IISER Mathematics - Complex Numbers
Exam Duration: 45 Mins Total Questions : 30
The relation a+bi < c + id is meaningful only when
- (a)
a=0, c=0
- (b)
a=0, d=0
- (c)
b=0, c=0
- (d)
b=0, d=0
The smallest positive integral value of \(n\) for which \({ \left| \cfrac { 1+i }{ 1-i } \right| }^{ n }=1\), is
- (a)
4
- (b)
8
- (c)
12
- (d)
14
If a,b,x are real numbers such that \(\left( \cfrac { 1-ix }{ 1+ix } \right) =a-ib\) then
- (a)
a2 - b2 = 0
- (b)
a2 + b2 = 0
- (c)
a2 + b2 = 1
- (d)
a2 - b2 = 1
The most general value of satisfying the equation \((cos{ \ \theta }+isin\ { \theta })(cos{ \ 2\theta }+isin\ 2{ \theta })(cos{ \ 3\theta }+isin\ 3{ \theta })....(cos{ \ n\theta }+isin\ n{ \theta })=1,\) is
- (a)
\(2k\pi \)
- (b)
\(2(k+1)\pi \)
- (c)
\(\cfrac { 4k\pi }{ n(n+1) } \)
- (d)
NONE OF THESE
For all complex numbers z1, z2 satisfying \(\left| { z }_{ 1 } \right| =12\) and \(\left| { z }_{ 2 }-3-4i \right| =5\) the minimum value of \(\left| { z }_{ 1 }-{ z }_{ 2 } \right| \), is
- (a)
0
- (b)
2
- (c)
7
- (d)
12
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
Let a, b, c be three cube roots of unity, the value of
\(\left| \begin{matrix} { e }^{ a } & { e }^{ 2a } & { e }^{ 3a }-1 \\ { e }^{ b } & { e }^{ 2b } & { e }^{ 3b }-1 \\ { e }^{ c } & { e }^{ 2c } & { e }^{ 3c }-1 \end{matrix} \right| \) is
- (a)
(1 + a)3
- (b)
(1+b)3
- (c)
(a+b+c)3n,(n\(\epsilon\)N)
- (d)
(a+2b+3c)2n,(n\(\epsilon\)1)
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) \(\overset { n-1 }{ \underset { r=0 }{ \Sigma } } { \alpha }^{ kr }\)=\(\begin{cases} n,\quad if\quad k\quad is\quad multiple\quad of\quad n \\ 0,\quad if\quad k\quad is\quad not\quad multiple\quad of\quad n \end{cases}\)
If \(\omega \) be non real complex cube root of unity, then the value
- (a)
\(\left( \frac { -1+i\sqrt { 3 } }{ 2 } \right) \)
- (b)
\(\left( \frac { 1-i\sqrt { 3 } }{ 2 } \right) \)
- (c)
\(\left( \frac { -1-i\sqrt { 3 } }{ 2 } \right) \)
- (d)
\(\left( \frac { 1+i\sqrt { 3 } }{ 2 } \right) \)
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
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.
If \({ z }_{ 1 }=\sqrt { 3 } +i,\quad i=\sqrt { -1 } \) , then the value of \(\sqrt { \left| { z }_{ 2 }-{ z }_{ 3 } \right| ^{ 2 }+\left| { z }_{ 2 }+{ z }_{ 3 } \right| ^{ 2 } } \) is equal to
- (a)
0
- (b)
2
- (c)
4
- (d)
6
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.
