CHAPTER
2 Vector and Calculus
Quadratic Equation Sine law
−b ± b 2 − 4ac sin A sin B sin C
Roots of ax2 + bx + c = 0 are x = = =
2a a b c
b
Sum of roots x1 + x2 = − a Cosine law
c b2 + c2 − a 2 c2 + a 2 − b2 a 2 + b2 − c2
=cos A = , cos B = , cos C
Product of roots x1x2 = a 2bc 2ca 2ab
A
Binomial Approximation
If x << 1, then (1 + x)n ≈ 1 + nx and (1 – x)n ≈ 1 – nx A
c b
Logarithm
log mn = log m + log n
log m/n = log m – log n B C
B a C
log mn = n log m
logem = 2.303 log10m Maxima and Minima of a Function y = f(x)
log 2 = 0.3010 dy d2y
For maximum value = 0 & 2 = − ve
Componendo and Dividendo law dx dx
p a p+q a+b dy d2y
=If = then For minimum value = 0 & 2 = + ve
q b p −q a −b dx dx
Arithmetic Progression-AP Formula Average of a Varying Quantity
a, a + d, a + 2d, a + 3d, …, a + (n – 1)d, x2 x2
If y = f(x) then < y=
>= y
∫=
x1
ydx ∫
x1
ydx
here d = common difference x2
− x1
n ∫ dx x
x1
2
Sum of n terms Sn = [2a + (n – 1)d]
2 To convert an angle from degree to radian, we should multiply
Note: it by p/180° and to convert an angle from radian to degree, we
n(n + 1) should multiply it by 180°/p.
(i) 1 + 2 + 3 + 4 + 5 … + n =
2 By help of differentiation, if y is given, we can find dy/dx and
n(n + 1)(2n + 1) by help of integration, if dy/dx is given, we can find y.
(ii) 12 + 22 + 32 + … + n2 =
6 The maximum and minimum values of function A cos q + B
Geometrical Progression-GP Formula sin q are A2 + B 2 and − A2 + B 2 respectively.
a, ar, ar2, … here, r = common ratio
Parallelogram Law of Vector Addition
a (1 − r n )
Sum of n terms Sn = If two vectors are represented by two adjacent sides of a
1− r
parallelogram which are directed away from their common point
a then their sum (i.e. resultant vector) is given by the diagonal of the
Sum of ∞ terms S∞ =
1− r paralellogram passing away through that common point.
2 Vector and Calculus
Quadratic Equation Sine law
−b ± b 2 − 4ac sin A sin B sin C
Roots of ax2 + bx + c = 0 are x = = =
2a a b c
b
Sum of roots x1 + x2 = − a Cosine law
c b2 + c2 − a 2 c2 + a 2 − b2 a 2 + b2 − c2
=cos A = , cos B = , cos C
Product of roots x1x2 = a 2bc 2ca 2ab
A
Binomial Approximation
If x << 1, then (1 + x)n ≈ 1 + nx and (1 – x)n ≈ 1 – nx A
c b
Logarithm
log mn = log m + log n
log m/n = log m – log n B C
B a C
log mn = n log m
logem = 2.303 log10m Maxima and Minima of a Function y = f(x)
log 2 = 0.3010 dy d2y
For maximum value = 0 & 2 = − ve
Componendo and Dividendo law dx dx
p a p+q a+b dy d2y
=If = then For minimum value = 0 & 2 = + ve
q b p −q a −b dx dx
Arithmetic Progression-AP Formula Average of a Varying Quantity
a, a + d, a + 2d, a + 3d, …, a + (n – 1)d, x2 x2
If y = f(x) then < y=
>= y
∫=
x1
ydx ∫
x1
ydx
here d = common difference x2
− x1
n ∫ dx x
x1
2
Sum of n terms Sn = [2a + (n – 1)d]
2 To convert an angle from degree to radian, we should multiply
Note: it by p/180° and to convert an angle from radian to degree, we
n(n + 1) should multiply it by 180°/p.
(i) 1 + 2 + 3 + 4 + 5 … + n =
2 By help of differentiation, if y is given, we can find dy/dx and
n(n + 1)(2n + 1) by help of integration, if dy/dx is given, we can find y.
(ii) 12 + 22 + 32 + … + n2 =
6 The maximum and minimum values of function A cos q + B
Geometrical Progression-GP Formula sin q are A2 + B 2 and − A2 + B 2 respectively.
a, ar, ar2, … here, r = common ratio
Parallelogram Law of Vector Addition
a (1 − r n )
Sum of n terms Sn = If two vectors are represented by two adjacent sides of a
1− r
parallelogram which are directed away from their common point
a then their sum (i.e. resultant vector) is given by the diagonal of the
Sum of ∞ terms S∞ =
1− r paralellogram passing away through that common point.
