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12th Maths Weekly Test -3 (Complex Numbers)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -3 (Complex Numbers)-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The value of \(\sum _{ i=1 }^{ 13 }{ \left( { i }^{ n }+i^{ n-1 } \right) } \) is

1+ i
i
1
0

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \)/Then the complex number is

\(\cfrac { 1 }{ i+2 } \)
\(\cfrac { -1 }{ i+2 } \)
\(\cfrac { -1 }{ i-2 } \)
\(\cfrac { 1 }{ i-2 } \)

If z is a non zero complex number, such that 2iz^2₌\(\bar { z } \) then |z|is then |z| is

\(\cfrac { 1 }{ 2 } \)
1
2
3

If \(\left| z-\cfrac { 3 }{ z } \right| =2\) then the least value |z| is

1
2
3
5

The solution of the equation |z|-z=1+2i is

\(\cfrac { 3 }{ 2 } -2i\)
\(-\cfrac { 3 }{ 2 } +2i\)
\(2-\cfrac { 3 }{ 2 } i\)
\(2+\cfrac { 3 }{ 2 } i\)

If z is a complex number such that \(z\varepsilon C/R\quad \)and \(z+\cfrac { 1 }{ z } \epsilon R\) then|z| is

0
1
2
3

If \(\cfrac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

\(\cfrac { 1 }{ 2 } \)
1
2
3

The principal argument of \(\cfrac { 3 }{ -1+i } \)

\(\cfrac { -5\pi }{ 6 } \)
\(\cfrac { -2\pi }{ 3 } \)
\(\cfrac { -3\pi }{ 4 } \)
\(\cfrac { -\pi }{ 2 } \)

If (1+i)(1+2i)(1+3i)(1+ni)=x+iy , then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \)

1
i
x²+y²
1+n²

The principal argument of the complex number \(\cfrac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is

\(\cfrac { 2\pi }{ 3 } \)
\(\cfrac { \pi }{ 6 } \)
\(\cfrac { 5\pi }{ 6 } \)
\(\cfrac { \pi }{ 2 } \)

The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 1 }{ 4 } }\) is

-2
-1
1
2

The value of \(\left( \cfrac { 1+3\sqrt { i } }{ 1-\sqrt { 3i } } \right) ^{ 10 }\)

\(cis\cfrac { 2\pi }{ 3 } \)
\(cis\cfrac { 4\pi }{ 3 } \)
\(-cis\cfrac { 2 }{ 3 } \)

The value of (1+i) (1+i²) (1+i³) (1+i⁴) is

2
0
1
i

If, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to

1
-1
i
0

If a=cosθ + i sinθ, then \(\frac { 1+a }{ 1-a } \) =

cot \(\frac { \theta }{ 2 } \)
cot θ
i cot \(\frac { \theta }{ 2 } \)
i tan\(\frac { \theta }{ 2 } \)

The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is

16
8
4
2

If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| =

\(\frac { 1 }{ 13 } \)
\(\frac { 1 }{ 5} \)
\(\frac { 1 }{ 12 } \)
none of these

If z=\(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) =

0
\(\frac{1}{2}\)
cot\(\frac { \theta }{ 2 } \)
\(\frac{1}{2}\)cot\(\frac { \theta }{ 2 } \)

The amplitude of \(\frac{1}{i}\) is equal to

0
\(\frac { \pi }{ 2 } \)
-\(\frac { \pi }{ 2 } \)
\(\pi \)

The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \) =1 lies on

circle x²+y² =1
x-axis
y-axis
the lines x+y=1

Part - B

Generic 2 - Answer All Question

Simplify the following
i¹⁹⁴⁷+i¹⁹⁵⁰

Given the complex number z=2+3i, represent the complex numbers in Argand diagram
z, iz , and z+iz

If z_i z_i =1−3i, = −4 , and z₃ = 5 , show that (z₁+z₂)+z₃=z₁+(z₂+z₃)

If z₁=2+5i, z₂=-3-4i, and z₃=1+i, find the additive and multiplicate inverse of z₁,z₂ and z₃

If z=x+iy , find the following in rectangular form.
\(Re\left( \cfrac { 1 }{ z } \right) \)

The complex numbers u,v, and w are related by \(\cfrac { 1 }{ u } =\cfrac { 1 }{ v } +\cfrac { 1 }{ w } \) If v=3−4i and w=4+3i, find u in rectangular form.

Find the least value of the positive integer n for which \(\left( \sqrt { 3 } +i \right) ^{ n }\)
real

Find the modulus of the following complex numbers
\(\cfrac { 2i }{ 3+4i } \)

Which one of the points10 − 8i , 11+ 6i is closest to1+ i .

If \(\left| z-\cfrac { 2 }{ z } \right| =2\) show that the greatest and least value of |z| are \(\sqrt { 3 } +1\) and \(\sqrt { 3 } -1\) respectively.

Part - C

Generic 3 - Answer All Question

Explain the falacy:

Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.

Find the locus of z if Re\(\left( \frac { z+1 }{ z-i } \right) \) =0 where z=x+iy.

If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) =p+iq, show that p²+q² =\(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).

Given the complex number z=2+3i, represent the complex numbers in Argand diagram
z, iz , and z+iz

If z₁ =3,z₂ =-7i, and z₃ =5+4i, show that z₁ (z₂ +z₃ )=z₁ z₂ +z₁ z₃

If z₁ =2− i and z₂ =-4+3i , find the inverse of z₁ z₂ and \(\frac { { z }_{ 1 } }{ { z }_{ 1 } } \)

Prove the following properties z is real if and only if z = \(\overline { z } \)

If z₁ =3+4i,z₂ =5-12i, and z₃ =6+8 , find |z₁ |,|z₂ |,|z₃ |,|z₁ +z₂ |,|z₂ -z₂ |,and|z₁ +z₃ |

Show that the points \(1,\frac { -1 }{ 2 } +i\frac { \sqrt { 3 } }{ 2 } ,and\frac { -1 }{ 2 } -i\frac { \sqrt { 3 } }{ 2 } \) are the vertices of an equilateral triangle.

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