For any positive integer $n$, define $f_n:\left(0, \infty \right)\to R$ as $f_n\left(x\right)={\sum }_{i=1}^n{\tan }^{-1}\left(\frac{1}{1+\left(x+j\right)\left(x+j-1\right)}\right)$ for all $x\in \left(0,\infty \right)$ . $($Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$$)$ Then, which of the following statement(s) is (are) TRUE?

A farmer $F_1$ has a land in the shape of a triangle with vertices at $P\left(\mathrm{0,\; 0}\right),Q\left(\mathrm{1,\; 1}\right)$ and $R\left(\mathrm{2,\; 0}\right).$ From this land, a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y=x^n, n>1$. If the area of the region taken away by the farmer $F_2$ is exactly $\mathrm{30\%}$ of the area of $∆PQR,$ then the value of $n$ is

A signal which can be green or red with probability \(\frac{4}{5}\) and \(\frac{1}{5}\) respectively, is received by station \(A\) and then transmitted to station B. The probability of each station receiving the signal correctly is \(\frac{3}{4}\). If the signal received at station \(B\) is green, then the probability that the original signal green is

Suppose four distinct positive numbers \(a_1, a_2, a_3\) and \(a_4\) are in GP. Let \(b_1=a_1\), \(b_2=b_1+a_2, b_3=b_2+a_3\) and \(b_4=b_3+a_4\).
Statement 1 The numbers \(b_1, b_2, b_3\) and \(b_4\) are neither in AP nor in GP.
Statement 2 The numbers \(b_1, b_2, b_3\) and \(b_4\) are in HP.

Let $\left|X\right|$ denote the number of elements in set $X.$ Let $S=\left\{1, 2, 3, 4, 5, 6\right\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S,$ then the number of ordered pairs $\left(A, B\right)$ such that $1\le \left|B\right|<\left|A\right|,$ equals

For every twice differentiable function $f:\mathbb{R}\to \left[-2, 2\right] \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\left(f\left(0\right)\right)}^2+{\left(f^{\prime }\left(0\right)\right)}^2=85$ , which of the following statement(s) is (are) TRUE?

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) and \(\mathbf{d}\) are the unit vectors such that \((\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1\) and \(\mathbf{a} \cdot \mathbf{c}=\frac{1}{2}\), then

Let \(f:(-1,1) \rightarrow I R\) be such that \(f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}\) for \(\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\). Then the value (s) of \(f\left(\frac{1}{3}\right)\) is (are)

The carboxyl functional group \((-\mathrm{COOH})\) is present in

In the experiment to determine the speed of sound using a resonance column:

Based on VSEPR theory, the number of 90 degree \(\mathrm{F}-\mathrm{Br}-\mathrm{F}\) angles in \(\mathrm{BrF}_5\) is

Consider a reaction \(A+R \rightarrow\) Product. The rate of this reaction is measured to be \(k[A][R]\). At the start of the reaction, the concentration of \(R,[R]_0\), is 10 -times the concentration of \(A,[A]_0\). The reaction can be considered to be a pseudo first order reaction with assumption that \(k[R]=k^{\prime}\) is constant. Due to this assumption, the relative error (in \%) in the rate when this reaction is \(40 \%\) complete, is _______
[ \(k\) and \(k^{\prime}\) represent corresponding rate constants]

For any two points \(M\) and \(N\) in the \(X Y\)-plane, let \(\overrightarrow{M N}\) denote the vector from \(M\) to \(N\), and \(\overrightarrow{0}\) denote the zero vector. Let \(P, Q\) and \(R\) be three distinct points in the \(X Y\)-plane. Let \(S\) be a point inside the triangle \(\triangle P Q R\) such that \(\overrightarrow{S P}+5 \overrightarrow{S Q}+6 \overrightarrow{S R}=\overrightarrow{0}\). Let \(E\) and \(F\) be the mid-points of the sides \(P R\) and \(Q R\), respectively. Then the value of \(\frac{\text { length of the line segment } E F}{\text { length of the line segment } E S}\) is ________