Paragraph: There are five students $S_1, S_2, S_3, S_4$ and $S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i, i = \mathrm{1,2},\mathrm{3,4},5.$ But, on the examination day, the five students are randomly allotted the five seats
Question : The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1,$ and NONE of the remaining students gets the seat previously allotted to him/her is

The letters of the word \(\mathrm{COCHIN}\) are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word \(\mathrm{COCHIN}\) is

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Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
The sum \(V_1+V_2+\ldots+V_n\) is

The product of all positive real values of $x$ satisfying the equation $x^{\left(16{\left({\log }_{5}x\right)}^3-68{\log }_{5}x\right)}=5^{-16}$ is

Let $M=\left(a_{ij}\right), i, j\in \{1,2,3\}$, be the $3\times 3$ matrix such that $a_{ij}=1$ if $j+1$ is divisible by $i$, otherwise $a_{ij}=0$. Then which of the following statements is(are) true?

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\).

Question: Consider the statements:

\(P\) : There exists some \(x \in \mathrm{R}\) such that \(f(x)+2 x\)

\(=2\left(1+x^{2}\right)\)

\(Q\) : There exists some \(x \in \mathrm{R}\) such that \(2 f(x)+1\) \(=2 x(1+x)\)

Then

Let $a$ and $b$ be positive real numbers such that $a>1$ and $b<a.$ Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ Suppose the tangent to the hyperbola at $P$ passes through the point $\left(1,0\right),$ and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$ -axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

Let ${\mathrm{E}}_1\left(\mathrm{r}\right),\mathrm{ }{\mathrm{E}}_2\left(\mathrm{r}\right)$ and ${\mathrm{E}}_3\left(\mathrm{r}\right)$ be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density $\mathrm{\lambda }$ , and an infinite plane with uniform surface charge density $\mathrm{\sigma }$ . If ${\mathrm{E}}_1\left({\mathrm{r}}_0\right)={\mathrm{E}}_2\left({\mathrm{r}}_0\right)={\mathrm{E}}_3\left({\mathrm{r}}_0\right)$ at a given distance ${\mathrm{r}}_0$ , then

Perpendiculars are drawn from points on the line $\frac{\text{x}+2}{2}=\frac{\text{y}+1}{-1}=\frac{\text{z}}{3}$  to the plane $x+y+z=3$. The feet of perpendiculars lie on the line

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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .Question:
The number of matrices \(A\) in \(A\) for which the system of linear equations \(A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) has a unique solution, is

For the reaction, \(\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)\), the plot of \(\ln \frac{p_{\mathbf{Z}}}{p^{\theta}}\) versus \(\frac{10^{4}}{T}\) is given below (in solid line), where \(p_{\mathbf{Z}}\) is the pressure (in bar) of the gas \(\mathbf{Z}\) at temperature \(T\) and \(p^{\theta}=1\) bar.

(Given, \(\frac{\mathrm{d}(\ln K)}{\mathrm{d}\left(\frac{1}{T}\right)}=-\frac{\Delta H^{\theta}}{R}\), where the equilibrium constant, \(K=\frac{p_{z}}{p^{\theta}}\) and the gas constant, \(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) )
The value of standard enthalpy, $∆{\mathrm{H}}^{\mathrm{o}}$ (in $\mathrm{kJ} {\mathrm{mol}}^{-1}$) for the given reaction is ___.

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A fair die is tossed repeatedly until a six is obtained. Let \(X\) denotes the number of tosses required.Question:
The conditional probability that \(X \geq 6\) given \(X>3\) equals

A container has a base of $50\mathrm{cm}\times 5\mathrm{cm}$ and height $50\mathrm{cm}$, as shown in the figure. It has two parallel electrically conducting walls each of area $50\mathrm{cm}\times 50\mathrm{cm}$. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant $3$at a uniform rate of $250{\mathrm{cm}}^3{\mathrm{s}}^{-1}$. What is the value of the capacitance of the container after $10$seconds?
[Given: Permittivity of free space ${\epsilon }_0=9\times {10}^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]