Statement 1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and Statement 2 The electrical potential of a sphere of radius \(R\) with charge \(Q\) uniformly distributed on the surface is given by \(\frac{Q}{4 \pi \varepsilon_0 R}\).

A thin spherical insulating shell of radius $\mathrm{R}$ carries a uniformly distributed charge such that the potential at its surface is ${\mathrm{V}}_0$ . A hole with a small area $\mathrm{\alpha }4\mathrm{\pi }{\mathrm{R}}^2\left(\mathrm{\alpha }\ll 1\right)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are \(\beta_{G}, \beta_{R}\) and \(\beta_{B}\), respectively. Then,

Which one of the following statement is WRONG in the context of X-rays generated from X-ray tube?

Most materials have the refractive index, \(n>1\). So, when a light ray from air enters a naturally occurring material, then by Snell's

law, \(\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}\), it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, \(\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}\), where \(c\) is the speed of electromagnetic waves in vacuum, \(v\) its speed in the medium, \(\varepsilon_{r}\) and \(\mu_{r}\) are the relative permittivity and permeability of the medium respectively. In normal materials, both \(\varepsilon_{r}\) and \(\mu_{r}\), are positive, implying positive \(n\) for the medium. When both \(\varepsilon_{r}\) and \(\mu_{r}\) are negative, one must choose the negative root of \(n\). Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since \(n\) is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
Question : Choose the correct statement.

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A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
If height \(h_2\) of water level is further decreased, then

In an insulated vessel, \(0.05 \mathrm{~kg}\) steam at \(373 \mathrm{~K}\) and \(0.45 \mathrm{~kg}\) of ice at \(253 \mathrm{~K}\) are mixed. Find the final temperature of the mixture (in kelvin).
\( \text { Given, }L_{\text {fusion }} =80 \mathrm{cal} / \mathrm{g}=336 \mathrm{~J} / \mathrm{g},\) \( L_{\text {vaporization }}=540 \mathrm{cal} / \mathrm{g}=2268 \mathrm{~J} / \mathrm{g}, \)
\( S_{\text {ice }} =2100 \mathrm{~J} / \mathrm{kg}, K=0.5 \mathrm{cal} / \mathrm{gK} \text { and } S_{\text {water }}=\) \(4200 \mathrm{~J} / \mathrm{kg}, K=1 \mathrm{cal} / \mathrm{gK} .\)

In the circuit shown, $L=1\text{μ}H,C=1\text{μ}FandR=1k\mathrm{\Omega }.$ They are connected in series with an a.c. source $V=V_0\sin \text{ω}t$ as shown. Which of the following options is/are correct?

A small mass \(m\) is attached to a massless string whose other end is fixed at \(P\) as shown in the figure. The mass is undergoing circular motion in the \(x-y\) plane with centre at \(O\) and constant angular speed \(\omega\). If the angular momentum of the system, calculated about \(O\) and \(P\) are denoted by \(\vec{L}_{O}\) and \(\vec{L}_{P}\) respectively, then

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_{1}and{n}_2\left(>n_1\right),$ as shown in the figure. A ray of light is incident with angle ${\theta }_i$ from medium 1 and emerges in medium 2 with refraction angle ${\theta }_f$ with a lateral displacement $\mathcal{l}$ . Which of the following statement(s) is (are) true?

The value of \(\frac{(5050) \int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^1\left(1-x^{50}\right)^{101} d x}\)

Let the function $f:\left[0,1\right]\to \mathrm{ℝ}$ be defined by $f\left(x\right)=\frac{4^x}{4^x+2}$. Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\cdots +f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$  is_________  

Let the function $f:\left(0,\pi \right)\to \mathrm{ℝ}$ be defined by $f\left(\theta \right)={\left(\sin \theta +\cos \theta \right)}^2+{\left(\sin \theta -\cos \theta \right)}^4$. Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \left\{{\lambda }_1\pi ,\dots ,{\lambda }_r\pi \right\},$ where $0<$${\lambda }_1<\cdots <{\lambda }_r<1.$ Then the value of  ${\lambda }_1+\cdots +{\lambda }_r$ is_______