Let \((x, y, z)\) be points with integer coordinates satisfying the system of homogeneous equations \(3 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0\). Then, the number of such points for which \(x^2+y^2+z^2 \leq 100\) is

If \(z\) is any complex number satisfying \(|z-3-2 i| \leq 2\), then the maximum value of \(|2 z-6+5 i|\) is

Let $\mathrm{\alpha }$ and $\mathrm{\beta }$ be the roots of ${\mathrm{x}}^2-\mathrm{x}-1=0,$ with $\mathrm{\alpha }>\mathrm{\beta }.$ For all positive integers $\mathrm{n},$ define
${\mathrm{a}}_{\mathrm{n}}=\frac{{\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}}{\mathrm{\alpha }-\mathrm{\beta }},\mathrm{n}\ge 1$
${\mathrm{b}}_1=1$ and ${\mathrm{b}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}+{\mathrm{a}}_{\mathrm{n}+1},\mathrm{n}\ge 2.$
Then which of the following options is/are correct?

Let \(f(x)=2+\cos x\) for all real \(x\).
Statement I For each real \(t\), there exists a point \(c\) in \([t, t+\pi]\) such that \(f^{\prime}(c)=0\).
Statement II \(f(t)=f(t+2 \pi)\) for each real \(t\).

Let $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

For \(r=0,1, \ldots, 10\), let \(A_r, B_r\) and \(C_r\) denote, respectively, the coefficient of \(x^r\) in the expansions of \((1+x)^{10}\), \((1+x)^{20}\) and \((1+x)^{30}\). Then \(\sum_{r=1}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right)\) is equal to

Let $f\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f\left(0\right)=0, f\left(\frac{\pi }{2}\right)=3$ and $f^{\prime }\left(0\right)=1.$ If $g\left(x\right)={\int }_x^{\frac{\mathrm{\pi }}{2}} \left[f^{\prime }\left(t\right)cosect-\cot t cosect f\left(t\right)\right]dt$ , for $x\in \left(0,\frac{\pi }{2}\right]$ then $\underset{x\to 0}{\lim }g\left(x\right)=$

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of $PQ$ lies in the plane $x+y=3$) lies on the z-axis. Let the distance of $P$ from the x-axis be $5$. If $R$ is the image of $P$ in the xy-plane, then the length of $PR$ is ______.

In dilute aqueous ${\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4$ , the complex diaquodioxalatoferrate (II) is oxidized by ${\mathrm{M}\mathrm{n}\mathrm{O}}_4^{-}$ . For this reaction, the ratio of the rate of change of $\left[{\mathrm{H}}^{+}\right]$ to the rate of change of $\left[{\mathrm{M}\mathrm{n}\mathrm{O}}_4^{-}\right]$ is

The total number of ${\mathrm{sp}}^2$ hybridised carbon atoms in the major product $\mathbf{P}$ (a non-heterocyclic compound) of the following reaction is

Extraction of metal from the ore cassiterite involves

The correct statement about the following disaccharide is

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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Taking the electronic charge as \(e\) and the permittivity as \(\varepsilon_0\), use dimensional analysis to determine the correct expression for \(\omega_p\).