TN Maths 12th, Chapter 2 Important Questions
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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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