Let $\alpha , \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations $x+2y+z=7$ $x+\alpha z=11$ $2x-3y+\beta z=\gamma$ Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	If $\beta =\frac{1}{2}\left(7\alpha -3\right)$ and $\gamma =28$ then the system has	$\left(1\right)$	a unique solution
$\left(\mathrm{Q}\right)$	If $\beta =\frac{1}{2}\left(7\alpha -3\right)$ and $\gamma \ne 28$, then the system has	$\left(2\right)$	no solution
$\left(\mathrm{R}\right)$	If $\beta \ne \frac{1}{2}\left(7\alpha -3\right)$ where $\alpha =1$ and $\gamma \ne 28$, then the system has	$\left(3\right)$	infinitely many solutions
$\left(\mathrm{S}\right)$	If $\beta \ne \frac{1}{2}\left(7\alpha -3\right)$ where $\alpha =1$ and $\gamma =28$, then the system has	$\left(4\right)$	$x=11, y=-2$ and $z=0$ as a solution
		$\left(5\right)$	$x=-15, y=4$ and $z=0$ as a solution

The correct option is:

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2+3z+4z^2}{2-3z+4z^2}$ is a real number, then the value of ${\left|z\right|}^2$ is ______.

Axis of a parabola is \(y=x\) and vertex and focus are at a distance \(\sqrt{2}\) and \(2 \sqrt{2}\) respectively, from the origin. Then, equation of the parabola is

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Consider the lines: \(L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}\)Question:
The shortest distance between \(L_1\) and \(L_2\) is

Let \(\vec{w}=\hat{i}+\hat{j}-2 \hat{k}\), and \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}\) be two vectors such that \(\vec{u} \times \vec{v}=\vec{w}\) and \(\vec{v} \times \vec{w}=\vec{u}\). Let \(\alpha, \beta, \gamma\) and \(t\) be real numbers such that \(\vec{u}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k},-t \alpha+\beta+\gamma=0, \alpha-t \beta+\gamma=0\), and \(\alpha+\beta-t \gamma=0\).
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	\(|\vec{v}|^2\) is equal to	(1)	0
(Q)	If \(\alpha=\sqrt{3}\),then \(\gamma^2\) is equal to	(2)	1
(R)	If \(\alpha=\sqrt{3}\),then \((\beta+\gamma)^2\) is equal to	(3)	2
(S)	If \(\alpha=\sqrt{2}\),then \(t+3\) is equal to	(4)	3
		(5)	5

Let \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of $D$ is

Let $\stackrel{r}{7\overbrace{5\dots 5}7}$ denote the $\left(r+2\right)$ digit number where the first and the last digits are $7$ and the remaining $r$ digits are $5$. Consider the sum $S=77+757+7557+\dots +\stackrel{98}{7\overbrace{5\dots 5}7}$. If $S= \frac{7{\displaystyle \stackrel{99}{\overbrace{5\dots 5 }}}7+m}{n}$, where $m$ and $n$ are natural numbers less than $3000$, then the value of $m+n$ is

The maximum number of possible isomers (including stereoisomers) which may be formed on mono-bromination of $1$-methylcyclohex- $1$-ene using ${\mathrm{Br}}_2$ and $\mathrm{UV}$ light is ___.

An infinitely long thin wire, having a uniform charge density per unit length of \(5 \mathrm{nC} / \mathrm{m}\), is passing through a spherical shell of radius \(1 \mathrm{~m}\), as shown in the figure. A \(10 \mathrm{nC}\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points \(\mathrm{P}\) and \(\mathrm{R}\), in Volt, is _______ .
[Given: In SI units \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9, \ln 2=0.7\). Ignore the area pierced by the wire.]

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An aqueous solution of a mixture of two inorganic salts, when treated with dilute \(H C\) ?, gave a precipitate \((\boldsymbol{P})\) and a filtrate \((\boldsymbol{Q})\). The precipitate \(\boldsymbol{P}\) was found to dissolve in hot water. The filtrate \((\boldsymbol{Q})\) remained unchanged, when treated with \(\mathrm{H}_{2} \mathrm{~S}\) in a dilute mineral acid medium. However, it gave a precipitate \((\boldsymbol{R})\) with \(H_{2} S\) in an ammoniacal medium. The precipitate \(\boldsymbol{R}\) gave a coloured solution \((\boldsymbol{S})\), when treated with \(\mathrm{H}_{2} \mathrm{O}_{2}\) in an aqueous \(\mathrm{NaOH}\) medium.

Question:
The precipitate \(\boldsymbol{P}\) contains

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.\) Then which of the following statements is TRUE?

One end of a taut string of length 3m along the x axis is fixed at x = 0. The speed of the waves in the string is $100\mathrm{ }\mathrm{m}{\mathrm{s}}^{-1}$ . The other end of the string is vibrating in the y direction so that stationary waves are set up in the string. The possible waveform(s) of these stationary waves is (are)

The correct order of acidity for the following compounds is :