Mathematics - Complex Numbers
Exam Duration: 45 Mins Total Questions : 30
The value of sum, \(\sum _{ n=1 }^{ 13 }{ ({ i }^{ n }+{ i }^{ n+1 }) } \), where \(i=\sqrt { -1 } \), equals
- (a)
\(i\)
- (b)
\(i-1\)
- (c)
\(-i\)
- (d)
0
If \(i=\sqrt { -1 } \), then \({ 4+5\left( -\cfrac { 1 }{ 2 } +\cfrac { i\sqrt { 3 } }{ 2 } \right) }^{ 334 }+3{ \left( -\cfrac { 1 }{ 2 } +\cfrac { i\sqrt { 3 } }{ 2 } \right) }^{ 365 }\) is equal to
- (a)
\(1-i\sqrt { 3 } \)
- (b)
\(-1+i\sqrt { 3 } \)
- (c)
\(i\sqrt { 3 } \)
- (d)
\(-i\sqrt { 3 } \)
Value of \(\sqrt { 1 } +\sqrt { -1 } \) is
- (a)
0
- (b)
\(\pm \sqrt { 2 } \)
- (c)
\(-i\sqrt { 2 } \)
- (d)
\((-2+2i)\)
If \(\left| z \right| =1\), the value of \(\left( \cfrac { z-1 }{ z+1 } \right) \) is
- (a)
purely real
- (b)
purely imaginary
- (c)
complex number
- (d)
0
If \({ (1+x) }^{ n }={ p }_{ 0 }+{ p }_{ 1 }+{ p }_{ 2 }x+{ p }_{ 3 }{ x }^{ 2 }+......+{ p }_{ n }{ x }^{ n }\), then value of \({ p }_{ 0 }-{ p }_{ 2 }+{ p }_{ 4 }-...\) is
- (a)
\({ 2 }^{ n/2 }\)
- (b)
\({ 2 }^{ n/2 }\ cos\cfrac { n\pi }{ 4 } \)
- (c)
\({ 2 }^{ n/2 }\ sin\cfrac { n\pi }{ 4 } \)
- (d)
NONE OF THESE
The continued product of the four values of \({ cos\left( \cfrac { \pi }{ 3 } \right) +isin\left( \cfrac { \pi }{ 3 } \right) }^{ 3/4 }\) is
- (a)
\(-1\)
- (b)
\(1\)
- (c)
\(\cfrac { 3 }{ 2 } \)
- (d)
\(\cfrac { -1 }{ 2 } \)
If \({ \left( \frac { 1+i }{ 1-i } \right) }^{ m }=1\), then the least integral value of m is
- (a)
2
- (b)
4
- (c)
8
- (d)
None of these
If \(\frac { 3+2isin\theta }{ 1-2isin\theta } \) will be real, then \(\theta \) is equal to
- (a)
\(2n\pi \)
- (b)
\(n\pi +\frac { \pi }{ 2 } \)
- (c)
\(n\pi \)
- (d)
None of these
If Z2+Z+1=0, where Z is complex number, then the value of \({ \left( z+\frac { 1 }{ z } \right) }^{ 2 }+{ \left( { z }^{ 2 }+\frac { 1 }{ { z }^{ 2 } } \right) }^{ 2 }+{ \left( { z }^{ 3 }+\frac { 1 }{ { z }^{ 3 } } \right) }^{ 2 }+.........+{ \left( { z }^{ 6 }+\frac { 1 }{ { z }^{ 6 } } \right) }^{ 2 }\) is
- (a)
54
- (b)
6
- (c)
12
- (d)
18
If Z1 and Z2 are two non-zero complex numbers such that |Z1 + Z2| = |Z1| + |Z2|, then arg(Z1)-arg(Z2) is equal to
- (a)
\(-\frac { \pi }{ 2 } \)
- (b)
0
- (c)
\(-\pi \)
- (d)
\(\frac { \pi }{ 2 } \)
The equation z-1 n-1=0 has n roots which are called the nth roots of unity. The n, nth roots of unity are \(1,\alpha ,{ \alpha }^{ 2 },....{ \alpha }^{ n-1 }\) which are in GP, where \(\alpha =cos\left( \frac { 2\pi }{ n } \right) +i\quad sin\left( \frac { 2\pi }{ n } \right) ;i=\sqrt { -1 } \) then we have following results:
(i) \(\overset { n-1 }{ \underset { r=0 }{ \Sigma } } \alpha ^{ r }=0\quad or\quad \overset { n-1 }{ \underset { r=0 }{ \Sigma } } cos\left( \frac { 2\pi r }{ n } \right) =0\quad and\quad \overset { n-1 }{ \underset { r=0 }{ \Sigma } } sin\left( \frac { 2\pi r }{ n } \right) =0\)
