Polar Form of Complex Numbers

Polar Form of complex numbers-

In the previous post, we discussed the basics of complex numbers.This is the second and final post on the complex numbers.here we shall discuss, polar form of complex numbers
We consider complex numbers like vectors and every vector must have some magnitude and a certain direction. If we write a complex number in form of a point in the Cartesian plane/ordered a pair like x+iy=(x,y) then the distance of this point from the origin (0,0) is equal to the magnitude of our complex number while the angle it’s making with x-axis will show it’s direction.<img src="polar form of a complex number.jpg" alt="polar form of a complex number">

In the above right triangle, using Pythagoras theorem Read more

Permutations and Combinations-algebra tutors

Permutations and Combinations(part-2)

In my previous post, we discussed the fundamental principle of counting and various methods of permutations. In this post, I shall discuss combinations in details.

Meaning of Combination- If we are given a set of objects and we want to select a few objects out of this set, then we can do it by many different ways. These ways are known as combinations.
Example- If we are given three balls marked as B, W and R and we want to select two balls then we can select like this- BW, BR, WR.

These are known as the combination of this selection.

Combination of n different objects taken r at a time when repetition is not allowed– If repetition is not allowed the number of ways of selecting r objects out of a group of n objects is called  {}^n{c_r}

{}^n{c_r}=   \frac{{n!}}{{r!\left( {n - r} \right)!}}

In latest notation system  {}^n{c_r} is also known as C(n;r) or   \left( \begin{array}{l} n\\ \\ r \end{array} \right)

Properties of  \left( \begin{array}{l} n\\ \\ r \end{array} \right)– It’s a very useful and interesting Mathematical tool. It has following properties.

(i) {}^n{c_r}{}^n{c_{n - r}}

(ii)  {}^n{c_n} = {}^n{c_0} = 1

(iii)  {}^n{c_r} + {}^n{c_{r - 1}} = {}^{n + 1}{c_r}    known as Pascal’s law

(iv) r. {}^n{c_r} = n{}^{n - 1}{c_{r - 1}}

(v)  \frac{{{}^n{c_r}}}{{{}^n{c_{r - 1}}}} = \frac{{n - r + 1}}{r}

(vi) If n is even then we should put r=n/2 for maximum value of  {}^n{c_r} and if n is odd then  {}^n{c_r} is greatest when r= \frac{{{n^2} - 1}}{4}

(vii) In the expansions of {(1 + x)^n} if we put x=1 then

{}^n{c_0} + {}^n{c_1} + {}^n{c_2} + ....... + {}^n{c_n} = {2^n}

{}^n{c_0} + {}^n{c_2} + {}^n{c_4} + ......... = {2^{n - 1}}

{}^n{c_1} + {}^n{c_3} + {}^n{c_5} + ......... = {2^{n - 1}} Read more

Permutation and Combination

Permutations and Combinations-

‘Permutations and Combinations’ is the next post of my series Topics in IB Mathematics.It is very useful and interesting as a topic. It’s also very useful in solving problems of Probability. To understand Permutations and Combinations, we first need to understand Factorial.

Definition of Factorial-  If we multiply n consecutive natural numbers together, then the product is called factorial of n. Its shown by n! or by

for example :       n! = n(n - 1)(n - 2)(n - 3)..........3.2.1

Some Properties of Factorials-
(i) Factorials can only be calculated for positive integers at this level. We use gamma functions to define non-integer factorial that’s not required at this level
(ii) Factorial of a number can be written as a product of that number with the factorial of its predecessor    n! = n[(n - 1)(n - 2)(n - 3)..........3.2.1]

 = n(n - 1)!

(iii)  0! = 1  you can watch this video for the explanation.

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Quadratic equations and Quadratic Functions

Many IB mathematics tutors consider quadratic equations as a very important topic of ib maths. There are following ways to solve a quadratic equation

► Factorization method

►complete square method 

► graphical method

► Quadratic formula method

<img src="ib mathematics tutors.png" alt="ib mathematics tutors>

The quadratic formula is the strongest method to solve a quadratic equation. In this article, I will use …… steps to prove the quadratic formula

Given equation: ax²+bx+c=0

Step-1: transfer constant term to the right side

ax²+bx=-c

Step-2: divide both sides by coefficient of x²

                                      x²+bx/a=-c/a
Step-3: write (coefficient of x/2)²     that is (b/2a)²=b²/4a²
Step-4: Add this value to both sides
                                x²+bx/a+b²/4a² =-c/a²+b²/4a²
                                (x+b/2a)²=b²-4ac/4a²
now, take square root on both sides

                                  x+b/2=±√b²-4ac/a²

                                   x=-(b/2a)±√b²-4ac/2a

                                   x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}

This formula is known as quadratic formula, we can put values of a, b and c  from any equation and find the value of x (the variable) by directly using this formula.

