The value of
\(6+\log _{\frac{3}{2}}\)\(\left(\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}}} \cdots}}\right)\) is

Let $f:\mathbb{R}\to \mathbb{R}\mathbb{ }\mathrm{a}\mathrm{n}\mathrm{d} g:\mathbb{R}\to \mathbb{R}$ be two non-constant differentiable functions. If $f^{\prime }\left(x\right)=\left(e^{\left(f\left(x\right)\right)-g\left(x\right)}\right)g^{\prime }\left(x\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} x\in \mathbb{R}$, and $f\left(1\right)=g\left(2\right)=1$ then which of the following statement(s) is (are) TRUE?

Let \(S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\)\(\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}\), and \(T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}\). Then which of the following statements is (are) TRUE?

Let $l_1,l_2...l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1,w_2$, $\dots ,w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1{d}_2=10$. For each $i=1,2,3..........100$, let $R_i$ be a rectangle with length $l_i$, width $w_i$ and area $A_i$.If $A_{51}-A_{50}=1000$, then the value of $A_{100}-A_{90}$ is _______.

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $I$ be the $3\times 3$ identity matrix. Let $E$ and $F$ be two $3\times 3$ matrices such that $(I-EF)$ is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is (are) TRUE ?

Let \(\omega\) be the complex number \(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\). Then the number of distinct complex number \(z\) satisfying \(\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=0\) is equal to

If \(f^{\prime \prime}(x)=-f(x)\), where \(f(x)\) is a continuous double differentiable function and \(g(x)=f^{\prime}(x)\). If \(F(x)=\left(f\left(\frac{x}{2}\right)\right)^2+\left(g\left(\frac{x}{2}\right)\right)^2\) and \(F(5)=5\), then \(F(10)\) is

Consider the following four compounds I, II, III, and IV.

Choose the correct statement(s).

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The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
In a CO molecule, the distance between \(\mathrm{C}\) (mass \(=12 \mathrm{amu}\) ) and \(\mathrm{O}\) (mass = \(16 \mathrm{amu}\) ), where \(1 \mathrm{amu}=\frac{5}{3} \times 10^{-27} \mathrm{~kg}\), is close to

The center of a disk of radius \(r\) and mass \(m\) is attached to a spring of spring constant \(k\), inside a ring of radius \(R>r\) as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as \(T=\frac{2 \pi}{\omega}\). The correct expression for \(\omega\) is ( \(g\) is the acceleration due to gravity):

Consider an $\mathrm{LC}$ circuit, with inductance $L=0.1 \mathrm{H}$ and capacitance $C={10}^{-3} \mathrm{F}$, kept on a plane. The area of the circuit is $1 {\mathrm{m}}^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_0+\beta \mathrm{t}$ with $\beta =0.04 \mathrm{T} {\mathrm{s}}^{-1}$. The maximum magnitude of the current in the circuit is____$\mathrm{m} \mathrm{A}$.

A transparent thin film of uniform thickness and refractive index ${\mathrm{n}}_1=1.4$ is coated on the convex spherical surface of radius R at one end of a long solid glass cylinder of refractive index ${\mathrm{n}}_2=1.5$ , as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance f₁ from the film, while rays of light traversing from glass to air get focused at distance f₂ from the film. Then

An infinitely long wire, located on the \(z\)-axis, carries a current \(I\) along the \(+z\)-direction and produces the magnetic field \(\vec{B}\). The magnitude of the line integral \(\int \vec{B} \cdot \overrightarrow{d l}\) along a straight line from the point \((-\sqrt{3} a, a, 0)\) to \((a, a, 0)\) is given by
[ \(\mu_0\) is the magnetic permeability of free space.]