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FIROZ AHMADʼS MATHEMATICS

MATHEMATICS
Mob
...
Sc
...
Ed, M
...

Exericies-I


...


Exericies-III


...
Sc
...
Ed, M
...


Ram Rajya More, Siwan (Bihar)

1

FIROZ AHMADʼS MATHEMATICS

THINGS TO REMEMBER
 Bionomial Theorem (For Positive integer)
If n is a positive integer and x, a  R, then
(x + a)n = nC0xn + nC1xn–1a + nC2xn–2a2 +
...
+ (–1)n an
Coefficients nC0, nC0, nC0,
...

r!n – r !

General Term of Binomial Theorem
Let the general term in the expansion of (x + a)n is (r + 1)th term
...
The sum of indices of x and a in each term is n
...
The number of terms in (x + a)n is (n + 1)
...
The coefficient of terms equidistant from the beginning and the end are equal
...
(i) (x + a)n + (x – a)n = 2[nC0 xn + nC0 xn – 2 a2 +
...
]
5
...

2

6
...


 Middle Term in Binomial Expansion
1
...

2 


Tn  nC n / 2 x n / 2 a n / 2
2

1

 n 1 
 n 1 
2
...



Tn 1  C n1 x
n

2

and
Ram Rajya More, Siwan (Bihar)

a

n 1
2

2

Tn3  nC n1 x
2

M
...
(Mat hs), B
...
Phil (M at hs)

n 1
2

n 1
2

a

n 1
2

2

2

FIROZ AHMADʼS MATHEMATICS

 Greatest Term in the Expansion of (x + a)n
If Tr and Tr + 1 be the rth and (r + 1)th terms in the expansion of (x +a)n, then
n
Tr 1
Cr x n r a r
 n
Tr
Cr 1 x n r 1a r 1

 n  r 1 a

=
r

x
n  r 1 a
1
r
r



Tr 1
1
Tr



(n – r + 1) | a | > r | x |



n  1 | a |



r<

Let

n  1 | a |  I  f

| x||a|

| x||a|

(Where I is an integer and 0 < f < 1)
If f = 0, then Tr and Tr + 1 are equal and greatest and if 0 < f < 1 then Tr + 1 will be greatest
...
If nCr = nCs , then either r = s or r + s = n
2
...
+ nCn = 2n

3
...


n

C0 + nC2 + nC4 +
...
= 2n – 1
C0 + nC1 + nC2 + nC3 +
...


n

6
...


n+1

Cr + 1 =

n 1 n

...
nCr + 2nCr + 3nCr +
...
2n – 1
9
...
= 0
10
...
+ (n + 1) nCn = (n + 2)2n – 1
11
...
+ Cn – rCn =
12
...
 C n2 

2n!
n  r !n  r !

2n !
n!2

0
,

13
...
 
n/2 n
 1 C n / 2 ,
M
...
(Mat hs), B
...
Phil (M at hs)

Ram Rajya More, Siwan (Bihar)

3

FIROZ AHMADʼS MATHEMATICS

n

14
...
  1n nCn2   12 1
...


...

2!

3!

eg, The expansion (2 + 3x) upto four terms in decreasing power of x, is as follows
–5

2  3 x 5  3x1 
 

1
=
243x 5

2 

3 x  

5

2
3


 2   5 6   2   5 6  7   2 
  
  
...




=

1
243x 5

=

1  1 10 1 20 1 280 1


...
7

...

5

243  x
3 x
3 x
27 x


General Term in the Expansion of (1 + x)n
General term in the expansion of (1 + x)n is as follows
Tr 1 

nn  1n  2
...
n  r  1 x r 
...

2!
3!
r!
r

1
...


1  x  n  1  nx  nn  1 x 2  nn  1n  2 x3 
...
n  r  1 x r 
...


1  x n  1  nx  nn  1 x 2 
...
n  r  1 x r 
...

5
...

7
...

9
...
+ (–1)r xr +
...
+ xr +
...
+ (–1)r (r + 1)xr +
...
+ (r + 1)xr +
...

(1 – x)–3 = 1 + 3x + 6x2 +
...
Sc
...
Ed, M
...
a2r2
...
, ak  R, then (a1 + a2 +
...
 r n r !r !
...
nCr xn – r ar

 General term in the expansion of (1 – x)n is
Tr + 1 = nCr x r

 General term in the expansion of (1 – x)n is
Tr + 1 = (–1)r nCr x r

 When there are two middle terms in the expansion, then their binomial coefficients are equal
...

 If the value of x is so small than on leaving square and higher powers of x, we get
...

 The number of terms in the expansion of (a1 + a2 +
...
yn2
...


n+k–1

n!
where n = n1 + n2 + n3
...


M
...
(Mat hs), B
...
Phil (M at hs)

Ram Rajya More, Siwan (Bihar)

5



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