A conducting wire of parabolic shape, initially $\mathrm{y}={\mathrm{x}}^2,$ is moving with velocity $\overrightarrow{\mathrm{V}}={\mathrm{V}}_0\widehat{\mathrm{i}}$ in a non-uniform magnetic field $\overrightarrow{\mathrm{B}}={\mathrm{B}}_0\left(1+{\left(\frac{\mathrm{y}}{\mathrm{L}}\right)}^{\mathrm{\beta }}\right)\widehat{\mathrm{k}},$ as shown in figure. If ${\mathrm{V}}_0,{\mathrm{B}}_0,\mathrm{L}$ and $\mathrm{\beta }$ are positive constants and $\mathrm{\Delta }\mathrm{\varphi }$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

Two small particles of equal masses start moving in opposite directions from a point \(A\) in a horizontal circular orbit. Their tangential velocities are \(v\) and \(2 v\) respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at \(A\), these two particles will again reach the point \(A\) ?

Two identical plates P and Q , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures \(\mathrm{T}_{\mathrm{P}}\) and \(\mathrm{T}_{\mathrm{Q}}\), respectively, with \(\mathrm{T}_{\mathrm{Q}}<\mathrm{T}_{\mathrm{P}}\), as shown in Fig. 1. The radiated power transferred per unit area from P to Q is \(W_0\). Subsequently, two more plates, identical to P and Q, are introduced between P and Q, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from P to Q (Fig. 2) in the steady state is \(W_S\), then the ratio \(\frac{W_0}{W_S}\) is ________.

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When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If \(r=5 \times 10^{-4} \mathrm{~m}, \rho=10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), \(g=10 \mathrm{~ms}^{-2}, T=0.11 \mathrm{Nm}^{-1}\), the radius of the drop when it detaches from the dropper is approximately

Statement 1 The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.
and Statement 2 In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.

The circuit shown in the figure contains an inductor \(L\), a capacitor \(C_0\), a resistor \(R_0\) and an ideal battery. The circuit also contains two keys \(\mathrm{K}_1\) and \(\mathrm{K}_2\). Initially, both the keys are open and there is no charge on the capacitor. At an instant, key \(K_1\) is closed and immediately after this the current in \(R_0\) is found to be \(I_1\). After a long time, the current attains a steady state value \(I_2\). Thereafter, \(\mathrm{K}_2\) is closed and simultaneously \(\mathrm{K}_1\) is opened and the voltage across \(C_0\) oscillates with amplitude \(V_0\) and angular frequency \(\omega_0\).

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

A block of base \(10 \mathrm{~cm} \times 10 \mathrm{~cm}\) and height \(15 \mathrm{~cm}\) is kept on an inclined plane. The coefficient of friction between them is \(\sqrt{3}\). The inclination \(\theta\) of this inclined plane from the horizontal plane is gradually increased from \(0^{\circ}\). Then,

The IUPAC name of \(\mathrm{C}_6 \mathrm{H}_5 \mathrm{COCl}\) is

Let \(\theta, \varphi \in[0,2 \pi]\) be such that \(2 \cos \theta(1-\sin \varphi)\) \(=\sin ^{2} \theta ~\times\) \(\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1, \tan\) \((2 \pi-\theta)>0\) and \(-1 < \sin \theta < -\frac{\sqrt{3}}{2}\), then \(\varphi\) cannot satisfy

In the $\text{xy - plane}$, the region $y>0$ has a uniform magnetic field $B_1\widehat{k}$ and the region $y<0$ has another uniform magnetic field $B_2\widehat{k}$. A positively charged particle is projected from the origin along the positive $\text{y}-\text{axis}$ with speed $v_0=\pi \mathrm{m}{s}^{-1}$ at $t=0$, as shown in the figure. Neglect gravity in this problem. Let $t=T$ be the time when the particle crosses the $\text{x}-\text{axis}$ from below for the first time. If $B_2=4B_1$, the average speed of the particle, in $\mathrm{m}{\mathrm{s}}^{-1}$, along the $\text{x}-\text{axis}$ in the time interval $T$ is ________.

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$\overrightarrow{r}=-\left(t+p\right)\hat{i}+t\hat{j}+\left(1+p\right)\hat{k}$
where $t,p$ are real parameters and $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10\hat{i}+15\hat{j}+20\hat{k}$ and $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ respectively, then which of the following is/are TRUE?

Let $f:R\to R$ be a differentiable function with $f\left(0\right)=1$ and satisfying the equation $f\left(x+y\right)=f\left(x\right)f^{\text{'}}\left(y\right)+f^{\text{'}}\left(x\right)f\left(y\right)$ for all $x, y\in R$. Then, the value of ${\log }_e\left(f\left(4\right)\right)$ is 

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields \(\vec{E}=E_{0} \hat{j}\) and \(\vec{B}=B_{0} \hat{j}\). At time \(t=0\), this charge has velocity \(\vec{v}\) in the in the \(x\)-y plane, making an angle \(\theta\) with the \(x\)-axis. Which of the following option(s) is (are) correct for time \(t>0\) ?