12th Maths Monthly Test - 2 ( Complex Numbers)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Complex Numbers)-Aug 2020

Time : 1.30 Hours

Part - A

Generic 2 - Answer All Question

Evaluate the following if z=5−2i and w= −1+3i
z+w

Simplify the following
i¹⁹⁴⁸-i^-1869

Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3 \right| \)

Solve the equation z³+27=0 .
If z =1, show that \(2\le \left| { z }^{ 2 }-3 \right| \le 4\)

Show that \(\left( \cfrac { \sqrt { 3 } }{ 2 } +\cfrac { i }{ 2 } \right) ^{ 5 }+\left( \cfrac { \sqrt { 3 } }{ 2 } -\cfrac { i }{ 2 } \right) ^{ 3 }=-\sqrt { 3 } \)

Find the modulus of the following complex number
\(\cfrac { 2-i }{ 1+i } +\cfrac { 1-2i }{ 1-i } \)

Show that \(\left( \cfrac { 19-7i }{ 9+i } \right) ^{ 15 }+\left( \cfrac { 20-5i }{ 7-6i } \right) ^{ 15 }\) is real

Find the rectangular form of the complex numbers
\(\left( cos\cfrac { \pi }{ 6 } +isin\cfrac { \pi }{ 6 } \right) \left( cos\cfrac { \pi }{ 12 } +isin\cfrac { \pi }{ 12 } \right) \)

If \(2cosa=x+\cfrac { 1 }{ x } \) and \(2cos\beta =y+\cfrac { 1 }{ y } \), show that \(xy-\cfrac { 1 }{ xy } =2isin\left( \alpha +\beta \right) \).

Find the modulus of the following complex numbers
(1-i)¹⁰
Given the complex number z=2+3i, represent the complex numbers in Argand diagram
z, iz , and z+iz

Obtain the Cartesian form of the locus of z=x+iy in each of the following cases
|z+i|=|z-1|

Represent the complex numbe 1 + i\(\sqrt { 3 } \) in polar form.

Find the square roots of
−6+8i

If z=x+iy and arg \(\left( \cfrac { z-i }{ z+2 } \right) =\cfrac { \pi }{ 4 } \), then show that x²+y²+3x-3y+2=0

Find the modulus and principal argument of the following complex numbers:

\(-\sqrt { 3 } -i\)

If z₁=3,z₂=-7i, and z₃=5+4i, show that z₁(z₂+z₃)=z₁z₂+z₁z₃

Simplify the following
\(\sum _{ n=1 }^{ 12 }{ { i }^{ n } } \)

Find z^−1, if z=(2+3i)(1− i).

If z₁=3-2i and z₂=6+4i, find \(\cfrac { { z }_{ 1 } }{ z_{ 2 } } \)

Show that the following equations represent a circle, and, find its centre and radius
|3x-6+12i|=8

Obtain the Cartesian equation for the locus of z=x+iy in each of the following cases:
|z-4|²-|z-1|²=16

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

Prove the following properties
\(Ro\left( z \right) =\cfrac { z+\bar { z } }{ 2 } \) and Im\(\left( z \right) =\cfrac { z-\bar { z } }{ 2i } \)

Represent the complex number −1−i

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

Part - B

Generic 5 - Answer All Question

Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }+\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is real ii) \(\left( \cfrac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \cfrac { 8+i }{ I+2i } \right) ^{ 15 }\) is purely imaginary.

Simplify: (1+i)¹⁸

If z₁ ,z₂ and z₃ are three complex numbers such that |z₁ |=1,|z₂ |=2|z₃ |=3 and |z₁ +z₂ +z₃ |=1,show
that |9z₁ z₂ +4z₁ z₂ +z₂ z₃ |=6

Show that \({ \left( 2+i\sqrt { 3 } \right) }^{ 10 }+{ \left( 2-i\sqrt { 3 } \right) }^{ 10 }\) is real

ii) \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ I+2i } \right) ^{ 15 }\) is purely imaginary.

