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12th Maths Weekly Test -4 (Complex Numbers)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -4 (Complex Numbers)-Aug 2020

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is

0
1
-1
i

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

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

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

1
2
3
5

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

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

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

z₁,z₃ and z₃ are complex number such that z1+z2+z3=0 and |z₁|=|z₂|=|z₃|=1 then z₁²+z₂²+z₂³ is

3
2
1
0

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

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

If z=x+iy is a complex number such that |z+2|=|z−2|, then the locus of z is

real axis
imaginary axis
ellipse
circle

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

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

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

−110°
−70°
70°
110°

If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \) ,then (A,B) equals

(1,0)
(−1,1)
(0,1)
(1,1)

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

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

If \(\alpha \) and \(\beta \) are the roots of x²+x+1=0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\)

-2
-1
1
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

If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right| \) =3k, then k is equal to

1
-1
\(\sqrt { 3i } \)
\(-\sqrt { 3i } \)

If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)

1
2
3
4

If \(\sqrt { a+ib } \) =x+iy, then possible value of \(\sqrt { a-ib }\) is

x²+y²
\(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)
x+iy
x-iy

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

1
-1
i
0

If z=cos\(\frac { \pi }{ 4 } \)+i sin\(\frac { \pi }{ 6 } \), then

|z| =1, arg(z) =\(\frac { \pi }{ 4 } \)
|z| =1, arg(z) =\(\frac { \pi }{ 6 } \)
|z|=\(\frac { \sqrt { 3 } }{ 2 } \), arg(z)=\(\frac { 5\pi }{ 24 } \)
|z| =\(\frac { \sqrt { 3 } }{ 2 } \), arg (z) =tan^-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

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 a = 1+ i, then a² equals

1-i
2i
(1+i)(1-i)
i-1

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

If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if

a > b > 0
a < b < 0
b < a < 0
b > a > 0

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