Question 1: Palette 1/15

JEE Advanced 2020 MSQ

For non-negative integers s and r, let

$\left( {\matrix{ s \cr r \cr } } \right) = \left\{ {\matrix{ {{{s!} \over {r!(s - r)!}}} & {if\,r \le \,s,} \cr 0 & {if\,r\, > \,s} \cr } } \right.$

For positive integers m and n, let

$g(m,\,n) = \sum\limits_{p = 0}^{m + n} {{{f(m,n,p)} \over {\left( {\matrix{ {n + p} \cr p \cr } } \right)}}} $

where for any non-negative integer p,

$f(m,n,p) = \sum\limits_{i = 0}^p {\left( {\matrix{ m \cr i \cr } } \right)\left( {\matrix{ {n + i} \cr p \cr } } \right)\left( {\matrix{ {p + n} \cr {p - i} \cr } } \right)} $

Then which of the following statements is/are TRUE?

A

g(m, n) = g(n, m) for all positive integers m, n

B

g(m, n + 1) = g(m + 1, n) for all positive integers m, n

C

g(2m, 2n) = 2g(m, n) for all positive integers m, n

D

g(2m, 2n) = (g(m, n))² for all positive integers m, n

Question 2: Palette 2/15

JEE Advanced 2014 MCQ

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

Question 3: Palette 3/15

JEE Advanced 2010 MCQ

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

A

$\left( {{B_{10}} - {C_{10}}} \right)$

B

${A_{10}}\left( {{B^2}_{10}{C_{10}}{A_{10}}} \right)$

C

$0$

D

${{C_{10}} - {B_{10}}}$

Question 4: Palette 4/15

JEE Advanced 2005 MCQ

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 $\left( {\matrix{ n \cr r \cr } } \right) = {}^n{C_r}$

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

$\left( {\matrix{ {31} \cr {10} \cr } } \right)$

Question 5: Palette 5/15

JEE Advanced 2004 MCQ

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

$\left( {\sqrt 3 ,2} \right]$

Question 6: Palette 6/15

JEE Advanced 2003 MCQ

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

${}^{12}{C_6} + 2$

Question 7: Palette 7/15

JEE Advanced 2002 MCQ

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

Question 8: Palette 8/15

JEE Advanced 2001 MCQ

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

$6/\left( {n - 5} \right)$

Question 9: Palette 9/15

JEE Advanced 2000 MCQ

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

$\left( {\matrix{ {n + 2} \cr r \cr } } \right)$

Question 10: Palette 10/15

JEE Advanced 1999 MCQ

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

Question 11: Palette 11/15

JEE Advanced 1998 MCQ

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

Question 12: Palette 12/15

JEE Advanced 1992 MCQ

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

Question 13: Palette 13/15

JEE Advanced 1986 MCQ

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

A

0

B

${\left( { - 1} \right)^{n/2}}\left( {n + 1} \right)$

C

${\left( { - 1} \right)^{n/2}}\left( {n + 2} \right)$

D

${\left( { - 1} \right)^n}n$

Question 14: Palette 14/15

JEE Advanced 1983 MCQ

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

Question 15: Palette 15/15

JEE Advanced 1983 MCQ

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