JEE Mathematics - Complex Numbers Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
The value of \(\left| \cfrac { 1 }{ 1+3i } -\cfrac { 1 }{ 1-3i } \right| \) is
- (a)
\(\cfrac { 9 }{ 25 } \)
- (b)
\(\cfrac { -3 }{ 5 } \)
- (c)
\(\cfrac { 3 }{ 5 } \)
- (d)
None of these
If arg.\(\left| \cfrac { 2-z }{ 2+z } \right| =\cfrac { \pi }{ 6 } \), then locus of \(z\) is a
- (a)
a straight line
- (b)
circle
- (c)
parabola
- (d)
NONE OF THESE
The points z1,z2,z3,z4 in the complex plane are the vertices of the parallelogram taken in order, if and only if
- (a)
z1+z4 = z2+z3
- (b)
z1+z3 = z2+z4
- (c)
z1+z2 = z3+z4
- (d)
NONE OF THESE
Value of \(\sqrt { 1 } +\sqrt { -1 } \) is
- (a)
0
- (b)
\(\pm \sqrt { 2 } \)
- (c)
\(-i\sqrt { 2 } \)
- (d)
\((-2+2i)\)
Multiplying a complex number by \( i\) rotates the radius vector representing a complex number through
- (a)
\({ 45 }^{ \circ }\)
- (b)
\({ 90 }^{ \circ }\)
- (c)
\({ 135 }^{ \circ }\)
- (d)
\({ 180 }^{ \circ }\)
The value of \({ \quad (sin\quad \theta +icos\quad \theta ) }^{ n }\) is
- (a)
\({ \quad (sin\ n\theta +icos\ n\theta ) }^{ n }\)
- (b)
\({ \quad (sin\ n\theta -icos\ n\theta ) }^{ n }\)
- (c)
\(cos(\cfrac { n\pi }{ 2 } -n\theta )+sin(\cfrac { n\pi }{ 2 } -n\theta )\)
- (d)
None of these
The complex cube roots of unity are in
- (a)
A.P.
- (b)
G.P.
- (c)
H.P.
- (d)
NONE OF THESE
If \({ \left( \cfrac { 1+i }{ 1-i } \right) }^{ x }=1\), then
- (a)
x=4n, where n is any positive integer
- (b)
x=2n, where n is any positive integer
- (c)
x=4n+1, where n is any positive integer
- (d)
x=2n+1, where n is any positive integer
If the imaginary part of \(\left( \cfrac { 2z+1 }{ iz+1 } \right) \) is -2, then the locus of the point represented by z on the argand diagram is
- (a)
a circle
- (b)
a straight line
- (c)
a parabola
- (d)
NONE OF THESE
The points representing complex number z, on the argand diagram, satisfying the relation \(\left| { z }-3 \right| =\left| { z }-5 \right| \); lie on
- (a)
straight line
- (b)
circle
- (c)
ellipse
- (d)
NONE OF THESE
If \(\left| { z } \right| =1\), then minimum value of \(\left| { z } \right| +\left| { z }-1 \right| \), is
- (a)
4
- (b)
3
- (c)
2
- (d)
1
If the value of the sum \(\sum _{ n=1 }^{ 13 }{ \left( { i }^{ n }+{ i }^{ n+1 } \right) } \) where \(i=\sqrt { -1 } \) is
- (a)
i
- (b)
i-1
- (c)
-i
- (d)
0
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)
\(i\)
If |Z| = 1 and \(W=\frac { z-1 }{ z+1 } (where,Z\neq -1)\) , then Re(w) is
- (a)
\(0\)
- (b)
\(-\frac { 1 }{ { \left| z+1 \right| }^{ 2 } } \)
- (c)
\(\left| \frac { z }{ z+1 } \right| .\frac { 1 }{ { \left| z+1 \right| }^{ 2 } } \)
- (d)
\(\frac { \sqrt { 2 } }{ { \left| z+1 \right| }^{ 2 } } \)
The amplitude of \(sin\frac { \pi }{ 5 } +i\left( 1-cos\frac { \pi }{ 5 } \right) \) is
- (a)
\(\frac { \pi }{ 5 } \)
- (b)
\(\frac { 2\pi }{ 5 } \)
- (c)
\(\frac { \pi }{ 10 } \)
- (d)
\(\frac { \pi }{ 15 } \)
Let Z be a purely imaginary number such that Im(Z)<0, then arg(z) is equal to
- (a)
\(\pi \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
0
- (d)
\(-\frac { \pi }{ 2 } \)
The locus of Z given by \(\left| \frac { z-1 }{ z-i } \right| =1\) , is
- (a)
a circle
- (b)
an ellipse
- (c)
a straight line
- (d)
