Of the three independent events ${\text{E}}_1$, ${\text{E}}_2$ and ${\text{E}}_3$, the probability that only ${\text{E}}_1$ occurs is $\alpha$, only ${\text{E}}_2$ occurs is $\beta$ and only  ${\text{E}}_3$ occurs is $\gamma$. Let the probability $p$ that none of events ${\text{E}}_1$, ${\text{E}}_2$ or ${\text{E}}_3$ occurs satisfy the equations $\left(\alpha -2\beta \right)p=\alpha \beta$ and $\left(\beta -3\gamma \right)p = 2\beta \gamma$. All the given probabilities are assumed to lie in the interval $\left(\text{0, 1}\right)$.
Then   $\frac{{\text{Probability of occurrence of E}}_1}{{\text{Probability of occurrence of E}}_3}=$ ​

Equation of common tangent of \(y=x^2, y=-x^2+4 x-4\) is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line \(4 x-\) \(5 y=20\) to the circle \(x^{2}+y^{2}=9\) is

In a study about a pandemic, data of $900$ persons was collected. It was found that $190$ persons had symptom of fever, $220$ persons had symptom of cough, $220$ persons had symptom of breathing problem, $330$ persons had symptom of fever or cough or both, $350$ persons had symptom of cough or breathing problem or both, $340$ persons had symptom of fever or breathing problem or both, $30$ persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these $900$ persons, then the probability that the person has at most one symptom is ______.

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 $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three non - coplanar unit vectors such that the angle between every pair of them is $\frac{\pi }{3}.$ If $\overrightarrow{a} \times \overrightarrow{b}+ \overrightarrow{b} \times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+r\overrightarrow{c}$ , where p, q and r are scalars, then the value of $\frac{p^2+2q^2+r^2}{q^2}$ is _______

If the function \(f(x)=x^3+e^{\frac{x}{2}}\) and \(g(x)=f^{-1}(x)\), then the value of \(g^{\prime}(1)\) is

The freezing point (in \({ }^{\circ} \mathrm{C}\) ) of solution containing \(0.1 \mathrm{~g}\) of \(\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]\) (molecular weight 329\()\) in \(100 \mathrm{~g}\) of water \(\left(K_f=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\) is

Reaction of \(\mathrm{Br}_2\) with \(\mathrm{Na}_2 \mathrm{CO}_3\) in aqueous solution gives sodium bromide and sodium bromate with evolution of \(\mathrm{CO}_2\) gas. The number of sodium bromide molecules involved in the balanced chemical equation is

A compound ${\mathrm{H}}_2\mathrm{X}$ with molar weight of 80g is dissolved in a solvent having density of $0.4\mathrm{ }\mathrm{g}\mathrm{m}{\mathrm{l}}^{–1}$ . Assuming no change in volume upon dissolution, the molality of a 3.2 molar solution is

The function $\mathit{\text{f}}\left(x\right)=2\left|x\right|+\left|x+2\right|-\left|\left|x+2\right|-2\left|x\right|\right|$ has a local minimum or a local maximum at $x=$

List I contains four combinations of two lenses ( 1 and 2 ) whose focal lengths (in cm ) are indicated in the figures. In all cases, the object is placed 20 cm from the first lens on the left, and the distance between the two lenses is 5 cm . List II contains the positions of the final images.

	List - I		List - II
(I)		(P)	Final image is formed at 7.5 cm on the right side of lens 2.
(II)		(Q)	Final image is formed at 60.0 cm on the right side of lens 2.
(III)		(R)	Final image is formed at 30.0 cm on the left side of lens 2.
(IV)		(S)	Final image is formed at 6.0 cm on the right side of lens 2.
		(T)	Final image is formed at 30.0 cm on the right side of lens 2.

Which one of the following options is correct?

For the reaction, \(\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)\), the plot of \(\ln \frac{p_{\mathbf{Z}}}{p^{\theta}}\) versus \(\frac{10^{4}}{T}\) is given below (in solid line), where \(p_{\mathbf{Z}}\) is the pressure (in bar) of the gas \(\mathbf{Z}\) at temperature \(T\) and \(p^{\theta}=1\) bar.

(Given, \(\frac{\mathrm{d}(\ln K)}{\mathrm{d}\left(\frac{1}{T}\right)}=-\frac{\Delta H^{\theta}}{R}\), where the equilibrium constant, \(K=\frac{p_{z}}{p^{\theta}}\) and the gas constant, \(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) )
The value of $∆{\mathrm{S}}^{\mathrm{o}}$ (in $\mathrm{kJ} {\mathrm{mol}}^{-1}$) for the given reaction, at $1000 \mathrm{K}$ is ___.