Axiomatic Approach

If S is the sample space of a random experiment and E is the power set of S, then a real valued function from E to [0,1] is probability function if it satisfies the following axioms:

1. Axiom of Positivity

   P(E) ≥ 0 if E ∈ f(s)

2. Axiom of Certainty

   P(S) = 1, when S is the sample space

3. Axiom of Union

   P(A∪B) = P(A) + P(B) for every exclusive events A and B

Continue Reading >>Axiom of Positivity, Certainity and Union

Probability

If S is the sample space of a random experiment, 'E' is any event, then probability of E is denoted by P(E) and is defined as

P(E) = (Number of Simple Events in E / Total Number of Simple Events in S.)

Note that, P(φ) = 0 and P (S) = 1

Continue Reading >>What is Probability?

Go here for Matrices EAMCET Part 1

Homogeneous Equations

If the system is AX=0 and

1. |A| ≠ 0 then the system has unique solution, that is the zero solution.


2. |A| = 0 then the system has atleast 1 non-zero solution. In-fact the system has infinite number of solutions.


3. If the number of equations in a homogeneous system is less than number of unknowns then system has infinite number of solutions.

Non-Homogeneous System

Suppose the given non-homogeneous system is AX = B, find Δ, Δ1, Δ2, Δ3 and

1. If Δ ≠ 0 the system has unique solution . In the above case the given equations represent three planes intersecting in a point.
   


2. If Δ = 0, Δ1 ≠ 0, ( Δ2, Δ3 = 0) then the system has no solution. In this case the given equations represent three planes forming a triangular prism or parallel planes, not coincidental.
   
     
   


3. If Δ= 0, Δ1 = 0 ( Δ2, Δ3 = 0), then the system has infinite number of solutions. In this case the given equations represent three planes passing through a line or a coincidental planes.
   
   

Suppose the given non-homogeneous system is

Consider two pairs of the above equations eliminate the same unknown. Suppose the equations thus obtained are

when z is eliminated.

1. If
   


2. If
   
   then the system has no solutions.


3. If
   

Continue Reading >>Matrices EAMCET Part 2 - Homogeneous Equations and Non-Homogeneous System

Here are some of the most important points from matrices to be understood and remembered by EAMCET aspirants.

1. Product of two upper triangular matrices is a upper triangular matrix.


2. Determinant of triangular matrix is the product of the elements in the principle diagonal.


3. (i) trace (KA) = K tr(A)
   (ii) tr (A + B) = tr(A) + tr(B)
   (iii) tr (AB) = tr (BA)


4. If A is a skew symmetric matrix then,
   1-> All the elements in the principle diagonal are equal to zero.
   2-> |A|≠ 0 then its order is an odd number.


5. If A, B are symmetric matrices, K is a scalar then
   (i) KA is symmetric.
   (ii) (A + B) and (A - B) are symmetric.
   (iii) AB, BA are symmetric if and only if AB = BA.


6. For any square matrix A, (A + A^T) is symmetric and (A - A^T) is skew symmetric.
   
7. A is a matrix of order m x n then A.A^T, A^T.A are symmetric matrices.
   
   


8. The only non-singular idempotent matrix is the unit matrix. That is if A² = A and A ≠ I then |A| = 0
   


9. If A, B are square matrices of the same type satisfying the condition AB = A, BA = B then
   (i) Both A and B are idempotent matrices
   (ii) For any natural number 'n', (A+B)ⁿ = 2^(n-1) (A+B)
   


10. If A is an involutary matrix then,
    


11. If A is an orthogonal matrix then,
    
    Determinant of A is ±1. If the rays are in right handed system then the value of determinant is +1 and if they are in left handed system then the value is -1.


12. Every nilpotent matrix is a singular matrix.


13. If A is a nilpotent matrix of index 3 then,
    


14. If A is involutary matrix and A ≠ I, then the determinant of |I + A| = 0.
    Proof :-
    

Continue Reading >>Matrices EAMCET - Part 1

Some Important Matrices Determinants to be Remembered for competitive exams:-

Continue Reading >>Some Important Determinants to be Remembered

We aim to measure the resistance of a given resistor using Kelvin Double Bridge and determine its tolerance. Kelvin Double Bridge is nothing but a modification of Wheatstone bridge. It is used for measuring of low resistance to a good precision. It compares two ratio arms P,Q and p,q and hence is called 'double bridge'.

