Q: The modulus of the complex number z such that |z+3-i|=1 and (z)= is equal to

Let z=x+i y Given, |z+3-i|=1 aligned & |(x+3)+(y-1) i|=1 \\ & (x+3)^2+(y-1)^2 =1 \\ & (x+3)^2+(y-1)^2=1 (i) aligned Above is the equation of circle with centre (-3,1) and radius =1 Also, (z)= ^ -1 | y x |= | y x |= =0 y=0 By Eq. (i), aligned & x & =-3 \\ & z & =-3+0 i \\ & Modulus of z & =3 aligned

Q: The modulus of (1+i 3 )(2+2 i) ( 3 -i) is

aligned & (1+i 3 )(2+2 i) 3 -i = 2+2 3 i+2 i-2 3 3 -i \\ & = (2-2 3 )+(2 3 +2) i 3 -i 3 +i 3 +i \\ & = 2 3 -6+2 i-2 3 i+6 i+2 3 i-2 3 -2 3+1 \\ & = 8 i-8 4 =-2+2 i \\ & Modulus = (-2)^2+(2)^2 =2 2 aligned

Q: The smallest positive integral value of n such that [ 1+ 8 +i 8 1+ 8 -i 8 ]^n is purely imaginary, is equal to

[ 1+ 8 +i 8 1+ 8 -i 8 ]^n= [ 1+ +i 1+ -i ]^n [ = 2 - 8 ] = [ 2 ^2 2 +2 i 2 2 2 ^2 2 -2 i 2 2 ]^n = [ 2 +i 2 2 -i 2 ]_ - ^n= (e^ 2 i 2 ]^n=e^ i n =e^ i n ( 3 8 ) = 3 n 8 +i 3 n 8 For n=4 we get imaginary part.

Q: If z is a complex number such that z^ 2 =( z )^ 2 , then

Let z=x+i y, then its conjugate z =x-i y Given that z^ 2 =( z )^ 2 array l x^ 2 -y^ 2 +2 i x y=x^ 2 -y^ 2 -2 i x y \\ 4 i x y=0 array

Question 1

The points represented by the complex number 1+i,-2+3 i, 5 3 i on the argand plane are

Accepted Answer

According to question, Let find slope of A B, B C and C A Slope of A B= 1-3 1-(-2) = -2 3 Slope of B C= 3- ( 5 3 ) -2-0 = -2 3 Slope of C A= 5 3 -1 0-1 = -2 3 Since, slope of all lines are same, therefore, points are collinear.

Question 2

The modulus of the complex number z such that |z+3-i|=1 and (z)= is equal to

Accepted Answer

Let z=x+i y Given, |z+3-i|=1 aligned & |(x+3)+(y-1) i|=1 \ & (x+3)^2+(y-1)^2 =1 \ & (x+3)^2+(y-1)^2=1 (i) aligned Above is the equation of circle with centre (-3,1) and radius =1 Also, (z)= ^ -1 | y x |= | y x |= =0 y=0 By Eq. (i), aligned & x & =-3 \ & z & =-3+0 i \ & Modulus of z & =3 aligned

Question 3

If z, z ,-z,- z forms a rectangle of area 2 3 square units, then one such z is

Accepted Answer

Let a complex number, z=x+i y z = x -i y Then, vertices of rectangle for z, z ,-z,- z are (x, y),(x,-y),(-x,-y),(-x, y) Now, Area of rectangle =(2 x)(2 y)=4 x y It is given that, Area =2 3 =4 x y 2 x y= 3 x= 1 2 , y= 3 z= 1 2 + 3 i

Question 4

If z_1, z_2, . . z_n are complex numbers such that |z_1 |= |z_2 |= . .= |z_n |=1, then |z_1+z_2+ . .+z_n | is equal to

Accepted Answer

|z_1 |= |z_2 |= |z_3 |= .= |z_n |=1 z_1 z _1=z_2 z _2=z_3 z _3= =z_n z _n=1 z _1= 1 z , z _2= 1 z _2 , z _3= 1 z _3 , , z _ n = 1 z _ n Now, |z_1+z_2+z_3+ .+z_n | = | z _1+ z _2+ + z _n | | z _1+ z _2+ z _3+ .+ z _n |= | 1 z_1 + 1 z_2 + 1 z_3 + + 1 z_n |

