JEE Main Mathematics - Binomial Theorem and Its Application
Exam Duration: 60 Mins Total Questions : 30
If the third term in the expansion of \(\left( x+{ x }^{ log{ }_{ 10 }^{ { }^{ x } } } \right) ^5\)is \(10^6\),then x equals
- (a)
1
- (b)
10
- (c)
102
- (d)
103
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)
\(3^n+\frac { 1 }{ 2 } \)
The number of the expansion of (a+b+c)n is
- (a)
n+1
- (b)
n+3
- (c)
\((n+1)(n+2)\over2\)
- (d)
None of these
If the absolute term in the expansion of \(_{ }\left( 5x-\frac { \lambda }{ { x }^{ 2 } } \right)^{10}\)is 405,then value of 1 is
- (a)
\(\pm 1\)
- (b)
\(\pm 2\)
- (c)
\(\pm 3\)
- (d)
None of these
The number of integral terms in the expansion of \(\left( \sqrt { 35 } +\sqrt [ 8 ]{ 5 } \right) _{ }^{ 256 }{ }\)is
- (a)
32
- (b)
33
- (c)
34
- (d)
35
If in the expansion \(\left[ \frac { 1 }{ a } +{ a }^{ log_{ 10 }a } \right] ^{ 5 }\) of , the value of the third term is 1000, then the value of a is
- (a)
10
- (b)
100
- (c)
1000
- (d)
99
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 a and b are the coefficient of xr and xn-r, respectively in the expansion of (1+x)n, then
- (a)
a = b
- (b)
a + b = n2
- (c)
a =nb
- (d)
a + b = 2n/2
If (1+x)n = nC0 + nC1x+.... + nCnxn, then the value of \(\overset { n }{ \underset { r=0 }{ \Sigma } } \quad \overset { n }{ \underset { s=0 }{ \Sigma } } ^{ n }{ C }_{ r }.^{ n }{ C }_{ s }\) is
- (a)
2n
- (b)
2n+1
- (c)
22n+1
- (d)
22n
Statement I \(\overset { n }{ \underset { r=0 }{ \Sigma } } (r+1).^{ n }{ C }_{ r }=(n+2)2^{ n }-1\)
Statement II \(\overset { n }{ \underset { r=0 }{ \Sigma } } (r+1)^{ n }{ C }_{ r }.x^{ r }=(1+x)^{ n }+nx(1+x)^{ n-1 }\)
- (a)
Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I
- (b)
Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
- (c)
Statement I is true, Statement II is false
- (d)
Statement I is true, Statement II is true
The coefficient of the middle term in the binomial expansion in powers of x of (1+\(\alpha \)x) and of (1-\(\alpha \)x)6 is the same, if is equal to
- (a)
\(-\frac { 5 }{ 3 } \)
- (b)
\(\frac { 10 }{ 3 } \)
- (c)
\(-\frac { 3 }{ 10 } \)
- (d)
\(\frac { 3 }{ 5 } \)
(115)96 - (96)115 is divided by
- (a)
15
- (b)
17
- (c)
19
- (d)
21
The sum \(\overset { m }{ \underset { i=0 }{ \Sigma } } \sum _{ i=0 }^{ m }{ \left( \begin{matrix} 10 \\ i \end{matrix} \right) } \left( \begin{matrix} 20 \\ m-i \end{matrix} \right) \) (Where \(\left( \begin{matrix} p \\ q \end{matrix} \right) =0\) if \(p<q\) is maximum when m is
- (a)
5
- (b)
10
- (c)
15
- (d)
20
If S be the sum of coefficients in the expansion of \(\left( \alpha x+\beta y-\gamma z \right) ^{ n }\) where \(\left( \alpha ,\beta ,\gamma >0 \right) \), then the value of \(\underset { n\rightarrow \infty }{ lim } \frac { S }{ \left\{ { S }^{ \frac { 1 }{ n } }+1 \right\} } \)
- (a)
\(_{ e }{ \left( \frac { \alpha \beta }{ \gamma } \right) }\)
- (b)
\(_{ e }{ \left( \frac { \alpha +\beta -\gamma }{ \alpha +\beta -\gamma +1 } \right) }\)
- (c)
\(\frac { \alpha \beta }{ \gamma } \)
- (d)
0
The term independent of x in the expansion of \(\left( \sqrt { \left( \frac { x }{ 3 } \right) } +\sqrt { \left( \frac { 3 }{ { 2x }^{ 2 } } \right) } \right) ^{ 10 }\) is
- (a)
5/12
- (b)
1
- (c)
10C1
- (d)
none of these
