**COMPLEX NUMBERS**

**COMPLEX NUMBERS PAST PAPERS KARACHI BOARD (XI)**

**By Sir Farhan Jaffri**

Real Complex Numbers

2008.

Q.1. (a) (ii). Express X2 + y2=9 in terms of conjugate co-ordinates

(iii) If Z1= 1 +I and Z2=3+2i, evaluate |Z1 – 4Z2|.

(b) (i): Find the real and imaginary parts of i(3+2i).

(ii) Find the multiplicative inverse of the complex no, (3,5)

2007.

Q.1 (a)(ii) If Z1= 1 +I and Z2=3+2i, evaluate |5Z1 – 4Z2|

(iii) Solve the complex equations (x,y).(2,3)=(-4,7)

(b) Separate the following into real and imaginary parts

(1+2i)/(3-4i) + 2/5

2006.

Q.1. (b) Show that (a,b).(a/a2+b2, -b/a2+b2) = (1,0)

Q.1. ( c ) If z=(x,y), then show that Z.Z’ =|Z|2

2005.

Q.1. (a) (ii). Solve the complex equation, (x + 2yi)2 = xi

2004.

Q.1. (a) (ii) If Z1 and Z2 are complex numbers, verify that | Z1. Z2|=| Z1|| Z2|

(iii) Solve the complex equations (x,y).(2,3)=(-5,8)

2003.

Q.1. (a)(ii). If Z1= 1 +I and Z2=3+2i, evaluate |5Z1 – 4Z2|

(iii) Separate (7-5i)/(4+3i) into real and imaginary parts.

(iv) Find the additive and multiplicative inverse of (3,-4)

2002.

Q.1. (a) (ii) Find the multiplicative inverse of (√3+i)/( √3-i), separating the real and imaginary parts.

(iii) Solve the complex equations (x,y).(2,3)=(5,8)

2001.

Q.1. (a) (ii) Define modulus and the conjugate of complex numbers Z= x – iy

(iii) If Z= (1+i)/(1-i), then show that Z.Z’=|Z|2 verify that

(1+3i)/(3-5i) + -4/17 = -4/17 + 7i/17

2000.

Q.1. (a) (ii) Separate into real and imaginary parts (1+2i)/(2-i) and hence find |(1+2i)/(2-i)|.

(iii) By using the definition of multiplicative inverse of two ordered pairs, find the multiplicative inverse of (5,2) and solve the equation (2,3).(x,y)=(-4,7)

1999.

Q.1.(a)(ii) Divide 4+I by 3-4i.

(iii)Prove that (3/25, -4/25) is a multiplicative inverse of (3,4)

(iv) Multiply (-3,5) by (2,1)

1998.

Q.1.(a)(ii) Solve the complex equation (x + 2yi)2 = xi

(iii) Find the additive and multiplicative inverse of (2-3i).

(iv) Is there a complex number whose additive and multiplicative inverse are equal?

1997.

Q.1.(a)(ii) If Z1 and Z2 are complex numbers, verify that | Z1. Z2|=| Z1|| Z2|

1996.

Q.1. (b)(iv). The multiplicative identity in C is ___________.

Q.1. © What is the imaginary part of [(2+7i)’]2.

1995.

Q.1.(a) Show that (1-i)4 is a real number.

Q.1. (b) Find the additive and multiplicative inverse of (1,-3)

1994.

Q.1.(b) If Z1 = 1-I and Z2=3+2i evaluate (i) [(Z1)’]2 (ii) Z1/Z2

1993.

Q.1.(a) If Z1 and Z2 are complex numbers, verify that | Z1. Z2|=| Z1|| Z2|

1992.

Q.1. (a) (ii) Simplify (x,3y).(2x-y)

Q.1. (a) (iii) show that Z = 1±i, satisfies the equation Z2-2Z+2=0

1991.

Q.1.(a) (ii) Express X2+Y2=9 in terms of conjugate co-ordinates

Q.1. (b)(i) Solve the complex equation (X + 3i)2 = 2yi