Mathematics - Complex Numbers 1
Exam Duration: 45 Mins Total Questions : 30
If 1,\(\omega ,{ \omega }^{ 2 }\) are the cube roots of unity, 0 represents the origin, \(\omega \rightarrow P,{ \omega }^{ 2 }\rightarrow Q\) then
- (a)
OP>OQ
- (b)
OP
- (c)
OP=OQ
- (d)
NONE OF THESE
\(If\quad { z }_{ 1 }=(\cos { { \theta }_{ 1 } } +i\sin { { \theta }_{ 1 }) } ,\\ \quad \quad{ z }_{ 2 }=(\cos { { \theta }_{ 2 } } +i\sin { { \theta }_{ 2 }) } ,\\ { \quad \quad z }_{ 3 }=(\cos { { \theta }_{ 3 } } +i\sin { { \theta }_{ 3 }) } and\\ \quad \quad { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 }=0,\)
then the value of \(\cfrac { 1 }{ { z }_{ 1 } } +\cfrac { 1 }{ { z }_{ 2 } } +\cfrac { 1 }{ { z }_{ 3 } } \) is
- (a)
\(cos({ \theta }_{ 1 }+{ \theta }_{ 2 }+{ \theta }_{ 3 })+isin({ \theta }_{ 1 }+{ \theta }_{ 2 }+{ \theta }_{ 3 })\)
- (b)
\(cos({ \theta }_{ 1 }+{ \theta }_{ 2 }+{ \theta }_{ 3 })-isin({ \theta }_{ 1 }+{ \theta }_{ 2 }+{ \theta }_{ 3 })\)
- (c)
\(({ cos\theta }_{ 1 }+cos{ \theta }_{ 2 }+cos{ \theta }_{ 3 })-i(sin{ \theta }_{ 1 }+sin{ \theta }_{ 2 }+{ sin\theta }_{ 3 })\)
- (d)
0
The sum of the fifth powers of the value of (1)1/5 is
- (a)
\(0\)
- (b)
\(1\)
- (c)
\(5\)
- (d)
NONE OF THESE
The value of
\(\sum _{ k=1 }^{ 6 }{ \left[ sin\left( \cfrac { 2\pi k }{ 7 } \right) -icos\left( \cfrac { 2\pi k }{ 7 } \right) \right] } \), is
- (a)
\(-1\)
- (b)
\(0\)
- (c)
\(-i\)
- (d)
\(i\)
If complex numbers \({ z }_{ 1 },{ z }_{ 2 },{ z }_{ 3 }\) are in A.P., then these lie on
- (a)
a circle
- (b)
a straight line
- (c)
a parabola
- (d)
an ellipse
If \({ \left( \frac { 1-i }{ 1+i } \right) }^{ 100 }=a+ib\) , then
- (a)
a=2,b=-1
- (b)
a=1,b=0
- (c)
a=0,b=1
- (d)
a=-1,b=2
\(\sqrt { i } -\sqrt { -i } \) is equal to
- (a)
\(i\sqrt { 2 } \)
- (b)
\(\frac { 1 }{ i\sqrt { 2 } } \)
- (c)
\(0\)
- (d)
\(i\)
The real part of \(\frac { 1 }{ 1-cos\theta +isin\theta } \) is equal to
- (a)
\(\frac { 1 }{ 4 } \)
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
\(tan\frac { \theta }{ 2 } \)
- (d)
\(\frac { 1 }{ 1-cos\theta } \)
If |Z1| = |Z2| = ........... = |Zn| = 1, then the value of |Z1 + Z2 + Z3 + ....... + Zn| is
- (a)
1
- (b)
|Z1| + |Z2| +.........|Zn|
- (c)
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}a\left| \frac { 1 }{ { Z }_{ 1 } } +\frac { 1 }{ { Z }_{ 2 } } +.....\frac { 1 }{ { Z }_{ n } } \right| \)
- (d)
None of the above
If \(\alpha \) is an nth root of unity, then \(1+2\alpha +3{ \alpha }^{ 2 }+....n{ \alpha }^{ n-1 }\) equals
- (a)
\(\frac { n }{ 1-\alpha } \)
- (b)
\(\frac { -n }{ 1-\alpha } \)
- (c)
\(\frac { -n }{ { \left( 1-\alpha \right) }^{ 2 } } \)
- (d)
None of these
Let A (z1), B(z2), C(z3) be the vertices of an equilateral triangle ABC such that \(\left| z_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =2\) A circle is inscribed in the triangle ABC which touches the sides AB, BC and CA at D (z4), E(z5) and F(z6) respectively. P(z) be any point on its incircle other than D, E, F.
