Koundinya-Mathematics Teachers Hub

Sunday, 25 December 2011

This should challenge you!!!

Delete 100 digits in the number

12345678910111213141516171819. . . . 96979899100

so that the resultant number has

(a) The greatest possible value;
(b) The least possible value.

Solve this!!!

For x > 1 determine the sum of the infinite series

${\color{Yellow} \frac{x}{x+1}+\frac{x^{2}}{(x+1)(x^{2}+1)}+\frac{x^{4}}{(x+1)(x^{2}+1)(x^{4}+1)}+...}$

For
${\color{Yellow} 0\leq t\leq 1}$
Show that

${\color{blue} t\leq tant\leq t+t^{3}}$

Now, for
${\color{blue} 0\leq x\leq \Pi}$
and
${\color{Yellow} 0\leq y\leq 1}$
Show that

${\color{blue} ysinx\leq tan(ysinx)\leq ysinx+(ysinx)^{3}}$

Finally, show that the limit
${\color{Yellow} L = \lim_{y\rightarrow 0}\frac{1}{y}\int_{0}^{\Pi }tan(ysinx)dx = 2}$

Friday, 23 December 2011

Try this!!!

In how many ways can a cube be painted with six different colours with one colour on each face?

Use of complex numbers in Integration

Twin Integrals :
To find
${\color{black} \int e^{ax} cos(bx)dx }$
and
${\color{black} \int e^{ax} sin(bx)dx }$

We begin with
${\color{black} \int e^{kx} dx = \frac{e^{kx}}{k}}$
Let k = a + ib
then,
${\color{Yellow} \int e^{(a+ib)x} dx = \frac{e^{(a+ib)x}}{a+ib}}$
${\color{black} \therefore \int e^{ax}(cos(bx) +isin(bx))dx= \frac{e^{ax}}{a^{2}+b^{2}}(cos(bx)+isin(bx))(a-ib)}$
${\color{black} \int e^{ax}(cos(bx) +isin(bx))dx= \frac{e^{ax}}{a^{2}+b^{2}}(acos(bx)+bsin(bx))+i\frac{e^{ax}}{a^{2}+b^{2}}(asin(bx)-bcos(bx))}$
Equating Real and Imaginary parts
${\color{black} \int e^{ax}cos(bx)dx= \frac{e^{ax}}{a^{2}+b^{2}}(acos(bx)+bsin(bx))}$
and
${\color{black} \int e^{ax}sin(bx)dx= \frac{e^{ax}}{a^{2}+b^{2}}(asin(bx)-bcos(bx))}$

Conic Sections and Eccentricity(e)

Let us examine the process of Limit e ---> 1of a conic

The X shaped lines that define the surface of the cone are called generators and the vertical line passing through the intersection of the generator is called axis

We have conic sections formed when a double right circular cone is intersected by a plane.
General equation of a conic is b²x²+a²y² = a²b²

where, b² = a²|1-e²|

When e = 0 , we get the conic section- 'Circle'
Note: Here the plane cuts the double cone at right angle with the axis of the cone

When e < 1 , we get the conic section- 'Ellipse'

Note: Here the plane cuts the double cone at an inclination with the axis of the cone

When e = 1 , we get the conic section- 'Parabola'
Note: Here the plane lies parallel to the generator of the double cone

When e > 1 , we get the conic section- 'Hyperbola'
Note : Here the plane cuts bottom and top cones of the double cone

So as Eccentricity(e) varies the conic section varies as shown below

Key Note: Can we conclude the process in the following form

when e = 0 we have a perfectly symmetrical figure which is a circle.

As e moves from 0 to 1, the circle gets deformed. As e approaches 1 the deformation becomes deeper and deeper and as e equals 1 there is an explosion instantaneously and the conic section becomes a parabola. Further as e moves from 1 instaneously another explosion results into a hyperbola.