- Information
- AI Chat
Binomial theorem and logarithm
Mathematics class 11 (041)
University of Kerala
Recommended for you
Preview text
BINOMIAL THEOREM
In NDA exam, generally 1- questions are asked from this chapter which are based on term, term independent of z, number/ sum general An of terms, middle terms and combinational identities. algebraic expression consisting of two dissimilar terms them is calleda with positive or negativesign between binomial erpression, e. (x + a),
+ and x -> etc., are called binomial expressions.
y
Binomial Theorem for Any Positive Integer n The formula by which any power of a binomial expression can be expanded is known as binomial theorem. Binomial theorem for any positive integer n is (a + b" = C(n, 0)a" +C(%, 1) a"-' b+C(% 2) a"- * b* +... +C (7, n-1) ab"-+C(n n) b" OR (a+b=
"Cg a'b°
- "Ca"'b+
"C2a"- *b+..."Cab+ "C,a'b
ie. (a+b)" -
####### "C,(a)(b) where, C(%r)=
n! r!(n-7)! rn; ,reN
[here, the cocfficients "Co "C"C, are called the Binomial coeficients.] Special Cases
Puting a = l and b =x, we get (1+x)" = "C%+ "Cx+ "C,x+... + "C,x" = "C, r
Putting a = l and b= -x we get (1 -x)" = "C - "Cjx + "C,x* + +(-1*"C,x -1)" "C,
General Term In the expansion of (a + b" the general term is T,. = "C, a"'b'. In the binomial expansion of (a + by", the nh term from the end is l(n+1)-7 +1) = (n-r +2xh term from the beginning.
Middle Term
In the expansion of (a +b)", the middle term is ;+1h term i if n is cven. If n is odd, the two middle
terms are h and h terms.
NDA/NA Pathfinder 94
Greatest Coefficient (v) If n is odd,
then number of terms in (xt a)"+(x-a)"
It n is even, the greatest coetficient is "Cn/2 and if n is odd, and (x+a)" -(x - a)" are cqual to
the greatest coetficient is "C,- or "C,.1 both being cqual. (V1) If n is even, then number
of terms in the expansion of 2
EXAMPLE 1. The middle term in the expansion of (x a"+(x-a)" are,and that in the
IS expansion (r + a)" - (x - a)" are
a. C b. C d. 18 C0 EXAMPLE 3. If (1 +ax)"
=1 + 8x +24x t..., then c find a and n. Sol. b. Here 'n' is even. d. 2, a. 2,8 b. 8, 24 c. 8, 4 . Middle term is term. Sol. d. We have, (1+ ax" = "C% (ax)"+"C, (ax)'+ "C, (a+ nn a +
. Middle term is 10th term.
ho C- = 1+n ax +
Comparing with the given series, we get nn-? t
EXAMPLE 2. The greatest coefficient in the 2n 1+8x + 24x +.. = 1+ n ax+ 2! expansion ofx s 2! Hence, na = 8, a = 24 a =*,n (n-1) 13.. (2n 1)2" n 2 n (n) 32 (n - 1) = 24n 4 (n -1) = 3n n= n d. None of these
####### a- n 4 2
Method for Finding the Independent Term or Constant Term
Sol. a. Since the middle term has greatest coefficient, greatest coefticient = coefficient of the middle term
2n (2n)! 2n (2n-10(2n- 2(2n-3)...
4.3-1 In the expansion of (a + b)", the term which is free form
i. which have x" is known as independent term. To find independent term put the index of x obtained in general term equal to zero and find the value of r to calculate the required term.
n n!n! n!n
((2-D(2n -3)...3{2n(2n-2(2n-4)... 4) 2 n!n! [1:35.. (2n-D]2[n(n- D(n-2)..-1] n!n! EXAMPLE 4. The coefficient of the term 1-3.. (2-) 2'n!_1-3.5..-) 2" independen 10
of x in the expansion
n!n! of+ is
n!
a. 10 b. 252 C. 256 d. 20 Properties of Binomial Theorem Sol. b. In the expansion of (i) The number of terms | in the Binomial expansion is n+1, that is one more than the index n.
the general term
(ii) In any term, the sum of the indices of a and bis equal
####### T1C,Wa
10C,2 (a)=OC,x7=1C, (d
to n. (ii) T+1 in the expansion of (1+x)"
is equal to
n(n-1)(n-2)..(7-7+1),
5-r 5-
For independent term, r!
(iv) Sum of the binomial coefficients in the expansion of (a+b" is 2" (Put a =b=1).
r-5 0 r= Thus,T5+1="C5(x)** = °Cs = 252
In LOGARITHM
NDA
exam,
1 question
can
be asked
from
this
chapter
which
is based
on properties
of
If logarithm. a is positive
real
number
other
than
1 and
x is a rational
number
such
that
a = N, then
we say
that
logarithm
of N to base
a is x, written
as log
N=x.
Thus,
a* = N
log
=x, where
a>0,
a # and
N>0.
102
100
log
100
=
It e is also
known
as fundamental
logarithmic
identity.
When
base
is 'e then
the logarithmic
function
is
called
natural
or Napierian
logarithmic
function
and
when
base
is 10, then
it is called
common
logarithmic
function.
Note
e is the base
of natural
logarithm
(Napier
logarithm) log
- = loge
Log
of negative
integers
are not defined,
log 0 is not defined.
