Mathematics - Binomial Theorem and Its Application
Exam Duration: 45 Mins Total Questions : 30
If \(_{ }^{ n }{ C }_{ r-1 }=36,_{ }^{ n }{ C }_{ r }=84,_{ }^{ n }{ C }_{ r+1 }=126,\)then
- (a)
r=3,r=12
- (b)
r=3,r=9
- (c)
r=3,r=11
- (d)
None of these
If \({a}_n=\begin{matrix} n \\ \sum \\ r=0 \end{matrix}\frac { 1 }{ _{ }^{ n }{ C }_{ r }^{ } } \\ \),then \(\begin{matrix} n \\ \sum \\ r=0 \end{matrix}\frac { r }{ _{ }^{ n }{ C }_{ r }^{ } } \\ \) is equal to
- (a)
\((n-1)a_n\)
- (b)
\(n a_n\)
- (c)
\(\frac { n }{ 2 } { a }_{ n }^{ }\)
- (d)
None of these
The value of \(2C_0+\frac { { 2 }^{ 2 } }{ 2 } +C_1+\frac { { 2 }^{ 4 } }{ 2 } C_3+....\frac { { 2 }^{ 11 } }{ 2 } C_10\) is
- (a)
\(\frac { { 3 }^{ 11 }-1 }{ 11 } \)
- (b)
\(\frac { { 2}^{ 11 }-1 }{ 11 } \)
- (c)
\(\frac { { 11 }^{ 3 }-1 }{ 11 } \)
- (d)
\(\frac { { 11 }^{ 2 }-1 }{ 11 } \)
The sum of the coefficients in the expansion of\((1+2x-x^2)^{2143}\)is
- (a)
1
- (b)
\(2^{2143}\)
- (c)
\(3^{2143}\)
- (d)
0
In the expansion of (1+x)50,the sum of the coefficients of odd power of x is
- (a)
0
- (b)
249
- (c)
250
- (d)
251
If (1-x+x2)n=a0+a1x+a2x2+.......+a2nx2n,then a0+a2+a4+....a2n equals
- (a)
\(3^n+1\over2\)
- (b)
\(3^n-1\over2\)
- (c)
\(1-3^n\over2\)
- (d)
\(3^n+\frac { 1 }{ 2 } \)
Which of the following is middle term in the expansion (1+x)2n ?
- (a)
\(\frac { 1.3.5...(2n-1) }{ n! } { 2 }^{ n }.{ x }^{ n }\)
- (b)
\(\frac { 1.2.3.4....(n+1) }{ (n+1)! } \)
- (c)
\(\frac { 1.2.3.4...n }{ n! } \)
- (d)
None of the above
If the middle term of \(\left( \frac { 1 }{ x } +\ xsin\quad x \right) ^{ 10 }\) is equal to \({ 7 }\frac { 7 }{ 8 } \), then value of x is
- (a)
\({ 2n\pi + }\frac { \pi }{ 6 } \)
- (b)
\({ n\pi + }\frac { \pi }{ 6 } \)
- (c)
\({ n\pi +(-1)^{ n } }\frac { \pi }{ 6 } \)
- (d)
\({ n\pi +(-1)^{ n } }\frac { \pi }{ 3 } \)
The coefficients of x and x (p and q are positive integers) in the expansion of (1+x)p+q are
- (a)
equal
- (b)
equal with opposite signs
- (c)
reciprocal of each other
- (d)
None of the above
The greatest value of the term independent in the expansion of (a sin x+a-1cos x)10, where \(x\epsilon R\) is
- (a)
\({ 7 }\frac { 7 }{ 17 } \)
- (b)
\(7\frac { 7 }{ 8 } \)
- (c)
\(7\frac { 8 }{ 7 } \)
- (d)
\(\frac { 8 }{ 63 } \)
For natural numbers m, n, if (1-y)m(1+y)n = 1+a1y+a2y2+....and a1=a2=10, then (m, n ) is
- (a)
(35, 20)
- (b)
(45, 35)
- (c)
(35. 45)
- (d)
(20, 45)
If the coefficient of x in equals the coefficient of x in , then a and b satisfy the relation
- (a)
ab = 1
- (b)
a/b = 1
- (c)
a+b = 1
- (d)
a-b = 1
(115)96 - (96)115 is divided by
- (a)
15
- (b)
17
- (c)
19
- (d)
21
If n-1Cr = (k2 - 3). nCr+1, then k belongs to
- (a)
(\(-\infty \), -2]
- (b)
[2, \(\infty \))
- (c)
\(\left[ -\sqrt { 3 } ,\sqrt { 3 } \right] \)
- (d)
(\(\sqrt { 3 } \), 2]
The sum of coefficients of the two middle terms in the expansion of (1 + X)2n-1 is equal to
- (a)
(2n-1)Cn
- (b)
(2n-1)Cn+1
- (c)
2nCn-1
- (d)
2nCn
If (1 + x)n = C0 + C1x + C2x2 + ... + Cnxn, then the value of \(\sum _{ k=0 }^{ n }{ \left( k+1 \right) ^{ 2 } } .{ C }_{ k }\) is
- (a)
2n-3(n2 + 5n + 4)
- (b)
2n-2(n2 + 5n + 4)
- (c)
2n-2(5n + 4)
- (d)
none of these
