Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
Title: Inter based maths theury
Description: The notes are required to inter based on compdation leave exams
Description: The notes are required to inter based on compdation leave exams
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
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 n1 x
n
2
and
Ram Rajya More, Siwan (Bihar)
a
n 1
2
2
Tn3 nC n1 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
...
1n nCn2 12 1
...
...
2!
3!
eg, The expansion (2 + 3x) upto four terms in decreasing power of x, is as follows
–5
2 3 x 5 3x1
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
nn 1n 2
...
n r 1 x r
...
2!
3!
r!
r
1
...
1 x n 1 nx nn 1 x 2 nn 1n 2 x3
...
n r 1 x r
...
1 x n 1 nx nn 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
Title: Inter based maths theury
Description: The notes are required to inter based on compdation leave exams
Description: The notes are required to inter based on compdation leave exams