How to Prepare for GATE 2023 -
First, get a clear idea of the GATE Syllabus and GATE Exam Pattern.
Give more mock tests.
Avoid reading any new topics before 60 days of the exam.
Be confident in your preparation strategy.
Purchase resourceful online practice test series.
Refer to standard writer's books and make notes.
Nothing is difficult if you are really concentrated and committed. With proper support, plan, education, and time management, candidates can clear the GATE exam.
First, get a clear idea of the GATE Syllabus and GATE Exam Pattern.
Give more mock tests.
Avoid reading any new topics before 60 days of the exam.
Be confident in your preparation strategy.
Purchase resourceful online practice test series.
Refer to standard writer's books and make notes.
Nothing is difficult if you are really concentrated and committed. With proper support, plan, education, and time management, candidates can clear the GATE exam.
🌀Notes on Special Theory of Relativity:-
Frame of reference:- A system of coordinates whose axes can be suitably chosen is said to be a frame of reference.
Inertial frame:- A frame of reference either at rest or moving with a uniform velocity (zero acceleration) is known as inertial frame.
Non-inertial or accelerated frame:- It is a frame of reference which is either having a uniform linear acceleration or is being rotated with a uniform speed.
Galilean transformation:-
(a) Transformation of position:-
x' = x – vt
y' = y
z' = z
t' = t
Inverse transformation,
x = x'+ vt '
y = y'
z = z'
t = t'
(b) Transformation of distance:-
l ' = l
Here, l = x2-x1 is the length of rod as observed in frame S.
Anything which remains unchanged when observed from the two Galilean frames of reference is known as Galilean invariant.
(c) Transformation of velocity:-
u'= u– v
Inverse transformation:- u = u'+ v
(d) Transformation of acceleration:-
a' = a
Thus, the acceleration of body, as observed by two observers siting in two inertial frames, is same. Hence, acceleration is said to be Galilean invariant.
Law of conservation of momentum:-
It states that the total momentum of an isolated system (no external force) always remains constant.
In S frame:- m1u1 + m2u2 = m1v1 + m2v2
In S ' frame:- m1u'1 + m2 u'2 = m1 v'1 + m2 v'2
Thus, the law of conservation of momentum is valid in S ' also, indicating that the law is Galilean invariant.
Ether and velocity of light:- c' = c ±v
Postulates of special theory of relativity:-
(a) The laws of physical phenomenon are same when stated in terms of two systems of reference in uniform translator motion relative to each other.
(b) The velocity of light in vacuum is constant, independent not only of the direction of propagation but also of the relative velocity of the source and the observer.
Lorentz transformation equations:-
x' = [x - vt]/√[1- v2/c2]
y'= y
z'= z
t' = [t – [(v/c 2) x]]/√[1- v2/c2]
Lorentz inverse transformation equations:-
x= [x' + vt] /√[1- v2/c2]
y = y'
z= z'
t= [t' + [(v/c 2) x']]/√[1- v2/c2]
Length contraction:-
l = l0√[1- v2/c2]
So, l<l0
Here, l0 is the proper or original length in S frame and l is the relativistic length in S' frame.
Time Dilation:-
??t = ??t0/√ [1-v2/c2]
So, ?t > ??t0
Here, ??t0 is the proper or original time in S frame and ?t is the relativistic time in S' frame.
?Thus, the time interval as observed by the moving observer appears to be lengthened.
Frequency:- If f 0 is the natural frequency of a process in frame S, then the frequency f as observed from S ' given by,
f = f0√[1- v2/c2]
Relativistic velocity addition theorem:-
ux' = [ux – v]/[1- [(v/c2)ux]]
uy' = [uy√[1- v2/c2]]/[1- [(v/c2)ux]]
uz' = [uz√[1- v2/c2]]/[1- [(v/c2)ux]]
Relativistic variation of mass:-
m= m0/√[1-v2/c2]
Rest mass energy:-
E0 = m0c2
Total energy (Mass-Energy Equivalence):-
E= mc2 = [m0/√[1-v2/c2]] c2
Kinetic energy:-
EK= E - E0 = (m - m0) c2
Here c is the speed of light, m0 is the rest mass and m is the relativistic mass.
Relativistic momentum:-
p= mv
= [m0/√[1-v2/c2]] v
Frame of reference:- A system of coordinates whose axes can be suitably chosen is said to be a frame of reference.
Inertial frame:- A frame of reference either at rest or moving with a uniform velocity (zero acceleration) is known as inertial frame.
Non-inertial or accelerated frame:- It is a frame of reference which is either having a uniform linear acceleration or is being rotated with a uniform speed.
Galilean transformation:-
(a) Transformation of position:-
x' = x – vt
y' = y
z' = z
t' = t
Inverse transformation,
x = x'+ vt '
y = y'
z = z'
t = t'
(b) Transformation of distance:-
l ' = l
Here, l = x2-x1 is the length of rod as observed in frame S.
Anything which remains unchanged when observed from the two Galilean frames of reference is known as Galilean invariant.
(c) Transformation of velocity:-
u'= u– v
Inverse transformation:- u = u'+ v
(d) Transformation of acceleration:-
a' = a
Thus, the acceleration of body, as observed by two observers siting in two inertial frames, is same. Hence, acceleration is said to be Galilean invariant.
Law of conservation of momentum:-
It states that the total momentum of an isolated system (no external force) always remains constant.
In S frame:- m1u1 + m2u2 = m1v1 + m2v2
In S ' frame:- m1u'1 + m2 u'2 = m1 v'1 + m2 v'2
Thus, the law of conservation of momentum is valid in S ' also, indicating that the law is Galilean invariant.
Ether and velocity of light:- c' = c ±v
Postulates of special theory of relativity:-
(a) The laws of physical phenomenon are same when stated in terms of two systems of reference in uniform translator motion relative to each other.
(b) The velocity of light in vacuum is constant, independent not only of the direction of propagation but also of the relative velocity of the source and the observer.
Lorentz transformation equations:-
x' = [x - vt]/√[1- v2/c2]
y'= y
z'= z
t' = [t – [(v/c 2) x]]/√[1- v2/c2]
Lorentz inverse transformation equations:-
x= [x' + vt] /√[1- v2/c2]
y = y'
z= z'
t= [t' + [(v/c 2) x']]/√[1- v2/c2]
Length contraction:-
l = l0√[1- v2/c2]
So, l<l0
Here, l0 is the proper or original length in S frame and l is the relativistic length in S' frame.
Time Dilation:-
??t = ??t0/√ [1-v2/c2]
So, ?t > ??t0
Here, ??t0 is the proper or original time in S frame and ?t is the relativistic time in S' frame.
?Thus, the time interval as observed by the moving observer appears to be lengthened.
Frequency:- If f 0 is the natural frequency of a process in frame S, then the frequency f as observed from S ' given by,
f = f0√[1- v2/c2]
Relativistic velocity addition theorem:-
ux' = [ux – v]/[1- [(v/c2)ux]]
uy' = [uy√[1- v2/c2]]/[1- [(v/c2)ux]]
uz' = [uz√[1- v2/c2]]/[1- [(v/c2)ux]]
Relativistic variation of mass:-
m= m0/√[1-v2/c2]
Rest mass energy:-
E0 = m0c2
Total energy (Mass-Energy Equivalence):-
E= mc2 = [m0/√[1-v2/c2]] c2
Kinetic energy:-
EK= E - E0 = (m - m0) c2
Here c is the speed of light, m0 is the rest mass and m is the relativistic mass.
Relativistic momentum:-
p= mv
= [m0/√[1-v2/c2]] v
🔰Notes on Trigonometric Equations and Identities🔰
A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).
In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).
Periods of various trigonometric functions are listed below:
1) sin x has period 2π
2) cos x has period 2π
3) tan x has period π
4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a
5) tan (ax+b) and cot (ax+b) have π/a as their period
6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a
7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period
Sum and Difference Formulae of Trigonometric Ratios
1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)
2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)
3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)
4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)
5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]
6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]
7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)
8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)
9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]
10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]
Double or Triple -Angle Identities
1) sin 2x = 2sin x cos x
2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1
3) tan 2x = 2 tan x / (1-tan 2x)
4) sin 3x = 3 sin x – 4 sin3x
5) cos3x = 4 cos3x – 3 cosx
6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)
For angles A, B and C, we have
1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC
2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC
3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C
4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]
List of some other trigonometric formulas:
1) 2sinAcosB = sin(A + B) + sin (A - B)
2) 2cosAsinB = sin(A + B) - sin (A - B)
3) 2cosAcosB = cos(A + B) + cos(A - B)
4) 2sinAsinB = cos(A - B) - cos (A + B)
5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]
6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]
7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]
8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]
9) tanA ± tanB = sin (A ± B)/ cos A cos B
10)cot A ± cot B = sin (B ± A)/ sin A sin B
Method of solving a trigonometric equation:
1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.
2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.
3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.
Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a
2) sin θ = 0 ⇒ θ = nπ
3) cosθ = 0 ⇒ θ = (2n + 1)π/2
4) tan θ = 0 ⇒ θ = nπ
5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]
6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]
7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]
8) sinθ = 1 ⇒ θ= (4n + 1)π/2
9) sin θ = -1 ⇒ θ = (4n - 1) π /2
10) sin θ = -1 ⇒ θ = (2n +1) π /2
11) |sinθ| = 1⇒ θ =2nπ
12) cosθ = 1 ⇒ θ =(2n + 1)
13) |cosθ| = 1⇒ θ =nπ
A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).
In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).
Periods of various trigonometric functions are listed below:
1) sin x has period 2π
2) cos x has period 2π
3) tan x has period π
4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a
5) tan (ax+b) and cot (ax+b) have π/a as their period
6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a
7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period
Sum and Difference Formulae of Trigonometric Ratios
1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)
2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)
3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)
4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)
5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]
6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]
7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)
8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)
9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]
10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]
Double or Triple -Angle Identities
1) sin 2x = 2sin x cos x
2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1
3) tan 2x = 2 tan x / (1-tan 2x)
4) sin 3x = 3 sin x – 4 sin3x
5) cos3x = 4 cos3x – 3 cosx
6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)
For angles A, B and C, we have
1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC
2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC
3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C
4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]
List of some other trigonometric formulas:
1) 2sinAcosB = sin(A + B) + sin (A - B)
2) 2cosAsinB = sin(A + B) - sin (A - B)
3) 2cosAcosB = cos(A + B) + cos(A - B)
4) 2sinAsinB = cos(A - B) - cos (A + B)
5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]
6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]
7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]
8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]
9) tanA ± tanB = sin (A ± B)/ cos A cos B
10)cot A ± cot B = sin (B ± A)/ sin A sin B
Method of solving a trigonometric equation:
1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.