IB Mathematics tutors can also explain the concept of conjugate roots with the help of quadratic formula. In a quadratic equation,
                                                              ax²+bx+c=0
if a, b and c are all rational numbers and one root of the quadratic equation is a+√b then the second root will automatically become a-√b. that can be understood easily as we use one +ve and one -ve sign in quadratic formula.
These types of roots are called Conjugate Roots.
If  a & b  are  the  roots  of  the  quadratic  equation  ax² + bx + c = 0,  then;
(i)  \alpha {\rm{ }} + {\rm{ }}\beta {\rm{ }} = \frac{{--b}}{a}  (ii)  \alpha {\rm{ }}{\rm{.}}\beta {\rm{ }} = \frac{c}{a}     (iii)  \alpha {\rm{ - }}\beta {\rm{ }} = \frac{{\sqrt D }}{a}
Nature  Of  Roots:
(a) Consider the quadratic equation ax² + bx + c = 0  where a, b, c  \in  R & a \ne 0 then
(i) D > 0   \Leftrightarrow  roots  are  real & distinct  (unequal).
(ii) D = 0  \Leftrightarrow  roots  are  real & coincident  (equal).
(iii) D < 0 \Leftrightarrow  roots  are  imaginary
(B) Consider the quadratic equation ax2+ bx + c = 0 where a, b, c  \in  Q & a \ne 0 then
 If  D > 0  &  is a perfect  square , then  roots  are  rational & unequal.
A quadratic  equation  whose  roots  are  a & b  is  (x – a)(x – b) = 0  i.e.   x2 – (a + b) x + a b = 0 i.e.
                          x2 – (sum of  roots) x +  product  of  roots = 0
Consider  the  quadratic  expression , y = ax² + bx + c  , a, b, c  \in  R & a \ne 0  then
(i) The graph between x, y  is always a  parabola.  If a > 0  then the shape of the parabola is concave upwards &  if a < 0  then the shape of the parabola is concave downwards.
Common  Roots  Of  2  Quadratic  Equations  [Only  One  Common  Root]-   Let  \alpha
be  the  common  root  of  ax² + bx + c = 0  &  a’x2 + b’x + c’ = 0
Therefore    \;a{\alpha ^2}{\rm{ }} + {\rm{ }}b\alpha {\rm{ }} + {\rm{ }}c{\rm{ }} = {\rm{ }}0{\rm{ }}\;and{\rm{ }}\;a{\alpha ^2} + {\rm{ }}b\alpha {\rm{ }} + {\rm{ }}c{\rm{ }} = {\rm{ }}0
 If we solve above pair by cramer’s rule we get

                                            \frac{{{\alpha ^2}}}{{bc' - cb'}} = \frac{\alpha }{{ac' - c'a}} = \frac{1}{{ab' - a'b}}

This will give us                    \alpha = \frac{{bc' - cb'}}{{ac' - c'a}} = \frac{{ac' - c'a}}{{ab' - a'b}}

                                            {(ac' - c'a)^2} = (ab' - a'b)(bc' - cb')

Every pair of the quadratic equation whose coefficients fulfils the above condition will have one root in common.

The condition that a quadratic function-  
                                           f(x , y) = ax² + 2 hxy + by² + 2 gx + 2 fy + c  may be  resolved  into  two  linear  factors  is  that     abc{\rm{ }} + {\rm{ }}2{\rm{ }}fgh{\rm{ }} - a{f^2} - b{g^2} - c{h^2}{\rm{ }} = {\rm{ }}0{\rm{ }}\; or
                                                                            \left| {\begin{array}{ccccccccccccccc} a&h&g\\ h&b&f\\ g&f&c \end{array}} \right| = 0
Reducible Quadratic Equations-These are the equations which are not quadratic in their initial condition but after some calculations, we can reduce them into quadratic equations
(i) If the power of the second term is exactly half to the power of the first term and the third term is a constant, these types of equations can be reduced to quadratic equations.
i.e., x + \sqrt x - 6 = 0 , {x^6} + {x^3} - 6 = 0{e^{2x}} + {e^x} - 6 = 0 ,{a^{2x}} + {a^x} - 6 = 0{\left( {2x + \frac{3}{{2x}}} \right)^2} + \left( {2x + \frac{3}{{2x}}} \right) - 6 = 0  all these equations can easily be reduced into quadratic equations by applying the method of substitution.