(ii) \({ z }^{ n }-1=\prod _{ r=0 }^{ n-1 }{ (z-\alpha ^{ r }) } \)
(iii) \(\prod _{ r=0 }^{ n-1 }{ { \alpha }^{ 2 } } =(-1)^{ n-1 }\)
(iv) \(\overset { n-1 }{ \underset { r=0 }{ \Sigma } } { \alpha }^{ kr }\)=\(\begin{cases} n,\quad if\quad k\quad is\quad multiple\quad of\quad n \\ 0,\quad if\quad k\quad is\quad not\quad multiple\quad of\quad n \end{cases}\)
The algebraic sum of perpendicular distance from the points \(1,\alpha ,{ \alpha }^{ 2 },{ \alpha }^{ 3 },.....{ \alpha }^{ n-1 }\) to the line \(\bar { a } .z+a\bar { z } +b=0\) , (where a is complex number and b is real) is equal to
- (a)
\(\frac { n }{ 2\left| a \right| } \)
- (b)
\(\frac { n\left| b \right| }{ 2a } \)
- (c)
\(\frac { nb }{ \left| a \right| } \)
- (d)
\(\frac { nb }{ 2\left| a \right| } \)
The area of the triangle on the argand plane formed by the complex numbers -z, iz, z-iz, is (where i=\(\sqrt{-1})\)then value of \(x_1.x_2. ... \infty\), is
- (a)
1
- (b)
-1
- (c)
-i
- (d)
i
Consider the following statements:
\(S_1:-8=2i\times4i=\sqrt{(-4)}\times\sqrt{(-16)}\)
\(S_2:=\sqrt{(-4)}\times\sqrt{(-16)}=\sqrt{(-4)\times(-16)}\)
\(S_3:\sqrt{(-4)\times(-16)}=\sqrt{64}\)
\(S_4:\sqrt{64}=8\)
of these statements, the incorrect one is
- (a)
S1 only
- (b)
S2 only
- (c)
S3 only
- (d)
none of these
The value of \(\sqrt i+\sqrt{(-i)}\)(where \(i=\sqrt{-1}\)) is
- (a)
0
- (b)
\(\sqrt{2}\)
- (c)
-i
- (d)
i
\(\sin^{-1}\left\{\frac{1}{i}(z-1)\right\}\), where z is non real and \(i=\sqrt{-1}\), can be the angle of a triangle if
- (a)
Re(z)=1, Im(z)=2
- (b)
Re(z)=1,\(-1\le Im(z)\le1\)
- (c)
Re(z)+Im(z)=0
- (d)
none of the above
Let \(A=\frac{2}{\sqrt{3}}e^{i\pi/2}\), B=\(\frac{2}{\sqrt3}e^{-i\pi/6}\) ,C=\(\frac{2}{\sqrt3}e^{-i5\pi/6}\)(where \(i=\sqrt{-1}\) be three points forming a triangle ABC in the argand plane. Then \(\triangle\)ABC is
- (a)
equilateral
- (b)
isosceles
- (c)
scalene
- (d)
none of these
If \(|a_i|<1,\lambda_i\ge0\) for i=1,2,3,...,n and \(\lambda_1+\lambda_2+\lambda_3+...+\lambda_1=1\)then the value of \(|\lambda_1a_1+\lambda_2a_2+...+\lambda_na_n|\) is
- (a)
=1
- (b)
<1
- (c)
>1
- (d)
none of these
If z satisfies |z-1|<|z+3| then \(\omega =2z+3-i\) (where \(i=\sqrt{-1})\) satisfies
- (a)
\(|\omega-5-i|<|\omega+3+i|\)
- (b)
\(|\omega-5|<|\omega+3|\)
- (c)
\(Im(i\omega)>1\)
- (d)
\(|arg (\omega-1)|<\pi/2\)
The common roots of the equations z3+(1+i)z2+(1+i)z+i=0, (where \(i=\sqrt{-1}\)) and z1993+z1994+1=0 are
- (a)
1
- (b)
\(\omega\)
- (c)
\(\omega^2\)
- (d)
\(\omega^{981}\)
If \(x^2+1=0\Rightarrow x^2=-1 \) or \(x=\pm\sqrt{-1}=\pm i\) (iota) is called the imaginary unit.
