jee- mains formulas
DESCRIPTION
PCM formulas useful for JEE-MainsTRANSCRIPT
MATHS FORMULAS
In an x-y Cartesian coordinate system,
equation of the circle is:
(x-a)2 + (y-b)2 = r2
sin 0° = 0
cos 0° = 1 tan 0° = 0
sin 30° = 1/2 cos 30° = (√3)/2
tan 30° = 1/(√3) sin 45° = 1/(√2)
cos 45° = 1/(√2) tan 45° = 1
sin 60° = (√3)/2 cos 60° = 1/2
tan 60° = √3 sin 90° = 1
cos 90° = 0 tan 90° = ∞
sin = 1/cosec cos = 1/sec
tan = 1/cot sin/cos = tan
sin2 + cos2 = 1 1 + tan2 = sec2
1 + cot2 = cosec2 sin( 90° - ) = cos
cos( 90° - ) = sin tan( 90° - ) = cot
sin( 90° + ) = cos cos( 90° + ) = - sin
tan( 90° + ) = - cot sin( 180° - ) = sin
cos( 180° - ) = - cos tan( 180° - ) = - tan
sin( 180° + ) = - sin cos( 180° + ) = - cos
tan( 180° + ) = tan Pythagorean Theorem:
The square of the length of the
hypotenuse of a right triangle is
equal to the sum of the squares of
the legs
a2 = b2 + c2
a,b - two sides of the triangle
connected by the right angle
c - hypotenuse of the triangle
circumference of a circle circumference of a circle = 2 . π . r
where,
π = PI = 22/7
r = radius of circle
Area of Triangle Area of Triangle = (1/2) . b . h
where,
h = height of triangle
b = the length of the base of triangle
Area of rectangle Area of rectangle = l . b
where,
l = length of rectangle
b = width of rectangle
Area of circle Area of circle = π . r 2
where,
π = PI = 22/7
r = radius of circle
Area of trapezoid Area of trapezoid = (1/2) . (height). (base
one + base two)
Area of Ellipse Area of Ellipse = π . r1 . r2
where,
r1 = major radius
r2 = minor radius
Area of Cylinder (surface area) Area of Cylinder (surface area) =
2 . π . r . h
where,
r = radius of cylinder
h = length of cylinder
Area of Cone (surface area) Area of Cone (surface area) =
π . r . l
where,
r = radius of cone
l = length of side of the cone
Volume of cylinder Volume of cylinder = π . r 2 . h
where,
π = PI = 22/7
r = radius of cylinder
h = length of cylinder
Volume of sphere Volume of sphere = (4/3) . π . r 3
where,
π = PI = 22/7
r = radius of sphere
Volume of Cone Volume of Cone = (1/3) . π . r 2 . h
where,
π = PI = 22/7
r = radius of Cone
h = height of cone
sin(-x) = -sin(x)
cosec(-x) = -cosec(x) cos(-x) = cos(x)
sec(-x) = sec(x) tan(-x) = -tan(x)
cot(-x) = -cot(x) sin( + ) = sin cos + cos
sin
sin( - ) = sin cos - cos sin cos( + ) = cos cos - sin sin
cos( - ) = cos cos + sin sin tan( + ) =
(tan + tan) (1 - tan tan)
tan( - ) = (tan - tan) (1 + tan tan)
sin2 = 2 sin cos
cos2 = cos2 - sin2 = 2cos2 - 1 = 1 -
2sin2 tan2 =
(2 tan) (1 - tan2)
sin3 = 3sin - 4sin3 cos3 = 4cos3 - 3cos
For any triangle ABC with side lengths
a,b,c
Law of Cosines c2 = a2 + b2 - 2 a b cos C
b2 = c2 + a2 - 2 c a cos B
a2 = b2 + c2 - 2 b c cos A
Law of Sines sinA/a = sinB/b = sinC/c
cos θ cos β = cos(θ - β) + cos(θ + β) 2
sin θ sin β = cos(θ - β) - cos(θ + β)
2
sin θ cos β = sin(θ + β) + sin(θ - β) 2
sin3 θ = 3sin θ - sin 3θ 4
cos3 θ = 3cos θ + cos 3θ 4
sin3 θ . cos3 θ = 3sin 2θ - sin 6θ 32
sin4 θ = 3 - 4 cos 2θ + cos 4θ 8
cos4 θ = 3 + 4 cos 2θ + cos 4θ 8
sin4 θ . cos4 θ = 3 - 4 cos 4θ + cos 8θ 128
sin5 θ = 10 sin θ - 5 sin 3θ + sin 5θ 16
cos5 θ = 10 cos θ + 5 cos 3θ + cos 5θ 16
sin5 θ . cos5 θ = 10 sin 2θ - 5 sin 6θ + sin 10θ 512
Quadratic Equation For the equation: a x 2 + b x + c = 0 the value of x will be
x = - b ± √ (b2 - 4 a c) 2 a
(a + b)2 = a2 + 2 a b + b2 (a - b)2 = a2 - 2 a b + b2
(a + b) . (a - b) = (a2 - b2) Arithmetic progression:
Arithmetic progression is a
sequence of numbers such that the
difference of any two successive
members of the sequence is a
constant.
