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1

### 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
2

### 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
3

### 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
A
0
B
$${\left( { - 1} \right)^{n/2}}\left( {n + 1} \right)$$
C
$${\left( { - 1} \right)^{n/2}}\left( {n + 2} \right)$$
D
$${\left( { - 1} \right)^n}n$$
4

### 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

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