Logarithmic
function
is positive
as well as negative
but exponential
function
is always
positive.
The
base
of a logarithm
is never
taken
as 0, negative
number
and 1.
Properties
of
Logarithm
Let
m and
n be positive
number
such
that
a, b, c>
and
a, b, c#
) log.
ai) log
1=
ii)
loge
a= log,
a. loge b
(iv) log
a log
c
log,
(v) log,
a log.
b=
(vi)
log.(m.
n)=
log.
m+
log. n
(vi)
log
log.
m-
log.
"n
(vii)
log.
m" =n loga
m
(ix) log,
(m2)
= E
log,,
m
(xi)
a og*
=x og4,
x>0,
e>
x) a oBa " =n
(xii)
log=-logi/ab
EXAMPLE
- What
is the
value
of (log,
625)/
(log16e925)
?
b. 1
C. 2
d. 4
log
625
log
625
log 254
log 5 21og
5
log
25
4 log 5
####### 2log
log 13
2 log 5
Sol.
d. We
have,
log
13
log 169
Iog 13
2log
13
MATHEMATICS
Logarithm
103
EXAMPLE
- If (log,
x)*
log,
- <2, then
which
one
Logarithmic
Inequalities
of the following
is correct?
- Ifa>
1,p>
log.
p>
- If0<a<
1,p>
If a> 1,0<p<
If p>a>
If a> p>
If0<a<p<
a. 0cxb:
x<
c. 3«x<
d
log.
p<
Sol.
b.
(log 3 a + log , *<
2
(log,
*)+
(log 3 *) - <
log. log. p<
p>
0<
log,
P<
0< log.
P<
log
P>
a>m',
if m>
a<m,
if o <m<
(log
3 *+ 2) (log 3
x-1)
<-2<log
*<
32x<
3
<x<
Characteristic and Mantissa
- If0<p<a<
In log
N the integral
part
of N is called
the
characteristic
and
decimal part of N
is called
the
- If log m a>b
mantissa.
####### Ja<m',
####### if
####### m>
a>mb,
if 0 <m<
- If log,m
a<b
) (a) If N>1,
then
the characteristic
of log
N is one
less than the
number
of digits in integral part
of N.
(b) If 0<N<1,
then
the characteristic
of logio
N is
greater
than
the
immediately
after
the decimal
point
and the
first
significant digit
and
is a negative integer.
()
(a) Insert decimal point in antilog of a number When
characteristic is n
then
insert the decimal
point
after
(
- 1)th digit.
- log,
a> log,
b=a2b
if base'p'
is positive
and>
1
or as bif base p
is positive and <
1, i. 0
<p<
In other words, if base
is greater than 1 then
inequality remains same and
if base
is positive but
less than
1, then the sign of inequality
is reversed.
- For dx)>
1, log o) flx)
log oe) &{*)
fx)
glx)>
For 0< (x)<
1, log o) f(x)
log ) g(r)
O< fx)s
g(r)
- For d(x)>
1, log
et) flx)2a
» f(x)
a)
- For0<
Ax)<
1, logoe)
fx)
a s 0< flx)s
a
one
number
of zeros
(b) When
characteristic is , then insert the decimal
point
such
that
the first
significant
digit
is at th
place
EXAMPLE
- If 2log, N
=p, log,
2N=q
and q
-p=
then
find
the value of
N?
EXAMPLE
- If logo.
(x-1)
logo,
(x + 1), then x
belongs
to the
interval
a. 512
b. 536
C. 548
d. 560
a. (1, 21
b. (-, 2 c.
[2,
d. None of
these
So.
We are given
that,
2loga
N =p
Sol.
c. loga
(*- 02 logo
(x- 1)
For log to
be defined x
-1>
» *>
From
Eq. (i), logo
(x-)
2 loga
(x-)
logo
(x-D
2 logo
(r -)
-1sx
- ..)
log
2N=
-P
4
rom
Eq. (i),
8P = N
»23P
= N
.(10)
and
.(iv) .iin)
From
Eq. (i,
2N=2 =
N==29-
...(v)
2
From
Eqs.
(iv) and (v), 2*P = (29-)
23P = 229-
3p = 2q-
2q -3p=
2...(vi)
On solving
Eqs. (ii) and (vi), we get p = and q = 10
-1(1-E
-) s
x 2.
xe [2, oo)
1-a-iso
EXAMPLE
- If log,
(r*
slog,
(4x-11),
then
N 20-=
= 512
a. 4<rs
b. *-
or x>
EXAMPLE
- If logo2=0,
then
what
is the
number
of digits
in 20*?
C. -1SxS
d. x-1or
*> 5
Sol.
a.
5 4x -
base
= e> 1]
a. 81
b. 82
C 83
d. 84
- d. Let ax= 2064,
Taking
log on both
sides,
we get
*-4x
- 50
(x-
(x+
) <0»
-1SxS
..)
Also,
> 0
>x < -
or *>
ii)
O =
log
20 = 64x
=83.
Number
of digits
in 20
=83+
1 =
And
4x-11>0x>
4
..(ii)
From
Eqs. (i), (i) and (ii),
we
get
4 <xs
Binomial theorem and logarithm
Course: Mathematics class 11 (041)
University: University of Kerala
