1.Algebra Booster

Complex Numbers 4.87
58. Let z = cos q + i sin q
z cos q + isin
q
fi =
2
1 - z 1 - ( cos q + isin
q) 2
cos q + isin
q
=
1 - (cos2q
+ isin 2 q)
cos q + isin
q
=
(1 -cos 2 q) -isin 2q
cos q + isin
q
=
2
2sin q - i 2sinqcosq
(cos q + isin q)
=
–2 isin q(cos q + isin q)
1
=
–2 isin
q
i
=
2sinq
z
Thus, the complex number lies on the imaginary
axis.
1 - z
2
59.
Y
X¢
Qz ( 2)
90°
O
Y¢
1
3
Pz ( 1)
1 1
È Êpˆ Êpˆ˘
Here, z1 = Í6+ 2 cos Á ,5+
2sin
Ë
˜ Á ˜
4¯ Ë 4¯˙
Î
˚
= (7, 6)
By rotation theorem, about the origin
z2-0 z2-0
ip/2 ip/2
= e = e = i
z1-0 z1-0
z2= iz1= i(7 + 6 i) = - 6 + 7 i = (-6, 7)
3 3
60. Given zz + zz = 350
fi
fi
2 2
zz◊ z + zz ◊ z = 350
2 2 2 2
|| z ◊ z + || z ◊ z = 350
2 2 2
fi | z| ( z + z ) = 350
fi (x 2 + y 2 )(2x 2 – 2y 2 ) = 350
fi (x 2 + y 2 )(x 2 – y 2 ) = 175
fi (x 2 + y 2 )(x 2 – y 2 ) = 25 ¥ 7
fi (x 2 + y 2 ) = 25, (x 2 – y 2 ) = 7
It is possible only when x = 4, y = 3.
Thus, the area of the rectangle = (2 ¥ 4) ¥ (2 ¥ 3)
= 48 sq unit.
61. Given,
15
Â
m=
1
2m-1
Im( z )
X
3 5 29
= Im( z) + Im( z ) + Im( z ) + … + Im( z )
= sin q + sin 3q + sin 5 q + … + sin 29q
= sin q + sin( q + 2 q) + sin( q + 2.2 q)
+º+ sin( q + (15 – 1)2 q)
Ê15.2q
ˆ
sin Á
Ë
˜
2 ¯ Ê Ê2q
ˆˆ
= ¥ sin q + (15 -1)
2q
Á ˜
Ê ˆ Á
Ë Ë 2 ¯ ˜
¯
sin Á
Ë
˜
2 ¯
sin (15 q)
= ¥ sin (15 q)
sin ( q)
2
sin (15 q)
=
sin ( q)
2
sin (15 ¥ 2°)
=
sin (2°)
2
sin (30°)
=
sin (2°)
1
=
4sin(2°)
62. (A) z = 0 clealrly satisfies |z – i|z|| = |z + i|z||
For z π 0, we can write
z z
- i = + i
|| z || z
fi
z
is equidistant from –i and i
|| z
fi
z
lies on the real axis
|| z
fi z lies on the real axis
fi Im (z) = 0 and Im(z) £ 1.
(B) Given |z + 4| + |z – 4| = 10 represents an ellipse
whose foci are –4 and 4 and the length of the major
axis = 10.
Thus, 2ae = 8 and 2a = 10
fi ae = 4 and a = 5
fi e = 4/5.
(C) Given |w| = 2
fi w = 2e iq = 2(cos q + i sin q)
Now,
-iq
1 iq
e
z = w- = 2e
-
w 2
fi
3 5
x = cos q,
y = sin q
2 2
fi
2 2
x y
+ = 1
9/4 25/4
which represents an ellipse and its eccentricity
=
2
b 9 4
1- = 1- =
2
a 25 5
(D) Given |w| = 1
fi w = e iq