A cylindrical vessel of height \(500 \mathrm{~mm}\) has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it upto height \(H\). Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 \(\mathrm{mm}\). Find the fall in height (in \(\mathrm{mm}\) ) of water level due to opening of the orifice.
[Take atmospheric pressure \(=1.0 \times 10^5 \mathrm{Nm}^{-2}\), density of water \(=1000 \mathrm{~kg} \mathrm{~m}^{-3}\) and \(g=10\) \(\mathrm{ms}^{-2}\). Neglect any effect of surface tension.]

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \mathrm{mm}$. The circular scale has $100$ divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	$0$ division	$4$ divisions
Attempt-$1$: With wire	$4$ divisions	$20$ divisions
Attempt-$2$: With wire	$4$ divisions	$16$ divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of 1000 K is 0.4. It extracts 150 J of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance 10 . The hot reservoir of the heat pump is at a temperature of 300 K . Which of the following statements is/are correct:

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\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is \(\rho\). In equilibrium, the height \(H\) of the water column in the cylinder satisfies

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds \(v_1\) and \(v_2\), respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and \(v_1=3 m / s\), then \(v_2\) in \(m / s\) is \(\left( g =10 \frac{m}{ s ^2}\right)\)

 The diameter of a cylinder is measured using a vernier calipers with no zero error. It is found that the zero of the vernier scale lies between $5.10 \mathrm{cm}$ and $5.15 \mathrm{cm}$ of the main scale. The vernier scale has $50$  divisions equivalent to $2.45 \mathrm{cm}$. The ${24}^{\mathrm{th}}$ division of the vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is, 

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 _________.

A right-angled prism of refractive index ${\mu }_1$ is placed in a rectangular block of refractive index ${\mu }_2$, which is surrounded by a medium of refractive index ${\mu }_3$, as shown in the figure. A ray of light $e$ enters the rectangular block at normal incidence. Depending upon the relationships between ${\mu }_1\text{, }{\mu }_2$ and ${\mu }_3$, it takes one of the four possible paths $ef, eg, eh$ or $ei$.

Match the paths in List $\mathrm{I}$ with conditions of refractive indices in List $\mathrm{II}$ and select the correct answer using the codes given below the lists:

	List I		List II
A.	$e\to f$	P.	${\mu }_1>\sqrt{2}{\mu }_2$
B.	$e\to g$	Q.	${\mu }_2>{\mu }_1\text{and}{\mu }_2>{\mu }_3$
C.	$e\to h$	R.	${\mu }_1={\mu }_2$
D.	$e\to i$	S.	${\mu }_2<{\mu }_1<\sqrt{2}{\mu }_2\text{and}{\mu }_2>{\mu }_3$

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A circle \(C\) of radius 1 is inscribed in an equilateral \(\triangle P Q R\). The points of contact of \(C\) with the sides \(P Q, Q R, R P\) are \(D, E, F\) respectively. The line \(P Q\) is given by the equation
\(\sqrt{3} x+y-6=0\) and the point \(D\) is \(\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)\). Further, it is given that the origin and the centre of \(C\) are on the same side of the line \(P Q\).Question:
The equation of circle \(C\) is

The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.10 M). If ${\mathrm{\lambda }}_{{\mathrm{X}}^{-}}^0\approx {\mathrm{\lambda }}_{{\mathrm{Y}}^{-}}^0$ , the difference in their $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}$ values, $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{X}\right)-\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{Y}\right)$ , is (Consider degree of ionization of both acids to << 1)

A thin uniform annular disc (see figure) of mass \(M\) has outer radius \(4 R\) and inner radius \(3 R\). The work required to take a unit mass from point \(P\) on its axis to infinity is

One mole of an ideal gas is taken from \(a\) to \(b\) along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is \(W_s\) and that along the dotted line path is \(W_d\), then the integer closest to the ratio \(\frac{W_d}{W_s}\) is

At \(25^{\circ} \mathrm{C}\), the concentration of \(\mathrm{H}^{+}\)ions in \(1.00 \times 10^{-3} \mathrm{M}\) aqueous solution of a weak monobasic acid having acid dissociation constant \(\left(K_a\right)=4.00 \times 10^{-11}\) is \(\boldsymbol{X} \times 10^{-7} \mathrm{M}\). The value of \(\boldsymbol{X}\) is _____ .
Use: Ionic product of water \(\left(K_w\right)=1.00 \times 10^{-14}\) at \(25^{\circ} \mathrm{C}\)