Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow(0,4)\) be functions defined by \(f(x)=\log _e\left(x^2+2 x+4\right)\), and \(g(x)=\frac{4}{1+e^{-2 x}}\)
Define the composite function \(f \circ g^{-1}\) by \(\left(f \circ g^{-1}\right)(x)=\left(g^{-1}(x)\right)\), where \(g^{-1}\) is the inverse of the function \(g\).
Then the value of the derivative of the composite function \(f \circ g^{-1}\) at \(x=2\) is ________ .

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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
If \(f(-10 \sqrt{2})=2 \sqrt{2}\), then \(f^{\prime \prime}(-10 \sqrt{2})\) is equal to

Four fair dice \(D_{1}, D_{2}, D_{3}\) and \(D_{4}\); each having six faces numbered \(1,2,3,4,5\) and 6 are rolled simultaneously. The probability that \(D_{4}\) shows a number appearing on one of \(D_{1}, D_{2}\) and \(\mathrm{D}_{3}\) is

Let $\left|M\right|$ denote the determinant of a square matrix $M$. Let $g:\left[0,\frac{\pi }{2}\right]\to \mathrm{ℝ}$ be the function defined by $g\left(\theta \right)=\sqrt{f\left(\theta \right)-1}+\sqrt{f\left(\frac{\pi }{2}-\theta \right)-1}$ where $f\left(\theta \right)=\frac{1}{2}\left|\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta +\frac{\pi }{4}\right)& \tan \left(\theta -\frac{\pi }{4}\right)\\ \sin \left(\theta -\frac{\pi }{4}\right)& -\cos \frac{\pi }{2}& {\log }_e\left(\frac{4}{\pi }\right)\\ \cot \left(\theta +\frac{\pi }{4}\right)& {\log }_e\left(\frac{\pi }{4}\right)& \tan \pi \end{array}\right|$
Let $p\left(x\right)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g\left(\theta \right)$, and $p\left(2\right)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?

Let \(M\) and \(N\) be two \(3 \times 3\) non-singular skew-symmetric matrices such that \(M N=N M\). If \(P^T\) denotes the transpose of \(P\), then \(M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T\) is equal to

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$ Then which of the following statements is(are) TRUE?

Let \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p^{\prime}(0)\) is

The correct statement(s) about the compound given below is are

Regarding the molecular orbital (MO) energy levels for homonuclear diatomic molecules, the INCORRECT statement(s) is(are)

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Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
Assume that two deuteron nuclei in the core of fusion reactor at temperature \(T\) are moving towards each other, each with kinetic energy \(1.5 \mathrm{kT}\), when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature \(T\) required for them to reach a separation of \(4 \times 10^{-15} \mathrm{~m}\) is in the range

A trinitro compound, $1,3,5$-tris-($4$-nitrophenyl) benzene, on complete reaction with an excess of $\mathrm{Sn}/\mathrm{HCl}$ gives a major product, which on treatment with an excess of ${\mathrm{NaNO}}_2/\mathrm{HCl}$ at $0°\mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$, upon treatment with excess of ${\mathrm{H}}_2\mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$. Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$. The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$. The molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{g} {\mathrm{mol}}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{g} {\mathrm{mol}}^{-1}$.
The total number of carbon atoms and heteroatoms present in one molecule of \(S\) is
[Use: Molar mass (in \(\text{gmol} ^{-1}\) ): \(\text H =1,\text C =12,\text N=14,\text O =16,\) \(\text{Br} =80, \text{Cl} =35.5\) Atoms other than C and H are considered as heteroatoms]

Consider the following volume-temperature \((\mathrm{V}-\mathrm{T})\) diagram for the expansion of 5 moles of an ideal monoatomic gas.

Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence \(\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Z}\) is ________.
[Use the given data: Molar heat capacity of the gas for the given temperature range, \(\mathrm{C}_{\mathrm{V}, \mathrm{m}}=12 \mathrm{~J} \mathrm{~K}^{-1}\) \(\mathrm{mol}^{-1}\) and gas constant, \(\left.\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)

Both \(\left[\mathrm{Ni}(\mathrm{CO})_4\right]\) and \(\left[\mathrm{Ni}(\mathrm{CN})_4\right]^{2-}\) are diamagnetic. The hybridisations of nickel in these complexes, respectively, are