2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.
3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.
Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a
2) sin θ = 0 ⇒ θ = nπ
3) cosθ = 0 ⇒ θ = (2n + 1)π/2
4) tan θ = 0 ⇒ θ = nπ
5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]
6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]
7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]
8) sinθ = 1 ⇒ θ= (4n + 1)π/2
9) sin θ = -1 ⇒ θ = (4n - 1) π /2
10) sin θ = -1 ⇒ θ = (2n +1) π /2
11) |sinθ| = 1⇒ θ =2nπ
12) cosθ = 1 ⇒ θ =(2n + 1)
13) |cosθ| = 1⇒ θ =nπ
📖Notes on Thermal Properties of Matter
Heat:- Heat is the agent which produces in us the sensation of warmth and makes bodies hot. It is form of energy. The part of thermal energy which flows from one body to the other due to temperature difference is called heat.
Nature of heat:-
(a) The weight of a body remains the same weather it is heated or cooled.
(b) Heat flows from higher to lower temperature
(c) In any exchange of heat, heat lost by the hot body is equal to the heat gained by the cold body.
(d) Substances generally expand when heated
(e) A certain amount of heat known as latent heat is required to change the state of a body from solid to liquid or from liquid to gas without any change in temperature.
Thermal Energy:- In accordance to dynamical theory of heat the sum total of translational, vibrational and rotational energies of the molecules of a system is called the thermal energy of the system .
Unit of Heat:-
(a) Calorie (cal):- It is the amount of heat required to raise the temperature of 1 gram of water through 1ºC.
(b) Kilocalorie (kcal):- It is the amount of heat required to raise the temperature of 1 kilo gram of water through 1ºC.
Temperature:- It is defined as the degree of hotness of a body.
Zeroth Law of Thermodynamics:-
It states that the two systems (A and B) which are separately in equilibrium with a third system (C) must also be in equilibrium with each other.
Absoluter Zero of Temperature:-
(a) Charle’s law:- Vt = V0(1+ t/273)
(b) Gay Lussac’s law:- Pt = P0(1+ t/273)
(c) Absolute zero of temperature is defined as the temperature at which a gas has zero volume and exerts zero pressure. It is that temperature at which molecular motion ceases.
(d) C∝√T, C = √[c12 + c22 +…….+ cn2]/n
Absolute gas scale or absolute scale of temperature:- It is that scale of temperature whose zero (i.e. 0ºK) = -273ºC
A centigrade degree is exactly equal to the absolute or Kelvin’s degree.
Conversion of temperature from one scale to another:-
C/100 = (K-273)/100 = (F-32)/180 = Re/80 = (Rα-492)/180
Here C, K, F, Re and Rα are respectively, the temperatures of same both on centigrade, Kelvin, Fahrenheit, Reaumer and Rankin scale, respectively.
F = [(9/5)C ]+32
K = C+273
Linear Expansion (longitudinal expansion):-
When the expansion due to heating takes place only along one direction, the expansion is said to be one dimensional and linear.
Coefficient of linear expansion (α):- Coefficient of linear expansion of the material of a rod is defined as the change in length per unit length, at 0ºC, per degree centigrade rise of temperature.
α = lt-l0/l0t
Heat:- Heat is the agent which produces in us the sensation of warmth and makes bodies hot. It is form of energy. The part of thermal energy which flows from one body to the other due to temperature difference is called heat.
Nature of heat:-
(a) The weight of a body remains the same weather it is heated or cooled.
(b) Heat flows from higher to lower temperature
(c) In any exchange of heat, heat lost by the hot body is equal to the heat gained by the cold body.
(d) Substances generally expand when heated
(e) A certain amount of heat known as latent heat is required to change the state of a body from solid to liquid or from liquid to gas without any change in temperature.
Thermal Energy:- In accordance to dynamical theory of heat the sum total of translational, vibrational and rotational energies of the molecules of a system is called the thermal energy of the system .
Unit of Heat:-
(a) Calorie (cal):- It is the amount of heat required to raise the temperature of 1 gram of water through 1ºC.
(b) Kilocalorie (kcal):- It is the amount of heat required to raise the temperature of 1 kilo gram of water through 1ºC.
Temperature:- It is defined as the degree of hotness of a body.
Zeroth Law of Thermodynamics:-
It states that the two systems (A and B) which are separately in equilibrium with a third system (C) must also be in equilibrium with each other.
Absoluter Zero of Temperature:-
(a) Charle’s law:- Vt = V0(1+ t/273)
(b) Gay Lussac’s law:- Pt = P0(1+ t/273)
(c) Absolute zero of temperature is defined as the temperature at which a gas has zero volume and exerts zero pressure. It is that temperature at which molecular motion ceases.
(d) C∝√T, C = √[c12 + c22 +…….+ cn2]/n
Absolute gas scale or absolute scale of temperature:- It is that scale of temperature whose zero (i.e. 0ºK) = -273ºC
A centigrade degree is exactly equal to the absolute or Kelvin’s degree.
Conversion of temperature from one scale to another:-
C/100 = (K-273)/100 = (F-32)/180 = Re/80 = (Rα-492)/180
Here C, K, F, Re and Rα are respectively, the temperatures of same both on centigrade, Kelvin, Fahrenheit, Reaumer and Rankin scale, respectively.
F = [(9/5)C ]+32
K = C+273
Linear Expansion (longitudinal expansion):-
When the expansion due to heating takes place only along one direction, the expansion is said to be one dimensional and linear.
Coefficient of linear expansion (α):- Coefficient of linear expansion of the material of a rod is defined as the change in length per unit length, at 0ºC, per degree centigrade rise of temperature.
α = lt-l0/l0t
📖Notes on Thermal Properties of Matter
Expansion in two dimensions (Superficial expansion):-
When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.
Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.
β = St-S0/S0t
Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.
Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .
γ = Vt-V0/V0t
Relation between expansion coefficients:-
(a) Relation between α and β: β = 2α
(b) Relation between α and γ: γ= 3α
(c) Relation between β and γ: γ = 3/2 β
(d) α : β : γ = 1:2:3
Thermal expansion of liquids:-
(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.
γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)
(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.
γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)
Work and Heat:-
Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.
W∝H or W = JH
Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.
J = W/H
Value of J:- J = 4.2×107 erg cal-1 = 4.2 J cal-1
Specific heat capacity or specific heat (c):-
Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.
c = Q/mΔT
Unit:- kcal kg-1K-1 or J kg-1K-1
Dimension:- M0L2T-2K-1
Molar specific heat capacity(C):-
Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.
(a) C = Mc (Here M is the molecular weight of the substance)
(b) C = 1/n (dQ/dT)
Heat Capacity or Thermal Capacity:-
It is defined as the amount of heat required to raise the temperature of body through 1ºC.
Q = mcΔT
If ΔT = 1ºC, Q = heat capacity = mc
Unit:- kcal K-1 or JK-1
Water Equivalent:-
Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.
w = mc
Water equivalent of a body is equal to the product of its mass and its specific heat.
Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.
(a)Specific latent heat of fusion (Lf):-
Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.
(b) Specific latent heat of vaporization (Lv):-
Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.
Dimensional formula:- M0L2T-2
Unit:- kg cal kg-1 or J kg-1
Triple point of water = 273.16 K
Absolute zero = 0 K = -273.15ºC
For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)
For a resistance thermometer, Re = R0[1+αθ]
Expansion in two dimensions (Superficial expansion):-
When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.
Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.
β = St-S0/S0t
Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.
Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .
γ = Vt-V0/V0t
Relation between expansion coefficients:-
(a) Relation between α and β: β = 2α
(b) Relation between α and γ: γ= 3α
(c) Relation between β and γ: γ = 3/2 β
(d) α : β : γ = 1:2:3
Thermal expansion of liquids:-
(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.
γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)
(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.
γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)
Work and Heat:-
Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.
W∝H or W = JH
Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.
J = W/H
Value of J:- J = 4.2×107 erg cal-1 = 4.2 J cal-1
Specific heat capacity or specific heat (c):-
Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.
c = Q/mΔT
Unit:- kcal kg-1K-1 or J kg-1K-1
Dimension:- M0L2T-2K-1
Molar specific heat capacity(C):-
Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.
(a) C = Mc (Here M is the molecular weight of the substance)
(b) C = 1/n (dQ/dT)
Heat Capacity or Thermal Capacity:-
It is defined as the amount of heat required to raise the temperature of body through 1ºC.
Q = mcΔT
If ΔT = 1ºC, Q = heat capacity = mc
Unit:- kcal K-1 or JK-1
Water Equivalent:-
Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.
w = mc
Water equivalent of a body is equal to the product of its mass and its specific heat.
Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.
(a)Specific latent heat of fusion (Lf):-
Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.
(b) Specific latent heat of vaporization (Lv):-
Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.
Dimensional formula:- M0L2T-2
Unit:- kg cal kg-1 or J kg-1
Triple point of water = 273.16 K
Absolute zero = 0 K = -273.15ºC
For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)
For a resistance thermometer, Re = R0[1+αθ]
🔰Notes on Wave Optics
Huygens Principle:- Wave-front of a wave, at any instant , is defined as the locus of all the particles in the medium which are being disturbed at the same instant of time and are in the same phase of vibration.
(a) Each point on a wave front acts as a source of new disturbance and emits its own set of spherical waves called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light so long as they move in the same medium.
(b) The envelope or the locus of these wavelets in the forward direction gives the position of new wave front at any subsequent time.
Determination of Phase Difference:-
The phase difference between two waves at a point will depend upon
(a) The difference in path lengths of the two waves from their respective sources.
(b) The refractive index of the medium
(c) Initial phase difference between the source if any.
(d) Reflections, if any, in the path followed by waves.
