**COMPLEX NUMBERS**

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

**By Professor Masood Amir**

**Real Complex Numbers**

**2008. **

**Q.1. (a) (ii). Express X ^{2} + y^{2}=9 in terms of conjugate co-ordinates**

**(iii) If Z _{1}= 1 +I and Z_{2}=3+2i, evaluate |Z_{1} – 4Z_{2}|.**

**(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 Z _{1}= 1 +I and Z_{2}=3+2i, evaluate |5Z_{1} – 4Z_{2}|**

**(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/a ^{2}+b^{2}, -b/a^{2}+b^{2}) = (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 Z _{1} and Z_{2} are complex numbers, verify that | Z_{1}. Z_{2}|=| Z_{1}|| Z_{2}| **

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

**2003.**

**Q.1. (a)(ii). If Z _{1}= 1 +I and Z_{2}=3+2i, evaluate |5Z_{1} – 4Z_{2}|**

**(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 Z _{1} and Z_{2} are complex numbers, verify that | Z_{1}. Z_{2}|=| Z_{1}|| Z_{2}|**

**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 Z _{1} = 1-I and Z_{2}=3+2i evaluate (i) [(Z_{1})’]^{2} (ii) Z_{1}/Z_{2}**

**1993.**

**Q.1.(a) If Z _{1} and Z_{2} are complex numbers, verify that | Z_{1}. Z_{2}|=| Z_{1}|| Z_{2}|**

**1992.**

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

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

**1991****.**

**Q.1.(a) (ii) Express X ^{2}+Y^{2}=9 in terms of conjugate co-ordinates**

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