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1

IIT-JEE 2008

MCQ (More than One Correct Answer)
Let $${S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}} $$ and $${T_n} = \sum\limits_{k = 0}^{n - 1} {{n \over {{n^2} + kn + {k^2}}}} $$ for $$n$$ $$=1, 2, 3, ............$$ Then,
A
$${S_n} < {\pi \over {3\sqrt 3 }}$$
B
$${S_n} > {\pi \over {3\sqrt 3 }}$$
C
$${T_n} < {\pi \over {3\sqrt 3 }}$$
D
$${T_n} > {\pi \over {3\sqrt 3 }}$$
2

IIT-JEE 1999

MCQ (More than One Correct Answer)
For a positive integer $$n$$, let
$$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$$. Then
A
$$a\left( {100} \right) \le 100$$
B
$$a\left( {100} \right) > 100$$
C
$$a\left( {200} \right) \le 100$$
D
$$a\left( {200} \right) > 100$$
3

IIT-JEE 1993

MCQ (More than One Correct Answer)
For $$0 < \phi < \pi /2,$$ if
$$x = $$$$\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,\,\,\,\,z = \sum\limits_{n = 0}^{} {{{\cos }^{2n}}\phi {{\sin }^{2n}}\phi } } } \infty $$ then
A
$$xyz = xz + y$$
B
$$xyz = xy + z$$
C
$$xyz = x + y + z$$
D
$$xyz = yz + x$$
4

IIT-JEE 1988

MCQ (More than One Correct Answer)
If the first and the $$(2n-1)$$st terms of an A.P., a G.P. and an H.P. are equal and their $$n$$-th terms are $$a,b$$ and $$c$$ respectively, then
A
$$a = b = c$$
B
$$a \ge b \ge c$$
C
$$a + c = b$$
D
$$ac - {b^2} = 0$$

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