1_mathematics_reading_book.pdf - MATHEMATICS BOOK FOR 10 2...

MATHEMATICS
BOOK FOR 10+2
ENTRANCES
BOOKS FOR B.SC MATHEMATICS/STATISTICS
ENTRANCE EXAMS:
1. Challenge & Thrill of Pre-College Mathematics
2. Mathematics Olympiad by Rajeev Manocha
3. An excursion in Mathematics by Modak
4. Mathematical Circles by Fomin
5. Play with Graphs by Arihant
6. Maths Wit Volume 1 & 2 by S.Chatterjee
7. Plane Trigonometry by SL Loney
8. Coordinate Geometry by SL Loney
9. Essential Calculus early transcendentals by James Stewart
10. Test of Mathematics at 10+2 level by East-West Press

ALGEBRA
2
Arithmetic series
General ( k th) term, u
k
= a + ( k – 1) d
last ( n th) term, l = u
n
= a + ( n – l) d
Sum to n terms, S
n
= n ( a + l ) = n [2 a + ( n – 1) d ]
Geometric series
General ( k th) term, u
k
= a r k –1
Sum to n terms, S
n
=
=
Sum to infinity S
∞
= , – 1 < r < 1
Binomial expansions
When n is a positive integer
( a + b )
n
= a
n
+
( )
a
n –1
b +
( )
a
n –2
b
2
+ ... +
( )
a
n–r
b
r
+ ... b
n
, n ∈ H11934
where
( )
=
n
C
r
=
( )
+
( )
=
( )
General case
(1 + x )
n
= 1 + nx + x
2
+ ... + x
r
+ ... , | x | < 1,
n ∈ H11938
Logarithms and exponentials
e
x ln a
= a
x
log
a
x =
Numerical solution of equations
Newton-Raphson iterative formula for solving f( x ) = 0, x
n +1
= x
n
–
Complex Numbers
{ r (cos θ + j sin θ )}
n
= r
n
(cos n θ + j sin n θ )
e
j θ
= cos θ + j sin θ
The roots of z
n
= 1 are given by z = exp( j) for k = 0, 1, 2, ..., n –1
Finite series
∑
n
r =1
r
2
= n ( n + 1)(2 n + 1) ∑
n
r =1
r
3
= n
2
( n + 1)
2
1
–
4
1
–
6
2 π k
––––
n
f( x
n
)
––––
f'( x
n
)
log
b
x
–––––
log
b
a
n ( n – 1) ... ( n – r + 1)
–––––––––––––––––
1.2 ... r
n ( n – 1)
–––––––
2!
n + 1
r + 1
n
r + 1
n
r
n !
––––––––
r !( n – r )!
n
r
n
r
n
2
n
1
a
–––––
1 – r
a ( r
n
– 1)
––––––––
r – 1
a (1 – r
n
)
––––––––
1 – r
1
–
2
1
–
2
Infinite series
f( x ) = f(0) + x f'(0) + f"(0) + ... + f
( r )
(0) + ...
f( x ) = f( a ) + ( x – a )f'( a ) + f"( a ) + ... + + ...
f( a + x ) = f( a ) + x f'( a ) + f"( a ) + ... + f
( r )
( a ) + ...
e
x
= exp( x ) = 1 + x + + ... + + ... , all x
ln(1 + x ) = x – + – ... + (–1)
r +1
+ ... , – 1 < x H11088 1
sin x = x – + – ... + (–1)
r
+ ... , all x
cos x = 1 – + – ... + (–1) r + ... , all x
arctan x = x – + – ... + (–1)
r
+ ... , – 1 H11088 x H11088 1
sinh x = x + + + ... + + ... , all x
cosh x = 1 + + + ... + + ... , all x
artanh x = x + + + ... + + ... , – 1 < x < 1
Hyperbolic functions
cosh
2
x – sinh
2
x = 1, sinh2 x = 2sinh x cosh x , cosh2 x = cosh
2
x + sinh
2
x
arsinh x = ln( x + ), arcosh x = ln( x + ), x H11091 1
artanh x = ln
( )
, | x | < 1
Matrices
Anticlockwise rotation through angle θ , centre O:
( )
Reflection in the line y = x tan θ :
( )
cos 2 θ sin 2 θ
sin 2 θ –cos 2 θ
cos θ –sin θ
sin θ cos θ
1 + x
–––––
1 – x
1
–
2
x 2 1 –
x 2 1 +
x 2 r +1
––––––––
(2 r + 1)
x 5
––
5
x 3
––
3
x
2 r
––––
(2 r )!
x
4
––
4!
x
2
––
2!
x
2 r +1
––––––––
(2 r + 1)!
x
5
––
5!
x
3
––
3!
x
2 r +1
––––––
2 r + 1
x
5
––
5
x
3
––
3
x
2 r
––––
(2 r )!
x
4
––
4!
x
2
––
2!
x 2 r +1
––––––––
(2 r + 1)!
x 5
––
5!
x 3
––
3!
x
r
––
r
x
3
––
3
x
2
––
2
x
r
––
r !