Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SP{S}_1=\alpha$, with $\alpha <\frac{\pi }{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_{1}P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $S{P}_1$, and $\beta =S_{1}P$. Then the greatest integer less than or equal to $\frac{\beta \delta }{9}\sin \frac{\alpha }{2}$ is _____ .

Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by
\(f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases}\) And
\(g(n)=\left\{\begin{array}{cc}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n <0\end{array}\right.\)
Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\).
Then which of the following statements is (are) TRUE?

If the distance between the plane \(A x-2 y+z=d\) and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \quad\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\), then \(|d|\) is

Let S be the set of all non - zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1 and x_2$ satisfying the inequality $\left|x_1-x_2\right|<1.$ Which of the following intervals is(are) a subset(s) of S?

Consider three planes
\[
P_1: x-y+z=1, P_2: x+y-z=-1
\]
and \(P_3: x-3 y+3 z=2\)
Let \(L_1, L_2\) and \(L_3\) be the lines of intersection of the planes \(P_2\) and \(P_3, P_3\) and \(P_1, P_1\) and \(P_2\), respectively.
Statement 1 Atleast two of the lines \(L_1, L_2\) and \(L_3\) are non-parallel.
Statement 2 The three planes do not have a common point.

Let $P_1$ and $P_2$ be two planes given by $P_1:10x+15y+12z-60=0,$ $P_2:-2x+5y+4z-20=0$. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_1$ and $P_2$?

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Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

The value of \(r\) is

In the complex acetylbromidodicarbonylbis(triethylphosphine)iron (II), the number of $\mathrm{F}\mathrm{e}-\mathrm{C}$ bond (s) is

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

	List-I		List-II
(I)	${10}^{-3} \mathrm{kg}$ of water $100 °\mathrm{C}$ is converted to steam at the same temperature, at a pressure of ${10}^5\mathrm{Pa}$. The volume of the system changes from ${10}^{-6}{\mathrm{m}}^3$ to ${10}^{-3}{\mathrm{m}}^3$ in the process. Latent heat of water $=2250 \mathrm{kJ} {\mathrm{kg}}^{-1}$	(P)	$2 \mathrm{kJ}$
(II)	$0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 \mathrm{K}$ undergoes an isobaric expansion to volume $3 V$. Assume $8.0 \mathrm{J} {\mathrm{mol}}^{–1} {\mathrm{K}}^{–1}$	(Q)	$7 \mathrm{kJ}$
(III)	One mole of a monoatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3}{\mathrm{m}}^3$ and pressure $2 \mathrm{kPa}$ to volume $\frac{V}{8}$.	(R)	$4 \mathrm{kJ}$
(IV)	Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \mathrm{kJ}$ of heat and undergoes isobaric expansion.	(S)	$5 \mathrm{kJ}$
		(T)	$3 \mathrm{kJ}$

Which one of the following options is correct?

Considering the reaction sequence given below, the correct statement(s) is(are)

Among the triatomic molecules/ions, $\mathrm{B}\mathrm{e}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{N}}_3^{-},\mathrm{ }{\mathrm{N}}_2\mathrm{O},\mathrm{ }{\mathrm{N}\mathrm{O}}_2^{+},\mathrm{ }{\mathrm{O}}_3,\mathrm{S}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{I}\mathrm{C}\mathrm{l}}_2^{-},\mathrm{ }{\mathrm{I}}_3^{-}$ and $\mathrm{X}\mathrm{e}{\mathrm{F}}_2$ , the total number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution from the d-orbital(s) is
[Atomic number: S = 16, Cl = 17, I = 53 and Xe = 54]

The slope of the tangent to the curve ${\left(y-x^5\right)}^2=\text{x}{\left(1+{\text{x}}^2\right)}^2$ at the point (1, 3) is ________.

Let $f^{\prime }\left(x\right)= \frac{192x^3}{2+{\sin }^4\pi x}$ for all $x\in \mathbb{ }\mathbb{R}$ with $f\left(\frac{1}{2}\right)=0.$ If $m \le \underset{\frac{1}{2}}{\overset{1}{\int }}f\left(x\right)dx \le M,$ then the possible values of m and M are