a parabola
If \(\alpha \quad and\quad \beta \) are imaginary cube roots of unity, then \({ \alpha }^{ 4 }+{ \beta }^{ 4 }+\frac { 1 }{ \alpha \beta } \) is equal to
- (a)
3
- (b)
0
- (c)
1
- (d)
None of these
If \(Z\neq 1\) and \(\frac { { Z }^{ 2 } }{ Z-1 } \) is real, then the point represented by the complex number z lies
- (a)
either on the real axis or on a circle passing through the origin
- (b)
on a circle with centre at the origin
- (c)
either on the real axis or on a circle not passing through the origin
- (d)
on the imaginary axis
If z lies on the circle centred at origin. If area of the triangle whose vertices are z, \(\omega z\) and \(z+ \omega z\) where \(\omega\) is the cube root of unity, is \(4\sqrt{3}\) sq unit. Then radius of the circle is
- (a)
1 unit
- (b)
2 unit
- (c)
3 unit
- (d)
4 unit
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
If a,b,c,p,q,r are six complex numbers, such that \(\frac{p}{a}+\frac{q}{b}+\frac{r}{c}=1+i\) and \(\frac{a}{p}+\frac{b}{q}+\frac{c}{r}=0\), where \(i=\sqrt{-1}\) then value of \(\frac{p^2}{a^2}+\frac{q^2}{b^2}+\frac{c^2}{r^2}\) is
- (a)
0
- (b)
-1
- (c)
2i
- (d)
-2i
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
Let z1=6+i and z2=4-3i (where \(i=\sqrt{-1})\). Let z be a complex number such that \(arg\left(\frac{z-z_1}{z_2-z}\right)=\frac{\pi}{2}\), then z satisfies
- (a)
|z-(5-i)|=5
- (b)
|z-(5-i)|=\(\sqrt{5}\)
- (c)
|z-(5+i)|=5
- (d)
|z-(5+i)|=\(\sqrt{5}\)
If \(z=(\lambda+3)+i\sqrt{(5-\lambda^2)}\); then the locus of z is (where i=\(\sqrt{-1})\)
- (a)
a straight line
- (b)
a circle
- (c)
an ellipse
- (d)
a parabola
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)
\(\ge 1\)
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 \(\alpha =cos\left( \frac { 2\pi }{ 7 } \right) +isin\left( \frac { 2\pi }{ 7 } \right) \) then equation whose roots are \(\alpha +{ \alpha }^{ 2 }+{ \alpha }^{ 4 }\quad and\quad { \alpha }^{ 3 }+{ \alpha }^{ 5 }+{ \alpha }^{ 6 }\) is
- (a)
z2-z-2=0
- (b)
z2-z+2=0
- (c)
z2+z-2=0
- (d)
z2+z+2=0
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
The complex number z satisfies the condition \(\left|z-\frac{25}{z}\right|=24\). The maximum distance from the origin of co-ordinates to the point z is
- (a)
25
- (b)
30
- (c)
32
- (d)
none of these
The value of the expression \(2(1+\omega)(1+\omega^2)+3(2\omega+1)(2\omega^2+1)+4(3\omega+1)(3\omega^2+1)+..+(n+1)(n\omega+1)(n\omega^2+1)\) is (\(\omega\) is the cube root of unity)
- (a)
\(\frac{n^2(n+1)^2}{4}\)
- (b)
\(\left(\frac{n(n+1)}{2}\right)^2+n\)
- (c)
\(\left(\frac{n(n+1)}{2}\right)^2-n\)
- (d)
none of these
If \(\sum _{ k=0 }^{ 200 }{ i^{ k }+\prod _{ p=1 }^{ 50 }{ { i }^{ p } } } =x+iy\) where i=\(\sqrt{-1}\)) then (x,y) is
- (a)
(0,1)
- (b)
(1,-1)
- (c)
(2,3)
- (d)
(4,8)
If X be the set of all complex numbers z such that |z|=1 and define relation R on X by z1 R z2 is \(|arg\ z_1-arg\ z_2|=\frac{2\pi}{3}\) then R is
- (a)
reflexive
- (b)
symmetric
- (c)
transitive
- (d)
anti-symmetric
\(\sin^{-1}\left\{\frac{1}{i}(z-1)\right\}\), 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
If \(\alpha\) is a complex constant such that \(\alpha z^2+z+\bar\alpha=0\) has a real root, then
- (a)
\(\alpha+\bar\alpha=1\)
