This is to inform all that I teach mathematics online on the WizIq platform and have introduced a new course namely Quadratic Equations & Inequalities.
This course is suitable for students who follow the curriculum for CBSE, IGCSE, K-12 , Plus 1 & Plus2 etc.
The link is given below:
http://www.wiziq.com/course/53618-quadratic-equations
You may enroll for the course and be an expert in solving quadratic equations and inequalities.
I am preparing courses on 3-D Geometry, Trigonometry, Matrices and Differential Equations etc and they will be introduced in due course.
You will be informed of any new introduction through this blog.
Monday, 28 April 2014
Saturday, 1 March 2014
CONTINUITY AND DIFFERENTIABILITY: A SIMPLE INTERPRETATION
Functions are a very important concept in mathematics. Many theories have been developed on functions and they help us to solve a whole lot of problems in science and engineering.
Continuity and differentiability are two very important attributes associated with functions. Certain functions are continuous from a certain point to a certain point, but not differentiable throughout. Fine. But what do we mean by continuity and differentiability?
Let us first talk about the graph of a function.
Graph of a function? What is that?
Graph of a function is nothing but a drawing to represent it. We can always draw the graph of a function. It can be in the shape of a curve that can be drawn by hand on a piece of paper or in the form of a straight line or broken lines or a combination of curves and straight line segments or even a scattering of isolated points.
If we can draw the graph of a function from point A to point B without lifting the pencil off the paper, we say that the function is continuous from A to B. That means, the drawing that represents a continuous function has no gaps in it. If there is a gap then the function is not continuous!
Well, that is fair enough. Now, what is the meaning of differentiability?
If the drawing (graph) of a function is continuous from point A to point B and has no sharp bend or sharp turning on it, we say that the function is differentiable throughout between A and B.
But if the graph has gaps or sharp bends or sharp turnings, the function is said to be not differentiable there. In other words a function is not differentiable at the gaps or sharp bends or sharp turnings of the graph.
To summarize, a function is said to be
If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
Continuity and differentiability are two very important attributes associated with functions. Certain functions are continuous from a certain point to a certain point, but not differentiable throughout. Fine. But what do we mean by continuity and differentiability?
Let us first talk about the graph of a function.
Graph of a function? What is that?
Graph of a function is nothing but a drawing to represent it. We can always draw the graph of a function. It can be in the shape of a curve that can be drawn by hand on a piece of paper or in the form of a straight line or broken lines or a combination of curves and straight line segments or even a scattering of isolated points.
If we can draw the graph of a function from point A to point B without lifting the pencil off the paper, we say that the function is continuous from A to B. That means, the drawing that represents a continuous function has no gaps in it. If there is a gap then the function is not continuous!
Well, that is fair enough. Now, what is the meaning of differentiability?
If the drawing (graph) of a function is continuous from point A to point B and has no sharp bend or sharp turning on it, we say that the function is differentiable throughout between A and B.
But if the graph has gaps or sharp bends or sharp turnings, the function is said to be not differentiable there. In other words a function is not differentiable at the gaps or sharp bends or sharp turnings of the graph.
To summarize, a function is said to be
- continuous throughout if there is no gap in the drawing (graph) of the function
- differentiable throughout if there is no gap, no sharp bends or no sharp turnings in the drawing.
If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
DIVISIBILITY BY 11
Well, if you see a number how do you know if it is divisible by 11?
This is the trick.
Here let us find the sums of alternate digits.
Let us find the sums of alternate digits
Let us find the sums of alternate digits
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
This is the trick.
- Find the sums of alternate digits
- Find the difference between the two sums
- If that difference can be divided by 11without a remainder, the original number is also divisible by 11
Here let us find the sums of alternate digits.
- 5+8+2=15
- 3+1=4
- Difference=15-4=11
- 11 is divisible by 11. So 53812 is also divisible by 11.
Let us find the sums of alternate digits
- 5+8+8+6=27
- 4+0+1=5
- Difference=27-5=22
- 22 is divisible by 11; so is 5480816
Let us find the sums of alternate digits
- 3+9+7=19
- 4+8+5=17
- Difference=19-7=2
- 2 is not divisible by 11. So 349875 can not be divided by 11 without leaving a remainder.
