JEE Main Mathematics - Complex Numbers
Exam Duration: 60 Mins Total Questions : 30
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 the cube roots of unity are \(1,\omega ,{ \omega }^{ 2 }\), then the roots of (x+1)3+8=0 are
- (a)
\(-1,1+2\omega ,1+2{ \omega }^{ 2 }\)
- (b)
\(-1,-1-2\omega ,-1-2{ \omega }^{ 2 }\)
- (c)
\(-1,-1,-1\)
- (d)
None of these
If |Z1| = |Z2| and \(arg\left( \frac { { Z }_{ 1 } }{ { Z }_{ 2 } } \right) =\pi \) , then Z1 + Z2 is equal to
- (a)
0
- (b)
Purely imaginary
- (c)
purely real
- (d)
None of the above
If Z1, Z2,....... Z20 are the roots of the equation \(\sum _{ r=1 }^{ 20 }{ \frac { 1 }{ { Z }_{ r }-1 } } \) is
- (a)
10
- (b)
-5
- (c)
-10
Observe the following columns.
Column I | Column II |
---|---|
A. If |z-2i|+|z-7i| = K, then locus of z is an ellipse, if K= | p. 7 |
B. If |(2z-3) / (3z-2)| = K, then locus of z is a circle, if 2/3 is a point inside circle and 3/2 is outside the circle, if K= | q. 8 |
C. If |z=3|-|z-4i| = K, then locus of Z is a hyperbola if K is | r. 2 |
If |z-(3+4i)| =\((k/50)a\overline { z } +a\overline { z } +b|\) , where a=3+4i, then locus of z is a hyperbola with k= | s. 4 |
t. 5 |
- (a)
A B C D (p,q) (p,q,s,t,r) (r,s) (q,r) - (b)
A B C D (p,q) (p,q,r,s,t) (r,s) (p,q) - (c)
A B C D (r,s) (p,q,t,s,r) (p,q) (q,r) - (d)
None of the above
If |Z2 - 1|=|Z|2 + 1, then Z lies on
- (a)
the real axis
- (b)
the imaginary axis
- (c)
a circle
- (d)
an ellipse
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)
\(\left( \frac { 1+i\sqrt { 3 } }{ 2 } \right) \)
If \(\omega\) is a complex cube root of unity, then \(\cos[\{(1-\omega)(1-\omega ^2)+...+(10-\omega)(10-\omega^2)\}\frac{\pi}{900}]\)
- (a)
-1
- (b)
0
- (c)
1
- (d)
\(\sqrt{3}/2\)
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 (AB)2+(BC)2+(CA)2 is equal to
- (a)
9
- (b)
18
- (c)
27
- (d)
36
The least positive integer n for which \((\frac{1+i}{1-i})^n=\frac{2}{\pi}\sin^{-1}\frac{1+x^2}{2x}\),where x>0 and\(i=\sqrt{-1}\) is
- (a)
2
- (b)
4
- (c)
8
- (d)
12
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
If \(z\ne 0\), then \(\int _{ x=0 }^{ 100 }{ \left[ arg\left| z \right| \right] } \)dx is (where [.] denotes the greatest integer functions)
- (a)
0
- (b)
10
- (c)
100
- (d)
not defined
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 the circles \(z\bar { z } +\bar { a_{ 1 } } z+a_{ 1 }\bar { z } +{ b }_{ 1 }=0\) and \(z\bar { z } +\bar { a_{ 2 } } z+a_{ 2 }\bar { z } +{ b }_{ 2 }=0\) \((where\quad { b }_{ 1 },{ b }_{ 2 },\in R)\) intersect orthogonally, then
- (a)
\({ a }_{ 1 }{ a }_{ 2 }+\bar { { a }_{ 1 } } \bar { { a }_{ 2 } } ={ b }_{ 1 }+{ b }_{ 2 }\)
- (b)
\({ a }_{ 1 }{ a }_{ 2 }-\bar { { a }_{ 1 } } \bar { { a }_{ 2 } } ={ b }_{ 1 }-{ b }_{ 2 }\)
- (c)
\({ a }_{ 1 }\bar { a_{ 2 } } +\bar { { a }_{ 1 } } { a }_{ 2 }={ b }_{ 1 }+{ b }_{ 2 }\)
- (d)
\({ a }_{ 1 }{ a }_{ 2 }-\bar { { a }_{ 1 } } { a }_{ 2 }={ b }_{ 1 }-{ b }_{ 2 }\)
