JEE Mathematics - Binomial Theorem and Its Application Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
If \(_{ }^{ 24}{ C }_{ 2r }=_{ }^{ 24 }{ C }_{ 2r-4 }\), then r =
- (a)
24
- (b)
14
- (c)
10
- (d)
7
The value of the expression \(^{47}C_4+\overset { 5 }{ \underset { j=1 }{ \sum { } } } ^{52-j}C_3\),is equal to
- (a)
\(^{47}C_5\)
- (b)
\(^{52}C_5\)
- (c)
\(^{52}C_4\)
- (d)
None of these
Given positive integers r>1,n>2 and the coefficients of (3r)th and (r+2)th terms in the expansion of (1+x)2n are equal,then
- (a)
n=2r
- (b)
n=3r
- (c)
n=2r-1
- (d)
None of these
The expansion \(3C_0-17C_1+31C_2-45C_3+...\) upto (n+1) term is, where \(C_{ r }^{ }{ }=_{ }^{ r }{ C }_{ r }^{ }\)
- (a)
0
- (b)
1
- (c)
14
- (d)
-1
If the coefficients of second,third and fourth terms in the expansion of(1+x)n are in A.P., then n=
- (a)
7
- (b)
9
- (c)
11
- (d)
13
Let \(R=\left( 5\sqrt { 5 } +11 \right) _{ }^{ 2n-1 }\)and f=R-[R],where [ ] denotes the greatest integer function,then Rf is equal to
- (a)
42n+1
- (b)
42n
- (c)
42n-1
- (d)
None of these
The number of distinct terms in the expansion of(a+b)100+(a-b)100,is
- (a)
50
- (b)
51
- (c)
101
- (d)
202
The unit digit in the number (183)!+3183
- (a)
3
- (b)
6
- (c)
7
- (d)
9
The expansion of polynomial of degree
- (a)
5
- (b)
7
- (c)
9
- (d)
6
If the coefficients of the (r-1) th, rth and (r+1)th terms in the expansion of (1+x)n are in the ratio 1:3:5, then the value of r is
- (a)
2
- (b)
3
- (c)
4
- (d)
5
The number of terms in the expansion of (a+b+c)n, where \(n\epsilon N\) is
- (a)
\(\frac { (n+1)(n+2) }{ 2 } \)
- (b)
n+1
- (c)
n+2
- (d)
(n+1)n
In the expansion of \(\left( { x }^{ 3 }-\frac { 1 }{ { x }^{ 2 } } \right) ^{ 15 }\) the constant term is
- (a)
15C9
- (b)
0
- (c)
10C9
- (d)
-15C9
In the coefficient of x7 and x8 are equal in \(\left( 2+\frac { x }{ 3 } \right) ^{ n }\), then n is
- (a)
56
- (b)
55
- (c)
45
- (d)
15
If the coefficient of 2nd, 3rd and the 4th terms in the expansion of (1+x)n are in AP, then value of n is a multiple of
- (a)
43
- (b)
17
- (c)
114
- (d)
14
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
The number of integral terms in the expansion of \(\left( \sqrt { 3 } +\sqrt [ 8 ]{ 5 } \right) ^{ 256 }\) is
- (a)
32
- (b)
33
- (c)
34
- (d)
35
Number of terms in the expansion of \(\left( \frac { { x }^{ 3 }+1+{ x }^{ 6 } }{ { x }^{ 3 } } \right) ^{ \Sigma n }\) (Where \(n\varepsilon N\)) is
- (a)
\(\Sigma n+1\)
- (b)
\(^{ \Sigma n+2 }{ { C }_{ 2 } }\)
- (c)
2n+1
- (d)
n2 + n +1
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]
If \(\left( 1+x \right) ^{ n }=\sum _{ r=0 }^{ n }{ { a }_{ r }{ x }^{ r } } \) and \({ b }_{ r }=1+\frac { { a }_{ r } }{ { a }_{ r }-1 } \) and \(\overset { n }{ \underset { r=1 }{ \Pi } } { b }_{ r }=\frac { \left( 100 \right) ^{ 100 } }{ 100! } \)
- (a)
99
- (b)
100
- (c)
101
- (d)
102
The coefficients of x2y2, yzt2 and xyzt in the expansion of (x + y + z + t)4 are in the ratio
- (a)
4: 2: 1
- (b)
1: 2: 4
- (c)
2: 4: 1
- (d)
1: 4: 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 7 divides \({ 32 }^{ { 32 }^{ 32 } }\), the remainder is
- (a)
1
- (b)
0
- (c)
4
- (d)
6
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
