• 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-

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  • 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

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  • Applications of Integration

    Applications of Integration In my previous posts, we discussed Definite and Indefinite Integrations. Now we shall learn about Applications of Derivatives. Initially, we shall discuss “Area Under Curves”. Area Under

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  • Definite Integration-Topics in IB Mathematics

    Definite Integration In the previous post, we discussed indefinite integration. Now we shall discuss definite integration ► Definite Integration- We already know that      this c here is an integral constant. we

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  • Indefinite Integration-Topics in IB Mathematics

    Indefinite Integration After a long series on differentiation and ‘Application of derivatives‘, we shall now discuss Indefinite Integration. It consists of two different words indefinite and integration. First of all,

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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

Complex Numbers

Complex Numbers-

Complex numbers come into existence when the square of a number is negative because we know it very well that the square of a number will always be positive doesn’t matter whether the number is positive or negative.<img src="complex number world.jpg" alt="complex number world">

In cases like  {x^2} + 1 = 0 or  2{x^3} + 5x = 0 here, if we solve, we find the square of x=-1. We say that x is not real here. Generally, these types of cases are considered as Complex numbers. Complex numbers were first observed by mathematician Girolamo Cardano (1501-1575). In his book Ars Magna, he discussed the mechanics of complex numbers in details and thus he started Complex Algebra.

Standard form of Complex Numbers-

Complex numbers are defined as expressions of the form a + ib where a,b \in  R & i = \sqrt { - 1}

It is denoted by Z  i.e. z= a + ib.

‘a’  is called as real part of z= (Re z)

and ‘b’ is called as imaginary part of z =(Im z).

i or IOTA- iota is a unique symbol. it’s the ninth letter of Latin alphabet. It’s used to denote imaginary numbers whose square root is -1.
Click here to download the book “An Imaginary Tale The Story of i” a very interesting book on iota by Paul J. Nahin. Read more

Principle of Mathematical Induction

Principle of Mathematical Induction:-

<img src="principle of mathematical induction" alt="principle of mathematical induction">

we know that the first positive even integer is 2 = 2  \times 1
the second positive even integer is 4 =  {\rm{ }}2 \times 2

the third positive even integer is 6 = 2 \times 3
the fourth positive even integer is 8 = 2 \times 4

………………………………………………….

If we move using the same pattern then nth positive integer=2n Read more

How to Prepare for board Exams-A Few Tips

How to Prepare for board Exams?

In my previous post of this series, we discussed the advantages of CBSE board exams. Now I am suggesting a few tricks about How to Prepare for board exams.
There are a lot of post on the internet that suggests you how to eat, how to sleep and how to manage your stress during board exam but in this post, I shall only discuss the academic tips about How to Prepare for board exams. I am taking class 10 Mathematics as a base subject here but you can apply more or less same tricks on almost all your subjects.

Question Paper Break-Up<img src="how to prepare for board exams".jpg" alt="how to prepare for board exams">

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Increasing and Decreasing Functions

Increasing and decreasing functions

This is my third post in the series of “Applications of derivatives”. The previous two were based on “Tangent and Normal” and “Maxima and Minima”.In this post, we shall learn about increasing and decreasing functions. That is one more application of derivatives.

Increasing and Decreasing Functions- We shall first learn about increasing functions

Increasing Function-

(a) Strictly increasing function- A function f (x) is said to be a strictly increasing function on (a, b) if x1< x2  \Rightarrow f(x1) < f (x2) for all xl, x2 \in (a, b).Thus, f(x) is strictly increasing on (a, b) if the values of f(x) increase with the increase in the values of x.Refer to the graph in below-given figure  \Downarrow <img src="increasing decreasing function.jpg" alt="increasing decreasing function"> Read more

IB Mathematics HL SL-Maxima and Minima

In my previous post, we discussed how to find the equation of tangents and normal to a curve. There are a few more  Applications of Derivatives in IB Mathematics HL SL, ‘Maxima and Minima’ is one of them.