Suppose z₁,z₂, and z₃are the vertices of an equilateral triangle inscribed in the circle |z|= 2. If z₁ =1+i\(\sqrt { 3 } \) then find z₂ and z₃

Solve the equation z +8i=0,where Z \(\in \) C

Solve the equation z³+8i=0,where
Simplify: (1+i)¹⁸

Solve the equation z³ +27=0.

Prove that the values of \(4\sqrt { -1 } arr\pm \frac { 1 }{ \sqrt { 2 } } \) (1 ± i).Let z=(-1)

Find all cube roots of \(\sqrt { 3 } +i\)

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12th Maths Monthly Test - 2 ( Complex Numbers)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Complex Numbers)-Aug 2020

Part - A

Generic 2 - Answer All Question

Evaluate the following if z=5−2i and w= −1+3i z+w

Simplify the following i1948-i-1869

Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3 \right| \)

Solve the equation z3+27=0 .

If z =1, show that \(2\le \left| { z }^{ 2 }-3 \right| \le 4\)

Show that \(\left( \cfrac { \sqrt { 3 } }{ 2 } +\cfrac { i }{ 2 } \right) ^{ 5 }+\left( \cfrac { \sqrt { 3 } }{ 2 } -\cfrac { i }{ 2 } \right) ^{ 3 }=-\sqrt { 3 } \)

Find the modulus of the following complex number \(\cfrac { 2-i }{ 1+i } +\cfrac { 1-2i }{ 1-i } \)

Show that \(\left( \cfrac { 19-7i }{ 9+i } \right) ^{ 15 }+\left( \cfrac { 20-5i }{ 7-6i } \right) ^{ 15 }\) is real

Find the rectangular form of the complex numbers \(\left( cos\cfrac { \pi }{ 6 } +isin\cfrac { \pi }{ 6 } \right) \left( cos\cfrac { \pi }{ 12 } +isin\cfrac { \pi }{ 12 } \right) \)

If \(2cosa=x+\cfrac { 1 }{ x } \) and \(2cos\beta =y+\cfrac { 1 }{ y } \), show that \(xy-\cfrac { 1 }{ xy } =2isin\left( \alpha +\beta \right) \).

Find the modulus of the following complex numbers (1-i)10

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

Obtain the Cartesian form of the locus of z=x+iy in each of the following cases |z+i|=|z-1|

Represent the complex numbe 1 + i\(\sqrt { 3 } \) in polar form.

Find the square roots of −6+8i

If z=x+iy and arg \(\left( \cfrac { z-i }{ z+2 } \right) =\cfrac { \pi }{ 4 } \), then show that x2+y2+3x-3y+2=0

Find the modulus and principal argument of the following complex numbers: \(-\sqrt { 3 } -i\)

If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

Simplify the following \(\sum _{ n=1 }^{ 12 }{ { i }^{ n } } \)

Find z−1, if z=(2+3i)(1− i).

If z1=3-2i and z2=6+4i, find \(\cfrac { { z }_{ 1 } }{ z_{ 2 } } \)

Show that the following equations represent a circle, and, find its centre and radius |3x-6+12i|=8

Obtain the Cartesian equation for the locus of z=x+iy in each of the following cases: |z-4|2-|z-1|2=16

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

Prove the following properties \(Ro\left( z \right) =\cfrac { z+\bar { z } }{ 2 } \) and Im\(\left( z \right) =\cfrac { z-\bar { z } }{ 2i } \)

Represent the complex number −1−i

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

Part - B

Generic 5 - Answer All Question

Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }+\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is real ii) \(\left( \cfrac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \cfrac { 8+i }{ I+2i } \right) ^{ 15 }\) is purely imaginary.