If (1 + x)n = C0 + C1x C2X2 + ... + Cnxn, then the value of \({ C }_{ 0 }+\frac { 1 }{ 2 } { C }_{ 1 }+\frac { 1 }{ 3 } { C }_{ 2 }+...+\frac { 1 }{ \left( n+1 \right) } { C }_{ n }\) is
- (a)
\(\frac { { 2 }^{ n-1 } }{ \left( n+1 \right) } \)
- (b)
\(\frac { { 2 }^{ n+1 } }{ \left( n+1 \right) } \)
- (c)
\(\frac { { 2 }^{ n-1 }-1 }{ \left( n+1 \right) } \)
- (d)
\(\frac { { 2 }^{ n+1 }-1 }{ \left( n+1 \right) } \)
The first integral term in the expansion of \(\left( \sqrt { 3 } +\sqrt [ 3 ]{ 2 } \right) ^{ 9 }\) is
- (a)
2nd term
- (b)
3rd term
- (c)
4th term
- (d)
5th term
The sum of the last ten coefficients in the expansion of (1 + x) 19, when expanded in ascending powers of x is
- (a)
218
- (b)
219
- (c)
218 - 19C10
- (d)
none of these
If (1 + 2x + 3x2)10 = a0 + a1x + a2x2 + ... + a20x20, then a1 equals
- (a)
10
- (b)
20
- (c)
210
- (d)
420
If n> 3 and Cr stands for nCr then \(\sum _{ r=0 }^{ n }{ \left( -1 \right) ^{ r }\left( n-r \right) \left( n-r+1 \right) \left( n-r+2 \right) { C }_{ r } } \) is equal to
- (a)
4
- (b)
3
- (c)
0
- (d)
1
For \(1\le r\le n\) the value of nCr + n-1Cr + n-2Cr + ....+ rCr is
- (a)
nCr+1
- (b)
n+1Cr
- (c)
n+1Cr+1
- (d)
none of these
The value of \(\sum _{ i=0 }^{ n }{ \sum _{ j=1 }^{ n }{ ^{ n }{ C }_{ j } } } ,^{ j }{ C }_{ i },i\le j\quad is\)
- (a)
3n-1
- (b)
0
- (c)
2n
- (d)
none of these
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
The value of r for which
30Cr .20C0 + 30Cr-1 ,20C1 + .. , + 30C0 .20Cr is maximum, is
- (a)
10
- (b)
15
- (c)
20
- (d)
25
If n is a positive integer and \(\left( 3\sqrt { 3 } +5 \right) ^{ 2n+1 }=\alpha +\beta \) where \(\alpha \) is an integer and \(0<\beta <1\) then
- (a)
\(\alpha\) is an even integer
- (b)
\({ \left( \alpha +\beta \right) }^{ 2 }\) is divisible by 22n+1
- (c)
the integer just below \(\left( 3\sqrt { 3 } +5 \right) ^{ 2n+1 }\)divisible by 3
- (d)
\(\alpha\) is divisible by 10
In the expansion of \(\left( \sqrt [ 3 ]{ 4 } +\frac { 1 }{ \sqrt [ 4 ]{ 6 } } \right) ^{ 20 }\)
- (a)
the number of rational terms = 4
- (b)
the number of irrational terms = 19
- (c)
the middle term is irrational
- (d)
the number of irrational terms = 17
if (1 +x - 3x2)10 = 1 + a1x + a2X2 + ...+a20X20, then the expression a2 +a4 + a6 + .....a20 is equal to
- (a)
\(\frac { 3^{ 10 }+1 }{ 2 } \)
- (b)
\(\frac { { 3 }^{ 9 }+1 }{ 2 } \)
- (c)
\(\frac { { 3 }^{ 10 }-1 }{ 2 } \)
- (d)
\(\frac { { 3 }^{ 9 }-1 }{ 2 } \)
In the expression of (1 + x +x2 + x3)6, the coefficient of x14 is
- (a)
130
- (b)
120
- (c)
128
- (d)
125
Find the coefficients of x11 in the expansion of (1 - 2x + 3x2) (1 +x)11
- (a)
132
- (b)
169
- (c)
144
- (d)
184
The coefficient of xn in the expansion of (1 +x)2n and (1 + x)2n-1 are in the radio
- (a)
1:2
- (b)
1:3
- (c)
3:1
- (d)
2:1
Statement I : (x - 2y)5 =x5 - 10x4 y + 40x3 y2 - 80x2y3 + 80xy4 - 32y5
Statement II : (x -y)n = nCoXn -nC1xn-1y + nC2xn-2 y2 -nC3xn-3y3 + ..+ (-1)nn Cnyn
- (a)
If both Statement-I and Statement-II are true and Staternent-Il is the correct explanation of Statement -I.
- (b)
If both Statement-I and Statement-Il are true but Statement-II is not the correct
explanation of Statement -I. - (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement -I is false and Statement-II is true.