IISER Mathematics - Binomial Theorem and Its application
Exam Duration: 45 Mins Total Questions : 30
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 sum of the coefficients in the expansion of\((1+2x-x^2)^{2143}\)is
- (a)
1
- (b)
\(2^{2143}\)
- (c)
\(3^{2143}\)
- (d)
0
Cofficient of x2 in the expansion of(1+4x+x2)1/2 is
- (a)
-3
- (b)
-1
- (c)
2
- (d)
None of these
\(1+\frac { 2nx }{ 1+x } +\frac { n(n+1) }{ 2! } \frac { 4x_{ }^{ 2 }{ } }{ \left( 1+x \right) ^2 } ........\infty \) equals
- (a)
\(\left( \frac { 1-x }{ 1+x } \right) _{ }^{ n }{ }\)
- (b)
\(\left( \frac { 1+x }{ 1-x } \right) _{ }^{ n }{ }\)
- (c)
\(\left( \frac { 1-x }{ 1+x } \right) _{ }^{ 2n }{ }\)
- (d)
None of these
The value of \(\sum _{ r=0 }^{ n-1 }{ \frac { { { { { ^{ n }C } } } }_{ r } }{ { { { { ^{ n }C } } } }_{ r }+{ { { { ^{ n }C } } } }_{ r+1 } } } \) is
- (a)
\(\frac { n }{ 2 } \)
- (b)
\(\frac { n+1 }{ 2 } \)
- (c)
\(\frac { n-1 }{ 2 } \)
- (d)
\(2n\)
Determine which one of the following is true?
- (a)
9950 + 10050 > 10150
- (b)
9950 + 10050 < 10150
- (c)
9950 + 10050 = 10150
- (d)
None of the above
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
The coefficient of x in the expansion of (1+x+x2+x3)10 is
- (a)
990
- (b)
900
- (c)
890
- (d)
899
In the expansion of \(\left[ \sqrt { { { x }^{ \left( \frac { 1 }{ log\quad x+1 } \right) } } } +{ x }^{ 1/12 } \right] \), fourth term is equal to 200 and x>1, then the value of x is
- (a)
10-4
- (b)
10-2
- (c)
10-3
- (d)
10-5
The positive integer just greater than S = (1+0.0001)10000 is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
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
If x is so small that x3 and higher powers of x may be neglected, then \(\frac { { \left( 1+x \right) }^{ 3/2 }-{ \left( 1+\frac { 1 }{ 2 } x \right) }^{ 3 } }{ { \left( 1-x \right) }^{ 1/2 } } \) may be approximated as
- (a)
\(\frac { x }{ 2 } -\frac { 3 }{ 8 } { x }^{ 2 }\)
- (b)
\(-\frac { 3 }{ 8 } { x }^{ 2 }\)
- (c)
\(3x+\frac { 3 }{ 8 } { x }^{ 2 }\)
- (d)
\(1-\frac { 3 }{ 8 } { x }^{ 2 }\)
If m and n are any two odd positive integers with n < m, then the largest positive integer which divides all numbers of the form (m2 - n2), is
- (a)
4
- (b)
6
- (c)
8
- (d)
9
The coefficient of \({ x }^{ r }\{ 0\le r\le (n-1)\} \qquad \)in the expansion of \(\left( x+3 \right) ^{ n-1 }+\left( x+3 \right) ^{ n-2 }\left( x+2 \right) +\left( x+3 \right) ^{ n-3 }\left( x+2 \right) ^{ 2 }+.....+\left( x+2 \right) ^{ n-1 }\)is
- (a)
\(^{ n }{ C }_{ r }({ 3 }^{ r }-{ 2 }^{ n })\)
- (b)
\(^{ n }{ C }_{ r }({ 3 }^{ n-r }-{ 2 }^{ n-r })\)
- (c)
\(^{ n }{ C }_{ r }({ 3 }^{ r }-{ 2 }^{ n-r })\)
- (d)
none of these
If there is a term containing x2r in \(\left( x+\frac { 1 }{ { x }^{ 2 } } \right) ^{ n-3 }\), then
- (a)
n - 2r is a positive integral multiple of 3
- (b)
n - 2r is even
- (c)
11 - 2r is odd
- (d)
none of the abvoe
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
The coefficient of a10b7c3 in the expansion of (bc + ca + ab)10 is
- (a)
30
- (b)
60
- (c)
120
- (d)
240
The last digit of \({ 3 }^{ { 3 }^{ 4n } }+1,\quad n\varepsilon N\) is
- (a)
4C3
- (b)
8C7
- (c)
8
- (d)
4
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
The number of distinct terms in the expansion of (x1 + x2 + x3 + ... + xn)4 is
- (a)
n+1C4
- (b)
n+2C4
- (c)
n+3C4
- (d)
n+4C4
Compute (98)5
- (a)
9039007948
- (b)
9039207968
- (c)
9029007968
- (d)
9049007968
The value of the expansion \(\left( \sqrt { 3 } +1 \right) ^{ 5 }=\left( \sqrt { 3 } -1 \right) ^{ 5 }\) equals
- (a)
88
- (b)
40
- (c)
88\(\sqrt { 3 } \)
- (d)
40 \(\sqrt { 3 } \)
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 (1 + x + 2x2)6 = 1 + a1x + a2X2 + ... + a12X12, then the expression a2 + a4 + a6 + ... a12 has the value
- (a)
32
- (b)
63
- (c)
64
- (d)
31
Coefficient of x4 in (1 + x + x2 +x3)11 is
- (a)
330
- (b)
990
- (c)
900
- (d)
895
(C0 + C1) (C1 + C2) ....(Cn-1 + Cn) is equal to
- (a)
(C0C1C2..Cn-1)(n+1)
- (b)
(C0C1C2..Cn-1)(n+1)n
- (c)
\(\frac { \left( C_{ 0 }{ C }_{ 1 }{ C }_{ 2 }...{ C }_{ n-1 } \right) \left( n+1 \right) ^{ n } }{ n! } \)
- (d)
None of these
Find the value of r in the coefficients of (2r + 4)th term and (r -2) term in the expansion of (1 + x)18 are equal
- (a)
8
- (b)
6
- (c)
4
- (d)
9
Give the integers r > 1, n > 2 , and coefficients of (3r)th and (r + 2)th terms in the binomial expansion of (1 +x)2n are equal then,
- (a)
n = 2r
- (b)
n = 3r
- (c)
n = 2r + 1
- (d)
None of these
If the radio of the fifth term from the begining to the fifth term from the end in the expansion of \(\left( 4\sqrt { 2 } +\frac { 1 }{ 4\sqrt { 3 } } \right) \) is \(\sqrt { 6 } :1\) , then
Statement I : The value of n is 10
Statement Ii : \(\frac { { 2 }^{ \frac { n-4 }{ 4 } }.{ 3 }^{ -1 } }{ 2.{ 3 }^{ \frac { 4+n }{ 4 } } } =\sqrt { 6 } \)
- (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.
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
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
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