P, Q, p, q are the resistances in the ratio arms. G is a galvanometer of D'Arsonal type, used as a null detector. S is a small standard resistor, R is a resistance under measurement. Usually low resistance consists of four leads. Two of them are called as voltage leads and remaining as current leads. "r" is the resistance of connecting lead between R and S.

Under balanced conditions,

From the above equation, it is clear that the resistance of connecting leads "r" has no effect on the measurement if the two sets of ratio arms have equal ratios ie, P/Q = p/q.

The effect of thermo-electric Emf can be eliminated by making other measurement with battery terminals reversed and taking the average of the two readings can eliminate the effect of thermo-electric emfs.

Procedure for the measurement of low resistance R using Kelvin Double Bridge

1. Move the Galvanometer switch to increase position. This connects the built-in galvanometer to the circuit. If an external more sensitive galvanometer is available, connect it to the terminals marked "extgalv" and put the galvanometer switch in "EXT" position.


2. Four terminals are provided for connecting unknown resistance of the bridge circuit. They are labeled by "+C, +P, -C, -P". Here +C and -C constitute the current terminals. If the given unknown resistance is of four leads then connect the two potential leads to +P & -P and current leads to +C & -C with correct current polarity. If the unknown resistance has two terminals then the leads from +C and +P are connected to other terminals of unknown resistance.


3. Now, press the button on the panel and obtain the balance by varying the dials.


4. Under balanced conditions, the sum of two dials multiplied by multiplier sitting gives the value of unknown resistance.


5. Find the tolerance of the resistance and tabulate the results. The example results are as given in the tabular form below:
   

   S.No	For Kelvin Double Bridge Calculated Value	From Multimeter Theoretical Value(Ω)	% Tolerance
   1.	[1 + 0.001x0] x 100 = 100	100.2	0.199
   2.	[0 + 96 x 0.001] x 100 = 9.6	10.7	10.2
   3	[0.9 + 49 x 0.001] x 100 = 94.9	95.4	0.52

Precautions

• Press the push button immediately during the course of balance.


• A variable high resistance should be connected in series with galvanometer for initial adjustments in order to protect it from high currents. Once the balance point is reached, the resistance should be cut-off to increase the sensitivity.

Continue Reading >>Measuring Low Resistance Using Kelvin Double Bridge

Here we aim to measure the three phase reactive power 'Q' absorbed by the given 3Φ induction motor by using single wattmeter. The formula for the reactive power 'Q' absorbed by "Balanced Load" is given by where,

• Q = Total reactive power absorbed


• V_L = Line voltage of supply


• V_ph = Voltage per phase of supply


• I_L = Line current of supply


• I_ph = Phase current of supply


• Φ = power factor angle

The break load on the induction motor can be taken as a balanced load on the I.M. The measurement can be done using the conventional 'Dynamometer type of wattmeter'. Here the current coil of the wattmeter is connected in the R-phase, the pressure coil is connected between Y-phase and B-phases.

Assuming that, Q = total reactive power.



for delta connection,
The phase angle between I_R and V_YB is (90 - Φ)

Continue Reading >>How to Measure 3-Phase Reactive Power of a Induction Motor Using Single Wattmeter

Anderson Bridge:-

The Bridge is used to measure the self-inductance of a coil in terms of standard known capacitor. The bridge and its connection diagrams are as shown below:

Anderson Bridge

Connection Diagram

The first figure shows the Anderson Bridge circuit diagram and second figure shows the connections to be made in the kit. It indicates the headphones detector, Galvanometer G, 1KHz Supply, 3V D.C Supply and inductance.