\(\sum _{ r=1 }^{ n }{ \left( \sum _{ p=0 }^{ r-1 }{ ^{ n }{ { C }_{ r } }^{ r }{ { C }_{ p }{ 2 }^{ p } } } \right) } \) is equal to
- (a)
4n - 3n +1
- (b)
4n - 3n -1
- (c)
4n - 3n + 2
- (d)
4n - 3n
If the sum of the coefficients in the expansion of (x-2y+3z)n =128, then the greatest coefficient in the expansion of (1+x)n is
- (a)
35
- (b)
20
- (c)
10
- (d)
5
In the expansion of (3-x/4 + 35x/4)n the sum of the binomial coefficient is 64 and the term with the greatest binomial coefficients exceeds the third by (n - 1) the value of x must be
- (a)
0
- (b)
1
- (c)
2
- (d)
3
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 (2 - 2x + x2)9
- (a)
number of distinct terms is 10
- (b)
coefficient of x^4 is 97
- (c)
sum of coefficient is 1
- (d)
number of distinct terms is 55
The last digit of \({ 3 }^{ { 3 }^{ 4n } }+1,\quad n\varepsilon N\) is
- (a)
4C3
- (b)
8C7
- (c)
8
- (d)
4
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
If (1 + x)n = C0 + C1x + C2x2 + C3x3 +..........+ Cnxn, then the value of \(\underset { 0\le i< }{ \Sigma } \underset { j\le n }{ \Sigma } \) (i + j)(Ci + Cj + CiCj) is
(Where \(\lambda \) = \(\underset { 0\le i< }{ \Sigma } \underset { j\le n }{ \Sigma } \) CiCj)
- (a)
\(n(2^n+\lambda)\)
- (b)
\(n^2.2^n+\lambda \)
- (c)
\(n(n.2^n+\lambda )\)
- (d)
none of these
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
If (1+ x)n = C0 + C1x + C2x2 + C3x3 + +Cnxn and \(\overset { n }{ \underset { r=0 }{ \Sigma } } \frac { 1 }{ ^{ n }{ { C }_{ r } } } =a\) then the value of \(\underset { 0\le i< }{ \Sigma } \underset { j\le n }{ \Sigma } \left( \frac { i }{ ^{ n }{ { C }_{ i } } } +\frac { j }{ ^{ n }{ { C }_{ j } } } \right) \) in terms of a and n is
- (a)
\(\frac { na }{ 2 } \)
- (b)
\(\frac { { n }^{ 2 }a }{ 2 } \)
- (c)
\(\frac { { na }^{ 2 } }{ 2 } \)
- (d)
\(\frac { { n }^{ 2 }{ a }^{ 2 } }{ 2 } \)
If n is a positive integer and a1 ,a2, a3, ... , am ∊ C, then
(a1 ,a2, a3+......+am)n = \(\sum { \frac { n! }{ { n }_{ 1 }!{ n }_{ 2 }!{ n }_{ 3 }!...{ n }_{ m }! } . } { a }_{ 1 }^{ { n }_{ 1 } }.{ a }_{ 2 }^{ { n }_{ 2 } }.{ a }_{ 3 }^{ { n }_{ 3 } }....{ a }_{ 1 }^{ { n }_{ m } }\) where n1 , n2, n3, ... , nm are all non negative integers subject to the condition n1 + n2 + n3 + ... + nm = n
If coefficient of x20 in (1- x + x2)20 and in (1+ x - x2)20 are respectively a and b, then
- (a)
a = b
- (b)
a > b
- (c)
a < b
- (d)
a + b = 0
Evaluate : \(\left( { x }^{ 2 }- \right) \sqrt { 1-{ x }^{ 2 } } )^{ 4 }+\left( { x }^{ 2 }+\sqrt { 1-{ x }^{ 2 } } \right) ^{ 4 }\)
- (a)
2x8 + 12x6 - 14x4 + 4x2 + 2
- (b)
2x8 - 12x6 +14x4 - 4x2 + 2
- (c)
2x8 - 12x6 - 14x4 + 4x2 -2
- (d)
x8 - 6x6 + 7x4- 2x2 + 1
The total number of terms in the expansion of (x + a)51 - (x - a)51 after simplification is
- (a)
102
- (b)
25
- (c)
26
- (d)
23
If n \(\epsilon \) N, Then 121n - 25n + 1900 n - (-4)n is divisible by
- (a)
1904
- (b)
2000
- (c)
2002
- (d)
2006
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
Find the term independent of x in the expansion of \(\left( \sqrt [ 3 ]{ x } +\frac { 1 }{ 2\sqrt [ 3 ]{ x } } \right) ,\quad x>0\)
- (a)
\(18_{ { C }_{ 9 } }\frac { 1 }{ { 2 }^{ 9 }\quad } \)
- (b)