Also, i2=-1,i3=i2.i=(-1)i=-i and i4=(i2)2=(-1)2=1
ie, \(i^n+i^{n+1}+i^{n+2}+i^{n+3}=0\forall n\epsilon I(Interger) \) and x3-1=0\(\Rightarrow\)(x-1)(x2+x+1)=0
\(\Rightarrow (x-1)(x-\omega)(x-\omega^2)=0\)
\(\therefore x=1,\omega,\omega^2\) are the cube roots of unity. ie,\(\omega^n+\omega^{n+1}+\omega{n+2}=0\forall n\epsilon I(interger)\)
Now let z=a+ib if \(|a:b|=\sqrt{3}:1 \ or 1:\sqrt{3}\)
Then, convert z in terms of \(\omega,\ or\ \omega^2\) . Also \(|1-\omega|=|1-\omega^2|=\sqrt{3}\)
Let \(\omega=-\frac{1}{2}+i\frac{\sqrt3}{2}\), where \(i=\sqrt{-1}\) then the value of the determine
\(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & -1-{ \omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 4 } \end{matrix} \right| \) is
- (a)
\(3\omega\)
- (b)
\(3\omega(\omega-1)\)
- (c)
\(3\omega^2\)
- (d)
\(3\omega(1-\omega)\)
Let \(z=a+ib=re^{i\theta}\) where a,b,\(\theta\epsilon R\) and \(i=\sqrt{1}\)
Then, \(r=\sqrt{(a^2+b^2)}=|z|\) and \(\theta=\tan^{-1}(\frac{b}{a})=arg(z)\)
Now, \(|z|^2=a^2+b^2=(a+ib)(a-ib)=z\bar z\Rightarrow\frac{1}{z}=\frac{\bar z}{|z|^2}\)
and \(|z_1z_2z_3...z_n|=|z_1||z_2||z_3|..|z_n|\)
If \(|f(z)|=1, then f(z) \) is called unimodular. In this case f(z) can always be expressed as \(f(z)=e^{i\alpha},\alpha\epsilon R\)
Also, \(e^{i\alpha}+e^{i\beta}=e^{i(\frac{\alpha+\beta}{2})}2\cos\left(\frac{\alpha-\beta}{2}\right)\) and \(e^{i\alpha}-e^{i\beta}=e^{i(\frac{\alpha+\beta}{2})}2i\sin\left(\frac{\alpha-\beta}{2}\right)\) , where \(\alpha,\beta,\epsilon R\)
If |z1|=|z2|=...|zn|=1, then the value of |z1+z2+z3+...+zn| is equal to
- (a)
1
- (b)
|z1|+|z2|+|z3|+...+|zn|
- (c)
\(|\frac{1}{z_1}+\frac{z}{z_2}+\frac{1}{z_3}+...+\frac{1}{z_n}|\)
- (d)
n
Let \(z=a+ib=re^{i\theta}\) where a,b,\(\theta\epsilon R\) and \(i=\sqrt{1}\)
Then, \(r=\sqrt{(a^2+b^2)}=|z|\) and \(\theta=\tan^{-1}(\frac{b}{a})=arg(z)\)
Now, \(|z|^2=a^2+b^2=(a+ib)(a-ib)=z\bar z\Rightarrow\frac{1}{z}=\frac{\bar z}{|z|^2}\)
and \(|z_1z_2z_3...z_n|=|z_1||z_2||z_3|..|z_n|\)
If \(|f(z)|=1, then f(z) \) is called unimodular. In this case f(z) can always be expressed as \(f(z)=e^{i\alpha},\alpha\epsilon R\)
Also, \(e^{i\alpha}+e^{i\beta}=e^{i(\frac{\alpha+\beta}{2})}2\cos\left(\frac{\alpha-\beta}{2}\right)\) and \(e^{i\alpha}-e^{i\beta}=e^{i(\frac{\alpha+\beta}{2})}2i\sin\left(\frac{\alpha-\beta}{2}\right)\) , where \(\alpha,\beta,\epsilon R\)
