Physics
(www.tiwariacademy.com)
(Mathematical Tools)
Physical constants
(Class 11)
 Mass of an electron (Me)
= 9.1x10-31kg.
 Mass of a proton ( Mp)
= 1.6725 x 10-27kg.
 Mass of a neutron (Mn)
= 1.6746 x 10-27kg.
 Charge of an electron (e)
= -1.6 x 10-19c
 Speed of light in vacuum (c)
= 3 x108 m/sec.
 Planck Constant (h)
= 6.6256 x 10-34 J x sec.
 Universal Gravitation constant (G)
= 6.67 x 10-11 Nm2/ kg2.
 Avogadro Number (NA)
= 6.023 x1023 mol-1.
 Boltzmann constant (K)
= 1.38 x 10-23 J/K
 Stefan Constant (σ)
= 5.67 x 10-8 W m-2 K-4.
 Wien Displacement Constant (b)
= 2.898 x10-3 m K
 Solar Constant (S)
= 1.388 x 103 W m-2
 Mass of the sun (MS)
= 2x 1030kg.
 Mass of the earth (ME)
= 5.98x 1024kg
 Radius of the earth (Re)
= 6400 Km. = 6.4x106m.
 Density of earth
= 5.522x103kg/m3.
 Average angular velocity of the earth
= 7.29x 10-5rad./sec
 Average distance between the sun and earth = 1.5x1011m.
 Average distance between moon and the earth =3.84x 108m.
 Magnetic Moment of the earth
= 6.4x 1021 Amp. X m2.
Conversion Coefficients









1Light year
1 A.U.
1Å
1 Pound
1 Fermi
1 C.S.L.
1 Shake
1ev
1 Horse Power
= 9.46x1015m.
= 1.496 x 1011m.
= 10-10m.
= 0.4536kg = 453.6 gm
= 10-15m.
= 1.4 x Mass of the sun.
= 10-8sec
= 1.6 x10-19 Joule.
= 746 Watt.
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Physics
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(Mathematical Tools)
(Class 11)
Quadratic Equation
An equation of second degree is called a quadratic equation.
It is of the form
ax2 + bx + c =0
The roots of a quadratic equation are
X = -b ±(b2 + 4ac)1/2
2a
Binomial Theorem
If n is any integer, positive or negative or a fraction and x is any real number,
then
( 1+x)n = 1 + nx + n(n-1)x2 + .. .
Ị2
If IxI <<1 , then (1+x)n = 1+ nx.
Mensuration
 Area of a circle
= 𝜋 r2 =π D2/4
 Surface area of a sphere
= 4πr2 = πD2
 Volume of a sphere
= 4/3 πr3
 Surface area of a cylinder
= 2πr(r+l)
 Volume of a cylinder
=π r2 l
 Curved surface area of a cone = πrl
 Volume of a cone
= 1/3πr2h
 Surface area of a cube
= 6x(side)2
 Volume of a cube
= (side)3
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Physics
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(Mathematical Tools)
(Class 11)
Fundamental Trigonometric relations
Cosec =
Sec =
Cot =
Tan =
1
Sinθ
1
Cosθ
Cosθ
Sinθ
1
=
Tanθ
Sinθ
Cosθ
Sin2 + Cos2 =1
1+ tan 2 =Sec2
1+ Cot2 = Cosec2
Sin(A+B) = SinACosB + CosASinB
Cos (A + B) = CosACosB – SinASinB
Sin (A – B) = SinACosB – CosASinB
Cos (A – B) = CosACosB + SinASinB
TanA+TanB
Tan(A + B) = 1−TanATanB
Sin 2A = 2SinACosA
Cos 2A = 2Cos2A – 1=1- 2Sin2A = Cos2A – Sin2A
Sin (A+B) + Sin(A-B) = 2 SinACosB
Cos(A+B) + Cos (A-B) = 2CosACosB
CosC + CosD = 2Cos
(C+D)
2
Cos
(C−D)
)
2
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Physics
(www.tiwariacademy.com)
(Mathematical Tools)
(Class 11)
Logarithms
Logamn = Logam + Logan
𝑚
Loga ( ) = Logam - Logan
𝑛
Logam = Logbm x Logab
Log10103 = Log101000 = 3
Loga1 = 0
Logaa = 1
Average Values
< Sinθ> = 0
< Sin2 >= ½
, < Cosθ> = 0
< Cos2 θ > = ½
Approximate Values
If angle ( ) small then
0
Sinθ
Cos
1
Tan
Differential Formulae
 Differentiation of a constant is zero
𝑑𝑐
𝑑𝑥



d(cy)
dx
𝑑(𝑥 𝑛 )
=0
= cdy
dx
= 𝑛𝑥 𝑛−1
𝑑𝑥
d[f(x)±g(x)]
dx
=
df(x)
dx
± dg(x)
dx
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Physics
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(Mathematical Tools)
(Class 11)

d{f(x)g(x)}
dx
𝑓(𝑥)

{ ( )}
𝑑𝑥 𝑔 𝑥
𝑑












𝑑𝑦
𝑑𝑥
𝑑𝑥
𝑑𝑢𝑛
𝑑𝑥
dx
𝑑𝑓(𝑥)
𝑑𝑔(𝑥)
−𝑓(𝑥)
𝑑𝑥
𝑑𝑥
{𝑔(𝑥)}2
= 𝑒𝑥
= 𝑛𝑢𝑛−1
=
𝑑𝑢
𝑑𝑥
1
𝑥
= 𝑎 𝑥 𝑙𝑜𝑔𝑒 𝑎
𝑑𝑥
𝑑𝑙𝑜𝑔𝑎𝑥
𝑑𝑥
𝑑(sin 𝑥)
𝑑𝑥
𝑑(tan 𝑥)
𝑑𝑥
𝑑(cot 𝑥)
1
= 𝑙𝑜𝑔𝑒 𝑎
𝑑𝑥
𝑑(cos 𝑥)
𝑥
= cos 𝑥
= − sin 𝑥
= 𝑠𝑒𝑐 2 𝑥
= −𝑐𝑜𝑠𝑒𝑐 2 𝑥
𝑑𝑥
𝑑(𝑐𝑜𝑠𝑒𝑐𝑥)
𝑑𝑥
𝑑(𝑠𝑒𝑐𝑥)
𝑑𝑥
𝑔(𝑥)
g(x)df(x)
𝑑𝑢 𝑑𝑥
𝑑𝑒 𝑥
𝑑𝑥
𝑑 (𝑎𝑥 )
=
dx
+
𝑑𝑦 𝑑𝑢
=
𝑑𝑙𝑜𝑔𝑒𝑥
f(x)dg(x)
=
= −cosec 𝑥 𝑐𝑜𝑡 𝑥
= sec 𝑥 𝑡𝑎𝑛 𝑥
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Physics
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(Mathematical Tools)
(Class 11)
Integral Formulae
 ∫dx =x +c
 ∫x
n+1
dx =
Where c = constant
𝑥 𝑛+1
𝑛+1
+𝐶
 ∫dx/x = logex +c
 ∫Sinx dx = - Cosx +c
 ∫Sin ax dx = -Cos ax
a
 ∫ Cos x dx = Sin x +c
 ∫Sec2x dx = tan x +c
 ∫ Cosec2x dx = -Cot x +c
 ∫ Sec x tan x dx = Sec x +c
 ∫Cosec x Cot x dx = - Cosec x + c
 ∫ ex dx = ex +c
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