1=2 + 5i , z2 = -3 - 4i nad z3 = 1 + i , Find the additive and multiplicative Inverse of z1 ,z2 and z3.

11. Simplify: (1+i/1-i)³ - (1–i/1+i)³ into rectangular form.

12. If (z+3/z–5i)= (1+4i/2), find the complex numbers z in the rectangular form.

13. If z1 = 2 – I and z2 = -4 + 3i , find the inverse of z1 z2 and z1/z2

14. Which one of the points i , -2 + i , and 3 is farthest from of the origin?

15.If z = 2 show that 3 ≤ z + 3 + 4i ≤ 7

16. If z = 3 show that 7 ≤ z + 6 – 8i ≤ 13

17. If z = 1 show that 2 ≤ z2 – 3 ≤ 4

 

18. If z = 2 show that 8 ≤ z + 6 + 8i ≤ 12

19. Which one of the points 10 – 8i , 11 + 6i is closest to 1 + i

20. Find the square root of

i) 4 +3i

ii) -6 + 8i

iii) -5 – 12i

iv) 6 – 8i

                                                        __

21. Show that the equation z²= z has four solutions.

                                                       ___

22. Show that the equation z³+2z = 0 has five solutions.

23. Show that the z + 2 – i < 2 represents interior points of a circle. find its centre and radius.

5 Mark Questions

1. Find the least value of the positive integer n for which (√3 + i )^n

i) Real

ii) Purely Imaginary

2. Show that :

i) (2 + i√3)¹⁰+(2 - i√3)¹⁰ is real

ii) (2 + i√3)¹⁰–(2 - i√3)¹⁰ is purely imaginary

iii) (19+9i/5–3i)¹⁵ - (8+i/1+2i)¹⁵ is purely imaginary.

iv) (19–7i/9+i)¹² + (20–5i/7–6i)¹²is real.

3. Show that the points i , (-1/2)+ (i√3/2) and (-1/2)-(i√3/2) are the vertices of an equilateral triangle.

4. If z = x + iy is a complex number such that Im (2z+1/i^z+1)= 0 , show that the locus of z is 2x²+2y²+x–2y =0.

5. If z = x + iy is a complex number such that |(z-4i)/(z+4i)| = 1 show that the locus of z is real axis.

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