Words of length$10$ are formed using the letters A, B, C, D, E, F, G, H, I, J. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, $\frac{y}{9x}=$

The number of $4$-digit integers in the closed interval $\left[2022,4482\right]$ formed by using the digits $0,2,3,4,6,7$ is ______.

If $g\left(x\right)={\int }_{\sin x}^{\sin 2x}{\sin }^{-1}tdt,$  then

Let \(P(6,3)\) be a point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). If the normal at the point \(P\) intersects the \(X\)-axis at \((9,0)\), then the eccentricity of the hyperbola is

Let \(\omega \neq 1\) be a cube root of unity and \(S\) be the set of all non-singular matrices of the form \(\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\), where each of \(a, b\) and \(c\) is either \(\omega\) or \(\omega^2\). Then, the number of distinct matrices in the set \(S\) is

Let \(\mathbb{R}^3\) denote the three-dimensional space. Take two points \(P=(1,2,3)\) and \(Q=(4,2,7)\). Let \(\operatorname{dist}(X, Y)\) denote the distance between two points \(X\) and \(Y\) in \(\mathbb{R}^3\). Let
\(\begin{gathered}S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} .\end{gathered}\)
Then which of the following statements is (are) TRUE?

Let $\mathrm{X}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{Y}$ be two arbitrary, $3\mathrm{ }\times 3$ , non - zero, skew - symmetric matrices and Z be an arbitrary $3\mathrm{ }\times 3$ , non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

A small object of uniform density rolls up a curved surface with an initial velocity \(v\). It reaches up to a maximum height of \(\frac{3 v^2}{4 g}\) with respect to the initial\(
\text { position. The object is }
\)

Young’s modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck’s constant $h$ and the speed of light $c$, as $Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$. Which of the following is the correct option?

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Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
The vapour pressure of the solution \(M\) is

Among the following, the correct statement(s) for electrons in an atom is(are)

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question : The relation between $\left[{ϵ}_0\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[{\mu }_0\right]$ is

Two vectors $\overrightarrow{A} \mathrm{a}\mathrm{n}\mathrm{d} \overrightarrow{B}$ are defined as $\overrightarrow{A}=a\widehat{i} and \overrightarrow{B}=a\left(\cos \omega t \widehat{i}+\sin \omega t\widehat{j}\right)$, where a is a constant and $\omega =\frac{\pi }{6} rad s^{-1}.$ If $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{3}\left|\overrightarrow{A}-\overrightarrow{B}\right|$ at time $t=\tau$ for the first time, the value of $\tau$ in seconds, is_______