For null detection which occurs when the bridge is in balanced condition, it can be shown that the self inductance is given by L = CR(Q + 2r)

Procedure for Anderson's Bridge:-

1. Make connections as shown in the connection diagram.


2. Here Galvanometer is the detector and 3V D.C supply is the source to the unknown inductance.


3. Adjust the dial R to get balance point with the galvanometer.


4. Using the resistance dial 'S' for fine balancing, note down the values of R


5. Replace D.C supply with A.C supply of 1KHz frequency. Replace the galvanometer with headphones for getting null point.


6. Set standard capacitance 'C' at position 0.1mF and adjust resistance dial 'r' till you get balance point as indicated by minimum sound in the headphones. Note the values resistance 'r' and calculate unknown inductance using the formula L = CR(Q + 2r). The range of Q factor for Anderson Bridge is 1 < Q < 10


7. Repeat above steps choosing different values of inductance and capacitance and tabulate the results.

Here in the Anderson Bridge, the term Bridge Sensitivity (S_B) is defined as the deflection of the galvanometer for unit fractional change in the unknown quantity. And besides vibrational galvanometers, other 2 types of detectors that are used most are headphones and tunable amplifier detectors.

Advantages of Anderson Bridge:-

• Fixed capacitor is used instead of variable capacitor as in case of maxwell's bridge.


• The bridge is used for accurate determination of inductance in terms of capacitance.

Disadvantages of Anderson Bridge:-

• Bridge is more complex.


• An additional junction point increased the difficulty of shielding the bridge.


• Complex balance equations.

Continue Reading >>How to Find the Value of Self-Inductance and its Inherent Resistance of a Given Coil Using Anderson Bridge

Google's most popular social networking website - Orkut has launched private scraps feature that enables users to send scraps secretly for a completely private conversation. This new feature on Orkut takes away the burden of using testimonials for sending private messages, thus making secret scraps more fun and interesting for selected friends.

All you have to do to send a private scrap to your friend is, simply compose your message first and before posting it, click the drop-down menu named visible to: and select the option only me and him.

By selecting this option only the person you're sending the scrap will be able to see it and other users simply won't see anything, thus keeping your scrap completely a secret.

And from now-on-wards every piece of content on Orkut will be labeled with a privacy level so that you can easily know who else is seeing it along with you. Orkut has provided three levels of privacy for any shared content:

1. 

   Visible to all the web

   This indicates that any orkut user can see the content posted. And if you add a comment to it, your comment will also be publicly visible.


2. 

   Visible only to friends

   This privacy level indicates that only the friends of the person who initially posted that content will be able to see it. And when you add a comment to it, your comment will show up in your profile page only for those friends that have access to the original content. Others simply won’t see anything.


3. 

   Visible only to selected friends

   This privacy level means that only a few users were selected to see the content posted. If the person who posted it has selected you to see the content then only you can see it and any comment you add will only be visible in your profile page to those selected friends that are able to see the initial content posted.

Continue Reading >>Orkut Launches Private Scraps and Privacy Levels for Any Shared Content

According to Addition Theorem on Probability, for any two elements A, B
P(A∪B) = P(A) + P(B) - P(A∩B)

Proof :-

Expressing A∪B as the union of two mutually exclusive events we get
(A∪B) = A ∪ (B-A)
P(A∪B) = P(A ∪ (B-A))
By applying axiom of union we get,
P(A∪B) = P(A) + P(B-A) since A, B-A are exclusive
= P(A) + P(B - (A∩B))
= P(A) + P(B) - P(A∩B) since P(B-A) = P(B) - P(A) when every A⊆B

note: If you find any errors then do correct in comment section below.
Therefore P(A∪B) = P(A) + P(B) - P(A∩B)

Continue Reading >>Proof of Addition Theorem on Probability Through Axiomatic Approach

World's largest maker of cell phones - Nokia has recently unveiled Bicycle Charger Kit that can charge your phone with free and eco-friendly energy generated by you when you cycle.

This new gadget from Nokia will be well received in those areas of the world where bicycles are still an essential transportation means and it can also save some money on charging costs.

