1
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an arithmetic progression with $$a_{1}=7$$ and common difference 8. Let $$T_{1}, T_{2}, T_{3}, \ldots$$ be such that $$T_{1}=3$$ and $$T_{n+1}-T_{n}=a_{n}$$ for $$n \geq 1$$. Then, which of the following is/are TRUE ?

A
$$T_{20}=1604$$
B
$$\sum\limits_{k=1}^{20} T_{k}=10510$$
C
$$T_{30}=3454$$
D
$$\sum\limits_{k=1}^{30} T_{k}=35610$$
2
JEE Advanced 2013 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$$ Then $${S_n}$$can take value(s)
A
1056
B
1088
C
1120
D
1332
3
IIT-JEE 2008 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
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
$${T_n} > {\pi \over {3\sqrt 3 }}$$
4
IIT-JEE 1999
MCQ (More than One Correct Answer)
+3
-0.75
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
$$a\left( {200} \right) > 100$$
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