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Amit Rai

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Logic Gates- Type of Logic Gate, Number System

 


A logic gate is a digital circuit which is based on certain logical relationship between the input and the output voltages of the circuit.

  1. The logic gates are built using the semiconductor diodes and transistors.
    Each logic gate is represented by its characteristic symbol.
  2. The operation of a logic gate is indicated in a table, known as truth table. This table contains all possible combinations of inputs and the corresponding outputs.

  3. A logic gate is also represented by a Boolean algebraic expression. Boolean algebra is a method of writing logical equations showing how an output depends upon the combination of inputs. Boolean algebra was invented by George Boole.



TYPES OF LOGIC GATES:

OR Gate :

The out of an OR gate will be one if at least value of one input  is 1.

Logic Symbol of OR Gate

  

The Boolean expression of OR gate is Y = A + B, read as Y equals A 'OR' B.

Truth table representation of Two input OR Gate.

 

A

B

Y

0

0

0

1

0

1

0

1

1

1

1

1

AND gate :

The output of AND will be 1 if the value of all input is 1.

Logical Symbol of AND Gate

The Boolean expression of AND gate is Y = A.B

 

Truth table for AND Gate

 

A

B

Y

1

1

1

1

0

0

0

1

0

0

0

1




Not Gate :

The output of a NOT gate attains the state 1 if and only if the input does not attain the state 1.

Logical Symbol of NOT gate.



Truth Table of Not Gate

 

A

Y

1

0

0

1

COMBINATION OF GATES :

The three basis gates (OR, AND and NOT) when connected in various combinations give us logic gates such as NAND, NOR gates, which are the universal building blocks of digital circuits .

 

NAND Gate :

Logical Symbol of NAND Gate:




Truth Table of NAND Gate:

A

B

Y

0

0

1

0

1

1

1

0

1

1

1

0

 

NOR Gate :

Logical Symbol of NOR Gate



Truth table of NOR Gate:

 

A

B

Y

0

0

1

0

1

0

1

0

0

1

1

0

 

XOR AND XNOR GATES

The Exclusive - OR gate (XOR gate)

The output of a two-input XOR gate attains the state 1 if one and only one input attains the state 1.

Logical Symbol of XOR Gate




Truth table of XOR Gate



A

B

Y

0

0

0

0

1

1

1

0

1

1

1

0




Exclusive - NOR gate (XNOR gate)

The output is in state 1 when its both inputs are the same that is, both 0 or both 1.

Logical Symbol :

Truth Table



A

B

Y

0

0

1

0

1

0

1

0

0

1

1

1




LAWS OF BOOLEAN ALGEBRA:

Basic OR, AND, and NOT operations are given below

 

OR

AND

NOT

A+0 = A

A.0 = 1

A + A` = 1

A+1 = 1

A.1 = A

A.A` = 0

A+A = A

A.A = A

A``.A = A

 

Boolean algebra obeys commutative, associative and distributive laws as given below :

Commutative laws :

         A+B=B+A;

       A.B = B.A

Associative laws :

A + (B + C) = (A + B) + C

A. (B . C) = (A. B) . C

Distributive laws :

A. (B + C) = A.B + A.C

Some other useful identities :

(i) A + AB = A
(iii) A.(A+ B)=A
(v) A+(B.C)=(A+B).(A+C)

De Morgan's theorem :

 

                                    

NUMBER SYSTEMS

 

Decimal Number system


The base of this system is 10 and in this system 10 numbers [0,1,2,3,4,5,6,7,8,9] are used.

Ex. 1396, 210.75 are decimal numbers.

Binary Number System

The base of this system is 2 and in this system 2 numbers (0 and 1) are used.

Ex. 1001, 1101.011 are Binary numbers.

Binary to decimal conversion







Example : 

  Convert binary number 1011.01 into decimal number.

Solution : 

Example : 

Convert binary number 1000101.101 into decimal number.

Solution : 

 

 

 

 

 

 

 

 

 

GEOMETRIC PROGRESSION - Concept, Properties with example.

GEOMETRIC PROGRESSION (G.P.)

G.P. is a sequence of non zero numbers each of the succeeding term is equal to the preceeding term multiplied by a constant. Thus in a GP the ratio of successive terms is constant. This constant factor is called the COMMON RATIO of the sequence & is obtained by dividing any term by the immediately previous term

is a GP with 'a' as the first term & 'r' as common ratio.

1. 

2. Sum of first n terms ;

   

3.Sum of infinite G.P ;

 

PROPERTIES OF GP :

 

  1. If each term of a G.P. be multiplied or divided by the some non-zero quantity, then the resulting sequence is also a G.P.

  2. Three consecutive terms of a GP : a/r, a, ar ;

  3. If a, b, c are in G.P. then b*b = ac.

  4. If each term of a G.P. be raised to the same power, then resulting sequence is also a G.P.

If the terms of a given G.P. are chosen at regular intervals, then the new sequence is also a G.P.

 
 
 
 
Example : 
 
 
 
Solution : 
 
 
 
 
Example :
 
 
 
 
Solution :
 
 
Example : 
 

A number consists of three digits which are in G.P. the sum of the right hand and left hand digits exceeds twice the middle digit by 1 and the sum of the left hand and middle digits is two third of the sum of the middle and right hand digits. Find the numbers.