Reflection of plane wave at plane surface (Laws of reflection):-
a) The incident ray, the reflected ray and normal to the reflecting surface at the point of incidence, all lie in one plane and that plane is perpendicular to the reflecting surface.
b) The angle of incidence is equal to the angle of reflection.
So, ∠i = ∠r
This signifies angle of incidence is equal to the angle of reflection.
Refraction of light:-
Refraction is the phenomena by virtue of which a wave going from one medium to another undergoes a change in velocity.
(a) The sine of the angle between the incident ray and the normal bears a constant ratio to the sine of the angle between refracted ray and the normal.
sin i/sin r = v1/v2 = 1µ2 = constant
Here, v1 and v2 are the velocities of sound in first and second medium respectively.1µ2 is the refractive index of the second medium with respect to first.
(b) The incident ray, the refracted ray and the normal to the refracting surface lie in the same plane.
Interference:- The modification in the distribution of light energy obtained by the superposition of two or more waves is called interference.
Principle of superposition:- It states that a number of waves travelling, simultaneously, in a medium behave independent of each other and the net displacement of the particle, at any instant, is equal to the sum of the individual displacements due to all the waves.
Displacement equation:- y = R sin 2π/λ (vt+x/2)
Amplitude:- R = 2a cos πx/λ
Intensity:- I = K4a2 cos2 (πx/λ) [I = KR2]
Maxima:- A point having maximum intensity is called maxima.
x = 2n (λ/2)
A point will be a maxima if the two waves reaching there have a path difference of even multiple of λ/2.
Imax = 4Ka2 = 4i (Here, i = Ka2)
Minima:- A point having minimum intensity is called a minima.
x = (2n+1) (λ/2)
A point will be a minima if the two waves reaching there have a path difference of odd multiple of λ/2.
Imin = K. 4a2×0 = 0
Condition for constructive interference:-
Path difference = (2n)λ/2
Phase difference = (2n)π
Condition for destructive interference:-
Path difference = (2n+1)λ/2
Phase difference = (2n+1)π
Coherent Sources:- Coherent sources are the sources which either have no phase difference or have a constant difference of phase between them.
Conditions for interference:-
(a) The two sources should emit, continuously, waves of same wavelength or frequency.
(b) The amplitudes of the two waves should be either or nearly equal
(c) The two sources should be narrow.
(d) The sources should be close to each other.
(e) The two sources should be coherent one.
Young’s double slit experiment:-
Path difference, x = yd/D
Maxima, y = nλD/d
Here, n = 0,1,2,3….
Minima, y = (2n+1) λD/d
Here, n = 0,1,2,3….
Fringe Width:- It is the distance between two consecutive bright and dark fringes.
β = λD/d
Displacement of fringes due to the introduction of a thin transparent medium:-
(a) Shift for a particular order of fringes:-
?y = (β/λ) (µ-1)t
(b) Shift across a particular point of observation:-
µ = (mλ/t) +1
Lloyd’s single mirror:-
?λ = β .2a/D
Power of lens:- P = 100/f
Huygens Principle:- Wave-front of a wave, at any instant , is defined as the locus of all the particles in the medium which are being disturbed at the same instant of time and are in the same phase of vibration.
(a) Each point on a wave front acts as a source of new disturbance and emits its own set of spherical waves called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light so long as they move in the same medium.
(b) The envelope or the locus of these wavelets in the forward direction gives the position of new wave front at any subsequent time.
Determination of Phase Difference:-
The phase difference between two waves at a point will depend upon
(a) The difference in path lengths of the two waves from their respective sources.
(b) The refractive index of the medium
(c) Initial phase difference between the source if any.
(d) Reflections, if any, in the path followed by waves.
Reflection of plane wave at plane surface (Laws of reflection):-
a) The incident ray, the reflected ray and normal to the reflecting surface at the point of incidence, all lie in one plane and that plane is perpendicular to the reflecting surface.
b) The angle of incidence is equal to the angle of reflection.
So, ∠i = ∠r
This signifies angle of incidence is equal to the angle of reflection.
Refraction of light:-
Refraction is the phenomena by virtue of which a wave going from one medium to another undergoes a change in velocity.
(a) The sine of the angle between the incident ray and the normal bears a constant ratio to the sine of the angle between refracted ray and the normal.
sin i/sin r = v1/v2 = 1µ2 = constant
Here, v1 and v2 are the velocities of sound in first and second medium respectively.1µ2 is the refractive index of the second medium with respect to first.
(b) The incident ray, the refracted ray and the normal to the refracting surface lie in the same plane.
Interference:- The modification in the distribution of light energy obtained by the superposition of two or more waves is called interference.
Principle of superposition:- It states that a number of waves travelling, simultaneously, in a medium behave independent of each other and the net displacement of the particle, at any instant, is equal to the sum of the individual displacements due to all the waves.
Displacement equation:- y = R sin 2π/λ (vt+x/2)
Amplitude:- R = 2a cos πx/λ
Intensity:- I = K4a2 cos2 (πx/λ) [I = KR2]
Maxima:- A point having maximum intensity is called maxima.
x = 2n (λ/2)
A point will be a maxima if the two waves reaching there have a path difference of even multiple of λ/2.
Imax = 4Ka2 = 4i (Here, i = Ka2)
Minima:- A point having minimum intensity is called a minima.
x = (2n+1) (λ/2)
A point will be a minima if the two waves reaching there have a path difference of odd multiple of λ/2.
Imin = K. 4a2×0 = 0
Condition for constructive interference:-
Path difference = (2n)λ/2
Phase difference = (2n)π
Condition for destructive interference:-
Path difference = (2n+1)λ/2
Phase difference = (2n+1)π
Coherent Sources:- Coherent sources are the sources which either have no phase difference or have a constant difference of phase between them.
Conditions for interference:-
(a) The two sources should emit, continuously, waves of same wavelength or frequency.
(b) The amplitudes of the two waves should be either or nearly equal
(c) The two sources should be narrow.
(d) The sources should be close to each other.
(e) The two sources should be coherent one.
Young’s double slit experiment:-
Path difference, x = yd/D
Maxima, y = nλD/d
Here, n = 0,1,2,3….
Minima, y = (2n+1) λD/d
Here, n = 0,1,2,3….
Fringe Width:- It is the distance between two consecutive bright and dark fringes.
β = λD/d
Displacement of fringes due to the introduction of a thin transparent medium:-
(a) Shift for a particular order of fringes:-
?y = (β/λ) (µ-1)t
(b) Shift across a particular point of observation:-
µ = (mλ/t) +1
Lloyd’s single mirror:-
?λ = β .2a/D
Power of lens:- P = 100/f
📖Notes on Thermal Properties of Matter
Expansion in two dimensions (Superficial expansion):-
When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.
Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.
β = St-S0/S0t
Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.
Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .
γ = Vt-V0/V0t
Relation between expansion coefficients:-
(a) Relation between α and β: β = 2α
(b) Relation between α and γ: γ= 3α
(c) Relation between β and γ: γ = 3/2 β
(d) α : β : γ = 1:2:3
Thermal expansion of liquids:-
(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.
γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)
(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.
γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)
Work and Heat:-
Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.
W∝H or W = JH
Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.
J = W/H
Value of J:- J = 4.2×107 erg cal-1 = 4.2 J cal-1
Specific heat capacity or specific heat (c):-
Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.
c = Q/mΔT
Unit:- kcal kg-1K-1 or J kg-1K-1
Dimension:- M0L2T-2K-1
Molar specific heat capacity(C):-
Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.
(a) C = Mc (Here M is the molecular weight of the substance)
(b) C = 1/n (dQ/dT)
Heat Capacity or Thermal Capacity:-
It is defined as the amount of heat required to raise the temperature of body through 1ºC.
Q = mcΔT
If ΔT = 1ºC, Q = heat capacity = mc
Unit:- kcal K-1 or JK-1
Water Equivalent:-
Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.
w = mc
Water equivalent of a body is equal to the product of its mass and its specific heat.
Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.
(a)Specific latent heat of fusion (Lf):-
Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.
(b) Specific latent heat of vaporization (Lv):-
Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.
Dimensional formula:- M0L2T-2
Unit:- kg cal kg-1 or J kg-1
Triple point of water = 273.16 K
Absolute zero = 0 K = -273.15ºC
For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)
For a resistance thermometer, Re = R0[1+αθ]
Expansion in two dimensions (Superficial expansion):-
When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.
Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.
β = St-S0/S0t
Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.
Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .
γ = Vt-V0/V0t
Relation between expansion coefficients:-
(a) Relation between α and β: β = 2α
(b) Relation between α and γ: γ= 3α
(c) Relation between β and γ: γ = 3/2 β
(d) α : β : γ = 1:2:3
Thermal expansion of liquids:-
(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.
γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)
(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.
γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)
Work and Heat:-
Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.
W∝H or W = JH
Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.
J = W/H
Value of J:- J = 4.2×107 erg cal-1 = 4.2 J cal-1
Specific heat capacity or specific heat (c):-
Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.
c = Q/mΔT
Unit:- kcal kg-1K-1 or J kg-1K-1
Dimension:- M0L2T-2K-1
Molar specific heat capacity(C):-
Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.
(a) C = Mc (Here M is the molecular weight of the substance)
(b) C = 1/n (dQ/dT)
Heat Capacity or Thermal Capacity:-
It is defined as the amount of heat required to raise the temperature of body through 1ºC.
Q = mcΔT
If ΔT = 1ºC, Q = heat capacity = mc
Unit:- kcal K-1 or JK-1
Water Equivalent:-
Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.
w = mc
Water equivalent of a body is equal to the product of its mass and its specific heat.
Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.
(a)Specific latent heat of fusion (Lf):-
Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.
(b) Specific latent heat of vaporization (Lv):-
Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.
Dimensional formula:- M0L2T-2
Unit:- kg cal kg-1 or J kg-1
Triple point of water = 273.16 K
Absolute zero = 0 K = -273.15ºC
For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)
For a resistance thermometer, Re = R0[1+αθ]
🔰Notes on Wave Optics
Huygens Principle:- Wave-front of a wave, at any instant , is defined as the locus of all the particles in the medium which are being disturbed at the same instant of time and are in the same phase of vibration.
