Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2-\alpha x+\beta$ are real and equal, 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 \(\alpha\) and \(\beta\) be the roots of \(x^2-6 x-2=0\) with \(\alpha>\beta\). If \(a_n=\alpha^n-\beta^n\) for \(n \geq 1\), then the value of \(\frac{a_{10}-2 a_8}{2 a_9}\) is

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\mathrm{ℝ}\to \mathrm{ℝ}$ be functions satisfying $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+f\left(x\right)f\left(y\right)\text{ and }f\left(x\right)=xg\left(x\right)$ for all $x,y\in \mathrm{ℝ}.$ If $\underset{x\to 0}{\lim }g\left(x\right)=1,$ then which of the following statements is/are TRUE?

Let $\alpha$ be a positive real number. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\left(\alpha ,\infty \right)\to \mathrm{ℝ}$ be the functions defined by $f\left(x\right)=\sin \left(\frac{\pi x}{12}\right)$ and $g\left(x\right)=\frac{2{\log }_e\left(\sqrt{x}-\sqrt{\alpha }\right)}{{\log }_e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha }}\right)}$.
Then the value of $\underset{x\to {\alpha }^{+}}{\lim }f\left(g\left(x\right)\right)$ is ______.

Let \(P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}\) and \(Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}\) be two sets. Then,

Three lines are given by $\overrightarrow{\mathrm{r}}=\mathrm{\lambda }\hat{\mathrm{i}}, \lambda \in \mathrm{R}$
$\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right),\mathrm{\mu }\in \mathrm{R}$ and
$\overrightarrow{\mathrm{r}}=\mathrm{\nu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+\widehat{\mathrm{k}}\right),\mathrm{\nu }\in \mathrm{R}.$
Let the lines cut the plane $\mathrm{x}+\mathrm{y}+\mathrm{z}=1$ at the points $\mathrm{A},\mathrm{ }\mathrm{B}$ and $\mathrm{C}$ respectively. If the area of the triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is $\mathrm{\Delta }$ then the value of ${\left(6\mathrm{\Delta }\right)}^2$ equals_________

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Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question:

The average current in the steady state registered by the ammeter in the circuit will be

One mole of a monatomic ideal gas undergoes the cyclic process \(\mathrm{J} \rightarrow \mathrm{K} \rightarrow \mathrm{L} \rightarrow \mathrm{M} \rightarrow \mathrm{J}\), as shown in the P-T diagram.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
[\(\mathcal{R}\) is the gas constant.]

List-I	List-II
(P) Work done in the complete cyclic process	(1) \(R T_0-4 R T_0 \ln 2\)
(Q) Change in the internal energy of the gas in the process JK	(2) \(0\)
(R) Heat given to the gas in the process KL	(3) \(3 K T_0\)
(S) Change in the internal energy of the gas in the process MJ	(4) \(-2 R T_0 \ln 2\)
	\((5)-3 R T_0 \ln 2\)

One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $\frac{T_A}{T_B}$?

A perfectly reflecting mirror of mass $\mathrm{M}$ mounted on a spring constitutes a spring-mass system of angular frequency $\mathrm{\Omega }$ such that $\frac{4\mathrm{\pi }\mathrm{M}\mathrm{\Omega }}{\mathrm{h}}={10}^{24}{\mathrm{m}}^{-2}$ with $\mathrm{h}$ as Planck's constant. $\mathrm{N}$ photons of wavelength $\mathrm{\lambda }=8\mathrm{\pi }\times {10}^{-6}\mathrm{m}$ strike the mirror simultaneously at normal incidence such that the mirror gets displaced by $1\mathrm{\mu }\mathrm{m}.$ If the value of $\mathrm{N}$ is $\mathrm{x}\times {10}^{12},$ then the value of $\mathrm{x}$ is________.
[Consider the spring as massless]

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The nuclear charge \((\mathrm{Ze})\) is non-uniformly distributed within a nucleus of radius \(R\). The charge density \(\rho(r)\) [charge per unit volume] is dependent only on the radial distance \(r\) from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question:
The electric field within the nucleus is generally observed to be linearly dependent on \(r\). This implies

The percentage of \(p\)-character in the orbitals forming \(\mathrm{P}-\mathrm{P}\) bonds in \(\mathrm{P}_4\) is