For example: Suppose a1, a2, a3, a4,
...... , an-1, an are in sequence of
arithmetic progression
Then the first term of an arithmetic
series is a1 and assume that the
common difference of successive
members is d, then the nth term of
the sequence is:
an = a1 + (n - 1).d
The sum of all the components of
an arithmetic series is:
Sn = a1 + a2 + a3 + ....... + an-1 + an
i.e. Sn = (n).(a1 + an)/2
Geometric progression: geometric
progression OR geometric series is a
sequence of numbers such that the quotient
of any two successive members of the
sequence is a constant. The ratio of two
successive number is called common ratio.
The constant ratio must not be equal to 0.
Example of geometric series :
ar1, ar2, ar3, . ....... ., arn-1, arn
The nth term of the geometric series can be
defined as:
an = a r(n - 1)
r is called common ratio and n must be
greater than 0:
Logarithms: logb 1 = 0
logb b = 1
logb(X . Y) = logbX + logbY
logb(X / Y) = logbX - logbY
logb(Xn) = n . logbX
logmn . lognm = 1
dC = 0
dx
d(Cu) = C
du dx dx
d(u + v) =
du +
dv dx dx dx
d(u . v) = u
dv + v
du dx dx dx
d ( u
) = v (du/dx) - u (dv/dx)
dx v v2
d ( u n ) = n u n-1
du dx dx
d ( x n ) = n x n-1
dx
d ( C u ) = C u ln(C)
du dx dx
d ( e u ) = e u
du dx dx
d( ln(u)) =
1 du dx u dx
d( sin(u)) = cos(u)
du dx dx
d( cos(u)) = - sin(u)
du dx dx
d( tan(u)) = sec2(u)
du dx dx
d( cosec(u)) = - cosec(u) . cot(u)
du dx dx
d( sec(u)) = sec(u) . tan(u)
du dx dx
d( cot(u)) = - cosec2(u)
du dx dx
d(sin-1u) =
1 du
dx
dx
d(cos-1u) =
-1 du
dx
dx
d(tan-1u) =
1 du dx 1 + u2 dx
d(cot-1u) =
- 1 du dx 1 + u2 dx
d(sec-1u) =
1 du
dx |u| dx
d(cosec-1u) =
-1 du
dx |u| dx
Chemistry formulas
Ideal Gas law
PV = nRT
n = number of moles
R = universal gas constant = 8.3145
J/mol K
Combined Gas law
P1V1 =
P2V2
T1 T2
Boyle's law
P1V1 = P2V2
Charles law
V1 =
V2
T1 T2
Gay-Lussac law
P1 =
P2
T1 T2
Diffusion: Rate at which two gases mix
Graham's law of diffusion
The rate of diffusion of a gas is inversely
proportional to the square root of their density
or the molar mass of the gas.
Effusion: Rate at which a gas escapes
thru pin hole
Graham's law of effusion
The rate of effusion of a gas is inversely
proportional to the square root of either
the density or the molar mass of the gas.
Solution: Solution is a homogeneous mixture
of two or more substances.
Solute is a substance that is dissolved in the
solution.
Solvent is the substance that dissolves the
solute. Solvent is present in greater amount.
Concentration is the ratio of solute and
solvent.
Concentration can be measured using
molarity, molality and mole fraction.
Molarity (M) = moles of solute liters of solution
Molality (m) = moles of solute
kg of solution
Unit of Molarity (M) : mol/L : moles per litre
Unit of Molality (M) : mol/kg : moles per kg
Mole fraction: Mole fraction of a
component in solution is the number of
moles of that component divided by the
total number of moles of all
components in the solution. Mole-fraction (Xa)= molesa
Dilution: Siluting a solution means adding
more solvent in solution without the addition of
more solute.