Simplify: (1+i)18

If z1 ,z2 and z3 are three complex numbers such that |z1 |=1,|z2 |=2|z3 |=3 and |z1 +z2 +z3 |=1,show that |9z1 z2 +4z1 z2 +z2 z3 |=6

Show that \({ \left( 2+i\sqrt { 3 } \right) }^{ 10 }+{ \left( 2-i\sqrt { 3 } \right) }^{ 10 }\) is real ii) \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ I+2i } \right) ^{ 15 }\) is purely imaginary.

Suppose z1,z2, and z3 are the vertices of an equilateral triangle inscribed in the circle |z|= 2. If z1 =1+i\(\sqrt { 3 } \) then find z2 and z3

Solve the equation z +8i=0,where Z \(\in \) C

Solve the equation z3+8i=0,where

Simplify: (1+i)18

Solve the equation z3 +27=0.

Prove that the values of \(4\sqrt { -1 } arr\pm \frac { 1 }{ \sqrt { 2 } } \) (1 ± i).Let z=(-1)

Find all cube roots of \(\sqrt { 3 } +i\)

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Tamilnadu Stateboardszs - Class

Evaluate the following if z=5−2i and w= −1+3i
z+w

Simplify the following
i¹⁹⁴⁸-i^-1869

Solve the equation z³+27=0 .

Find the modulus of the following complex number
\(\cfrac { 2-i }{ 1+i } +\cfrac { 1-2i }{ 1-i } \)

Find the rectangular form of the complex numbers
\(\left( cos\cfrac { \pi }{ 6 } +isin\cfrac { \pi }{ 6 } \right) \left( cos\cfrac { \pi }{ 12 } +isin\cfrac { \pi }{ 12 } \right) \)

Find the modulus of the following complex numbers
(1-i)¹⁰

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

Obtain the Cartesian form of the locus of z=x+iy in each of the following cases
|z+i|=|z-1|

Find the square roots of
−6+8i

If z=x+iy and arg \(\left( \cfrac { z-i }{ z+2 } \right) =\cfrac { \pi }{ 4 } \), then show that x²+y²+3x-3y+2=0

Find the modulus and principal argument of the following complex numbers:

\(-\sqrt { 3 } -i\)

If z₁=3,z₂=-7i, and z₃=5+4i, show that z₁(z₂+z₃)=z₁z₂+z₁z₃

Simplify the following
\(\sum _{ n=1 }^{ 12 }{ { i }^{ n } } \)

Find z^−1, if z=(2+3i)(1− i).

If z₁=3-2i and z₂=6+4i, find \(\cfrac { { z }_{ 1 } }{ z_{ 2 } } \)

Show that the following equations represent a circle, and, find its centre and radius
|3x-6+12i|=8

Obtain the Cartesian equation for the locus of z=x+iy in each of the following cases:
|z-4|²-|z-1|²=16

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

Prove the following properties
\(Ro\left( z \right) =\cfrac { z+\bar { z } }{ 2 } \) and Im\(\left( z \right) =\cfrac { z-\bar { z } }{ 2i } \)

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

Simplify: (1+i)¹⁸

If z₁ ,z₂ and z₃ are three complex numbers such that |z₁ |=1,|z₂ |=2|z₃ |=3 and |z₁ +z₂ +z₃ |=1,show
that |9z₁ z₂ +4z₁ z₂ +z₂ z₃ |=6

Show that \({ \left( 2+i\sqrt { 3 } \right) }^{ 10 }+{ \left( 2-i\sqrt { 3 } \right) }^{ 10 }\) is real

ii) \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ I+2i } \right) ^{ 15 }\) is purely imaginary.

Suppose z₁,z₂, and z₃are the vertices of an equilateral triangle inscribed in the circle |z|= 2. If z₁ =1+i\(\sqrt { 3 } \) then find z₂ and z₃

Solve the equation z³+8i=0,where

Simplify: (1+i)¹⁸

Solve the equation z³ +27=0.