CLASS XI MATHEMATICS COMPLEX NUMBERS

INDIAN SCHOOL MUSCAT SENIOR SECTION DEPARTMENT OF MATHEMATICS CLASS X1 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 1.

Find the value of x and y if 1

1

x = - 4 , y = 20/3

3

𝑥 − 4 𝑦 + 4 𝑦𝑖 = −3 + 5𝑖 Explain the fallacy.... 3

2.

− 1 = 𝑖 x 𝑖 = √−1 x √−1 = √(−1)(−1) = √1 = 1 𝑧−3𝒾

3.

If z is a complex number and | 𝑧+3𝒾 | = 1, then show that z is purely real.

4.

Write in polar form the complex number 𝑖 a) 1+2 1−3𝑖

1 √2

(cos

3𝜋 3𝜋 + 𝑖 sin ) 4 4

b) - √3 + 𝑖 5.

Find the amplitude of a)

6.

7. 8.

1 + cos 𝑥 + 𝑖 𝑠𝑖𝑛𝑥

b)

1 + 𝑐𝑜𝑠𝑥 − 𝑖 𝑠𝑖𝑛𝑥

(2𝑖+1)2 (𝑖+1)3

z2

If z = -5 + 4i , show that + 10z + 41 = 0 and hence find the value of z4 + 9z3 + 35z2 – z + 4 Write z2 in the form of p + i q where z = If a + i b = and

𝑏 𝑎

=

𝑐+𝑖 𝑐−𝑖 2𝑐

𝑐 2 −1

4 𝑖 3 −1 2𝑖+1

.

77 36 + 𝑖 25 25

, a, b, c ∈ R, show that a2 +b2 = 1

. 1+𝑏+𝑎𝑖

9.

If |𝑎 + 𝑖𝑏| = 1 , then show that

10.

Solve the following quadratic equations a) x2 + 4ix – 4 = 0 b) x2 – (7 – i) x + (18 - i) = 0

11.

(2+𝑖) 4 (1−𝑖 )7

a) 1 b) 20

1+𝑏−𝑎𝑖

= 𝑏 + 𝑎𝑖

Find the square root of – 16 – 30 i. Hence solve the following quadratic equation: 2x2 – (3 + 7i) x + (9 i - 3) = 0

a) - 2 i , -2 i b) 4 – 3 i or 3 + 2i 3–5i

𝟑+𝒊 , 𝟑𝒊 𝟐