If z1 ad z2 are complex numbers satisfying \(\left|\frac{z_1+z_2}{z_1-z_2}\right|=1\) and \(arg\left(\frac{z_1-z_2}{z_1+z_2}\right)\ne m\pi(m\epsilon I),\) then \(\frac{z_1}{z_2}\) is
- (a)
zero
- (b)
a rational number
- (c)
a positive real number
- (d)
a purely imaginary
Let a quadratic equation az2+bz+c=0 where a,b, c \(\epsilon\) R and a\(\ne\)0. If one root of this equation is p+iq, then other must be the conjugate p- iq and vice-versa.(p,q\(\epsilon\) R and i=\(\sqrt{-1}),\) But if a,b, c are not real, then roots az2+bz+c=0 are not conjugate to each other.
i.e., if one root is real, then other may be non real , Now, combining both cases we can say that az2+bz+c=0 where a,b, c \(\epsilon\) C and a\(\ne\) 0.
If \(\alpha\) is non real complex number and \(x^2+\alpha x+\bar\alpha=0\) has a real root \(\gamma\), then
- (a)
\(\gamma=\alpha+\bar\alpha\)
- (b)
\(\gamma=\alpha-\bar\alpha\)
- (c)
\(\gamma=1\)
- (d)
\(\gamma=|\alpha-\bar\alpha|\)
The value of x3+7x2-x+16 when x=1+2i is,
- (a)
17-24i
- (b)
-17+24i
- (c)
-17-24i
- (d)
17+24i
If z1=2+3i and z2=1+z2=1+2i. Then \(\frac{z_1}{z_2}\) is equal to
- (a)
\(\frac{8}{5}-\frac{1}{5}i\)
- (b)
\(\frac{8}{5}+\frac{1}{5}i\)
- (c)
-\(\frac{8}{5}-\frac{1}{5}i\)
- (d)
-\(\frac{8}{5}+\frac{1}{5}i\)
If z(2-i)=(3+i), then z20 is equal to
- (a)
210
- (b)
-210
- (c)
220
- (d)
-220
Conver z=\(\frac { i-1 }{ cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } } \) in the polar forms
- (a)
\(cos\frac { \pi }{ 12 } -isin\frac { \pi }{ 12 } \)
- (b)
\(\sqrt { 2 } \left( cos\frac { 5\pi }{ 12 } +isin\frac { 5\pi }{ 12 } \right) \)
- (c)
\(cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \)
- (d)
\(cos\frac { 5\pi }{ 12 } -isin\frac { 5\pi }{ 12 } \)
What is the polar form of the complex number (i25)3?
- (a)
\(cos\frac { \pi }{ 3 } -isin\frac { \pi }{ 3 } \)
- (b)
\(\left( cos\left( \frac { \pi }{ 2 } \right) +isin\left( -\frac { \pi }{ 2 } \right) \right) \)
- (c)
\(cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } \)
- (d)
\(\left( cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } \right) \)
If z1 and z2 are two non-zero complex numbers such that |z1|z2|=|z1|+|z2|, then arg \(\left( \frac { { z }_{ 1 } }{ { z }_{ 2 } } \right) \)is equal to
- (a)
0
- (b)
-π
- (c)
-\(\frac{\pi}{2}\)
- (d)
\(\frac{\pi}{2}\)
If(\(\sqrt8\)+i)50=349(ab+ib), then a2+b2 is
- (a)
3
- (b)
8
- (c)
9
- (d)
\(\sqrt8\)
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