(a) Each point on a wave front acts as a source of new disturbance and emits its own set of spherical waves called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light so long as they move in the same medium.
(b) The envelope or the locus of these wavelets in the forward direction gives the position of new wave front at any subsequent time.
Determination of Phase Difference:-
The phase difference between two waves at a point will depend upon
(a) The difference in path lengths of the two waves from their respective sources.
(b) The refractive index of the medium
(c) Initial phase difference between the source if any.
(d) Reflections, if any, in the path followed by waves.
Reflection of plane wave at plane surface (Laws of reflection):-
a) The incident ray, the reflected ray and normal to the reflecting surface at the point of incidence, all lie in one plane and that plane is perpendicular to the reflecting surface.
b) The angle of incidence is equal to the angle of reflection.
So, ∠i = ∠r
This signifies angle of incidence is equal to the angle of reflection.
Refraction of light:-
Refraction is the phenomena by virtue of which a wave going from one medium to another undergoes a change in velocity.
(a) The sine of the angle between the incident ray and the normal bears a constant ratio to the sine of the angle between refracted ray and the normal.
sin i/sin r = v1/v2 = 1µ2 = constant
Here, v1 and v2 are the velocities of sound in first and second medium respectively.1µ2 is the refractive index of the second medium with respect to first.
(b) The incident ray, the refracted ray and the normal to the refracting surface lie in the same plane.
Interference:- The modification in the distribution of light energy obtained by the superposition of two or more waves is called interference.
Principle of superposition:- It states that a number of waves travelling, simultaneously, in a medium behave independent of each other and the net displacement of the particle, at any instant, is equal to the sum of the individual displacements due to all the waves.
Displacement equation:- y = R sin 2π/λ (vt+x/2)
Amplitude:- R = 2a cos πx/λ
Intensity:- I = K4a2 cos2 (πx/λ) [I = KR2]
Maxima:- A point having maximum intensity is called maxima.
x = 2n (λ/2)
A point will be a maxima if the two waves reaching there have a path difference of even multiple of λ/2.
Imax = 4Ka2 = 4i (Here, i = Ka2)
Minima:- A point having minimum intensity is called a minima.
x = (2n+1) (λ/2)
A point will be a minima if the two waves reaching there have a path difference of odd multiple of λ/2.
Imin = K. 4a2×0 = 0
Condition for constructive interference:-
Path difference = (2n)λ/2
Phase difference = (2n)π
Condition for destructive interference:-
Path difference = (2n+1)λ/2
Phase difference = (2n+1)π
Coherent Sources:- Coherent sources are the sources which either have no phase difference or have a constant difference of phase between them.
Conditions for interference:-
(a) The two sources should emit, continuously, waves of same wavelength or frequency.
(b) The amplitudes of the two waves should be either or nearly equal
(c) The two sources should be narrow.
(d) The sources should be close to each other.
(e) The two sources should be coherent one.
Young’s double slit experiment:-
Path difference, x = yd/D
Maxima, y = nλD/d
Here, n = 0,1,2,3….
Minima, y = (2n+1) λD/d
Here, n = 0,1,2,3….
Fringe Width:- It is the distance between two consecutive bright and dark fringes.
β = λD/d
Displacement of fringes due to the introduction of a thin transparent medium:-
(a) Shift for a particular order of fringes:-
?y = (β/λ) (µ-1)t
(b) Shift across a particular point of observation:-
µ = (mλ/t) +1
Lloyd’s single mirror:-
?λ = β .2a/D
Power of lens:- P = 100/f
Huygens Principle:- Wave-front of a wave, at any instant , is defined as the locus of all the particles in the medium which are being disturbed at the same instant of time and are in the same phase of vibration.
(a) Each point on a wave front acts as a source of new disturbance and emits its own set of spherical waves called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light so long as they move in the same medium.
(b) The envelope or the locus of these wavelets in the forward direction gives the position of new wave front at any subsequent time.
Determination of Phase Difference:-
The phase difference between two waves at a point will depend upon
(a) The difference in path lengths of the two waves from their respective sources.
(b) The refractive index of the medium
(c) Initial phase difference between the source if any.
(d) Reflections, if any, in the path followed by waves.
Reflection of plane wave at plane surface (Laws of reflection):-
a) The incident ray, the reflected ray and normal to the reflecting surface at the point of incidence, all lie in one plane and that plane is perpendicular to the reflecting surface.
b) The angle of incidence is equal to the angle of reflection.
So, ∠i = ∠r
This signifies angle of incidence is equal to the angle of reflection.
Refraction of light:-
Refraction is the phenomena by virtue of which a wave going from one medium to another undergoes a change in velocity.
(a) The sine of the angle between the incident ray and the normal bears a constant ratio to the sine of the angle between refracted ray and the normal.
sin i/sin r = v1/v2 = 1µ2 = constant
Here, v1 and v2 are the velocities of sound in first and second medium respectively.1µ2 is the refractive index of the second medium with respect to first.
(b) The incident ray, the refracted ray and the normal to the refracting surface lie in the same plane.
Interference:- The modification in the distribution of light energy obtained by the superposition of two or more waves is called interference.
Principle of superposition:- It states that a number of waves travelling, simultaneously, in a medium behave independent of each other and the net displacement of the particle, at any instant, is equal to the sum of the individual displacements due to all the waves.
Displacement equation:- y = R sin 2π/λ (vt+x/2)
Amplitude:- R = 2a cos πx/λ
Intensity:- I = K4a2 cos2 (πx/λ) [I = KR2]
Maxima:- A point having maximum intensity is called maxima.
x = 2n (λ/2)
A point will be a maxima if the two waves reaching there have a path difference of even multiple of λ/2.
Imax = 4Ka2 = 4i (Here, i = Ka2)
Minima:- A point having minimum intensity is called a minima.
x = (2n+1) (λ/2)
A point will be a minima if the two waves reaching there have a path difference of odd multiple of λ/2.
Imin = K. 4a2×0 = 0
Condition for constructive interference:-
Path difference = (2n)λ/2
Phase difference = (2n)π
Condition for destructive interference:-
Path difference = (2n+1)λ/2
Phase difference = (2n+1)π
Coherent Sources:- Coherent sources are the sources which either have no phase difference or have a constant difference of phase between them.
Conditions for interference:-
(a) The two sources should emit, continuously, waves of same wavelength or frequency.
(b) The amplitudes of the two waves should be either or nearly equal
(c) The two sources should be narrow.
(d) The sources should be close to each other.
(e) The two sources should be coherent one.
Young’s double slit experiment:-
Path difference, x = yd/D
Maxima, y = nλD/d
Here, n = 0,1,2,3….
Minima, y = (2n+1) λD/d
Here, n = 0,1,2,3….
Fringe Width:- It is the distance between two consecutive bright and dark fringes.
β = λD/d
Displacement of fringes due to the introduction of a thin transparent medium:-
(a) Shift for a particular order of fringes:-
?y = (β/λ) (µ-1)t
(b) Shift across a particular point of observation:-
µ = (mλ/t) +1
Lloyd’s single mirror:-
?λ = β .2a/D
Power of lens:- P = 100/f
♦️Revision Notes on Arithmetic Progression♦️
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d
The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d]
If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l)
If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d]
If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.
If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.
If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.
Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P.
If nth term of a series is tn = An + B, then the series is in A.P.
If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.
In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d
Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.
In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.
The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.
The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P.
Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d
Hence, this gives ‘d’ = (b-a)/(n+1)
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Algebra Revision Notes on Arithmetic Progression
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d
The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d]
If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l)
If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d]
If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.
If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.
If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.
Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P.
If nth term of a series is tn = An + B, then the series is in A.P.
If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.
In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d
Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.
In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.
The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.
The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P.
Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d
Hence, this gives ‘d’ = (b-a)/(n+1)
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Algebra Revision Notes on Arithmetic Progression
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💎Notes on Gravitation and Projectile
Gravitation:-
Kepler’s first law (law of elliptical orbit):- A planet moves round the sun in an elliptical orbit with sun situated at one of its foci.
Kepler’s second law (law of areal velocities):- A planet moves round the sun in such a way that its areal velocity is constant.
Kepler’s third law (law of time period):- A planet moves round the sun in such a way that the square of its period is proportional to the cube of semi major axis of its elliptical orbit.Keplers Law of Planetary Motion
T2 ∝ R3
Here R is the radius of orbit.
T2 = (4π2/GM)R 3
Newton’s law of gravitation:-
Every particle of matter in this universe attracts every other particle with a forcer which varies directly as the product of masses of two particles and inversely as the square of the distance between them.
F= GMm/r2
Here, G is universal gravitational constant. G = 6.67 ´10 -11 Nm2 / kg2
Dimensional formula of G: G = Fr2/Mm =[MLT-2][L2]/[M2] = [M-1L3T-2]
Acceleration due to gravity (g):- g = GM/R2
Variation of g with altitude:- g' = g(1- 2h/R), if h<<R. Here R is the radius of earth and h is the height of the body above the surface of earth.
Variation of g with depth:- g' = g(1- d/R). Here g' be the value of acceleration due to gravity at the depth d.
Variation with latitude:-
At poles:- θ = 90°, g' = g
At equator:- θ = 0°, g' = g (1-ω2R/g)
Here ω is the angular velocity.
As g = GMe/Re2 , therefore gpole > gequator
Gravitational Mass:- m = FR2/GM
Gravitational field intensity:-
E = F/m
= GM/r2
Weight:- W= mg
Gravitational intensity on the surface of earth (Es):-
Es = 4/3 (πRρG)
Here R is the radius of earth, ρ is the density of earth and G is the gravitational constant.
Gravitational potential energy (U):- U = -GMm/r
(a) Two particles: U = -Gm1m2/r
(b) hree particles: U = -Gm1m2/r12 – Gm1m3/r13 – Gm2m3/r23
Gravitational potential (V):- V(r) = -GM/r
At surface of earth,
Vs= -GM/R
Here R is the radius of earth.
Escape velocity (ve):-
It is defined as the least velocity with which a body must be projected vertically upward in order that it may just escape the gravitational pull of earth.
ve = √2GM/R
or, ve = √2gR = √gD
Here R is the radius of earth and D is the diameter of the earth.