MiVi = MfVf
molesa + molesb .... Mi: Molarity of solution before diluting.
Vi: Volume of solution before diluting.
Mf: Molarity of solution after diluting.
Vf: Volume of solution after diluting.
Mole: Mole is the amount of substance
that contains same number of particles
as there are atoms in Carbon-12. One
mole of substance is Avogadro's
number (i.e. 6.023 x 1023).
One mole of gas has volume of 22.4 liter at
STP.
Relation between moles and grams
1 mole = molecular weight of substance
in grams.
Ionization Enthalpy: It is the energy needed to
remove an electron from an atom or molecule
(i.e from low state to n=∞). It is always
endothermic (i.e. positive).
OR
Ionization energy: energy needed to remove an
electron from an atom
Henderson-Hasselbalch equation:
where
[A-]: Concentration of conjugate base
[HA]: concentration of the acid
OR
pH = pKa + log10 [A-]
[HA]
pH = pKa + log10 [Conjugate Base]
[Acid]
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PHYSICS FORMULAS
Density is mass per unit volume
Density = mass / volume
velocity = displacement /
time
Force = rate of change of momentum Momentum = mass .
velocity
Power is rate of work done
Power = work / time
Unit of power is watt
Potential energy (P) PE = m.g.h
m = mass
g = acceleration due to gravity
(9.81m/s2)
h = height
Kinetic energy (P) P = (1/2).m.v2
m = mass
v = velocity
Gravity (Force due to gravity) Fg : Force of attraction
G : Gravitational constant
M1 : Mass of first object
M2 : Mass of second object
Fg = G M1 M2
r2
Acceleration due to
gravity at a depth 'd' from
earth surface is :
gd = g(1- d
) R
Acceleration due to gravity at height 'h'
from earth surface is :
h is very much smaller than R
gh = g(1- 2h
) R
Escape velocity Escape velocity from a
body of mass M and
radius r is
For example if you want
to calculate the escape
verlocity of sa object from
earth then,
M is dmass of earth
r is radius of earth
OPTICS
Index of refraction n = c/v
n - index of refraction
c - velocity of light in a vacuum
v - velocity of light in the given
Under constant
acceleration linear
motion v = final velocity
u = intitial velocity
a = acceleration
t = time taken to reach
velocity v from u
material
s = displacement
v = u + a t
s = ut + (1/2)a t 2
s = vt - (1/2)a t 2
v2 = u2 + 2 a s
Friction force (kinetic friction) When the object is moving then
Friction is defined as :
Ff = μ Fn
where
Ff = Friction force, μ= cofficient of
friction
Fn = Normal force
Linear Momentum Momentum = mass x
velocity
Capillary action
The height to which the liquid can be
lifted is given by:
h = 2γcosθ
ρgr
γ: liquid-air surface
tension(T)(T=energy/area)
θ: contact angle
ρ: density of liquid
g: acceleration due to gravity
r: is radius of tube
Simple harmonic motion Simple harmonic motion
is defined by:
d2x/dt2 = - k x
Time period of pendulum
Waves
f = 1
T
ω = 2 π
T
v = f . λ
where
ω = Angular frequency,
T=Time period, v = Speed
of wave, λ=wavelength
Doppler effect Relationship between
observed frequency f and emitted Resonance of a string frequency = f = nv
frequency f0:
where,
v=velocity of wave
vs=velocity of source. It is positive if
source of wave is moving away from
observer. It is negative if source of
wave is moving towards observer.
f = f0( v
) v + vs
where,
L: length of the string
n = 1, 2, 3...
2L
Resonance of a open tube of
air(approximate)
where,
L: length of the cylinder
n = 1, 2, 3...
v = speed of sound
Approximate frequency = f = nv
2L
Resonance of a open
tube of air(accurate)
where,
L: length of the cylinder
n: 1, 2, 3...
v: speed of sound
d:diameter of the
resonance tube
frequency = f = nv
2(L+0.8D)
Resonance of a closed tube of
air(approximate)
where,
L: length of the cylinder
n = 1, 2, 3...
v = speed of sound
Approximate frequency = f = nv
4L
Resonance of a closed
tube of air(accurate)
where,
L: length of the cylinder
n: 1, 2, 3...