\(\frac { { z }_{ 1 } }{ { z }_{ 3 } } \) is equal to
- (a)
\(1-i\sqrt { 3 } \)
- (b)
\(1+i\sqrt { 3 } \)
- (c)
\(\frac { -1+i\sqrt { 3 } }{ 2 } \)
- (d)
\(\frac { 1+i\sqrt { 3 } }{ 2 } \)
The number of solutions of the equation z2+|z|2=0, where \(z\epsilon C\)is
- (a)
one
- (b)
two
- (c)
three
- (d)
infinitely many
The general equation of straight line is \(\bar { a } z+a\bar { z } +b=0\) where a is complex number and b is real number. The real and complex slopes of the are \(-i\left( \frac { a+\bar { a } }{ a-\bar { a } } \right) \quad and\quad -\frac { a }{ \bar { a } } ,\) (where \(i=\sqrt { -1 } \)). If adding \(z\bar { z } \) in LHS (i), then (i) convert in general equation of circle \(z\bar { z } +a\bar { z } +\bar { a } z+b=0\)
with centre -a and radius \(\sqrt { \left| a \right| ^{ 2 }-b } \) if a=0, then circle \(\left| z \right| ^{ 2 }+b=0\)
which is defined only when b<0
If z1,z2 and z3 be three points on \(\left| z \right| =1\) . If \({ \theta }_{ 1 },\theta _{ 2 }\quad and\quad \theta _{ 3 }\) the arguments of z1,z2 and z3 respectively, the \(\Sigma cos(\theta _{ 1 }-\theta _{ 2 })\)
- (a)
\(\ge -\frac { 3 }{ 2 } \)
- (b)
\(\quad \le -\frac { 3 }{ 2 } \)
- (c)
\(\ge \frac { 3 }{ 2 } \)
- (d)
\(\ge 1\)
If z1 and z2 are any two complex numbers, then \(|z_1+\sqrt{(z1^2-z_2^2)|}+|z_1-\sqrt{(z_1^2-z_2^2)|}\) is equal to
- (a)
|z1|
- (b)
|z2|
- (c)
|z1+z2|
- (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
The centre of the circle represented by |z+1|=2|z-1| on the complex plane is
- (a)
0
- (b)
5/3
- (c)
1/3
- (d)
none of these
The equation |z+i|-|z-i|=k (where \(i=\sqrt{-1}\)) represents a hyperbola if
- (a)
-2<k<2
- (b)
k>2
- (c)
0<k<2
- (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 \(z_1\ne-z_2\) and \(|z_1+z_2|=|\frac{1}{z_1}+\frac{1}{z_2}|\) then
- (a)
at least one of z1 ,z2 is unimodular
- (b)
both z1,z2 are unimodular
- (c)
z1.z2 is unimodular
- (d)
none of the above
If z1=a+ib and z2=c+id(where \(i=\sqrt{-1})\) are the complex numbers such that |z1|=|z2|=1 and \(Re(z_1\bar z_2)=0\) then the pair of complex numbers,\(\omega_1=a+ic\) and \(\omega_2=b+id\) satisfy
- (a)
\( |\omega_1|=1\)
- (b)
\(|\omega_2|=1\)
- (c)
\(|\omega_1\bar\omega_2|=1\)
- (d)
\(Re(\bar\omega_1\omega_2)=0\)
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 \(|1-\omega|=|1-\omega^2|=\sqrt{3}\)
The complex numbers z1 ,z2 and z3 satisdfying \(\frac{z_1-z_3}{z_2-z_3}=\frac{1-i\sqrt3}{2}\) where i=\(\sqrt{-1}\)) are the vertices of a triangle which is
- (a)
of area zero
- (b)
right angle isosceles
- (c)
equilateral
- (d)
obtuse angled isosceles
\({ \left( \frac { 1-i }{ 1+i } \right) }^{ 2 }=\)
- (a)
1
- (b)
-1
- (c)
-\(\frac{1}{2}\)
- (d)
\(\frac{1}{\sqrt2}\)
Find the conjugate of \(\frac{(3-2i)(2+3i)}{(1+2i)(2-i)}\)
- (a)
\(\frac{63}{25}-\frac{16}{25}i\)
- (b)
\(\frac{63}{25}+\frac{16}{25}i\)
- (c)
-\(\frac{63}{25}+\frac{16}{25}i\)
- (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\(\sqrt{2}\)
If \(\frac{5z_2}{11z_1}\) is purely imaginary, then the value of \(\left| \frac { { 2 }z_{ 1 }+{ 3z }_{ 2 } }{ { 2z }_{ 1 }-{ 3z }_{ 2 } } \right| \)is
- (a)
\(\frac{37}{33}\)
- (b)
2
- (c)
1
- (d)
3
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)
-\(\frac{\pi}{4}\)
If z1=\(\sqrt3+i\sqrt3\)and z2=\(\sqrt3+i\) , then find the quadrant in which\(\left( \frac { { z }_{ 1 } }{ { z }_{ 2 } } \right) \) lies
- (a)
III
- (b)
II
- (c)
I
- (d)
IV
If z=reiθ, then |eiz|=
- (a)
1
- (b)
e2rsinθ
- (c)
ersinθ
- (d)
e-rsinθ
The square root of (7-24i) is
- (a)
±(3-5i)
- (b)
±(3+4i)
- (c)
±(3-4i)
- (d)
±(4-3i)