Escape velocity (ve) in terms of earth’s density:- ve = R√8πGρ/3
Orbital velocity (v0):-
v0 = √GM/r
If a satellite of mass m revolves in a circular orbit around the earth of radius R and h be the height of the satellite above the surface of the earth, then,
r = R+h
So, v0 = √MG/R+h = R√g/R+h
In the case of satellite, orbiting very close to the surface of earth, then orbital velocity will be,
v0 = √gR
Relation between escape velocity ve and orbital velocity v0 :- v0= ve/√2 (if h<<R)
Time period of Satellite:- Time period of a satellite is the time taken by the satellite to complete one revolution around the earth.
T = 2π√(R+h)3/GM = (2π/R)√(R+h)3/g
If h<<R, T = 2π√R/g
Height of satellite:- h = [gR2T2/4π2]1/3 – R
Energy of satellite:-
Kinetic energy, K = ½ mv02 = ½ (GMm/r)
Potential energy, U = - GMm/r
Total energy, E = K+U
= ½ (GMm/r) + (- GMm/r)
= -½ (GMm/r)
Gravitational force in terms of potential energy:- F = – (dU/dR)
Acceleration on moon:-
gm = GMm/Rm2 = 1/6 gearth
Here Mm is the mass of moon and Rm is the radius of moon.
Projectile:-
Projectile fired at angle α with the horizontal:- If a particle having initial speed u is projected at an angle α (angle of projection) with x-axis, then,
Time of Ascent, t = (u sinα)/g
Total time of Flight, T = (2u sinα)/g
Horizontal Range, R = u2sin2α/g
Maximum Height, H = u2sin2α/2g
Equation of trajectory, y = xtanα-(gx2/2u2cos2α)
Instantaneous velocity, V=√(u2+g2t2-2ugt sinα)
and
β = tan-1(usinα-gt/ucosα)
Projectile fired horizontally from a certain height:-
Equation of trajectory: x2 = (2u2/g)y
Time of descent (timer taken by the projectile to come down to the surface of earth), T = √2h/g
Horizontal Range, H = u√2h/g. Here u is the initial velocity of the body in horizontal direction.
Gravitation:-
Kepler’s first law (law of elliptical orbit):- A planet moves round the sun in an elliptical orbit with sun situated at one of its foci.
Kepler’s second law (law of areal velocities):- A planet moves round the sun in such a way that its areal velocity is constant.
Kepler’s third law (law of time period):- A planet moves round the sun in such a way that the square of its period is proportional to the cube of semi major axis of its elliptical orbit.Keplers Law of Planetary Motion
T2 ∝ R3
Here R is the radius of orbit.
T2 = (4π2/GM)R 3
Newton’s law of gravitation:-
Every particle of matter in this universe attracts every other particle with a forcer which varies directly as the product of masses of two particles and inversely as the square of the distance between them.
F= GMm/r2
Here, G is universal gravitational constant. G = 6.67 ´10 -11 Nm2 / kg2
Dimensional formula of G: G = Fr2/Mm =[MLT-2][L2]/[M2] = [M-1L3T-2]
Acceleration due to gravity (g):- g = GM/R2
Variation of g with altitude:- g' = g(1- 2h/R), if h<<R. Here R is the radius of earth and h is the height of the body above the surface of earth.
Variation of g with depth:- g' = g(1- d/R). Here g' be the value of acceleration due to gravity at the depth d.
Variation with latitude:-
At poles:- θ = 90°, g' = g
At equator:- θ = 0°, g' = g (1-ω2R/g)
Here ω is the angular velocity.
As g = GMe/Re2 , therefore gpole > gequator
Gravitational Mass:- m = FR2/GM
Gravitational field intensity:-
E = F/m
= GM/r2
Weight:- W= mg
Gravitational intensity on the surface of earth (Es):-
Es = 4/3 (πRρG)
Here R is the radius of earth, ρ is the density of earth and G is the gravitational constant.
Gravitational potential energy (U):- U = -GMm/r
(a) Two particles: U = -Gm1m2/r
(b) hree particles: U = -Gm1m2/r12 – Gm1m3/r13 – Gm2m3/r23
Gravitational potential (V):- V(r) = -GM/r
At surface of earth,
Vs= -GM/R
Here R is the radius of earth.
Escape velocity (ve):-
It is defined as the least velocity with which a body must be projected vertically upward in order that it may just escape the gravitational pull of earth.
ve = √2GM/R
or, ve = √2gR = √gD
Here R is the radius of earth and D is the diameter of the earth.
Escape velocity (ve) in terms of earth’s density:- ve = R√8πGρ/3
Orbital velocity (v0):-
v0 = √GM/r
If a satellite of mass m revolves in a circular orbit around the earth of radius R and h be the height of the satellite above the surface of the earth, then,
r = R+h
So, v0 = √MG/R+h = R√g/R+h
In the case of satellite, orbiting very close to the surface of earth, then orbital velocity will be,
v0 = √gR
Relation between escape velocity ve and orbital velocity v0 :- v0= ve/√2 (if h<<R)
Time period of Satellite:- Time period of a satellite is the time taken by the satellite to complete one revolution around the earth.
T = 2π√(R+h)3/GM = (2π/R)√(R+h)3/g
If h<<R, T = 2π√R/g
Height of satellite:- h = [gR2T2/4π2]1/3 – R
Energy of satellite:-
Kinetic energy, K = ½ mv02 = ½ (GMm/r)
Potential energy, U = - GMm/r
Total energy, E = K+U
= ½ (GMm/r) + (- GMm/r)
= -½ (GMm/r)
Gravitational force in terms of potential energy:- F = – (dU/dR)
Acceleration on moon:-
gm = GMm/Rm2 = 1/6 gearth
Here Mm is the mass of moon and Rm is the radius of moon.
Projectile:-
Projectile fired at angle α with the horizontal:- If a particle having initial speed u is projected at an angle α (angle of projection) with x-axis, then,
Time of Ascent, t = (u sinα)/g
Total time of Flight, T = (2u sinα)/g
Horizontal Range, R = u2sin2α/g
Maximum Height, H = u2sin2α/2g
Equation of trajectory, y = xtanα-(gx2/2u2cos2α)
Instantaneous velocity, V=√(u2+g2t2-2ugt sinα)
and
β = tan-1(usinα-gt/ucosα)
Projectile fired horizontally from a certain height:-
Equation of trajectory: x2 = (2u2/g)y
Time of descent (timer taken by the projectile to come down to the surface of earth), T = √2h/g
Horizontal Range, H = u√2h/g. Here u is the initial velocity of the body in horizontal direction.
Notes on Chemical Bonding
Chemical bond:-
Chemical bond is the attractive force which holds various constituents together in a molecule.
There are three types of chemical bonds: Ionic Bond, Covalent Bond, Co-ordinate Bond.
Octet Rule:
Atoms form chemical bonds in order to complete their octet i.e. eight electrons in their valence shell.
Lewis Structures:
Pair of bonded electrons is by means of a ‘dash’ (-) usually called a ‘bond’.
Lone pairs or ‘non-bonded’ electrons are represented by ‘dots’.
Electrons present in the last shell of atoms are called valence electrons.
Exceptions to the Octet Rule:
Species with odd number of electrons: NO, NO2,
Incomplete octet for the central atom: LiCl, BeH2 and BCl3
Expanded octet for the central atom: PF5, SF6 and H2SO4
Formal Charge:
Formal charge is the difference between the number of valence electrons in an isolated atom and number of electrons assigned to that atoms in Lewis structure.
Formal charge = [Total number of valence electrons in the free atom ) - (Total number of lone pairs of electrons) -1/2(Total number of shared electrons i.e. bonding electrons)]
Resonance:
For molecules and ions showing resonance it is not possible to draw a single Lewis structure.
All the properties of such species can only be explained by two or more Lewis structures. Example: Resonance of O3
Ionic Bonding:
Formation of Ionic Bond:
Formation of ionic bond takes place between a metal and a non-metal by transfer of electron.
Formation of gaseous cations
A(g) + I.E. → A+ (g) + e
Ionization Energy
Formation of gaseous anions
X(g) + e → X- (g) + E.A
Electron Affinity
Packing of ions of opposite charges to form ionic solids
A+ (g) + X- (g) →AX (s) +Energy
Lattice energy
Conditions required of formation of ionic bonds:
Low I.E of cation.
High E.A of anion.
High lattice energy.
Covalent Bonding:
Covalent bond is formed between two non-metals by sharing of electrons.
Electron pairs which participate in bonding are called bond pairs.
Electron pairs which do not participate in bonding are called lone pairs.
There could be single, double or triple covalent bonds between two elements depending on the number of electrons being shared.
VSEPR (Valence Shell Electron Pair Repulsion) Theory:
The shape of the molecule is determined by repulsions between all of the electron pairs present in the valence shell.
Order of the repulsion: Lone pair↔️ Lone pair > Lone pair↔️ Bond pair > Bond pair↔️ Bond pair.
Repulsion among the bond pairs is directly proportional to the bond order and electronegativity difference between the central atom and the other atoms.
Chemical bond:-
Chemical bond is the attractive force which holds various constituents together in a molecule.
There are three types of chemical bonds: Ionic Bond, Covalent Bond, Co-ordinate Bond.
Octet Rule:
Atoms form chemical bonds in order to complete their octet i.e. eight electrons in their valence shell.
Lewis Structures:
Pair of bonded electrons is by means of a ‘dash’ (-) usually called a ‘bond’.
Lone pairs or ‘non-bonded’ electrons are represented by ‘dots’.
Electrons present in the last shell of atoms are called valence electrons.
Exceptions to the Octet Rule:
Species with odd number of electrons: NO, NO2,
Incomplete octet for the central atom: LiCl, BeH2 and BCl3
Expanded octet for the central atom: PF5, SF6 and H2SO4
Formal Charge:
Formal charge is the difference between the number of valence electrons in an isolated atom and number of electrons assigned to that atoms in Lewis structure.
Formal charge = [Total number of valence electrons in the free atom ) - (Total number of lone pairs of electrons) -1/2(Total number of shared electrons i.e. bonding electrons)]
Resonance:
For molecules and ions showing resonance it is not possible to draw a single Lewis structure.
