JEE Advanced 2022 Paper 1 Online

MCQ (More than One Correct Answer)

-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 ?

$$T_{20}=1604$$

$$\sum\limits_{k=1}^{20} T_{k}=10510$$

$$T_{30}=3454$$

$$\sum\limits_{k=1}^{30} T_{k}=35610$$

JEE Advanced 2021 Paper 1 Online

MCQ (More than One Correct Answer)

-2

For any positive integer n, let S_n : (0, $$\infty$$) $$\to$$ R be defined by $${S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)} $$, where for any x $$\in$$ R, $${\cot ^{ - 1}}(x) \in (0,\pi )$$ and $${\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$. Then which of the following statements is (are) TRUE?

$${S_{10}}(x) = {\pi \over 2} - {\tan ^{ - 1}}\left( {{{1 + 11{x^2}} \over {10x}}} \right)$$, for all x > 0

$$\mathop {\lim }\limits_{n \to \infty } \cot ({S_n}(x)) = x$$, for all x > 0

The equation $${S_3}(x) = {\pi \over 4}$$ has a root in (0, $$\infty$$)

$$tan({S_n}(x)) \le {1 \over 2}$$, for all n $$\ge$$ 1 and x > 0

JEE Advanced 2013 Paper 1 Offline

MCQ (More than One Correct Answer)

-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)

1056

1088

1120

1332

IIT-JEE 2008 Paper 1 Offline

MCQ (More than One Correct Answer)

-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,

$${S_n} < {\pi \over {3\sqrt 3 }}$$

$${S_n} > {\pi \over {3\sqrt 3 }}$$

$${T_n} < {\pi \over {3\sqrt 3 }}$$

$${T_n} > {\pi \over {3\sqrt 3 }}$$

JEE Advanced Subjects