 

Solution : 

 
 
 
 

Sequence and Series - Definition of Arithmetic Progression with example.

DEFINITION

Sequence :

A succession of terms a1, a2, a3, a4........ formed according to some rule or law.

Examples are :

1, 4, 9, 16, 25,.....

–1, 1, –1, 1,........

x, 2x, 3x, 4x,......

1!, 2!, 3!, 4!,......

A finite sequence has a finite (i.e. limited) number of terms, as in the first example above. An infinite sequence has an unlimited number of terms, i.e. there is no last term, as in the second and third.

Series

The indicated sum of the terms of a sequence. In the case of a finite sequence 

the 

the corresponding series is 

This series has a finite or limited number of terms and is called a finite series.

2.ARITHMETIC PROGRESSION (A.P.)

A.P. is a sequence whose terms differ by a fixed number. This fixed number is called the common difference. If a is the first term & d the common difference, then A.P. can be written as

a, a + d, a + 2d, .............., a + (n – 1) d , ..........

where l is the last term.

Points to remember for Airthmetic Progression 

  1. nth term of an A.P. is of the form An + B i.e. a linear expression in 'n', in such a case the coefficient of n is the common difference of the A.P. i.e. A.

  2. Sum of first 'n' terms of an A.P. is of the form  i.e. a quadratic expression in 'n', in such case the common difference is twice the coefficient of . i.e. 2A

  3. Also nth term 
     

Example : 

If(x+1),3xand(4x+2)are first three terms of an A.P. then its 5th term is-

Solution

(x+1), 3x and (4x+2) are in AP then 

3x -(x+1)=(4x+2)-3x  i.e x=3

a =4 and d = 9-4 =5 

Tn = 4+ 4*5 = 24

Example :

The sum of first four terms of an A.P. is 56 and the sum of it's last four terms is 112. If its first term is 11 then find the number of terms in the A.P.

 

Solution : 

a + a + d + a + 2d + a + 3d = 56
4a + 6d = 56
44 + 6d = 56 (as a = 11)
6d = 12 hence d = 2
Now sum of last four terms.
a + (n – 1)d + a + (n – 2)d + a + (n – 3)d + a + (n – 4)d = 112

=>  4a + (4n – 10)d = 112  44 + (4n – 10)2 = 112

=>  4n – 10 = 34

=>  n = 11

 

Example :

The sum of first n terms of two A.Ps. are in ratio . Find the ratio of their 11th terms.

Solution :

 

 

What is Electric Charge? Explain properties of Charge with example?

ELECTRIC CHARGE

 

Charge is the property associated with matter due to which it produces and experiences electrical and magnetic effects. The excess or deficiency of electrons in a body gives the concept of charge.

 

Types of charge

(i) Positive charge : It is the deficiency of electrons as compared to proton.

(i) Negative charge : It is the excess of electrons as compared to proton.

SI unit of charge : ampere second i.e. Coulomb Dimension : [A T]

Practical units of charge are ampere hour (=3600 C) and faraday (= 96500 C)

  • Millikan calculated quanta of charge by 'Highest common factor' (H.C.F.) method and it is equal to charge of electron

     1 C = 3 *10^9 stat coulomb, 1 absolute - coulomb = 10 C, 1 Faraday = 96500 C

SPECIFIC PROPERTIES OF CHARGE

  • Charge is a scalar quantity : It represents excess or deficiency of electrons.

  • Charge is transferable : If a charged body is put in contact with an another body, then charge can be transferred to another body.

  • Charge is always associated with mass
    Charge cannot exist without mass though mass can exist without charge.

    • So the presence of charge itself is a convincing proof of existence of mass.

    • In charging, the mass of a body changes.

    • When body is given positive charge, its mass decreases.

    • When body is given negative charge, its mass increases.

  • Charge is quantised

    The quantization of electric charge is the property by virtue of which all free charges are integral multiple of a basic unit of charge represented by e. Thus charge q of a body is always given by

    q = ne n = positive integer or negative integer The quantum of charge is the charge that an electron or proton carries.
    Note : Charge on a proton = (–) charge on an electron = 1.6 10–19 C

  • Charge is conserved

    In an isolated system, total charge does not change with time, though individual charge may change i.e. charge can neither be created nor destroyed. Conservation of charge is also found to hold good in all types of reactions either chemical (atomic) or nuclear. No exceptions to the rule have ever been found.

  • Charge is invariant
    Charge is independent of frame of reference. i.e. charge on a body does not change whatever be its speed.

Example

When a piece of polythene is rubbed with wool, a charge of  is developed on polythene. What is the amount of mass, which is transferred to polythene.

Solution

From Q = ne, so the number of electron transffered 

Now mass of transffred electorn = n * mass of one electron

Example

 – particles (Nuclei of helium) per second falls on a neutral sphere, calculate time in which sphere gets charged by 

Solution 

Number of alpha – particles falling in t second = 

Charge on alpha – particle = +2e , So charge incident in time t = 

given charge is  : 

t=\frac {{10}^{-18}} {1.6*{10}^{-19}}=6.25s"/>

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