The coefficient of x53 in the expansion of \(\sum _{ m=0 }^{ 100 }{ ^{ 100 }{ { C }_{ m } } } \left( x-3 \right) ^{ 100-m }.{ 2 }^{ m }\) is
- (a)
100C47
- (b)
100C53
- (c)
-100C53
- (d)
-100C100
If m, n, r are positive integers and if r < m, r < n, then
mCr + mCr-1 ,nC1 + mCr-2. nC2 + ... + nCr = Coefficient of xr in (1 + x)m (1 + x)n
= Coefficient of xr in (1 + x)m+n
= m+nCr
If (1 + x)n = C0 + C1x +C2x2 + ..... + Cnxn and n is odd , then the value of is \({ C }_{ 0 }^{ 2 }-{ C }_{ 1 }^{ 2 }+{ C }_{ 2 }^{ 2 }-{ C }_{ 3 }^{ 2 }+....+\left( -1 \right) ^{ n }{ C }_{ n }^{ 2 }\) is
- (a)
0
- (b)
2nCn
- (c)
(-1)n 2nCn-1
- (d)
2nCn-2
If (1+ x + X2)n == a0 + a1x + a2x2 + .... + a2nx2n, then
- (a)
a0 - a2 + a4 - a6 + ........= 0, if n is odd
- (b)
a1 - a3 + a5 - a7 + .........= 0, if n is even
- (c)
a0 - a2 + a4 - a6 +........=0, if n = 4p, \(p\varepsilon { I }^{ + }\)
- (d)
a1 - a3 + a5 - a7 +......=0, if 4p + 1, \(p\varepsilon { I }^{ + }\)
S1 = \(\overset { n }{ \underset { i=j }{ \Sigma } } \) a1 + a2 + a3 + .......... +an
S2 = \(\underset { 1\le i< }{ \Sigma } \underset { j\le n }{ \Sigma } \) ai aj = a1a2 + a1a3 + ........... +an-1an
S3 = \(\underset { 1\le i< }{ \Sigma } \underset { j\le k }{ \Sigma } \underset { \le n }{ \Sigma } \) aiajak = a1a2a3 + a1a2a4 + ................ + an-2an-1an
...........................................................
Sn = a1 a2 a3 ......an
Then, (x + a1)(x + a2)(x + a3).........................(x + an) can be written as xn + S1 xn-1 + S2 xn-2 +.......+Sn
The coefficient of x6 (2 + x )3 (3 + x)2(5 + X)3 is in the expression
- (a)
78
- (b)
156
- (c)
312
- (d)
624
If p be the sum of the of add terms and Q and of even terms in the expansion of (x + a)n, then the value of [(x + a)2n - (x -a)2n] equals
- (a)
PQ
- (b)
2PQ
- (c)
4 PQ
- (d)
None of these
The Coefficients of three consective terms in the expansion of (1 + a)n are in the ratio 1 :7 : 42 find n
- (a)
45
- (b)
55
- (c)
40
- (d)
50
The sum of the coefficients of the first three terms in th expansion of \(\left( x-\frac { 3 }{ { x }^{ 2 } } \right) ^{ 3 },x\neq 0\), m being a natural number is 559, Find the coefficient of the term in th expansion containing x3
- (a)
-5940
- (b)
-5900
- (c)
-5945
- (d)
-5950
The coefficient of x-9 in the expansion \(\left( \frac { { x }^{ 2 } }{ 2 } \frac { 2 }{ x } \right) ^{ 9 }\) is
- (a)
512
- (b)
-512
- (c)
521
- (d)
251
Find the sixth of the expansion (y1/2 + x1/3)n, if the binomial coefficient of the third term from the end is 45.
- (a)
252y5/2, x5/3
- (b)
262y2/5, x5/3
- (c)
250y5/2, x5/3
- (d)
252y3/2, x5/3
State T For True and F for False,
(i) The sum of coefficients of the two middle terms in the expansion of (1 +x)2n-1 is equal to 2n-1Cn
(ii) In the expansion of \(\left( { x }^{ 2 }-\frac { 1 }{ { x }^{ 2 } } \right) ^{ 16 }\) , the value of constant term is 16C7
(iii) The ratio of the coefficient of xp and xq in the expansion of (1+ x)p+q is 1 : 1
(iv) The sum of the series \(\sum _{ r=10 }^{ 10 }{ ^{ 20 } } { C }_{ r }\), \({ 2 }^{ 19 }+\frac { ^{ 20 }C_{ 10 } }{ 2 } \)
- (a)
(i) (ii) (iii) (iv) F T T F - (b)
(i) (ii) (iii) (iv) F F F T - (c)
(i) (ii) (iii) (iv) F F T F - (d)
(i) (ii) (iii) (iv) T F T F