The value of (PA)2+(PB)2+(PC)2 is equal to
- (a)
9
- (b)
12
- (c)
15
- (d)
18
If\(\omega\ne1\) is a cube root of unity and \(x+y+z+\ne0\), then
\(\left| \begin{matrix} \frac { x }{ 1+\omega } & \frac { y }{ \omega +\omega ^{ 2 } } & \frac { z }{ \omega ^{ 2 }+1 } \\ \frac { y }{ \omega +\omega ^{ 2 } } & \frac { z }{ \omega ^{ 2 }+1 } & \frac { x }{ 1+\omega } \\ \frac { z }{ \omega ^{ 2 }+1 } & \frac { x }{ 1+\omega } & \frac { y }{ \omega +\omega ^{ 2 } } \end{matrix} \right| \)
- (a)
x2+y2+z2=0
- (b)
\(x+y\omega+z\omega^2=0\ or\ x=y=z\)
- (c)
\(x\ne y\ne z=0\)
- (d)
x=2y=3z
If \(a\quad cos\alpha +b\quad cos\beta +c\quad cos\gamma =0=a\quad sin\alpha +b\quad sin\beta +c\quad sin\gamma ;\quad a,b,c\quad \in R\quad -\pi ,<\alpha ,\beta ,\gamma \le \pi \) Then, let \(A={ e }^{ i\alpha },\quad B={ e }^{ i\beta }\quad and\quad C={ e }^{ i\gamma }\quad where\quad i=\sqrt { -1 } \)
\(\therefore \) \(aA+bB+cC=0\quad and\quad \frac { a }{ A } +\frac { b }{ B } +\frac { c }{ C } =0\)
we get \({ (aA) }^{ 3 }+{ (bB) }^{ 3 }+{ (cC) }^{ 3 }=3\quad abc\quad ABC,\)
\(\left( \frac { a }{ A } \right) ^{ 3 }+\left( \frac { b }{ B } \right) ^{ 3 }+\left( \frac { c }{ C } \right) ^{ 3 }=\frac { 3\quad abc }{ ABC } \quad and\quad aBC+bCA+cAB=0\)
If \({ sin }^{ n }\alpha +{ sin }^{ n }\beta +{ sin }^{ n }\gamma =\frac { 3 }{ 2 } \) and \(cos(\alpha +\beta )+cos(\beta +\gamma )+cos(\gamma +\alpha )=\lambda ,\)then the value of \((n,\lambda )\) is
- (a)
(3,2)
- (b)
(2,3)
- (c)
(2,0)
- (d)
(3,0)
If z then n is equal to1 and\(\bar z_1\)represent adjacent vertices of a rectangular polygon of n sides whose centre is origin and if\(\frac{Im(z_1)}{Re(z_1)}=\sqrt 2-1\)
- (a)
8
- (b)
16
- (c)
24
- (d)
32
Let A (z1), B (z2) and C (z3) be the vertices of a triangle ABC on the complex plane which is circumscribed by a circle \(\left| z \right| =1\) If the altitude of the triangle through the vertex A (z1) meets BC at D and circle \(\left| z \right| =1\) at P
The complex number associated with the point P is equal to
- (a)
\(-\frac { { z }_{ 1 }{ z }_{ 3 } }{ { z }_{ 3 } } \)
- (b)
\(\frac { { z }_{ 1 }{ z }_{ 2 } }{ { z }_{ 3 } } \)
- (c)
\(-\frac { { z }_{ 2 }{ z }_{ 3 } }{ { z }_{ 1 } } \)
- (d)
\(\frac { { z }_{ 2 }{ z }_{ 3 } }{ { z }_{ 1 } } \)
If \(\omega \) is a complex cube root of unity and \((1+\omega)^7=A+B\omega\) then A and B are respectively equal to
- (a)
0,1
- (b)
1,1
- (c)
1,0
- (d)
-1,1
If n is a positive integer but not a multiple of 3 and \(z=-1+i\sqrt{3}\) (where \(i=\sqrt{-1}\) ) then (z2n+2nzn+22n) is equal to