Maxima and Minima:-

1. A function f(x) is said to have a maximum at x = a if f(a) is greater than every other value assumed by f(x) in the immediate neighbourhood of x = a. Symbolically

 

\left. \begin{array}{l} f(a) > f(a + h)\\ f(a) > f(a - h) \end{array} \right] \Rightarrow x = a   gives maxima for a sufficiently small positive h.

Similarly, a function f(x) is said to have a minimum value at x = b if f(b) is least than every other value assumed by f(x) in the immediate neighbourhood at x = b. Symbolically

 

\left. \begin{array}{l} f(b) > f(b + h)\\ f(b) > f(b - h) \end{array} \right]  If x = b gives minima for a sufficiently small positive h.

 

<img src="IB Mathematics HL SL.jpg" alt="IB Mathematics HL SL">

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Applications of Derivatives in IB Mathematics

Applications of Derivatives in IB

Mathematics-

In my previous post, we discussed how to find the derivative of different types of functions as well as the geometrical meaning of differentiation. Here we are discussing  Applications of Derivatives in IB Mathematics
There are many different fields for the Applications of Derivatives. We shall discuss a few of them-

Slope and Equation of tangents to a curve- If We draw a tangent to a curve y=f(x) at a given point   ({x_1},{y_1}), then

The gradient of the curve at given point=the gradient of the tangent line  at given  point

and we already discussed that slope or gradient of the tangent at given point   ({x_1},{y_1})

m=  {\frac{{dy}}{{dx}}_{at({x_1},{y_1})}}

=f'({x_1})

Finally to find the equation of tangent we use the slope-point form of equation

y - {y_1} = m(x - {x_1})

The major part of this concept is also discussed in the previous post. We should also remember following points while solving these types of questions.

(i) If two lines are parallel to each other, their slopes are always equal
i.e     {m_1} = {m_2}
(ii) If two lines are perpendicular to each other, the product of their  slopes is always -1

{m_1}.{m_2} = - 1

(iii) If a line is passing through two points   ({x_1},{y_1}) and  ({x_2},{y_2})  then, slope of the line

m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

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Continuity of functions-IB Maths topics

Continuity of functions-

The word continuous means without any break or gap. Continuity of functions exists when our function is without any break or gap or jump . If there is any gap in the graph, the function is said to be discontinuous.

Graph of functions like sinx,cosx, secx, 1/x etc are continuous (without any gap) while greatest integer function has a break at every point(discontinuous).

1. A function f(x) is said to be continuous at x = c,  if  {\lim }\limits_{x \to c} f(x) = f(c) .

 

symbolically f is continuous at x = c if  {\lim }\limits_{x \to c - h} f(c + h) = {\lim }\limits_{x \to c - h} f(c - h) = f(c).

 

It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighborhood of x = a, not necessarily at x = a.

<img src="continuous functions.png" alt="continuous functions">

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How To Solve Limit Problems

How To Solve Limit Problems

 

In my previous post on limits, We have discussed some basic as well as advanced concepts of limits. Here we shall discuss different methods to solve limit questions. Based on the type of function, we can divide all our work into sections-:

Algebraic Limits- Problems of limits that involve algebraic functions are called algebraic limits. They can be further divided into following sections:-

Direct Substitution Method –Suppose we have to find. L = {\lim }\limits_{x \to a} f(x) we can directly substitute the value of the limit of the variable (i.e replace x=a) in the expression.

► If f(a) is finite then L=f(a)

► If f(a) is undefined then L doesn’t exist

► If f(a) is indeterminate  then this method fails

<img src="limit.png" alt="limit">

Example-1:- Find value of   {\lim }\limits_{x \to 2} (x²-5x+6) Read more

Limit, Continuity & Differentiability-IB Maths Topics

Limit of a function

Limit of a function f(x) is said to exist as, x \to a when

 {\lim }\limits_{x \to {a^ + }} f(x) = {\lim }\limits_{x \to {a^ - }} f(x) =   finite quantity.