All the properties of such species can only be explained by two or more Lewis structures. Example: Resonance of O3
Ionic Bonding:
Formation of Ionic Bond:
Formation of ionic bond takes place between a metal and a non-metal by transfer of electron.
Formation of gaseous cations
A(g) + I.E. → A+ (g) + e
Ionization Energy
Formation of gaseous anions
X(g) + e → X- (g) + E.A
Electron Affinity
Packing of ions of opposite charges to form ionic solids
A+ (g) + X- (g) →AX (s) +Energy
Lattice energy
Conditions required of formation of ionic bonds:
Low I.E of cation.
High E.A of anion.
High lattice energy.
Covalent Bonding:
Covalent bond is formed between two non-metals by sharing of electrons.
Electron pairs which participate in bonding are called bond pairs.
Electron pairs which do not participate in bonding are called lone pairs.
There could be single, double or triple covalent bonds between two elements depending on the number of electrons being shared.
VSEPR (Valence Shell Electron Pair Repulsion) Theory:
The shape of the molecule is determined by repulsions between all of the electron pairs present in the valence shell.
Order of the repulsion: Lone pair↔️ Lone pair > Lone pair↔️ Bond pair > Bond pair↔️ Bond pair.
Repulsion among the bond pairs is directly proportional to the bond order and electronegativity difference between the central atom and the other atoms.
Mathematics | Lagrange’s Mean Value Theorem
Difficulty Level : Easy
Last Updated : 16 Jul, 2021
Suppose
f:[a,b]\rightarrow R
be a function satisfying these conditions:
1) f(x) is continuous in the closed interval a ≤ x ≤ b
2) f(x) is differentiable in the open interval a < x < b
Then according to Lagrange’s Theorem, there exists at least one point ‘c’ in the open interval (a, b) such that:
f'(c)=\frac{f(b)-f(a)}{b-a}
In simple words, Lagrange’s theorem says that if there is a path between two points A(a, f(a)) and B(b, f(a)) in a 2-D plain then there will be at least one point ‘c’ on the path such that the slope of the tangent at point ‘c’, i.e., (f ‘ (c)) is equal to the average slope of the path, i.e.,
f'(c)=\frac{f(b)-f(a)}{b-a}
Example: Verify mean value theorem for f(x) = x2 in interval [2,4].
Solution: First check if the function is continuous in the given closed interval, the answer is Yes. Then check for differentiability in the open interval (2,4), Yes it is differentiable.
{f}'(x)=2x
f(2) = 4
and f(4) = 16
\frac{f(b)-f(a)}{b-a} = \frac{16-4}{4-2}=6
Mean value theorem states that there is a point c ∈ (2, 4) such that
{f}'(c)=6
But
{f}'(x)=2x
which implies c = 3. Thus at c = 3 ∈ (2, 4), we have
{f}'(c)= 6
Difficulty Level : Easy
Last Updated : 16 Jul, 2021
Suppose
f:[a,b]\rightarrow R
be a function satisfying these conditions:
1) f(x) is continuous in the closed interval a ≤ x ≤ b
2) f(x) is differentiable in the open interval a < x < b
Then according to Lagrange’s Theorem, there exists at least one point ‘c’ in the open interval (a, b) such that:
f'(c)=\frac{f(b)-f(a)}{b-a}
In simple words, Lagrange’s theorem says that if there is a path between two points A(a, f(a)) and B(b, f(a)) in a 2-D plain then there will be at least one point ‘c’ on the path such that the slope of the tangent at point ‘c’, i.e., (f ‘ (c)) is equal to the average slope of the path, i.e.,
f'(c)=\frac{f(b)-f(a)}{b-a}
Example: Verify mean value theorem for f(x) = x2 in interval [2,4].
Solution: First check if the function is continuous in the given closed interval, the answer is Yes. Then check for differentiability in the open interval (2,4), Yes it is differentiable.
{f}'(x)=2x
f(2) = 4
and f(4) = 16
\frac{f(b)-f(a)}{b-a} = \frac{16-4}{4-2}=6
Mean value theorem states that there is a point c ∈ (2, 4) such that
{f}'(c)=6
But
{f}'(x)=2x
which implies c = 3. Thus at c = 3 ∈ (2, 4), we have
{f}'(c)= 6
Mathematics | Introduction to Propositional Logic
What is Logic?
Logic is the basis of all mathematical reasoning, and of all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements
Which in Simple English means “There exists an integer that is not the sum of two squares”.
Importance of Mathematical Logic
The rules of logic give precise meaning to mathematical statements. These rules are used to distinguish between valid and invalid mathematical arguments.
Apart from its importance in understanding mathematical reasoning, logic has numerous applications in Computer Science, varying from design of digital circuits, to the construction of computer programs and verification of correctness of programs.
Propositional Logic
What is a proposition?
A proposition is the basic building block of logic. It is defined as a declarative sentence that is either true or false, but not both.
The Truth Value of a proposition is True (denoted as T) if it is a true statement, and False (denoted as F) if it is a false statement.
Some sentences that do not have a truth value or may have more than one truth value are not propositions.
To represent propositions, propositional variables are used. By Convention, these variables are represented by small alphabets such as .
The area of logic which deals with propositions is called propositional calculus or propositional logic.
It also includes producing new propositions using existing ones. Propositions constructed using one or more propositions are called compound propositions. The propositions are combined together using Logical Connectives or Logical Operators.
Truth Table
Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions which are joined together by Logical Connectives to form the given compound proposition. This compilation of all possible scenarios in a tabular format is called a truth table.
Most Common Logical Connectives-
1. Negation
2. Conjunction
3. Dis-junction
4. Exclusive Or
5. Implication
6. Bi-conditional or Double Implication
What is Logic?
Logic is the basis of all mathematical reasoning, and of all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements
Which in Simple English means “There exists an integer that is not the sum of two squares”.
Importance of Mathematical Logic
The rules of logic give precise meaning to mathematical statements. These rules are used to distinguish between valid and invalid mathematical arguments.
Apart from its importance in understanding mathematical reasoning, logic has numerous applications in Computer Science, varying from design of digital circuits, to the construction of computer programs and verification of correctness of programs.
Propositional Logic
What is a proposition?
A proposition is the basic building block of logic. It is defined as a declarative sentence that is either true or false, but not both.
The Truth Value of a proposition is True (denoted as T) if it is a true statement, and False (denoted as F) if it is a false statement.
Some sentences that do not have a truth value or may have more than one truth value are not propositions.
To represent propositions, propositional variables are used. By Convention, these variables are represented by small alphabets such as .
The area of logic which deals with propositions is called propositional calculus or propositional logic.
It also includes producing new propositions using existing ones. Propositions constructed using one or more propositions are called compound propositions. The propositions are combined together using Logical Connectives or Logical Operators.
Truth Table
Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions which are joined together by Logical Connectives to form the given compound proposition. This compilation of all possible scenarios in a tabular format is called a truth table.
Most Common Logical Connectives-
1. Negation
2. Conjunction
3. Dis-junction
4. Exclusive Or
5. Implication
6. Bi-conditional or Double Implication
Some theorems on Nested Quantifiers
Quantifiers are expressions that indicate the scope of the term to which they are attached, here predicates. A predicate is a property the subject of the statement can have.
For example, in the statement “the sum of x and y is greater than 5”, the predicate ‘Q’ is- sum is greater than 5,
and the statement can be represented as Q(x, y) where x and y are variables.
The scope of a quantifier or a quantification is the range in the formula that the quantifier engages in.
Types of quantification or scopes:
1. Universal(∀) – The predicate is true for all values of x in the domain.
2. Existential(∃) – The predicate is true for at least one x in the domain.
To know the scope of a quantifier in a formula, just make use of Parse trees. Two quantifiers are nested if one is within the scope of the other.
• Example-1:
∀x ∃y (x+y=5)
Here ‘∃’ (read as-there exists) and ‘∀’ (read as-for all) are quantifiers for variables x and y.
The statement can be represented as-
∀x Q(x)
Q(x) is ∃y P(x, y) Q(x)-the predicate is a function of only x because the quantifier applies only to variable x.
P(x, y) is (x + y = 5)
• Example-2
∀x ∀y ((x> 0)∧(y< 0) → (xy< 0))
(in English)
For every real number x and y, if x is positive and y is negative, implies xy is negative.
again,
∀x Q(x)
where Q(x) is ∀y P(x, y)
Example to convert a statement into a nested quantifiers formula:
“There is a pupil in this lecture who has taken at least one course in Discrete Maths.”
A statement consists of quantifiers and predicates, split it into it's two constituents.
Here x and y are the pupil and the course and their respective quantifiers are attached in front of them.
Write it down as-
For some x pupil, there exist a course in Discrete Maths such that x has taken y.
∃x ∃y P (x, y), where P (x, y) is "x has taken y".
• Theorem-1: The order of nested existential quantifiers can be changed without changing the meaning of the statement.
•
• Theorem-2: The order of nested universal quantifiers can be changed without changing the meaning of the statement.
•
• Example-3:
Assume P(x, y) is xy=8,
∃x ∃y P(x, y) domain: integers
Translates to-
There is an integer x for which there is an integer y such that xy = 8,
which is same as-
There is a pair of integers x, y for which xy = 8.
Meaning ∃x ∃y P(x, y) is equivalent to ∃y ∃x P(x, y).
Similarly,
Assume P(x, y) is (xy = yx).
∀x ∀y P(x, y) domain: real numbers
Translates to-
For all real numbers x, for all real numbers y, xy = yx or,
For every pair of real numbers x, y, xy = yx.
Again ∀x ∀y P(x, y) is equivalent to ∀y ∀x P(x, y).
However, when the nested quantifiers are not same, changing the order changes meaning of statement.
• Example-4:
Assume P(x, y, z) is (x + y = z).
∀x ∀y ∃z P(x, y, z) domain: real numbers
Translates to-
For all real numbers x and y there is a real number z such that x + y = z (True)
∀z ∃x ∃y P(x, y, z) domain: real numbers
There is a real number z such that for all real numbers x and y, x + y = z (False)
Negation of nested quantifiers:
• Theorem-3
To negate a sequence of nested quantifiers, you change each quantifier in the sequence to the other type and then negate the predicate.