Here, the free and environmentally friendly electricity for charging mobile phones comes basically from a Bottle Dynamo attached to the front bicycle wheel and is powered by the pedaling motion of the rider. The kit starts charging at walking speed (6 km/h) and stops when your speed reaches 50 km/h. The total charging time varies depending on the phone and the cycling speed. The faster the ride speed, the more battery life it generates. For example, a 10-minute journey at 10 kilometers per hour (six miles per hour) produces around 28 minutes of talk time or 37 hours of standby time.

The Nokia Bicycle Charger Kit consists mainly of 3 components:

• A Bottle Dynamo to generate electricity that fits to the front of the bike with a mounting bracket.


• Bicycle Charger DC-14 used as a charging station and rubberized phone holder to mount your phone securely to your bicycle and protect it from the vibrations of bumpy roads.
  

The Nokia Bicycle Charger Kit can be easily attached to any bicycle and is compatible with any Nokia phone which has 2 mm charging interface. It is also compatible with microUSB as long as you provide the cable. This Wonderful gadget is priced at about $18 and is will be made available from Nokia online and Nokia phone retailers by year's end.

Continue Reading >>Nokia Brings Out Bicycle Phone Charger Kit

Here is the complete list of all Nokia cell phone models with which the recently unveiled Nokia Bicycle Charger Kit is compatible:

Nokia 1200	Nokia 1202	Nokia 1203	Nokia 1208
Nokia 1650	Nokia 2630	Nokia 2690	Nokia 2730 Classic
Nokia 3109 Classic	Nokia 3110 Classic	Nokia 3250 XpressMusic	Nokia 3500 Classic
Nokia 5070	Nokia 5220 XpressMusic	Nokia 5230	Nokia 5235 Comes With Music
Nokia 5310 XpressMusic	Nokia 5500 Sport	Nokia 5610 XpressMusic	Nokia 5700 XpressMusic
Nokia 6070	Nokia 6080	Nokia 6120 Classic	Nokia 6121 Classic
Nokia 6124 Classic	Nokia 6151	Nokia 6220 Classic	Nokia 6233
Nokia 6234	Nokia 6300	Nokia 6301	Nokia 6303 classic
Nokia 7210 Supernova	Nokia 7500 Prism	Nokia 7610 Supernova	Nokia C5-00
Nokia E50	Nokia E51	Nokia E61	Nokia E61i & E65
Nokia E66 & E71	Nokia E75	Nokia E90 Communicator	Nokia N70
Nokia N70 Music Edition	Nokia N72	Nokia N73	Nokia N73 Music Edition
Nokia N77	Nokia N78	Nokia N79	Nokia N80
Nokia N80 Internet Edition	Nokia N81	Nokia N810 Internet Tablet	Nokia N810 WiMAX Edition
Nokia N81 8GB	Nokia N82	Nokia N91 8GB	Nokia N95
Nokia N95 8GB	Nokia X3	Nokia X6 16GB	Nokia N800 Internet Tablet

Continue Reading >>Nokia Bicycle Charger Kit Compatibility List

Developed by #1 provider of gaming experiences on social networks - Zynga, FarmVille is the most popular real-time farm simulation game in which you can manage a virtual farm by planting, growing and harvesting crops and trees, and raising livestock.

From the day it has been launched as an application on the popular social-networking site Facebook, FarmVille has become a huge time sink hole and additive to many, establishing itself as one of the most popular game applications on Facebook with over 82.4 million active users and over 23.9 million fans. The nature of the game is addictive because you have to come back each day to harvest your crops and increase your XP.

Recently FarmVille has come up with a good news for all its Farmers by introducing Southwestern items in the Market. The all new Desert Plains landscape lets you transform your farm into a American Southwestern style desert! It is made available in the Market for only 1,000 Coins.
More Southwestern items are made available now in the Market which includes the brand new Desert Tortoise, new Longhorned Cow and Appaloosa Horse, two new crops, the masterable Tomatillos, own Botanical Garden and much much more...

Everything grows in FarmVille!!
Play Free and see if you get FarmVille fever.

Continue Reading >>Southwestern Items Have Come to Farmville!

