A’level Numericals Help

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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)

 

 

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

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