So the negation of ∀x ∃y : P(x, y) is ∃x ∀y : ~P(x, y)
• Example-5:
“ ∃x at Cornell, x is at least 18 years old.”
To disagree with this, you’re negating the statement by flipping the ∃ to ∀ and then
negating the predicate:
“ ∀x at Cornell such that x is not at least 18 years old.”
Quantifiers are expressions that indicate the scope of the term to which they are attached, here predicates. A predicate is a property the subject of the statement can have.
For example, in the statement “the sum of x and y is greater than 5”, the predicate ‘Q’ is- sum is greater than 5,
and the statement can be represented as Q(x, y) where x and y are variables.
The scope of a quantifier or a quantification is the range in the formula that the quantifier engages in.
Types of quantification or scopes:
1. Universal(∀) – The predicate is true for all values of x in the domain.
2. Existential(∃) – The predicate is true for at least one x in the domain.
To know the scope of a quantifier in a formula, just make use of Parse trees. Two quantifiers are nested if one is within the scope of the other.
• Example-1:
∀x ∃y (x+y=5)
Here ‘∃’ (read as-there exists) and ‘∀’ (read as-for all) are quantifiers for variables x and y.
The statement can be represented as-
∀x Q(x)
Q(x) is ∃y P(x, y) Q(x)-the predicate is a function of only x because the quantifier applies only to variable x.
P(x, y) is (x + y = 5)
• Example-2
∀x ∀y ((x> 0)∧(y< 0) → (xy< 0))
(in English)
For every real number x and y, if x is positive and y is negative, implies xy is negative.
again,
∀x Q(x)
where Q(x) is ∀y P(x, y)
Example to convert a statement into a nested quantifiers formula:
“There is a pupil in this lecture who has taken at least one course in Discrete Maths.”
A statement consists of quantifiers and predicates, split it into it's two constituents.
Here x and y are the pupil and the course and their respective quantifiers are attached in front of them.
Write it down as-
For some x pupil, there exist a course in Discrete Maths such that x has taken y.
∃x ∃y P (x, y), where P (x, y) is "x has taken y".
• Theorem-1: The order of nested existential quantifiers can be changed without changing the meaning of the statement.
•
• Theorem-2: The order of nested universal quantifiers can be changed without changing the meaning of the statement.
•
• Example-3:
Assume P(x, y) is xy=8,
∃x ∃y P(x, y) domain: integers
Translates to-
There is an integer x for which there is an integer y such that xy = 8,
which is same as-
There is a pair of integers x, y for which xy = 8.
Meaning ∃x ∃y P(x, y) is equivalent to ∃y ∃x P(x, y).
Similarly,
Assume P(x, y) is (xy = yx).
∀x ∀y P(x, y) domain: real numbers
Translates to-
For all real numbers x, for all real numbers y, xy = yx or,
For every pair of real numbers x, y, xy = yx.
Again ∀x ∀y P(x, y) is equivalent to ∀y ∀x P(x, y).
However, when the nested quantifiers are not same, changing the order changes meaning of statement.
• Example-4:
Assume P(x, y, z) is (x + y = z).
∀x ∀y ∃z P(x, y, z) domain: real numbers
Translates to-
For all real numbers x and y there is a real number z such that x + y = z (True)
∀z ∃x ∃y P(x, y, z) domain: real numbers
There is a real number z such that for all real numbers x and y, x + y = z (False)
Negation of nested quantifiers:
• Theorem-3
To negate a sequence of nested quantifiers, you change each quantifier in the sequence to the other type and then negate the predicate.
So the negation of ∀x ∃y : P(x, y) is ∃x ∀y : ~P(x, y)
• Example-5:
“ ∃x at Cornell, x is at least 18 years old.”
To disagree with this, you’re negating the statement by flipping the ∃ to ∀ and then
negating the predicate:
“ ∀x at Cornell such that x is not at least 18 years old.”
Consensus Theorem in Digital Logic
Redundancy theorem is used as a Boolean algebra trick in Digital Electronics. It is also known as Consensus Theorem:
AB + A'C + BC = AB + A'C
The consensus or resolvent of the terms AB and A’C is BC. It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other.
The conjunctive dual of this equation is:
(A+B). (A'+C). (B+C) = (A+B). (A'+C)
In the second line, we omit the third product term BC. Here, the term BC is known as redundant term. In this way we use this theorem to simply the Boolean algebra. Conditions for applying Redundancy theorem are:
1. Three variables must present in the expression. Here A, B and C are used as variables.
2. Each variables is repeated twice.
3. One variable must present in complemented form.
After applying this theorem we can only take those terms which contains the complemented variable.
Proof – We can also prove it like this:
Y = AB + A'C + BC
Y = AB + A'C + BC.1
Y = AB + A'C + BC. (A + A')
Y = AB + A'C + ABC + A'BC
Y = AB (1 + C) + A’C (1 + B)
Y = AB + A'C
Example-1.
F = AB + BC' + AC
Here, we have three variables A, B and C and all are repeated twice. The variable C is present in complemented form. So, all the conditions are satisfied for applying this theorem.
After applying Redundancy theorem we can write only the terms containing complemented variables (i.e. C) and omit the Redundancy term i.e., AB.
.'. F = BC' + AC
Example-2.
F = (A + B). (A' + C). (B + C)
Three variables are present and all are repeated twice. The variable A is present in complemented form. Thus, all the three conditions of this theorem is satisfied.
After applying Redundancy theorem we can write only the terms containing complemented variables (i.e., A) and omit the Redundancy term i.e., (B + C).
.'. F = (A + B). (A' + C)
Consider the following equation:
Y = AB + A'C + BC
The third product term BC is a redundant consensus term. If A switches from 1 to 0 while B=1 and C=1, Y remains 1. During the transition of signal A in logic gates, both the first and second term may be 0 momentarily. The third term prevents a glitch since its value of 1 in this case is not affected by the transition of signal A.
Thus it is important to remove Logic Redundancy because it causes unnecessary network complexity and raises the cost of implementation.
Redundancy theorem is used as a Boolean algebra trick in Digital Electronics. It is also known as Consensus Theorem:
AB + A'C + BC = AB + A'C
The consensus or resolvent of the terms AB and A’C is BC. It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other.
The conjunctive dual of this equation is:
(A+B). (A'+C). (B+C) = (A+B). (A'+C)
In the second line, we omit the third product term BC. Here, the term BC is known as redundant term. In this way we use this theorem to simply the Boolean algebra. Conditions for applying Redundancy theorem are:
1. Three variables must present in the expression. Here A, B and C are used as variables.
2. Each variables is repeated twice.
3. One variable must present in complemented form.
After applying this theorem we can only take those terms which contains the complemented variable.
Proof – We can also prove it like this:
Y = AB + A'C + BC
Y = AB + A'C + BC.1
Y = AB + A'C + BC. (A + A')
Y = AB + A'C + ABC + A'BC
Y = AB (1 + C) + A’C (1 + B)
Y = AB + A'C
Example-1.
F = AB + BC' + AC
Here, we have three variables A, B and C and all are repeated twice. The variable C is present in complemented form. So, all the conditions are satisfied for applying this theorem.
After applying Redundancy theorem we can write only the terms containing complemented variables (i.e. C) and omit the Redundancy term i.e., AB.
.'. F = BC' + AC
Example-2.
F = (A + B). (A' + C). (B + C)
Three variables are present and all are repeated twice. The variable A is present in complemented form. Thus, all the three conditions of this theorem is satisfied.
After applying Redundancy theorem we can write only the terms containing complemented variables (i.e., A) and omit the Redundancy term i.e., (B + C).
.'. F = (A + B). (A' + C)
Consider the following equation:
Y = AB + A'C + BC
The third product term BC is a redundant consensus term. If A switches from 1 to 0 while B=1 and C=1, Y remains 1. During the transition of signal A in logic gates, both the first and second term may be 0 momentarily. The third term prevents a glitch since its value of 1 in this case is not affected by the transition of signal A.
Thus it is important to remove Logic Redundancy because it causes unnecessary network complexity and raises the cost of implementation.
Mathematics | Probability
Probability refers to the extent of occurrence of events. When an event occurs like throwing a ball, picking a card from deck, etc., then the must be some probability associated with that event.
In terms of mathematics, probability refers to the ratio of wanted outcomes to the total number of possible outcomes. There are three approaches to the theory of probability, namely:
1. Empirical Approach
2. Classical Approach
3. Axiomatic Approach
Basic Terminologies:
• Random Event :- If the repetition of an experiment occurs several times under similar conditions, if it does not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes, then such an experiment is called random event or a probabilistic event.
• Elementary Event – The elementary event refers to the outcome of each random event performed. Whenever the random event is performed, each associated outcome is known as elementary event.
• Sample Space – Sample Space refers to the set of all possible outcomes of a random event. Example, when a coin is tossed, the possible outcomes are head and tail.
• Event – An event refers to the subset of the sample space associated with a random event.
• Occurrence of an Event – An event associated with a random event is said to occur if any one of the elementary event belonging to it is an outcome.
• Sure Event – An event associated with a random event is said to be sure event if it always occurs whenever the random event is performed.
• Impossible Event – An event associated with a random event is said to be impossible event if it never occurs whenever the random event is performed.
• Compound Event – An event associated with a random event is said to be compound event if it is the disjoint union of two or more elementary events.
• Mutually Exclusive Events – Two or more events associated with a random event are said to be mutually exclusive events if any one of the event occurs, it prevents the occurrence of all other events. This means that no two or more events can occur simultaneously at the same time.
• Exhaustive Events – Two or more events associated with a random event are said to be exhaustive events if their union is the sample space.
Probability of an Event – If there are total p possible outcomes associated with a random experiment and q of them are favorable outcomes to the event A, then the probability of event A is denoted by P(A) and is given by
P (A) = q/p
The probability of non-occurrence of event A, i.e. P(A’) = 1 – P(A)
Note –
• If the value of P (A) = 1, then event A is called sure event.
• If the value of P (A) = 0, then event A is called impossible event.