Question 5

If |z_1 |=2, |z_2 |=3, |z_3 |=4 and |2 z_1+3 z_2+4 z_3 |= 4 , then absolute value of 8 z_2 z_3+27 z_3 z_1+ 64 z_1 z_2 equals

Accepted Answer

|8 z_2 z_3+27 z_3 z_1+64 z_1 z_2 |= |z_1 | |z_2 | |z_3 | | 8 z_1 + 27 z_2 + 64 z_3 | =(2)(3)(4) | 8 z _1 |z_1 |^2 + 27 z _2 |z_2 |^2 + 64 z _3 |z_3 |^2 | =24 |2 z _1+3 z _2+4 z _3 | =24 | 2 z_1+3 z_2+4 z_3 | =24 |2 z_1+3 z_2+4 z_3 |=(24)(4)=96

Question 6

The modulus of (1+i 3 )(2+2 i) ( 3 -i) is

Accepted Answer

aligned & (1+i 3 )(2+2 i) 3 -i = 2+2 3 i+2 i-2 3 3 -i \ & = (2-2 3 )+(2 3 +2) i 3 -i 3 +i 3 +i \ & = 2 3 -6+2 i-2 3 i+6 i+2 3 i-2 3 -2 3+1 \ & = 8 i-8 4 =-2+2 i \ & Modulus = (-2)^2+(2)^2 =2 2 aligned

Question 7

i^ 57 + 1 i^ 25 , when simplified has the value

Accepted Answer

i^ 57 + 1 i^ 25 = (i^4 )^ 14 . i+ 1 (i^4 )^6 .i =i+ 1 i =i-i=0

Question 8

If |w|=2, then the set of points z=w- 1 w is contained in or equal to the set of points z satisfying

Accepted Answer

We have, |w|=2 w=2( +i ) z=w- 1 w =2( +i )- 1 2 ( -i ) z= 3 2 + 5 2 i |z|= 9 4 + 25 4 = 17 2 < 3 Hence, option (d) is correct.

Question 9

The smallest positive integral value of n such that [ 1+ 8 +i 8 1+ 8 -i 8 ]^n is purely imaginary, is equal to

Accepted Answer

[ 1+ 8 +i 8 1+ 8 -i 8 ]^n= [ 1+ +i 1+ -i ]^n [ = 2 - 8 ] = [ 2 ^2 2 +2 i 2 2 2 ^2 2 -2 i 2 2 ]^n = [ 2 +i 2 2 -i 2 ]_ - ^n= (e^ 2 i 2 ]^n=e^ i n =e^ i n ( 3 8 ) = 3 n 8 +i 3 n 8 For n=4 we get imaginary part.

Question 10

If z is a complex number such that z^ 2 =( z )^ 2 , then

Accepted Answer

Let z=x+i y, then its conjugate z =x-i y Given that z^ 2 =( z )^ 2 array l x^ 2 -y^ 2 +2 i x y=x^ 2 -y^ 2 -2 i x y \ 4 i x y=0 array

Question 11

If z_ 1 = 2 [ 4 +i 4 ] and z_ 2 = 3 [ 3 +i 3 ] then |z_ 1 z_ 2 | is equal to m . Value of m is

Accepted Answer

z_ 1 = 2 [ 4 +i 4 ]= 2 [ 1 2 +i 1 2 ]=1+i |z_ 1 |= 2 and z_ 2 = 3 [ 3 +i 3 ]= 3 [ 1 2 +i 3 2 ] |z_ 2 |= 3 4 + 9 4 = 3 |z_ 1 z_ 2 |= |z_ 1 | |z_ 2 |= 2 3 =6

Question 12

If |z_ 1 |= |z_ 2 |= . . |z_ n |=1, then the value of |z_ 1 +z_ 2 + . z_ n |- | 1 z_ 1 + 1 z_ 2 + .+ 1 z_ n | is,

Accepted Answer

z_ 1 z _ 1 =z_ 2 z _ 2 = .=z_ n z _ n =1 z _ 1 = 1 z _ 1 , z _ 2 = 1 z _ 2 , z _ 3 = 1 z _ 3 , , z _ n = 1 z _ n |z_ 1 +z_ 2 + .+z_ n |- | 1 z_ 1 + 1 z_ 2 + .+ 1 z_ n | = |z_ 1 +z_ 2 + +z_ n |- | z _ 1 + z _ 2 + .+ z _ n |=0

Complex Number — Previous Year Questions & Solutions