12th Maths Monthly Test - 3 ( Complex Numbers)

St. Britto Hr. Sec. School - Madurai

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

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

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

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

Find the square roots of
−5−12i .

If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ =0

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

Represent the complex number −1−i

Simplify the following
\(\sum _{ n=1 }^{ 10 }{ { i }^{ n+50 } } \).

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.

Prove that the values of \(\sqrt [ 4 ]{ -1 } arr\quad \pm \cfrac { 1 }{ \sqrt { 2 } } \left( 1\pm i \right) \).Let z=(-1)

Find the principal argument Arg z , when z = \(\cfrac { -2 }{ 1+i\sqrt { 3 } } \)

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

If |z|= 3, show that \(7\le \left| z+6-8i \right| \le 13\).

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

If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when\(\theta =\cfrac { 2\pi }{ 3 } \).

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

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

Show that the following equations represent a circle, and, find its centre and radius
\(\left| 2z+2-4i \right| =2\)

Part - B

Generic 3 - Answer All Question

Show that \({ \left( \frac { 19-7i }{ 9+i } \right) }^{ 12 }+{ \left( \frac { 20-5i }{ 7-6i } \right) }^{ 12 }\) is real

Obtain the Cartesian form of the locus of z in
|2z-3-i|=3

Find the fourth roots of unity.

Find the product \(\cfrac { 3 }{ 2 } \left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) .6\left( cos\cfrac { 5\pi }{ 6 } +isin\cfrac { 5\pi }{ 6 } \right) \)in rectangular from

If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin

when θ = \(\frac { \pi }{ 3 } \)

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

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

Simplify the following:
i ¹⁷²⁹

If |z|= 3, show that 7 ≤ |z + 6 − 8i| ≤ 13.

Simplify the following:

i¹⁷²⁹

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

Part - C

Generic 5 - Answer All Question

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

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

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

Let z₁ ,z₂ , and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| \) = r > 0 and z₁ +z₂ +z₃ ≠ 0 prove

that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) =r

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

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

Simplify: \(\left( -\sqrt { 3 } +3i \right) ^{ 31 }\)

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.

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

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 Monthly Test - 3 ( Complex Numbers)

St. Britto Hr. Sec. School - Madurai

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

Part - A

Generic 2 - Answer All Question

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

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

Find the square roots of −5−12i .

If 1, ω, ω2 are the cube roots of unity show that (1+ω2)3 - (1+ω)3 =0

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

Represent the complex number −1−i

Simplify the following \(\sum _{ n=1 }^{ 10 }{ { i }^{ n+50 } } \).

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.

Prove that the values of \(\sqrt [ 4 ]{ -1 } arr\quad \pm \cfrac { 1 }{ \sqrt { 2 } } \left( 1\pm i \right) \).Let z=(-1)

Find the principal argument Arg z , when z = \(\cfrac { -2 }{ 1+i\sqrt { 3 } } \)

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

If |z|= 3, show that \(7\le \left| z+6-8i \right| \le 13\).

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

If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when\(\theta =\cfrac { 2\pi }{ 3 } \).

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

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

Show that the following equations represent a circle, and, find its centre and radius \(\left| 2z+2-4i \right| =2\)

Part - B

Generic 3 - Answer All Question

Show that \({ \left( \frac { 19-7i }{ 9+i } \right) }^{ 12 }+{ \left( \frac { 20-5i }{ 7-6i } \right) }^{ 12 }\) is real

Obtain the Cartesian form of the locus of z in |2z-3-i|=3

Find the fourth roots of unity.

Find the product \(\cfrac { 3 }{ 2 } \left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) .6\left( cos\cfrac { 5\pi }{ 6 } +isin\cfrac { 5\pi }{ 6 } \right) \)in rectangular from

If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when θ = \(\frac { \pi }{ 3 } \)

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

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

Simplify the following: i 1729

If |z|= 3, show that 7 ≤ |z + 6 − 8i| ≤ 13.

Simplify the following: i1729

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

Part - C

Generic 5 - Answer All Question

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

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

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

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

Simplify: \(\left( -\sqrt { 3 } +3i \right) ^{ 31 }\)

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.

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

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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Find the square roots of
−5−12i .

If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ =0

Simplify the following
\(\sum _{ n=1 }^{ 10 }{ { i }^{ n+50 } } \).

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

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

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

Show that the following equations represent a circle, and, find its centre and radius
\(\left| 2z+2-4i \right| =2\)

Obtain the Cartesian form of the locus of z in
|2z-3-i|=3

If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin

when θ = \(\frac { \pi }{ 3 } \)

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

Simplify the following:
i ¹⁷²⁹

Simplify the following:

i¹⁷²⁹

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.

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