- (a)
0
- (b)
-1
- (c)
1
- (d)
3 X 2n
If all the roots of z3+az2+bz+c=0 are of unit modulus, then
- (a)
\(|a|\le3\)
- (b)
\(|b|>3\)
- (c)
\(|c|\le3\)
- (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 \(\frac{z+1}{z+i}\) is a purely imaginary number,(where i=\(\sqrt{-1}\)) then z lies on a
- (a)
straight line
- (b)
circle
- (c)
circle with radius =\(1/\sqrt2\)
- (d)
circle passing through the origin
The reflection of the complex number \(\frac{2-i}{3+i}\) (where \(i=\sqrt{-1}\)) in the straight line \(z(1+i)=\bar {z}(i-1)\) is
- (a)
\(\frac{-1-i}{2}\)
- (b)
\(\frac{-1+i}{2}\)
- (c)
\(\frac{i(i+1)}{2}\)
- (d)
\(\frac{-1}{1+i}\)
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 \(|1-\omega|=|1-\omega^2|=\sqrt{3}\)
If \((\omega\ne1)\) is a cube root of unity and \(i=\sqrt{-1}\) , then
\(\left| \begin{matrix} 1 & 1+i+{ \omega }^{ 2 } & { \omega }^{ 2 } \\ 1-i & -1 & { \omega }^{ 2 }-1 \\ -i & -i+\omega -1 & -1 \end{matrix} \right| \) is equal to
- (a)
0
- (b)
4
- (c)
i
- (d)
\(\omega\)
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=i 2.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 \(|1-\omega|=|1-\omega^2|=\sqrt{3}\)
If \(\alpha,\beta\ and\ \gamma \) are the roots of x3-3x2+3x+7=0, then \(\sum((\alpha-1)/(\beta-1))\) is (where \(\omega\) is cube root of unity)
- (a)
0
- (b)
\(3\omega\)
- (c)
\(3/\omega\)
- (d)
\(2\omega^2\)
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 z =x+iy is a complex number with rationals z and y \(i=\sqrt{-1}\) and |z2n-1| is (\(n\epsilon N)\)
- (a)
an irrational number
- (b)
a rational number
- (c)
non terminating non recurring
- (d)
a positive real number
If iz3+z2-z+i=0, then |z| equals
- (a)
2
- (b)
1
- (c)
0
- (d)
None of these
Number of solutions of the equation |z|2+7\(\overline { z } \)=0
- (a)
1
- (b)
2
- (c)
4
- (d)
6
The modulus of \(\frac { \left( 1+i\sqrt { 3 } \right) \left( 2+2i \right) }{ \left( \sqrt { 3 } -i \right) } \) is
- (a)
2
- (b)
4
- (c)
3\(\sqrt{2}\)
- (d)
2\(\sqrt{2}\)
The arg \(\left( \frac { 3+i }{ 2-i } +\frac { 3-i }{ 2+i } \right) \) is equal to
- (a)
\(\frac{\pi}{2}\)
- (b)
0
- (c)
\(\frac{\pi}{4}\)
- (d)
-\(\frac{\pi}{4}\)
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)
\(cos\frac { 5\pi }{ 12 } -isin\frac { 5\pi }{ 12 } \)
tan \(\left[ ilog\left( \frac { a-ib }{ a+ib } \right) \right] \)is equal to
- (a)
ab
- (b)
\(\frac{2ab}{a^2-b^2}\)
- (c)
\(\frac{a^2-b^2}{2ab}\)
- (d)
\(\frac{2ab}{a^2+b^2}\)