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 If Q be the image of P about the line BC, then complex number associated with the point Q is equal to
- (a)
z1+z2+z3
- (b)
\(\frac { { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } }{ 2 } \)
- (c)
\(\frac { { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } }{ 3 } \)
- (d)
\(\frac { 2 }{ 3 } ({ z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 })\)
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
If R be the image of P about the origin (0), then the distance between the points Band R is equal to
- (a)
\(\left| { z }_{ 1 }-{ z }_{ 2 } \right| \)
- (b)
\(\left| { z }_{ 2 }-{ z }_{ 3 } \right| \)
- (c)
\(\left| { z }_{ 3 }-{ z }_{ 1 } \right| \)
- (d)
\(\left| { z }_{ 1 }-{ z }_{ 3 } \right| \)
If \(1,\omega \ and \ \omega^2\) are the three cube roots of unity, then the roots of the equation \((x-1)^3-8=0\) are
- (a)
\(-1,-1-2\omega,-1+2\omega^2\)
- (b)
\(3,2\omega,2\omega^2\)
- (c)
\(3,1+2\omega,1+2\omega^2\)
- (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
Let S be the set of complex number z which satisfy log1/3{log1/2(|z|2+4|z|+3)}<0, then S is (where i=\(\sqrt{-1})\)
- (a)
\(1\pm i\)
- (b)
3-i
- (c)
\(\frac{5}{2}+4i\)
- (d)
empty set
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
If z1,z2,z3,z4 are roots of the equation a0z4+a13+a22+a3z+a4=0 where a0,a1,a2,a3 and a4 are real then
- (a)
\(\bar z_1,\bar z_2,\bar z_3,\bar z_4\) are also roots of the equation
- (b)
z1 is equal to at least one of \(\bar z_1,\bar z_2,\bar z_3,\bar z_4\)
- (c)
\(-\bar z_1,-\bar z_2,-\bar z_3,-\bar z_4\) are also roots of the equation
- (d)
none of the above
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)
\(\omega\)
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 equations az2+bz+c=0 and z2+2z+3=0 have a common root where a,b,c \(\epsilon\)R, then a:b:c is
- (a)
2:3:1
- (b)
1:2:3
- (c)
3:1:2
- (d)
3:2:1
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 real part of \(\frac { { (1+i) }^{ 2 } }{ (3-i) } \)is
- (a)
\(\frac{1}{3}\)
- (b)
\(\frac{1}{5}\)
- (c)
-\(\frac{1}{3}\)
- (d)
None of these
Express each of the following in the form a+ib.
\(\\ (-i)(2i){ \left( -\frac { 1 }{ 8 } i \right) }^{ 3 }\\ \)
- (a)
0+i
- (b)
0-i
- (c)
\(\frac{1}{256}\)+0i
- (d)
0+\(\frac{1}{256}\)i
If complex numbers z lies in the interior or on the boundary of a circle of radius 3 units and centre (-4,0), find the greatest and least values of |z+1|.
- (a)
6,1
- (b)
6,0
- (c)
4,0
- (d)
4,1
The area of the triangle on the complex plane formed by the complex numbers z,-ix and z+iz is
- (a)
|2|2
- (b)
\({ |\overline { z } | }^{ 2 }\)
- (c)
\(\frac { { |\overline { z } | }^{ 2 } }{ 2 } \)
- (d)
None of these
\(\left| (1+i)\left( \frac { 2+i }{ 3+i } \right) \right| \) is equal to
- (a)
-\(\frac{1}{2}\)
- (b)
1
- (c)
-1
- (d)
\(\frac{1}{2}\)
What is the locus of z, if amplitude of z-2-3i is π/4?
- (a)
a straight line
- (b)
a circle
- (c)
a parabola
- (d)
a hyperbola