1
AIEEE 2004
+4
-1
Let $$S(K)$$ $$= 1 + 3 + 5... + \left( {2K - 1} \right) = 3 + {K^2}.$$ Then which of the following is true
A
Principle of mathematical induction can be used to prove the formula
B
$$S\left( K \right) \Rightarrow S\left( {K + 1} \right)$$
C
$$S\left( K \right) \ne S\left( {K + 1} \right)$$
D
$$S\left( 1 \right)$$ is correct
2
AIEEE 2004
+4
-1
The coefficient of the middle term in the binomial expansion in powers of $$x$$ of $${\left( {1 + \alpha x} \right)^4}$$ and $${\left( {1 - \alpha x} \right)^6}$$ is the same if $$\alpha$$ equals
A
$${3 \over 5}$$
B
$${10 \over 3}$$
C
$${{ - 3} \over {10}}$$
D
$${{ - 5} \over {3}}$$
3
AIEEE 2004
+4
-1
The coefficient of $${x^n}$$ in expansion of $$\left( {1 + x} \right){\left( {1 - x} \right)^n}$$ is
A
$${\left( { - 1} \right)^{n - 1}}n$$
B
$${\left( { - 1} \right)^n}\left( {1 - n} \right)$$
C
$${\left( { - 1} \right)^{n - 1}}{\left( {n - 1} \right)^2}$$
D
$$\left( {n - 1} \right)$$
4
AIEEE 2004
+4
-1
If $${S_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}}} \,\,and\,\,{t_n} = \sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}},\,}$$then $${{{t_{ n}}} \over {{S_n}}}$$ is equal to
A
$${{2n - 1} \over 2}$$
B
$${1 \over 2}n - 1$$
C
n - 1
D
$${1 \over 2}n$$
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