Example-Solve this equation and find x  {\left( {2x + \frac{3}{{2x}}} \right)^2} + \left( {2x + \frac{3}{{2x}}} \right) - 6 = 0 

Ans:

let y=  \left( {2x + \frac{3}{{2x}}} \right) then given equation will become

 

                                   {y^2} + y - 6 = 0  it’s a simple quadratic equation we can be easily factorised it and solve  so y=–3,2

so  \left( {2x + \frac{3}{{2x}}} \right)=-3

                                    \frac{{4{x^2} + 3}}{{2x}} = - 3

                               \begin{array}{l} 4{x^2} + 3 = - 6x\\ 4{x^2} - 6x + 3 = 0 \end{array}

the final equation can be solved using Quadratic formula and the same process can be repeated for  y=2

(ii) If a variable is added with its own reciprocal, then we get a quadratic equation i.e,x + \frac{1}{x} - 6 = 0\frac{{2x + 3}}{{x - 2}} + \frac{{x - 2}}{{2x + 3}} = 0{e^{2x}} + \frac{1}{{{e^x}}} - 6 = 0{a^{2x}} + \frac{1}{{{a^x}}} - 6 = 0 all these equations can be reduced into quadratic by replacing one term by any other variable.

 

Standard Form of a Quadratic Function-A quadratic function y=ax2+ bx + c can be

reduced into standard form    y = a{(x - h)^2} + k  by the method of completing the square. If we

draw the graph of this function we shall get a parabola with vertex (h,k). The parabola will be upward for a>0 and downward for a<0

 

Maximum and Minimum value of a quadratic function- If the function is in the form

y = a{(x - h)^2} + k Then ‘h’ is the input value of the function while ‘k’ is its output.

(i) If a>0 (in case of upward parabola) the minimum value of f is f(h)=k

(ii) If a<0 (in case of downward parabola)the maximum value of f is f(h)=k
If our function is in the form of  y=ax² + bx + c then vertex of the parabola V = \left( { - \frac{b}{{2a}},\frac{D}{{4a}}} \right)
The line passing through vertex and parallel to the y-axis is called the axis of symmetry.
The parabolic graph of a quadratic function is symmetrical about axis of symmetry.

 \Rightarrow  f(x) has a minimum value at vertex if a>0 and {f_{\min }} = - \frac{D}{{4a}}  at   x = - \frac{b}{{2a}}

 

 \Rightarrow   f(x) has a maximum value at vertex if a<0 and   {f_{\min }} = - \frac{D}{{4a}}  at   x = - \frac{b}{{2a}}

 

In the next post about quadratics, I shall discuss discriminant, nature of roots, relationships between the roots. In the meantime, you can download the pdf and solve practice questions.

 quadratic equation 200 questions.pdf

​​

 quadratic equation and function.pdf

   <img src="demo.png" alt="Smiley face">

Ib Maths Tutors-Types of functions(part-3)

Types of Functions-

IB Maths Tutors should give twenty-two hours for teaching functions and equations as per IBO recommendations. This is my third article on functions in the series of ib mathematics

IB Maths Tutors should give twenty hours in teaching functions and equations.This is my third article on functions in the series of ib mathematics

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here.

1.Greatest Integer Function–  This is an interesting function. It is defined as the largest     integer less than or equal to x

                                                         y = [x].

For all real numbers, x, this function gives the largest integer less
than or equal to x.

For example:   [1] = 1      [2.5] = 2      [4.7] = 4      [5.3] = 5
Beware!    [-2] = -2      [-2.6] = -3      [-4.1] = -5      [-6.5] = -7

domain=R
range=Z

<img src="ibmathstutors.jpg" alt="ibmathstutors">

greatest integer function

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IB Mathematics Tutors- types of mathematical function(part-1)

IB Mathematics Tutors should give a fair number of hours in teaching functions.This is my third article on functions in the series of ib mathematics

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here
1.Constant Function-
                                                F(x)=k
Domain= ℝ    

                                            Range    = {k}                                       

<img src="constant-function.jpg" alt="constant-function">

constant-function

                                   
Range    = {k}

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IB Mathematics (part-2)-Domain and Range of a function

In IB Mathematics both HL and SL, functions are one of the most important areas because they lie at the heart of much of mathematical analysis.here I am discussing domain and range of a function

Domain of a Function

Suppose I say that f is a real function.This means that for real input, the output should be real. For example

 F(x)=√x

if F is real, then x can only take non-negative values because only then the output will be real. Set of all real values of R is called domain

“Domain is the set of all possible inputs for which the output is real ”

In some cases, x is defined explicitly. for example,

                                     y=f(x)

                                       =x²;    1<x>2 here domain is defined explicitly as (1,2)

If no domain is mentioned explicitly, the domain will be assumed to be such that “F” produces real output

These are a few examples

  1. y=f(x)=√x
    Domain D= x≥0

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