12th Maths Weekly Test -1 (Complex Numbers)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (Complex Numbers)-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is

0
1
-1
i

The area of the triangle formed by the complex numbers z,iz, and z+iz in the Argand’s diagram is

\(\cfrac { 1 }{ 2 } \left| z \right| ^{ 2 }\)
|z|²
\(\cfrac { 3 }{ 2 } \left| z \right| ^{ 2 }\)
2|z|²

If \(z-2+i\le 2\) then the greatest value of |z| is

\(\sqrt { 3 } -2\)
\(\sqrt { 3 } +2\)
\(\sqrt { 5 } -2\)
\(\sqrt { 5 } +2\)

If |z|=1, then the value of \(\cfrac { 1+z }{ 1+z } \) is
z

\(\bar { z } \)

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

1
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
0

1

2

3

If |z_1|=1,|z₂|=2|z₃|=3 and |9z₁z₂+4z₁z₃+z₂z₃|=12, then the value of |z₁+z₂+z₃| is

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

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

The principal argument of (sin 40°+i cos40°)5 is

−110°
−70°
70°
110°

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

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

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

Prove the following properties z is real if and only if z \(\bar { z } \)
Find the modulus of the following complex numbers
\(\cfrac { 2i }{ 3+4i } \)

Part - C

Generic 3 - Answer All Question

Explain the falacy:

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

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

If z₁ =2+5i, z₂ =-3-4i, and z₃ =1+i, find the additive and multiplicate inverse of z₁ ,z₂ and z₃
Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

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

Part - D

Generic 5 - Answer All Question

If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) =64

Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.
If 2cosa = x + \(\frac { 1 }{ x } \) and 2cosβ = y + \(\frac { 1 }{ y } \) show that

i) \(\frac { x }{ y } +\frac { y }{ x } \) = 2cos(α − β).

ii) \(xy-\frac { 1 }{ xy } \) = 2isin(α + β)

iii) \(\frac { { x }^{ m } }{ { y }^{ n } } -\frac { { y }^{ n } }{ { x }^{ m } } \) = 2isin(mα − nβ)

iv) \({ x }^{ m }{ y }^{ n }+\frac { 1 }{ { x }^{ m }{ y }^{ n } } \) = 2cos(mα + nβ)

If z=x+iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) \) = 0 show that the locus of z is 2x² +2y² +x-2y=0

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12th Maths Weekly Test -1 (Complex Numbers)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (Complex Numbers)-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

in+in+1+in+2+in+3 is

The area of the triangle formed by the complex numbers z,iz, and z+iz in the Argand’s diagram is

If \(z-2+i\le 2\) then the greatest value of |z| is

If |z|=1, then the value of \(\cfrac { 1+z }{ 1+z } \) is

If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to

If |z1|=1,|z2|=2|z3|=3 and |9z1z2+4z1z3+z2z3|=12, then the value of |z1+z2+z3| is

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

The principal argument of (sin 40°+i cos40°)5 is

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

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

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

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

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

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

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

Part - C

Generic 3 - Answer All Question

Explain the falacy:

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

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

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

Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

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

Part - D

Generic 5 - Answer All Question

If 1, ω, ω2 are the cube roots of unity then show that (1+5ω2+ω4) (1+5ω+ω2) (5+ω+ω5) =64

Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.

If z=x+iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) \) = 0 show that the locus of z is 2x2 +2y2 +x-2y=0

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iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is

If |z_1|=1,|z₂|=2|z₃|=3 and |9z₁z₂+4z₁z₃+z₂z₃|=12, then the value of |z₁+z₂+z₃| is

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

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

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

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

If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) =64

If z=x+iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) \) = 0 show that the locus of z is 2x² +2y² +x-2y=0