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First Principle of Differentiation

First Formula of Differentiation

derivative 2nd formula – 2nd formula of differentiation

product rule of differentiation

quotient rule of differentiation

implicit functions – Derivatives of Implicit Functions

exponential functions – Exponential Function and Their Derivatives

Derivatives of trigonometric functions

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**Online Math Tutor Pakistan**

**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**

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**1992.Q.1.(a) (i) IF A={2,3}, B={3,4} AND C={4,5}, Prove that**

**Ax(BΠC) =(AxB)Π(AxC)**

**1992.Q.1(a)** **Let U = {0,1,2,3}, A={0,1), B={1,2}, C={2,3), then prove that **

**Ax(BUC) = (AxB)U(AxC)**

**1993. Q.1 (b) Let S={A,B,C,D}, where A={1}, B={1,2}, C={1,2,3} and D=φ**

**Construct the multiplication tables to show that U and ****Π are binary operations on S.**

**1994. Q.1. (a) IF U={1,2,3,4,5,6}, A={1,2,3}, B={1,3,5} and C={2,4,6}, Find A-C, and verify**

**That A Π B is a subset of A and A is a subset of AUB**

**1995. Q,1. (c) State De Morgan’s Law and verify it when A={3,4}, B={3,5} & U={1,2,3,4,5}**

**1996. Q.1. (d) Let A={0,1,2,4} Define a*b=a Ѵ a,b є A. Construct table for * in A**

**1997 Q.1. (a) Verify the property Ax(BUC)=(AxB)U(AxC) in the following sets**** A={a,b}, B={b,c}, C={c,d}**

**1998 Q.1. (a) Let A={0,2,4}, B={1,2} and C={3,4} then prove that Ax(BUC)=(AxB)U(AxC)**

**1999 Q.1. (a) Let U={1,2,3,4,5,6}, A={1,2,3,4}, B={1,3,4,5} **

** show that (A Π B)’ = A’ U B’**

**2000 Q.1. (a) Let U={2,3,4,5,6}, A={2,1}, B={3,4}, C={4,5} **

** Show that (B-C)’ = B’**

**2003 Q.1. (a) If A={2,3}, B={3,4}, C={c,f} and U={2,3,4,c,f} find (AxB)U(AxC) and B’-A’.**

**2004 Q.1. (a) If A={1,2,4}, B={2,3,4}, U={1,2,3,4,5} find (AUB)’ and **

** (A Π A’)’**

**2005 Q.1. (a) If A and B are the subsets of the universal set. Then prove that **

** AUB = AU(A’****n****B)**

**2006 Q.1. (a) If A and B are the subsets of the universal set. Then prove that **

** AUB = AU(A’****n****B)**

**2007 Q.1. (a) If U={a,b,c,d,e}, A={a,b,c}, B={b,c,d}, find (A****n****B)**

**2008 Q.1. (a) If A={2,3}, B={3,4}, C={4,5}, find Ax(BUC)**

Cont……………………..d

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By Masood Amir

**Calculus:**

Calculus is a tool used to measure change or variation of a function with respect to the independent variable.

**Differential Calculus**:

It is the branch of calculus used to measure change or variation of a function in a very small interval of time, the techniques use to measure such changes is called “*differentiation*”.

**Integral Calculus**:

It is the branch of calculus used to measure changes or variation over an interval of independent variable, e.g to find length of curve, the area of region and the volume of a solid in a specified period of time.

The technique used to measure such changes or variation is called “*Integration*” or “*Antiderivatives*”. It a reverse process of differentiation.

Mathematically, Integration is defined as “ If f’(x) represents the differential coefficient of f(x), then the problem of integration is given f’(x), find f(x) or given dy/dx, find y.

**Notation**: **”∫ ”** is used to show the integration, it is a symbol of “S” derived from the word “Sum”. i.e. Integration is a process in which we have to sum up the derivatives over a specified interval and to find the function.

**Techniques of Integration**:

As we know that integration is the reverse process of differentiation, our problem is to find the function f(x) or Y, when f’(X) or dy/dx is given.

**dy/dx = f’(X)**

**∫dy = ∫f ’(X)dx**

**Y= f(x) is our solution**

**Ist Formula of Integration (1st Rule of Integration)**

**Indefinite Integration:**

Ist Formula of Integration (Ist Rule of Integration):

**If y = x ^{n}**

**∫ dy = ∫ x ^{n}dx**

**y = x ^{n+1}/n+1** + C

**Why “C”****:**

In the process of differentiation, we eliminate constant, as the derivative of a constant is “*zero*”.

So, In functions like X** ^{n}**, X

Example: Solve **∫x ^{3}dx**

Solution: **x ^{3+1}**/

Example: Solve **∫(x ^{3 }+ x^{2} + 5x + 6)dx**

Solution:** ∫x ^{3}dx + ∫x^{2}dx + ∫5xdx + ∫6dx**

** X ^{4}/**

** **

**Worksheet # 1**

** **

**Find the Integral of the following:**

** **

**SET 1 **

a. **∫(x ^{3}-4x^{2}+5x-6)dx **

**∫(3x**^{5}-4x^{3}+3x^{2})dx

c. **∫(ax ^{5}-bx^{4})dx **

**∫(**^{x3/2}-5x^{4/3}+3x^{2})dx

e.**∫( ^{4}√3x^{2} -2x)dx **

**∫(3x + 5x**^{2}–x^{3}/2-0.4x^{4})dx

g. **∫(x(8x-****1/2****)dx **

**∫(2-x)(4+3x)dx****∫(x**^{-3}+ x^{-4})dx**∫(2x**^{3}-3)3x^{4}dx

k. **∫(4x ^{7} + 3x^{12} -5x^{8} + 2x -1)dx **

**∫(ax**^{3}– bx^{2}+ cx –d)dx**∫(1/x**^{3}+ 2/x^{2}-6)dx**∫(-3x**^{-8}+ 2√x)dx**∫((x**^{3}-5)(2x + 5)dx**∫(7x**^{-6}+ 5√x)dx

** **

** **

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**Methods of Differentiation – Derivatives Techniques – Question of Differentiation.**

Methods of Differentiation – Derivatives Techniques

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