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
Thursday, 27 February 2014
PERPENDICULAR LINES IN SPACE
Let us look at the following scenario.
P is a point and L is a line in space.
If we draw a perpendicular from P to L, will it always intersect L ?
What do you think is the answer?
For the answer click here
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
P is a point and L is a line in space.
If we draw a perpendicular from P to L, will it always intersect L ?
What do you think is the answer?
For the answer click here
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
Monday, 24 February 2014
LINES IN SPACE : AN INTERESTING QUESTION
We are familiar with geometry in two dimensions and have plotted points and drawn lines, curves and figures on a piece of paper.
A plane piece of paper presents a two-dimensional environment. That means only objects having no thickness can be created on a plane piece of paper. But wait! Artists have drawn mountains, brooks and horses on paper. And they are not flat objects. True. They use different intensities of shades and the techniques of perspectives to create the illusion of thickness on a flat surface.
In reality we live in a three -dimensional world. A natural progression therefore is to enquire about what happens in a world of three dimensions.
We talk about space.
Just as we plot points, lines and figures on a plane, we can have points, lines and figures in space.
Let us talk about lines in space.
Statement: Two non-intersecting lines in space are always parallel.
What do you think about this statement?
Please click here for the answer
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
A plane piece of paper presents a two-dimensional environment. That means only objects having no thickness can be created on a plane piece of paper. But wait! Artists have drawn mountains, brooks and horses on paper. And they are not flat objects. True. They use different intensities of shades and the techniques of perspectives to create the illusion of thickness on a flat surface.
In reality we live in a three -dimensional world. A natural progression therefore is to enquire about what happens in a world of three dimensions.
We talk about space.
Just as we plot points, lines and figures on a plane, we can have points, lines and figures in space.
Let us talk about lines in space.
Statement: Two non-intersecting lines in space are always parallel.
What do you think about this statement?
Please click here for the answer
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
Sunday, 23 February 2014
DIVISIBILITY BY 7 OF A LARGER NUMBER
I hope you have gone through my last post on "A TRICK TO CHECK DIVISIBILITY BY 7".
If the number is large like 6013 or 564743 having more than three digits, then the method is to apply the trick repeatedly.
Let us check if 6013 is divisible by 7.
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
If the number is large like 6013 or 564743 having more than three digits, then the method is to apply the trick repeatedly.
Let us check if 6013 is divisible by 7.
- Last digit=3
- Double it; we get 6
- Subtract 6 from 601, the answer is 595
- Is 595 divisible by 7? It is rather difficult to answer.
- So, take the lat digit of 595; that is 5
- Double it; we get 10
- Subtract 10 from 59; the answer is 49
- Is 49 divisible by 7? Yes!
- So, 595 is divisible by 7 and consequently 6013 is divisible by 7.
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
A TRICK TO CHECK DIVISIBILITY BY 7
There are many tricks in mathematics to solve a problem quickly and easily!
Question: Can you divide 406 by 7 without leaving a remainder?
This is the trick
Here, the given number is 406Question: Can you divide 406 by 7 without leaving a remainder?
This is the trick
- Take the last digit of the number
- Double it
- Subtract the number thus obtained from the given number without the last digit
- If you can divide the number obtained by 7, you can divide the given number also by 7
- Take the last digit. It is 6
- Double it; we get 12
- Subtract 12 from 40; we get 28
- 28 is divisible by 7. So 406 is also divisible by 7.
Which of the following numbers are divisible by 7?
(a) 441 (b) 623 (c) 474 (d) 4886 (e) 6249 (f) 444724 (g) 75328
Wait for the next post to be published soon to learn about the trick to check divisibility by 7 of a number having 4 or more digits.
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If you need online assistance in mathematics (CBSE, IGCSE, PLUS 1 & 2 ETC.) please contact me on raghavanckk@gmail.com Skype ID: raghavanckk Mobile: +27(0)737455575
I charge US$5-10 per hour depending on the topic/class.
I have more than 30 years of experience in teaching mathematics at O and A levels in different countries. At the moment I am also involved with a Mathematics Development Project under Rhodes University to improve the mathematics efficiency of High School students in South Africa.
I am conversant with the mathematics curricula of many countries.
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