v: speed of sound
d:diameter of the
resonance tube
frequency = f = nv
4(L+0.8D)
intensity of sound
intensity of sound =
Sound Power
area
Bragg's law nλ = 2d sinθ
where
I=intensity of interest in Wm-2
I0=intensity of interest in 10-12Wm-2
intensity of sound in decibel= 10log10 I
I0
dB = 10log10 I
I0
where
n = integer (based upon
order)
λ = wavelength
d = distance between the
planes
θ = angle between the
surface and the ray
de Broglie equation
where
p = momentum
λ = wavelength
h = Planck's constant
v = velocity
λ = h =
h
p mv
Relation between energy
and frequency E = hν
where
E = Energy
h = Planck's constant
ν = frequency
Davisson and Germer experiment
where
e = charge of electron
m = mass of electron
V = potential difference between the
plates thru which the electron pass
λ = wavelength
λ = h
Centripetal Force (F)
F = m v2
= m ω2 r r
Circular motion formula
v = ω r
Centripetal acceleration (a) = v2
r
Torque (it measures how
the force acting on the
object can rotate the
object) Torque is cross product of
radius and Force
Torque = (Force) X
(Moment arm) X sin θ
T = F L sin θ
whete θ = angle between
force and moment arm
Forces of gravitation F = G (m1.m2)/r
2
where G is constant. G = 6.67E - 11 N
m2 / kg2
Stefan-Boltzmann Law The energy radiated by a
blackbody radiator per
second = P
P = AσT4
where,
σ = Stefan-Boltzmann
constant
σ = 5.6703 × 10-8
watt/m2K4
Efficiency of Carnot cycle
η = 1 - Tc
Th
Ideal gas law P V = n R T
P = Pressure (Pa i.e.
Pascal)
V = Volume (m3)
n = number of of gas (in
moles)
R = gas constant (
8.314472 .m3.Pa.K-1mol-1]
)
T = Temperatue ( in
Kelvin [K])
Boyles law (for ideal gas) P1 V1 = P2V2
T (temperature is constant)
Charles law (for ideal
gas) V1
= V2
T1 T2
P (pressure is constant)
Translational kinetic energy K per
gas molecule (average molecular
kinetic energy:)
K = 3 k T
2
Internal energy of
monoatomic gas
K = 3 n R T
2
k = 1.38066 x 10-23 J/K Boltzmanns
constant
n = number of of gas (in
moles)
R = gas constant (
8.314472 .m3.Pa.K-1mol-1]
)
Root mean square speed of gas
V2rms =
3 k T
m
k = 1.38066 x 10-23 J/K Boltzmanns
constant
m = mass of gas
Ratio of specific heat (γ)
γ = Cp
Cv
Cp = specific heat capacity
of the gas in a constant
pressure process
Cv = specific heat capacity
of the gas in a constant
volume process
Internal entergy of ideal gas
Internal entergy of ideal gas (U) = cv
nRT
In Adiabatic process no
heat is gained or lost by
the system.
Under adiabetic condition
PVγ = Constant
TVγ-1 = Constant
where γ is ratio of specific
heat.
γ = Cp
Cv
Boltzmann constant (k)
k = R
Na
R = gas constant
Na = Avogadro's number.
Speed of the sound in gas
R = gas constant(8.314
J/mol K)
T = the absolute
temperature
M = the molecular weight
of the gas (kg/mol)
γ = adiabatic constant =
cp/cv
Capillary action The height to which the liquid can be
lifted is given by
h=height of the liquid lifted
T=surface tension
r=radius of capillary tube
h= 2T
ρrg
Resistance of a wire
R = ρL
A
ρ = rsistivity
L = length of the wire
A = cross-sectional area of
the wire
Ohm's law V = I . R
V = voltage applied
R = Resistance
I = current
Electric power (P) = (voltage applied)
x (current) P = V . I = I2 . R
V = voltage applied
R = Resistance
I = current
Resistor combination
If resistors are in series
then equivalent
resistance will be Req = R1 + R2 + R3 + . . . .
. . + Rn
If resistors are in
parallel then equivalent
resistance will be 1/Req = 1/R1 + 1/R2 + 1/R3
+ . . . . . . + 1/Rn
In AC circuit average power is :
Pavg = VrmsIrms cosφ
where,
Pavg = Average Power
Vrms = rms value of voltage
Irms = rms value of current
In AC circuit
Instantaneous power is :
PInstantaneous = VmIm sinωt
sin(ωt-φ)
where,
PInstantaneous =
Instantaneous Power
Vm = Instantaneous
voltage
Im = Instantaneous current
Capacitors Q = C.V
where
Q = charge on the capacitor
C = capacitance of the capacitor
V = voltage applied to the capacitor
Total capacitance (Ceq)
for PARALLEL
Capacitor
Combinations: Ceq = C1 + C2 + C3 + . . . .