• Also, P(A) + P(A’) = 1
Theorems:
• General – Let A, B, C are the events associated with a random experiment, then
1. P(A∪B) = P(A) + P(B) – P(A∩B)
2. P(A∪B) = P(A) + P(B) if A and B are mutually exclusive
3. P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C)- P(C∩A) + P(A∩B∩C)
4. P(A∩B’) = P(A) – P(A∩B)
5. P(A’∩B) = P(B) – P(A∩B)
• Extension of Multiplication Theorem – Let A1, A2, ….., An are n events associated with a random experiment, then
• P (A1∩A2∩A3 ….. An) = P (A1) P (A2/A1) P (A3/A2∩A1) ….. P (An/A1∩A2∩A3∩ ….. ∩An-1)
Probability refers to the extent of occurrence of events. When an event occurs like throwing a ball, picking a card from deck, etc., then the must be some probability associated with that event.
In terms of mathematics, probability refers to the ratio of wanted outcomes to the total number of possible outcomes. There are three approaches to the theory of probability, namely:
1. Empirical Approach
2. Classical Approach
3. Axiomatic Approach
Basic Terminologies:
• Random Event :- If the repetition of an experiment occurs several times under similar conditions, if it does not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes, then such an experiment is called random event or a probabilistic event.
• Elementary Event – The elementary event refers to the outcome of each random event performed. Whenever the random event is performed, each associated outcome is known as elementary event.
• Sample Space – Sample Space refers to the set of all possible outcomes of a random event. Example, when a coin is tossed, the possible outcomes are head and tail.
• Event – An event refers to the subset of the sample space associated with a random event.
• Occurrence of an Event – An event associated with a random event is said to occur if any one of the elementary event belonging to it is an outcome.
• Sure Event – An event associated with a random event is said to be sure event if it always occurs whenever the random event is performed.
• Impossible Event – An event associated with a random event is said to be impossible event if it never occurs whenever the random event is performed.
• Compound Event – An event associated with a random event is said to be compound event if it is the disjoint union of two or more elementary events.
• Mutually Exclusive Events – Two or more events associated with a random event are said to be mutually exclusive events if any one of the event occurs, it prevents the occurrence of all other events. This means that no two or more events can occur simultaneously at the same time.
• Exhaustive Events – Two or more events associated with a random event are said to be exhaustive events if their union is the sample space.
Probability of an Event – If there are total p possible outcomes associated with a random experiment and q of them are favorable outcomes to the event A, then the probability of event A is denoted by P(A) and is given by
P (A) = q/p
The probability of non-occurrence of event A, i.e. P(A’) = 1 – P(A)
Note –
• If the value of P (A) = 1, then event A is called sure event.
• If the value of P (A) = 0, then event A is called impossible event.
• Also, P(A) + P(A’) = 1
Theorems:
• General – Let A, B, C are the events associated with a random experiment, then
1. P(A∪B) = P(A) + P(B) – P(A∩B)
2. P(A∪B) = P(A) + P(B) if A and B are mutually exclusive
3. P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C)- P(C∩A) + P(A∩B∩C)
4. P(A∩B’) = P(A) – P(A∩B)
5. P(A’∩B) = P(B) – P(A∩B)
• Extension of Multiplication Theorem – Let A1, A2, ….., An are n events associated with a random experiment, then
• P (A1∩A2∩A3 ….. An) = P (A1) P (A2/A1) P (A3/A2∩A1) ….. P (An/A1∩A2∩A3∩ ….. ∩An-1)
Mathematics | Mean, Variance and Standard Deviation
Mean is average of a given set of data. Let us consider below example
2, 4, 4, 4, 5, 5, 7, 9
These eight data points have the mean (average) of 5:
(2 + 4 +4 + 4 +5 +5 +7 + 9)/8 = 5
Variance is the sum of squares of differences between all numbers and means.
Deviation for above example. First, calculate the deviations of each data point from the mean, and square the result of each:
Standard Deviation is square root of variance. It is a measure of the extent to which data varies from the mean.
Standard Deviation (for above data) = √4 = 2
Why mathematicians did chose a square and then square root to find deviation, why not simply take the difference of values?
One reason is the sum of differences becomes 0 according to the definition of mean. Sum of absolute differences could be an option, but with absolute differences, it was difficult to prove many nice theorems.
Coefficient of Variation = Standard Deviation/Mean * 100
* Value of standard deviation is 0 if all entries in input are same.
* If we add (or subtract) a number say 7 to all values in the input set, the mean is increased (or decreased) by 7, but the standard deviation doesn’t change.
* If we multiply all values in the input set by a number 7, both mean and the standard deviation is multiplied by 7. But if we multiply all input values with a negative number say -7, the mean is multiplied by -7, but the standard deviation is multiplied by 7.
* Standard deviation and variance is a measure that tells how spread out the numbers is. While variance gives you a rough idea of spread, the standard deviation is more concrete, giving you exact distances from the mean.
* Mean, median and mode are the measure of central tendency of data (either grouped or ungrouped).
Mean is average of a given set of data. Let us consider below example
2, 4, 4, 4, 5, 5, 7, 9
These eight data points have the mean (average) of 5:
(2 + 4 +4 + 4 +5 +5 +7 + 9)/8 = 5
Variance is the sum of squares of differences between all numbers and means.
Deviation for above example. First, calculate the deviations of each data point from the mean, and square the result of each:
Standard Deviation is square root of variance. It is a measure of the extent to which data varies from the mean.
Standard Deviation (for above data) = √4 = 2
Why mathematicians did chose a square and then square root to find deviation, why not simply take the difference of values?
One reason is the sum of differences becomes 0 according to the definition of mean. Sum of absolute differences could be an option, but with absolute differences, it was difficult to prove many nice theorems.
Coefficient of Variation = Standard Deviation/Mean * 100
* Value of standard deviation is 0 if all entries in input are same.
* If we add (or subtract) a number say 7 to all values in the input set, the mean is increased (or decreased) by 7, but the standard deviation doesn’t change.
* If we multiply all values in the input set by a number 7, both mean and the standard deviation is multiplied by 7. But if we multiply all input values with a negative number say -7, the mean is multiplied by -7, but the standard deviation is multiplied by 7.
* Standard deviation and variance is a measure that tells how spread out the numbers is. While variance gives you a rough idea of spread, the standard deviation is more concrete, giving you exact distances from the mean.
* Mean, median and mode are the measure of central tendency of data (either grouped or ungrouped).
Rolle’s Mean Value Theorem
Suppose f(x) be a function satisfying three conditions:
1) f (x) is continuous in the closed interval a ≤ x ≤ b
2) f (x) is differentiable in the open interval a < x < b
3) f (a) = f(b)
Then according to Rolle’s Theorem, there exists at least one point ‘c’ in the open interval (a, b) such that:
f ‘ (c) = 0
So, we can apply Rolle’s Theorem, according to which there exists at least one point ‘c’ such that:
f ‘ (c) = 0
Which means that there exists a point at which the slope of the tangent at that is equal to 0. We can easily see that at point ‘c’ slope is 0.
Similarly, there could be more than one points at which slope of tangent at those points will be 0.
Suppose f(x) be a function satisfying three conditions:
1) f (x) is continuous in the closed interval a ≤ x ≤ b
2) f (x) is differentiable in the open interval a < x < b
3) f (a) = f(b)
Then according to Rolle’s Theorem, there exists at least one point ‘c’ in the open interval (a, b) such that:
f ‘ (c) = 0
So, we can apply Rolle’s Theorem, according to which there exists at least one point ‘c’ such that:
f ‘ (c) = 0
Which means that there exists a point at which the slope of the tangent at that is equal to 0. We can easily see that at point ‘c’ slope is 0.
Similarly, there could be more than one points at which slope of tangent at those points will be 0.
Mathematics | PnC and Binomial Coefficients
Several Counting problems require finding the number of ways to arrange a certain number of distinct elements, where the relative order of these elements matter, other problems focus on finding the number of ways of selecting a particular number of elements from a set, where the order of the elements does not matter. Both types of problems are similar except for one crucial difference, that difference is order.
Permutation –
A permutation of a set of distinct objects is an ordered arrangement of these objects. A permutation is often also referred to as an arrangement. The relative order of the elements matters in an arrangement.
An ordered arrangement of elements of a set is called an r-permutation.
• Example 1 – How many permutations of the string “ABCDEFGH” have the string “ABC” as a substring?
• Solution – For “ABC” to be a substring, the letters A,B, and C must occur as a block. If we consider that block and the remaining 5 letters as objects, we have a total of 6 objects to arrange.
Therefore the number of strings having “ABC” as their substring = 6! = 720.
Combination –
A combination of a set of distinct objects is just a count of the number of ways a specific number of elements can be selected from a set of a certain size. The order of elements does not matter in a combination.
An unordered selection of elements from a set is called an r-combination.
Binomial Coefficients –
The -combinations from a set of n elements. This number is also called a binomial coefficient since it occurs as a coefficient in the expansion of powers of binomial expressions.
The binomial theorem gives a power of a binomial expression as a sum of terms involving binomial coefficients.
Several Counting problems require finding the number of ways to arrange a certain number of distinct elements, where the relative order of these elements matter, other problems focus on finding the number of ways of selecting a particular number of elements from a set, where the order of the elements does not matter. Both types of problems are similar except for one crucial difference, that difference is order.
Permutation –
A permutation of a set of distinct objects is an ordered arrangement of these objects. A permutation is often also referred to as an arrangement. The relative order of the elements matters in an arrangement.
An ordered arrangement of elements of a set is called an r-permutation.
• Example 1 – How many permutations of the string “ABCDEFGH” have the string “ABC” as a substring?
• Solution – For “ABC” to be a substring, the letters A,B, and C must occur as a block. If we consider that block and the remaining 5 letters as objects, we have a total of 6 objects to arrange.
Therefore the number of strings having “ABC” as their substring = 6! = 720.
Combination –
A combination of a set of distinct objects is just a count of the number of ways a specific number of elements can be selected from a set of a certain size. The order of elements does not matter in a combination.
An unordered selection of elements from a set is called an r-combination.
Binomial Coefficients –
The -combinations from a set of n elements. This number is also called a binomial coefficient since it occurs as a coefficient in the expansion of powers of binomial expressions.
The binomial theorem gives a power of a binomial expression as a sum of terms involving binomial coefficients.