A tangent \(P T\) is drawn to the circle \(x^{2}+y^{2}=4\) at the point \(P(\sqrt{3}, 1)\). A straight line \(L\), perpendicular to \(P T\) is a tangent to the circle \((x-3)^{2}+y^{2}-1\).

Question: A possible equation of \(L\) is

The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}\) determines a family of circles with

Let \(f(x)\) be a continuously differentiable function on the interval \((0, \infty)\) such that \(f(1)=2\) and \(\underset{t \rightarrow x}{\lim} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1\) for each \(x>0\). Then, for all \(x>0, f(x)\) is equal to

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20\%$ and plant $T_2$ produces $80\%$ of the total computers produced. $7\%$ of computers produced in the factory turn out to be defective. It is known that,  $P$(computer turns out to be defective given that it is produced in plant $T_1$) $=10P$(computer turns out to be defective given that it is produced in plant $T_2$). Where, $P\left(E\right)$ denotes the probability of an event $E.$ A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_2$ is

Three randomly chosen non negative integers $x,y,z$ are found to satisfy the equation  $x+y+z=10$ . Then the probability that $z$ is even, is
 

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 ______.

Let \(L=\lim _{x \rightarrow 0} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}\), \(a>0\). If \(L\) is finite, then

An \(\alpha\)-particle and a proton are accelerated from rest by a potential difference of \(100 \mathrm{~V}\). After this, their de Broglie wavelengths are \(\lambda_\alpha\) and \(\lambda_p\) respectively. The ratio \(\frac{\lambda_p}{\lambda_\alpha}\), to the nearest integer, is

The number of water molecule(s) directly bonded to the metal centre in \(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\) is

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate $\left({\mathrm{KMnO}}_4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\mathrm{\Delta }n=\left(n_1-n_2\right)\ll n_1.$ When they are connected by a tube of small length $l$ and cross-sectional area $S,{\mathrm{KMnO}}_4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order ${\left(\mathrm{\Delta }n\right)}^2$. Which of the following is/are correct? ($k_B$ is the Boltzmann constant)

The cubic unit cell structure of a compound containing cation $\mathrm{M}$ and anion $\mathrm{X}$ is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).

Match the statements/expressions given in Column I with the values given in Column II.