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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

A beaker of radius $r$ is filled with water$\left(\text{refractive index}\frac{4}{3}\right)$ up to a height $H$ as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of water level at the centre and at the circumference of the beaker is $h\left(h\ll H\text{,}h\ll r\right)$, as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature $R.$ Which of the following is/are correct? ($g$ is the acceleration due to gravity)

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $Y$ is _____.

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Two plane harmonic sound waves are expressed by the equations.
\(
\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}
\)
(All parameters are in MKS)
Question :
What is the speed of the sound?

An infinitely long thin non-conducting wire is parallel to the z - axis and carries a uniform line charge density $\lambda$ . It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $PQ$ subtends an angle ${120}^o$ at the centre $O$ of the spherical shell, as shown in the figure. The permittivity of free space is ${ϵ}_0$ . Which of the following statements is (are) true?

In an aluminum (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as shown in the figure. The electrical resistivities of Al and Fe are $2.7\times {10}^{-8}\mathrm{\Omega }\mathrm{}\mathrm{m}$ and $1.0\times {10}^{-7}\mathrm{\Omega }\mathrm{}\mathrm{m}$ , respectively. The electrical resistance between the two faces P and Q of the composite bar is

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In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).

Question :
Now consider the partition to be free to move without friction so that the pressure of gases in both compartments is the same. Then total work done by the gases till the time they achieve equilibrium will be

Let $F:R\to R$ be a function. We say that $f$ has:
 PROPERTY $1$  if $\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{\sqrt{\left|h\right|}}$ exists and is finite, and 
PROPERTY $2$  if $\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h^2}$ exists and is finite.
Then which of the following options is/are correct?

A resistance of \(2 \Omega\) is connected across one gap of a meter-bridge (the length of the wire is \(100 \mathrm{~cm}\) ) and an unknown resistance, greater than \(2 \Omega\), is connected across the other gap. When these resistances are interchanged, the balance point shifts by \(20 \mathrm{~cm}\). Neglecting any corrections, the unknown resistance is

Let \(a_1, a_2, a_3, \ldots, a_{11}\) be real numbers satisfying \(a_1=15,27-2 a_2>0\) and \(a_k=2 a_{k-1}-a_{k-2}\) for \(k=3,4, \ldots, 11\).
If \(\frac{a_1^2+a_2^2+\ldots+a_{11}^2}{11}=90\), then the value of \(\frac{a_1+a_2+\ldots+a_{11}}{11}\) is equal to

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Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

If \(s t=1\), then the tangent at \(P\) and the normal at \(S\) to the parabola meet at a point whose ordinate is

Three randomly chosen non negative integers $x,y,z$ are found to satisfy the equation  $x+y+z=10$ . Then the probability that $z$ is even, is
 

The correct order of acidity for the following compounds is :