A line with positive direction cosines passes through the point \(P(2,-1,2)\) and makes equal angles with the coordinate axes. The line meets the plane \(2 x+y+z=9\) at point \(Q\). The length of the line segment \(P Q\) equals

Which of the following inequalities is/are TRUE?

For $a\in R, \left|a\right|>1,$ let $\underset{n\to \infty }{\lim }\left(\frac{1+\sqrt[3]{2}+\dots +\sqrt[3]{n}}{n^{7/3}\left(\frac{1}{{\left(an+1\right)}^2}+\frac{1}{{\left(an+2\right)}^2}+\dots +\frac{1}{{\left(an+n\right)}^2}\right)}\right)=54.$ Then the possible value(s) of $a$ is/are:

Let \(f(x)=\frac{x}{\left(1+x^n\right)^{1 / n}}\) for \(n \geq 2\) and \(g(x)=\underbrace{(f \circ f o \ldots o f)}_{f \text { occurs } n \text { times }}(x)\). Then, \(\int x^{n-2} g(x) d x\) equals

Paragraph:
Read the following passage and answer the questions.
Question:
The value of \(\left[\begin{array}{lll}3 & 2 & 0\end{array}\right]\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]\) is

Let $\mathrm{f}\left(\mathrm{x}\right)=\sin \left(\frac{\mathrm{\pi }}{6}\sin \left(\frac{\mathrm{\pi }}{2}\sin \mathrm{x}\right)\right)$, for all $\mathrm{x}\in R$ and $\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{\pi }}{2}\sin \mathrm{x}$, for all $\mathrm{x}\in R$. Let $\left(\mathrm{f}o\mathrm{g}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)$ and $\left(\mathrm{g}o\mathrm{f}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$. Then which of the following is (are) true?

Let \(\mathbb{R}^2\) denote \(\mathbb{R} \times \mathbb{R}\). Let \(S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} .\)
Then which of the following statements is (are) TRUE?

The synthesis of 3-octyne is achieved by adding a bromoalkane into a mixture of sodium amide and an alkyne. The bromoalkane and alkyne respectively are

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius $1$.

(There are two questions based on $\mathrm{PARAGRAPH} " 1 "$, the question given below is one of them)
Let $\alpha$ be the area of the triangle $ABC$. Then the value of $(64\alpha {)}^2$ is

A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=\mathrm{}\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case will the particle move in a straight line with a constant velocity?

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of $D$ is

Upon treatment with ammoniacal ${\text{H}}_2\text{S}$, the metal ion that precipitates as a sulphide is

Consider a triangle \(A B C\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to vertices \(A, B\) and \(C\) respectively. Suppose \(a=6, b=10\) and the area of the triangle is \(15 \sqrt{3}\). If \(\angle A C B\) is obtuse and if \(r\) denotes the radius of the incircle of the triangle, then \(r^2\) is equal to