- (b)
\(\alpha+\bar\alpha=0\)
- (c)
\(\alpha+\bar\alpha=-1\)
- (d)
the absolute value of the real root is 1
If \(2\cos\theta=x+\frac{1}{x}\) and \(2\cos\phi=y+\frac{1}{y}\) then
- (a)
\(\frac{x}{y}+\frac{y}{x}=2\cos(\theta-\phi)\)
- (b)
\(x^my^n+\frac{1}{x^my^n}=2\cos(m\theta+n\phi)\)
- (c)
\(\frac{x^m}{y^n}+\frac{y^n}{x^m}=2\cos(m\theta-n\phi)\)
- (d)
xy+\(\frac{1}{xy}=2\cos(\theta+\phi)\)
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 \(i=\sqrt{-1}\), then \(4+5\left(-\frac{1}{2}+i\frac{\sqrt3}{2}\right)^{334}+3\left(-\frac{1}{2}+\frac{i\sqrt3}{2}\right)^{365}\) is equal to
- (a)
\(1-i\sqrt3\)
- (b)
\(-1+i\sqrt{3}\)
- (c)
\(i\sqrt3\)
- (d)
\(-i\sqrt3\)
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|=1,|z2|=2,|z3|=3 and |z1+z2+z3|=1, then |9z1z2+4z3z1+z2z3| is equal to
- (a)
6
- (b)
36
- (c)
216
- (d)
1296
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=-3i|,(where \(i=\sqrt{-1}\)) and \(arg\ z\epsilon(0,\pi/2)\), then \(\cot(arg\ z)-\frac{6}{z}\) is equal to
- (a)
0
- (b)
-i
- (c)
i
- (d)
\(\pi\)
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.
The condition that the equation az2 +bz+c=0 has both real roots where a,b,c are complex constants is
- (a)
\(\frac{a}{\bar a}=\frac{b}{\bar b}=\frac{c}{\bar c}\)
- (b)
\(\frac{a}{\bar a}=\frac{b}{\bar b}=-\frac{c}{\bar c}\)
- (c)
\(-\frac{a}{\bar a}=\frac{b}{\bar b}=\frac{c}{\bar c}\)
- (d)
\(\frac{a}{\bar a}=-\frac{b}{\bar b}=\frac{c}{\bar c}\)
The real values of x and y for wich the following equality hold, are respectively
(x4+2xi)-(3x2+iy)=(3-5i)+(1+2iy)
- (a)
2,3 or -2, 1/3
- (b)
1,3 or -1, 1/3
- (c)
2,1/3 or-2,3
- (d)
2,1/3 or -2, -1/3
If x=2+5i, then value of the expression x3-5x2+33x-49 equals
- (a)
-20
- (b)
10
- (c)
20
- (d)
-29
Express\((-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-i)\)in the form a+ib.
- (a)
\((1-2\sqrt{2})i\)
- (b)
\((1+2\sqrt{2})i\)
- (c)
\((-6-\sqrt{2})+(\sqrt1+2\sqrt{2})i\)
- (d)
\((-6+\sqrt{2})+(\sqrt1+2\sqrt{2})i\)
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
The modulus of the complex number \(z=\frac { (1-i\sqrt { 3 } )(cos\theta +isin\theta ) }{ 2(1-i)(cos\theta -isin\theta ) } \) is
- (a)
\(\frac{1}{2\sqrt{2}}\)
- (b)
\(\frac{1}{\sqrt{3}}\)
- (c)
\(\frac{1}{\sqrt{2}}\)
- (d)
2\(\sqrt{3}\)
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)
\(\left( cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } \right) \)
The argument of the complex number \(\left( \frac { i }{ 2 } -\frac { 2 }{ i } \right) \)is
- (a)
\(\frac{\pi}{4}\)
- (b)
\(\frac{3\pi}{4}\)
- (c)
\(\frac{\pi}{12}\)
- (d)
\(\frac{\pi}{2}\)
If z1,z2 and z3,z4 are two pairs of conjugate complex numbers, then arg\(\left( \cfrac { { z }_{ 1 } }{ { z }_{ 4 } } \right) \)+arg\(\left( \cfrac { { z }_{ 2 } }{ { z }_{ 3 } } \right) \) equals N degree. Find N.
- (a)
1
- (b)
2
- (c)
0
- (d)
3
Suppose z1, z2, z3 are the vertices of an equilateral triangle inscribed in the circle | z | =2.If then value of − z3 is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
If 1,\(\omega \),\(\omega \)2,\(\omega \)3..........,\(\omega \)n-1 are the n,nth roots of unity, then (1-\(\omega \))(1-\(\omega \)2).....(1-\(\omega \)n-1) equals M. Find M/n.
- (a)
0
- (b)
1
- (c)
2
- (d)
3