. . + Cn
Total capacitance (Ceq)
for SERIES Capacitor
Combinations: 1/Ceq = 1/C1 + 1/C2 + 1/C3
+ . . . . . . + 1/Cn
Parallel Plate Capacitor
C = κ ε0 A
d
where
C = [Farad (F)]
κ = dielectric constant
A = Area of plate
d = distance between the plate
ε0 = permittivity of free space (8.85 X
Cylindrical Capacitor
C = 2 π κ ε0 L
ln (b/a)
where
C = [Farad (F)]
κ = dielectric constant
L = length of cylinder [m]
a = outer radius of
conductor [m]
10-12 C2/N m2) b = inner radius of
conductor [m]
ε0 = permittivity of free
space (8.85 X 10-12 C2/N
m2)
Spherical Capacitor
C = 4 π κ ε0 a b
b - a
where
C = [Farad (F)]
κ = dielectric constant
a = outer radius of conductor [m]
b = inner radius of conductor [m]
ε0 = permittivity of free space (8.85 X
10-12 C2/N m2)
Magnetic force acting on
a charge q moving with
velocity v F = q v B sin θ
where
F = force acting on charge
q (Newton)
q = charge (C)
v = velocity (m/sec2)
B = magnetic field
θ = angle between V
(velocity) and B (magnetic
field)
Force on a wire in magnetic field (B)
F = B I l sin θ where
F = force acting on wire
(Newton)
I = Current (Ampere)
l = length of wire (m)
B = magnetic field
θ = angle between I (current) and B
(magnetic field)
In an RC circuit (Resistor-
Capacitor), the time
constant (in seconds) is:
τ = RC
R = Resistance in Ω
C = Capacitance in in
farads.
In an RL circuit (Resistor-inductor ),
the time constant (in seconds) is:
τ = L/R
R = Resistance in Ω
C = Inductance in henries
Self inductance of a
solenoid = L = μn2LA
n = number of turns per
unit length
L = length of the solenoid.
Mutual inductance of two solenoid two
long thin solenoids, one wound on top
of the other
M = μ0N1N2LA
N1 = total number of turns per unit
length for first solenoid
N2 = number of turns per unit length for
second solenoid
A = cross-sectional area
L = length of the solenoid.
Energy stored in capacitor
E = 1 C V 2
2
Coulomb's Law Like charges repel, unlike charges
attract.
F = k (q1 . q2)/r2
where k is constant. k = 1/(4 π ε0) ≈ 9 x
109 N.m2/C2
q1 = charge on one body
q2 = charge on the other body
r = distance between them
Calculator based upon Coulomb's Law
Ohm's law
V = IR
where
V = voltage
I = current
R = Resistence
Electric Field around a point charge
(q) E = k ( q/r2 )
where k is constant. k = 1/(4 π ε0) ≈ 9 x
109 N.m2/C2
q = point charge
r = distance from point charge (q)
Electric field due to thin
infinite sheet
E = σ
2 ε0
where
E = Electric field (N/C)
σ = charge per unit area
C/m2
ε0 = 8.85 X 10-12 C2/N m2
Electric field due to thick infinite
sheet
E = σ
ε0
where
E = Electric field (N/C)
σ = charge per unit area C/m2
ε0 = 8.85 X 10-12 C2/N m2
Magnetic Field around a
wire (B) when r is
greater than the radius
of the wire.
B = μ0 I
2 π r
where
I = current
r = distance from wire
and r ≥ Radius of the wire
Magnetic Field around a wire (B)
when r is less than the radius of the
wire.
B = μ0 I r
2 π R2
where
I = current
R = radius of wire
r = distance from wire
and r ≤ Radius of the wire (R)
Magnetic Field At the
center of an arc
B = μ0 I φ
4 π r
where
I = current
r = radius from the center
of the wire
Bohr's model
L = nh
2 π
where
L = angular momentum
n = principal quantum number =
1,2,3,...n
h = Planck's constant.
Emitting
Photons(Rydberg
Formula)
where
n1 < n2
E0 = 13.6 eV
Ephoton = E0( 1
- 1
) n1
2 n22
Half life of radioactive element
t1/2 = ln(2)
λ
Average life of
radioactive element
τ = 1
λ