A galvanometer gives full scale deflection with $0·006 A$ current. By connecting it to a $4990 \Omega$ resistance, it can be converted into a voltmeter of range $0-30 V$. If connected to a $\frac{2\mathrm{n}}{249}\mathrm{\Omega }$ resistance, it becomes an ammeter of range $0- 1·5 A$. The value of $n$ is

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: $\left[\mathrm{position}\right]=\left[X^{\alpha }\right];$ $\left[\mathrm{speed}\right]=\left[X^{\beta }\right]$; $\left[\mathrm{acceleration}\right]=\left[X^p\right]$;$\left[\mathrm{linear}\mathrm{momentum}\right]=\left[X^q\right]$; $\left[\mathrm{force}\right]=\left[X^r\right]$. Then

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m=0.4kg$ is at rest on this surface. An impulse of $1.0 N s$ is applied to the block at time $t=0$ so that it starts moving along the x-axis with a velocity $v\left(t\right)=v_0{e}^{-\frac{t}{\tau }}$ , where $v_0$ is a constant and $\tau =4$ . The displacement of the block, in meter, at $t=\tau$ is __________. Take $e^{-1}=0.37$

A thin ring of mass \(2 \mathrm{~kg}\) and radius \(0.5 \mathrm{~m}\) is rolling without slipping on a horizontal plane with velocity 1 \(\mathrm{m} / \mathrm{s}\). A small ball of mass \(0.1 \mathrm{~kg}\), moving with velocity \(20 \mathrm{~m} / \mathrm{s}\) in the opposite directions, hits the ring at a height of \(0.75 \mathrm{~m}\) and goes vertically up with velocity 10 \(\mathrm{m} / \mathrm{s}\). Immediately after the collision

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ${\mathrm{\omega }}_1$ and ${\mathrm{\omega }}_2$ and have total energies ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ , respectively. The variations of their momenta $\mathrm{p}$ with positions $\mathrm{x}$ are shown in the figures. If $\frac{\mathrm{a}}{\mathrm{b}}={\mathrm{n}}^2$ and $\frac{\mathrm{a}}{\mathrm{R}}=\mathrm{n}$ , then the correct equation(s) is (are)

A hollow pipe of length \(0.8 \mathrm{~m}\) is closed at one end. At its open end a \(0.5 \mathrm{~m}\) long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is \(50 \mathrm{~N}\) and the speed of sound is \(320 \mathrm{~ms}^{-1}\), the mass of the string is

A geostationary satellite above the equator is orbiting around the earth at a fixed distance \(r_1\) from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance \(r_2\) from the center of the earth, such that \(r_1=1.21 r_2\). The time period of the second satellite as measured from the geostationary satellite is \(\frac{24}{p}\) hours. The value of \(p\) is

Statement 1 There is a natural asymmetry between converting work to heat and converting heat to work.
Statement 2 No process is possible in which the sole result in the absorption of heat from a reservoir and its complete conversion into work.

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
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Question :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

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The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

When \(d \approx a\) but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height \(h\) above the loop. In that case

Columns 1, 2 and 3 contain starting materials, reaction conditions and type of reactions respectively.

Column 1	Column 2	Column 3
(I) Toulene	(i) \(NaOH\) \( /Br _2\)	(P) Condensation
(II) Acetophenone	(ii) \(B r_2 / h v\)	(Q) Carboxylation
(III) Benzaldehyde	(iii) \(\left( CH _3 CO \right)_2\) \(O / CH _3\) \(COOK\)	(R) Substitution
(IV) Phenol	(iv) \(NaOH\) \(/CO _2\)	(S) Haloform

The only CORRECT combination that gives two different carboxylic acids is

Paragraph: There are five students $S_1, S_2, S_3, S_4$ and $S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i, i = \mathrm{1,2},\mathrm{3,4},5.$ But, on the examination day, the five students are randomly allotted the five seats
Question : For $i = 1,2,3,4$ Let $T_i$ denote the event that the students $S_i$ and $S_{i+1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_1\cap T_2\cap T_3\cap T_4$ is

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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 :
While the piston is at a distance \(2 L\) from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is