 

 <img src="limit.png" alt="limit">

Fundamental Theorems On Limits :

Let    {\lim }\limits_{x \to {a^{}}} f(x) = l &   {\lim }\limits_{x \to {a^{}}} f(x) = l   If l & m exists then :

(i) f (x) ± g (x) = l ± m

 

(ii) f(x). g(x) = l. m

 

(iii)  {\lim }\limits_{x \to \infty } \frac{{f(x)}}{{g(x)}} = m  provided  m \ne 0

 

(iv)  {\lim }\limits_{x \to {a^{}}} kf(x) = k {\lim }\limits_{x \to {a^{}}} f(x)   where k is a constant.

 

(v)    {\lim }\limits_{x \to {a^{}}} f[g(x)] = f[ {\lim }\limits_{x \to {a^{}}} g(x)] = f(m)provided f is continuous at        g (x) = m

 

Standard Limits :

(a)  {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = 1 and {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = {\lim }\limits_{x \to 0} \frac{{{{\tan }^{ - 1}}x}}{x} = 1 {\lim }\limits_{x \to 0} \frac{{{{\sin }^{ - 1}}x}}{x} = 1 Where x is measured in radians

 

(b)  {\lim }\limits_{x \to 0} {(1 + x)^{\frac{1}{x}}}and {\lim }\limits_{x \to 0} {(1 + \frac{1}{x})^x} both are equal to e

 

(c) {\lim }\limits_{x \to a} f(x) = 1and {\lim }\limits_{x \to a} \theta (x) = \infty  then this will show that  {\lim }\limits_{x \to a} f{(x)^{ {\lim }\limits_{x \to a} \theta (x)}} = {e^{ {\lim }\limits_{x \to a} \theta (x)[f(x) - 1]}}

 

(d)  {\lim }\limits_{x \to a} f(x) = A > 0 and   {\lim }\limits_{x \to a} \theta (x) = B (a finite quantity) then    {\lim }\limits_{x \to a} f{(x)^{ {\lim }\limits_{x \to a} \theta (x)}} = {e^z}

 

where z= ^{ {\lim }\limits_{x \to a} \theta (x)\ln f(x)} = {e^{B\ln A}} = {A^B}

 

(e)  {\lim }\limits_{x \to 0} \frac{{{a^x} - 1}}{x} = \ln a where a>0. In particular  {\lim }\limits_{x \to 0} \frac{{{e^x} - 1}}{x} = 1

 

Indeterminant Forms:

\frac{0}{0},\frac{\infty }{\infty },0 \times \infty ,{0^\infty },{\infty ^0} etc are considered to be indeterminant values

We cannot plot \infty  on the paper. Infinity\infty is a symbol & not a number. It does not obey the laws of elementary algebra.

\infty +\infty =\infty

\infty ×\infty \infty

(a/\infty ) = 0 if a is finite v is not defined

a b =0,if & only if a = 0 or b = 0  and  a & b are finite.

Expansion of function like Binomial expansion, exponential & logarithmic expansion, expansion of sinx , cosx , tanx should be remembered by heart & are given below:

(i)  ex =1+x/1!+x3/3!+x4/4!……\infty

 

(ii)  ax=1+(xloga)/1!+ (xloga)2/2!+ (xloga)3/3!+ (xloga)4/4!+……….where a > 0

 

(iii)   ln(1-x)=x-x2/2+x3/3-x4/4……….    where -1 < x  1

 

(iv)  ln(1-x)=-x-x2/2-x3/3-x4/4……….     where  -1 x < 1

 

(v )  \sin x = x - \frac{{{x^3}}}{{3!}} + \frac{{{x^5}}}{{5!}} - \frac{{{x^7}}}{{7!}}.......

 

(vi) \cos x = 1 - \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}} - \frac{{{x^6}}}{{6!}}.......

 

(v)  \tan x = x + \frac{{{x^3}}}{3} + \frac{{2{x^5}}}{{5!}} - ..........

 

In next post, I will discuss various types of limit problems, their solutions and L’ Hospital’s rule.In the meantime, you can solve these basic questions from this PDF. This PDF is for beginners only. I will post difficult and higher level questions in the next post on this topic

 LimitsExercises.pdf

In my second post on limits, you can learn how to solve different types of questions on limits
Here is the link

http://ibelitetutor.com/blog/how-to-solve-limit-problems/

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

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