\({ 18 }_{ { C }_{ 8 } }\frac { 1 }{ { 2 }^{ 8 } } \)
- (c)
\({ 18 }_{ { C }_{ 7 } }\frac { 1 }{ { 2 }^{ 8 } } \)
- (d)
None of these
Find the Coefficient of x11 in the expansion of \(\left( { x }^{ 3 }-\frac { 2 }{ { x }^{ 2 } } \right) ^{ 12 }\)
- (a)
-25344
- (b)
-25250
- (c)
-25000
- (d)
24310
if the coefficients of x7 and x8 in \(\left( 2+\frac { x }{ 3 } \right) ^{ n }\) are equal then n is
- (a)
56
- (b)
55
- (c)
45
- (d)
15
The Sixth term in the expansion of \(\left[ { 2 }^{ log2\sqrt { { 9 }^{ x-1 }+7 } + }\frac { 1 }{ { 2 }^{ \frac { 1 }{ 5 } { log }_{ 2 }\left( { 3 }^{ x-1 }+1 \right) } } \right] \) is 84 then the number of value of x is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
The sum of the coefficients in the expansion of (1 - x)10 is
- (a)
0
- (b)
1
- (c)
102
- (d)
1024
The coefficient of x7 in the expansion of \(\left( \frac { { x }^{ 2 } }{ 2 } -\frac { 2 }{ x } \right) ^{ 9 }\) is
- (a)
-56
- (b)
14
- (c)
-14
- (d)
None of these
if the expansion of \(\left( \frac { 3\sqrt { x } }{ 7 } -\frac { 5 }{ 2x\sqrt { x } } \right) ^{ 13n }\) contains a term independent of x, then n should be a multiple of
- (a)
10
- (b)
5
- (c)
6
- (d)
4
If | x | < 1, then the coefficient of x6 in the expansions of (1 + x + x2)-3 is
- (a)
3
- (b)
6
- (c)
6
- (d)
12
The coefficient of the middle term in the expansion of (x + 2y)6 is
- (a)
6C3
- (b)
8(6C3)
- (c)
8(6C4)
- (d)
6C4
Statement I : The rth term from the end in the expansion of (x +a)n is nCn-r+1 xr-1 an-r+1
Statement II : The rth term from the end in the expansion of (x + a)n is (n - r + 2)th term from begining
- (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.
Statement -I : \(\left( \sqrt { 2 } +1 \right) ^{ 6 }+\left( \sqrt { 2 } -1 \right) ^{ 6 }=190\)
Statement -II : \(\left( x+1 \right) ^{ 6 }+\left( x-1 \right) ^{ 6 }=2({ x }^{ 6 }+15{ x }^{ 4 }+{ 15x }^{ 2 }+1)\)
- (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.
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.
If the coefficients of ar-1, ar and ar+1 in the expansion of (1 +a)n are in arithmetic progession, then n2 - n (4r +1) + 4r2 is equal to
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Match the following.
Column - I | Column - II |
---|---|
(i) (1 - 2x)5 | (P) \(\frac { { x }^{ 5 } }{ 243 } -\frac { { 5x }^{ 3 } }{ 81 } +\frac { 10x }{ 27 } -\frac { 10 }{ 9x } +\frac { 5 }{ { 3x }^{ 3 } } -\frac { 1 }{ { x }^{ 5 } } \) |
(ii) (2x - 3)6 | (q) x6 + 6x4 + 15x2 + 20 + \(\frac { 15 }{ { x }^{ 2 } } +\frac { 6 }{ { x }^{ 4 } } +\frac { 1 }{ { x }^{ 6 } } \) |
(iii) \(\left( \frac { x }{ 3 } -\frac { 1 }{ x } \right) ^{ 5 }\) | 1 - 10x + 40x2 - 80x3 + 80x4 - 32x5 |
(iv) \(\left( x+\frac { 1 }{ x } \right) ^{ 6 }\) | 64x6 - 576x5 + 2160x4 - 4320x3 +4860x2 -2916x +729 |
- (a)
(i) \(\rightarrow\) (p), (ii) \(\rightarrow\) (q), (iii) \(\rightarrow\) (r), (iv)\(\rightarrow\) (s)
- (b)
(i) \(\rightarrow\)(r), (ii) \(\rightarrow\)(s), (iii) \(\rightarrow\) (p), (iv) \(\rightarrow\) (q)
- (c)
(i) \(\rightarrow\) (q), (ii) \(\rightarrow\) (s), (iii) \(\rightarrow\) (p), (iv) \(\rightarrow\) (r)
- (d)
(i) \(\rightarrow\) (r), (ii) \(\rightarrow\)(s), (iii) \(\rightarrow\)(q), (iv) \(\rightarrow\) (p)