BHAI PARMANAND VIDYA MANDIR
MATHS ASSIGNMENT
CLASS – XI
COMPLEX NUMBERS
1. Solve the equation 2z = |z| + 2i
2. Find the value of x and y, if (1 + i)x – 2i + (2 – 3i)y + I = i
3+i
3–i
3. Find the conjugate of each of the following:
7−9𝑖
(a) (5 + √2i)2 (b) 2+𝑖 (c) √-3 + 4i2
4. Find the multiplicative inverse of each of the following:
(a) 2+3i
(b) (i+1) (i+2)
3-2i
(i-1) (i-2)
5. Find the value of x and y of each of the following:
(a) (3+i)x + (1-2i)y + 7i = 0
(b) (x-iy) (2+3i) = x+2i
1-i
______
(c) 3+i x2 = x2+y+4i
3𝑥
(d) √5 − 5 + 2√5 𝑖𝑦 = √2
6. Express each of the following in polar form:
(a) 3+4i (b)sin 60° + 𝑖 cos 60° (c)√3 𝑠𝑖𝑛45° + √3 cos45°i
(d) 2+2i
7. If
(e)
1+𝑖
(f)
1−𝑖
𝑧=
13
(1+𝑖)
1−𝑖 7
z1= 2+i, z2= 2-3i, z3 = 4+5i , evaluate:
𝑧 . 𝑧2
(a) Re ( 1
)
𝑧3
8. Show that if
9. If p+ iq =
(b)
𝑧−3𝑖
𝑧+3𝑖
(𝑎−𝑖)2
𝑧2 . 𝑧3
)
𝑧1
= 1, then z is a real number.
,show that p2 + q2 =
2𝑎 −𝑖
10. If x+ iy =
Im(
𝑎+𝑖𝑏
𝑐+𝑖𝑑
(𝑎 2 +1)2
4𝑎 2 +1
, then show that x- iy =
𝑎−𝑖𝑏
𝑐−𝑖𝑑
and hence prove that
𝑥2 + 𝑦2 =
11. Represent,
(cos 𝑥+i sinx ) (cos 𝑦 +𝑖 sin 𝑦 )
cot 𝑢+𝑖 (1+𝑖 tan 𝑣)
𝑎2 +𝑏 2
𝑐 2 + 𝑑2
in form of A+ iB
12. Represent,
1
1−𝑐𝑜𝑠𝑥 +2𝑖 𝑠𝑖𝑛𝑥
in the form of A+ iB
13. Express the following in the form of a+ ib:
(a) (-√ -1)31
(b) i35 + 1
i35
55
60
65
70
(c) i + i + i + i
14. Express
(1+𝑖)3
4+3𝑖
in the form x + iy:
15. Find the modulus of
(2−3𝑖)2
−1+5𝑖
16. Express the following complex number in the form a + ib. Also, find the conjugate and
modulus of each number.
(a)
5+4𝑖
(b)
4+5𝑖
17. Express the square of
18. Show that
√7+ √3𝑖
√7−√3𝑖
𝑖
1+𝑖
+
√3+√−1
2−√−1
in the form x + iy
√7−√3𝑖
√7+ √3𝑖
is real.
19. If z is a (non zero) complex number, then what is the amplitude of zz?
20. Find the modulus of
1+3𝑖 (2−5𝑖)
2−√6𝑖 −3+2√5𝑖
21. Find principal argument for following complex numbers
(i) -√3-i
(ii) √3 + i
(iii)1-√3i
22. Find (using properties of modulus and conjugate)
1
(i) 6 + 7𝑖 (2 − 3𝑖) (ii) 6 + 7𝑖 − 7 − 3𝑖
2+𝑖
2+𝑖
(iii) 𝑖
(iv) ( 𝑖 )
(v)
Can we say |Z1+Z2|=|Z1|+|Z2|
3
2
− 𝑖 + 2 + 7𝑖
23. Find real values of x and y for which the complex numbers -3+ix2y and x2+y+4i are
conjugate of each other.
(Ans. x=1,y = -4 or x= -1,y= -4)
24.If z is complex no. such that |z|=1, Prove that
conclusion if z =1.
𝑧−1
𝑧+1
is purely imaginary. What will be your