In an experiment to determine the focal length \((f)\) of a concave mirror by the \(u-v\) method, a student places the object pin \(A\) on the principal axis at a distance \(x\) from the pole \(P\). The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then

A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is$\frac{nq}{6{\epsilon }_0}$  (in $\mathrm{SI}$ units). The value of $n$ is _________.

Heater of an electric kettle is made of a wire of length L and diameter d. It takes 4 minutes to raise the temperature of 0.5 kg water by 40 K. This heater is replaced by a new heater having two wires of the same material, each of length L and diameter 2d. The way these wires are connected is given in the options. How much time in minutes will it take to raise the temperature of the same amount of water by 40K?

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Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

If two structures of same cross-sectional area, but different numerical apertures \(N A_{1}\) and \(N A_{2}\left(N A_{2} < N A_{1}\right)\) are joined longitudinally, the numerical aperture of the combined structure is

In terms of potential difference $V$ , electic current $I$ , permittivity ${\epsilon }_0$ , permeability ${\mu }_0$ and speed of light $c$ , the dimensionally correct equation(s) is (are)

At time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current $\mathrm{I}\left(\mathrm{t}\right)={\mathrm{I}}_0\cos \left(\mathrm{\omega }\mathrm{t}\right)$ , with ${\mathrm{I}}_0=1\mathrm{A}$ and $\mathrm{\omega }=500\mathrm{}$ rad/s starts flowing in it with the initial direction shown in the figure. At $\mathrm{t}=\frac{7\mathrm{\pi }}{6\mathrm{\omega }}$ , the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If $\mathrm{C}=20\mathrm{}\mathrm{\mu }\mathrm{F},\mathrm{}\mathrm{R}=10\mathrm{}\mathrm{\Omega }$ and the battery is ideal with of 50V, identify the correct statement(s).

Water is filled up to a height \(h\) in a beaker of radius \(R\) as shown in the figure. The density of water is \(\rho\), the surface tension of water is \(T\) and the atmospheric pressure is \(p_0\). Consider a vertical section \(A B C D\) of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude.

Let $X$ be the set of all five digit numbers formed using $1,2,2,2,4,4,0$. For example, $22240$ is in $X$ while $02244$ and $44422$ are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of $20$ given that it is a multiple of $5$. Then the value of $38 p$ is equal to

Let RS be the diameter of the circle $x^2+y^2=1$ where, $S$ is the point$(1, 0)$. Let $P$be a variable point (other than $R \& S$) on the circle and tangents to the circle at $S \& P$meet at the point$Q$. The normal to the circle at P intersects a line drawn through$Q$ parallel to $RS$ at point $E$ . Then, the locus of $E$ passes through the point (s):

Which of the following combinations will produce ${\text{H}}_2$ gas?

The maximum value of the expression \(\frac{1}{\sin ^2 \theta+3 \sin \theta \cos \theta+5 \cos ^2 \theta}\) is

Half-life of a radio active substance ' \(A\) ' is 4 days. The probability that a nucleus will decay in two half-lives is

The maximum value of the function \(f(x)=2 x^3-15 x^2+36 x-48\) on the set \(A=\left\{x \mid x^2+20 \leq 9 x\right\}\) is