Mathematical Induction and Binomial Theorem
14 Questions
MCQ (Single Correct)
2014
JEE Advanced
MCQ
JEE Advanced 2014 Paper 2 Offline
Coefficient of ${x^{11}}$ in the expansion of ${\left( {1 + {x^2}} \right)^4}{\left( {1 + {x^3}} \right)^7}{\left( {1 + {x^4}} \right)^{12}}$ is
A.
1051
B.
1106
C.
1113
D.
1120
2010
JEE Advanced
MCQ
IIT-JEE 2010 Paper 2 Offline
For $r = 0,\,1,....,$ let ${A_r},\,{B_r}$ and ${C_r}$ denote, respectively, the coefficient of ${X^r}$ in the expansions of ${\left( {1 + x} \right)^{10}},$ ${\left( {1 + x} \right)^{20}}$ and ${\left( {1 + x} \right)^{30}}.$
Then $\sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)} $ is equal to
Then $\sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)} $ is equal to
A.
$\left( {{B_{10}} - {C_{10}}} \right)$
B.
${A_{10}}\left( {{B^2}_{10}{C_{10}}{A_{10}}} \right)$
C.
$0$
D.
${{C_{10}} - {B_{10}}}$
2005
JEE Advanced
MCQ
IIT-JEE 2005 Screening
The value of $$\left( {\matrix{
{30} \cr
0 \cr
} } \right)\left( {\matrix{
{30} \cr
{10} \cr
} } \right) - \left( {\matrix{
{30} \cr
1 \cr
} } \right)\left( {\matrix{
{30} \cr
{11} \cr
} } \right) + \left( {\matrix{
{30} \cr
2 \cr
} } \right)\left( {\matrix{
{30} \cr
{12} \cr
} } \right)....... + \left( {\matrix{
{30} \cr
{20} \cr
} } \right)\left( {\matrix{
{30} \cr
{30} \cr
} } \right)$$
is where $\left( {\matrix{ n \cr r \cr } } \right) = {}^n{C_r}$
is where $\left( {\matrix{ n \cr r \cr } } \right) = {}^n{C_r}$
A.
$\left( {\matrix{
{30} \cr
{10} \cr
} } \right)$
B.
$\left( {\matrix{
{30} \cr
{15} \cr
} } \right)$
C.
$\left( {\matrix{
{60} \cr
{30} \cr
} } \right)$
D.
$\left( {\matrix{
{31} \cr
{10} \cr
} } \right)$
2004
JEE Advanced
MCQ
IIT-JEE 2004 Screening
If ${}^{n - 1}{C_r} = \left( {{k^2} - 3} \right)\,{}^n{C_{r + 1,}}$ then $k \in $
A.
$\left( { - \infty , - 2} \right)$
B.
$\left[ {2,\infty } \right)$
C.
$\left[ { - \sqrt 3 ,\sqrt 3 } \right]$
D.
$\left( {\sqrt 3 ,2} \right]$
2003
JEE Advanced
MCQ
IIT-JEE 2003 Screening
Coefficient of ${t^{24}}$ in ${\left( {1 + {t^2}} \right)^{12}}\left( {1 + {t^{12}}} \right)\left( {1 + {t^{24}}} \right)$ is
A.
${}^{12}{C_6} + 3$
B.
${}^{12}{C_6} + 1$
C.
${}^{12}{C_6}$
D.
${}^{12}{C_6} + 2$
2002
JEE Advanced
MCQ
IIT-JEE 2002 Screening
The sum $\sum\limits_{i = 0}^m {\left( {\matrix{
{10} \cr
i \cr
} } \right)\left( {\matrix{
{20} \cr
{m - i} \cr
} } \right),\,\left( {where\left( {\matrix{
p \cr
q \cr
} } \right) = 0\,\,if\,\,p < q} \right)} $ is maximum when $m$ is
A.
5
B.
10
C.
15
D.
20
2001
JEE Advanced
MCQ
IIT-JEE 2001 Screening
In the binomial expansion of ${\left( {a - b} \right)^n},\,n \ge 5,$ the sum of the ${5^{th}}$ and ${6^{th}}$ terms is zero. Then $a/b$ equals
A.
$\left( {n - 5} \right)/6$
B.
$\left( {n - 4} \right)/5$
C.
$5/\left( {n - 4} \right)$
D.
$6/\left( {n - 5} \right)$
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
For $2 \le r \le n,\,\,\,\,\left( {\matrix{
n \cr
r \cr
} } \right) + 2\left( {\matrix{
n \cr
{r - 1} \cr
} } \right) + \left( {\matrix{
n \cr
{r - 2} \cr
} } \right) = $
A.
$\left( {\matrix{
{n + 1} \cr
{r - 1} \cr
} } \right)$
B.
$2\left( {\matrix{
{n + 1} \cr
{r + 1} \cr
} } \right)$
C.
$2\left( {\matrix{
{n + 2} \cr
r \cr
} } \right)$
D.
$\left( {\matrix{
{n + 2} \cr
r \cr
} } \right)$
1999
JEE Advanced
MCQ
IIT-JEE 1999
If in the expansion of ${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$ the coefficients of $x$ and ${x^2}$ are $3$ and $-6$ respectively, then $m$ is
A.
6
B.
9
C.
12
D.
24
1998
JEE Advanced
MCQ
IIT-JEE 1998
If ${a_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}},\,\,\,then\,\,\,\sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}}} } $ equals
A.
$\left( {n - 1} \right){a_n}$
B.
$n{a_n}$
C.
${1 \over 2}n{a_n}$
D.
None of the above
1992
JEE Advanced
MCQ
IIT-JEE 1992
The expansion ${\left( {x + {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5}$ is a polynomial of degree
A.
5
B.
6
C.
7
D.
8
1986
JEE Advanced
MCQ
IIT-JEE 1986
If ${C_r}$ stands for ${}^n{C_r},$ then the sum of the series
${{2\left( {{n \over 2}} \right){\mkern 1mu} !{\mkern 1mu} \left( {{n \over 2}} \right){\mkern 1mu} !} \over {n!}}\left[ {C_0^2 - 2C_1^2 + 3C_2^2 - } \right......... + {\left( { - 1} \right)^n}\left( {n + 1} \right)C_n^2\mathop ]\limits^ \sim \,,$
where $n$ is an even positive integer, is equal to
where $n$ is an even positive integer, is equal to
A.
0
B.
${\left( { - 1} \right)^{n/2}}\left( {n + 1} \right)$
C.
${\left( { - 1} \right)^{n/2}}\left( {n + 2} \right)$
D.
${\left( { - 1} \right)^n}n$
1983
JEE Advanced
MCQ
IIT-JEE 1983
Given positive integers $r > 1,\,n > 2$ and that the coefficient of $\left( {3r} \right)$th and $\left( {r + 2} \right)$th terms in the binomial expansion of ${\left( {1 + x} \right)^{2n}}$ are equal. Then
A.
$n = 2r$
B.
$n = 2r + 1$
C.
$n = 3r$
D.
none of these
1983
JEE Advanced
MCQ
IIT-JEE 1983
The coefficient of ${x^4}$ in ${\left( {{x \over 2} - {3 \over {{x^2}}}} \right)^{10}}$ is
A.
${{{405} \over {256}}}$
B.
${{{504} \over {259}}}$
C.
${{{450} \over {263}}}$
D.
none of these