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QUESTION: 1

Find the result in the form a + ib of 2-√-25/1+√-16

Solution:

QUESTION: 2

Express the following in standard form :

Solution:

(3-4i) / (2-3i)*(2+3i) / (2+3i) = (6+9i-8i+12) / 13=(18/13)+(i/13)

QUESTION: 3

Find the real numbers x and y such that : (x + iy)(3 + 2i) = 1 + i

Solution:

(x + iy)(3 + 2i) = (1 + i)

x + iy = (1 + i)/(3 + 2i)

x + iy = [(1 + i) * (3 - 2i)] / [(3 + 2i)*(3 - 2i)]

x + iy = (3 + 3i - 2i + 2) / [(3)^{2} + (2)^{2}]

x + iy = (5 + i)/[ 9 + 4]

= (5 + i) / 13

=> 13x + 13iy = 5+i

13x = 5 13y = 1

x = 5/13 y = 1/13

QUESTION: 4

Express the following in standard form : (2 – √3i) (2 + √3i) + 2 – 4i

Solution:

Given: (2−√3i)(2+√3i) + 2 − 4i

(2−√3i)(2+√3i) = 7

⇒ 7 + 2 - 4i

⇒ 9 - 4i

QUESTION: 5

Find the reciprocal (or multiplicative inverse) of -2 + 5i

Solution:

-2+5i

multiplicative inverse of -2+5i is

1/(-2+5i)

=1/-2+5i *-2-5i/-2-5i

=-2-5i/(-2)^2 -(5i)^2

=-2-5i/4-(-25)

=-2-5i/4+25

=-2-5i/29

=-2/29 -5i/29

QUESTION: 6

Find the real numbers x and y such that : (x + iy)(3+2i) = 1 + i

Solution:

(x + iy) (3 + 2i)

= 3x + 2xi + 3iy + 3i*y = 1+i

= 3x-2y + i(2x+3y) = 1+i

= 3x-2y-1 = 0 ; 2x + 3y -1 = 0

on equating real and imaginary parts on both sides

on solving two equations

x= 5/13 ; y = 1/13

QUESTION: 7

Write in the simplest form: (i)^{-997}

Solution:

(i^-997)= 1/(i^997), 1/((i^4)^249) × i, since (i^4) = 1, (i^4)/i= (i^3)= -i

QUESTION: 8

Express the following in standard form : (8 - 4i) - (-2 - 3i) + (-10 + 3i)

Solution:

(8-4i)-(-2-3i)+(-10+3i)

=>8-4i+2+3i-10+3i

=>8+2-10-4i+3i+3i =>0+2i

QUESTION: 9

Express the following in standard form : (2-3i)^{2}

Solution:

(2-3i)^{2} = 4 + 9 (i)^{2} - 2.2.3i

= 4 - 9 - 12i since, i^{2} = -1

= - 5 - 12 i

QUESTION: 10

Find the reciprocal (or multiplicative inverse) of

Solution:

QUESTION: 11

Express the following in standard form : i^{20} + (1 - 2i)^{3}

Solution:

Given, i^{20} + (1 - 2i)^{3}

We knoe that i = √-1

i^{2} = -1

Now put the values in given equation

= i^{20} + (1 - 2i)^{3}

= ( i^{2})^{10} + { 1 - 8i^{3} - 6i + 12i^{2 }}

= 1 +1 - 8i^{3} - 6i + 12i^{2}

=1 +1 - 8i^{2}.i^{1} - 6i + 12i^{2}

=1 + 1 + 8i - 6i -12

= -10 + 2i

QUESTION: 12

Imaginary part of −i(3i + 2) is

Solution:

(-i)(3i) +2(-i) =-3(i^2)-2i =-3(-1)-2i =3-2i since i=√-1 =3+(-2)i comparing with a+bi,we get b=(-2)

QUESTION: 13

For a complex number a+ib, a-ib is called its

Solution:

This is called conjugate of complex no. z=a+ib. conjugate of z=a-ib

- sign is put before i

QUESTION: 14

Express the following in standard form :

Solution:

QUESTION: 15

The multiplicative inverse of 3 – 4i is

Solution:

Complete answers is in 3 steps:

1. Conjugate=3+4i

2. Modulus=√3^2 + 4^2 =5

3. Multiplicative inverse = conjugate/square of modulus =3+4i/5^2 =3+4i/25

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