Summary Physics short notes for neeet

Chapter 1

Basic Mathematics
Applied in Physics

QUADRATIC EQUATION
−b ± b 2 − 4ac
Roots of ax2 + bx + c = 0 are x =
2a
b c
Sum of roots x1 + x2 = − ; Product of roots x1x2 =
a a

BINOMIAL APPROXIMATION
If x << 1, then (1 + x)n ≈ 1 + nx & (1 – x)n ≈ 1 – nx

LOGARITHM
log mn = log m + log n
log m/n = log m – log n
log mn = n log m
logem = 2.303 log10m
log 2 = 0.3010

COMPONENDO AND DIVIDENDO LAW
p a p+q a+b
=If = then
q b p−q a −b

ARITHMETIC PROGRESSION-AP FORMULA
a, a + d, a + 2d, a + 3d, …, a + (n – 1)d, here d = common difference
n
Sum of n terms Sn = [2a + (n – 1)d], an = a1 + (n – 1)d
2

, NOTES


n(n + 1)
(i) 1 + 2 + 3 + 4 + 5 … + n =
2
n(n + 1)(2n + 1)
(ii) 12 + 22 + 32 + … + n2 =
6

GEOMETRICAL PROGRESSION-GP FORMULA
a, ar, ar2, … here, r = common ratio
a(1 − r n )
Sum of n terms Sn =
1− r
a
Sum of ∞ terms S∞ = , (r < 1)
1− r
TRIGONOMETRY
• 2p radian = 360° ⇒ 1 rad = 57.3°
1
• cosec q =
sin θ
1
• sec q =
cos θ
1
• cot q =
tan θ
• sin2q + cos2q = 1
• 1 + tan2q = sec2q
• 1 + cot2q = cosec2q
• sin (A ± B) = sin A cos B ± cos A sin B
• cos (A ± B) = cos A cos B  sin A sin B
tan A ± tan B
• tan (A ± B) =
1  tan A tan B
• sin 2A = 2 sin A cos A
• cos 2A = cos2 A – sin2 A = 1 – 2 sin2A = 2 cos2 A – 1
2 tan A
• tan 2A = , If A = B
1 − tan 2 A
sine law
sin A sin B sin C
= =
a b c
Hand Book (Physics) 2

, cosine law
b2 + c2 − a 2 c2 + a 2 − b2 a 2 + b2 − c2
= cos A = , cos B = , cos C
2bc 2ca 2ab
A

A

c b



B C
B a C

Approximation for small q
• sin q ≈ q
• cos q ≈ 1
• tan q ≈ q ≈ sin q
Differentiation
n dy
• y =x → = nx n −1
dx
dy 1
• y = lnx → =
dx x
dy
• y = sin x → = cos x
dx
dy
• y= cos x → =− sin x
dx
dy
• y= eαx +β → =αeαx +β
dx
dy dv du
• y = uv → =u + v (Product rule)
dx dx dx
dy df (g(x)) d(g(x))
y f (g(x)) → =
• = × (Chain rule)
dx dg(x) dx
dy
• y = k = constant ⇒ =0
dx
du dv
v −u
u dy dx dx
• y= ⇒ = (Division Rule)
v dx v2
3 Basic Mathematics Applied in Physics

, Integration
x n +1
∫x=
n
• dx + C, n ≠ −1
n +1
1
• ∫ x=
dx nx + C

• ∫ sin xdx =
− cos x + C

• ∫ cos xdx
= sin x + C
1 αx +β
∫ e=
αx +β
• dx e +C
α
(αx + β) n +1
• ∫ (α=
x + β) n dx +C
α(n + 1)

Maxima and Minima of a Function y = f(x)
dy d2 y
• For maxima = 0 & 2 = − ve
dx dx

dy d2 y
• For minima = 0 & 2 = + ve
dx dx
Average of a Varying Quantity

x2 x2


If y = f(x) then < y=
>= y
∫=
x1
ydx ∫
x1
ydx
x2
∫ dx x
x1
2 − x1


Formulae for Calculation of Area
• Area of a square = (side)2
• Area of rectangle = length × breadth
• Area of a triangle = 1/2 × base × height
• Area of a trapezoid = 1/2 × (distance between parallel sides)
× (sum of parallel sides)
• Area enclosed by a circle = p r2, (r = radius)
• Surface area of a sphere = 4p r2, (r = radius)
• Area of a parallelogram = base × height

Hand Book (Physics) 4