Since its launch in January 2005, BSNL's DataOne High Speed broadband service has grown from strength to strength with wide network extending to every nook and corner of India and has established itself as the leading ISP (Internet Service Provider) in India providing high-speed internet connection with fast bandwidth.
And as part of their ongoing efforts to better their Broadband internet service and serve customers in a much better way, BSNL has recently announced increase of speeds for various plans which are as follows:

BSNL Broadband Plans	Speed Increased
750 Unlimited	From 256 Kbps To 512 Kbps
Combo Unlimited 750	From 256 Kbps To 512 Kbps
Home Combo ULF1350	From 512Kbps To 2Mbps (Upto 15Gb and there after 512Kbps)

Enjoy Surfing!!

Continue Reading >>BSNL Broadband Announces Increase of Speeds for Various Plans

NTLDR is Missing is one of the annoying errors on Windows XP, the occurrence of which restricts users from logging into their systems instead displays the following error message on system boot up and often puts the users into an endless boot and restart sequence.

Test Disk NTLDR is missing Press Ctrl+Alt+Del to restart

NTLDR, abbreviated as NT-Loader is a boot loader file essential for loading NT-based operating system and resides in the root folder of a boot volume with standard file system of Windows - NTFS. To load NT-based OS successfully another boot file ntdetect.com must also be present in the root folder. If any of these essential system files are removed accidentally, then the stability of your system is compromised leading to errors on system boot up.

The other possible error logs encountered for 'NTLDR is Missing' error are as follow:

►►

Error #1516 Partition Improperly Dismounted. Please refer to the online user guide for instructions on how to resolve this is issue.

►►

Scandisk cannot check this drive now because the disk is not properly formatted, or a program such a disk utility has locked it. Format the disk or wait for the utility to finish and then restart your computer.

►►

System Error. Windows cannot read from drive C: If this is a network drive, make sure the network is working. If it is a local drive, check the disk.

►►

Drive C is either invalid, not writable, or does not have enough free space

Troubleshooting NTLDR Is Missing Error

For this process you need Windows XP bootable CD from which you'll have to copy ntldr and ntdetect.com boot files into your primary hard disk drive where your Windows OS is installed.

1. First, to enable boot from CD you need to change the boot priority options under system advanced setup. Depending on your computer manufacturer, press Delete or F5 to get to BIOS Setup Utility and then set the 1st Boot Device in Advanced Setup as CD/DVD. Save and Restart your computer having Windows XP bootable CD in your CD/DVD drive.


2. On system restart you'll see an option "Press any key to boot from CD...", just press any key on your keyboard to load essential files from bootable OS CD. And when the system boots, you are given the options to either restore your XP operating system or run a fresh installation copy. Press "R" to begin the restore process.


3. To continue with the process of Troubleshooting NTLDR is Missing error, refer
   How to Fix the NTLDR Is Missing Error @ eHow.com from step 2

Continue Reading >>Troubleshooting NTLDR is Missing Error on Windows Boot Up

In olden days, we used to prepare reports, documents and any other data manually. But now-a-days we have various applications to prepare those things like MS-Office, Adobe Pagemaker, WordPad, Latex etc. Among all these, Microsoft Office is the most widely used desktop application for document preparation and analysis. The most commonly used components of MS Office contain MS-Word, MS-Excel and MS-PowerPoint.

Microsoft Word is the premier application that allows you to create and edit text documents. In addition to text, MS-Word also allows you to insert pictures, tables, charts, drawing, and other features like cliparts etc., that will make your documents richer and more interactive.

The default WORD document includes the following layout tools:

• Title Bar shows the document name, for example "New Document" and the application software name "Microsoft Word"
• Menu Bar contains the list of menus available inside word document, and each menu contains a specific set of commands.
• Standard Toolbar contains a shortcut list of some of the most commonly used commands found inside the menu bar options.
• Formatting Toolbar contains a shortcut list of formatting options available inside the format menu
• Horizontal and Vertical Rulers are used for measurement purposes like any normal ruler, the default unit of measure is in inches.
• White Page Area is the space where you go type, edit and format your document.
• Status Bar shows the page number the user is working on, section numbers, present working page number out of the total pages found in the document, present line number, present column number etc.,

Continue Reading >>Layout Tools in the Default Word Document