A single slit diffraction experiment is performed to determine the slit width using the equation, \(\frac{b d}{D}=m \lambda\), where \(b\) is the slit width, \(D\) the shortest distance between the slit and the screen, \(d\) the distance between the \(m^{\text {th }}\) diffraction maximum and the central maximum, and \(\lambda\) is the wavelength. \(D\) and \(d\) are measured with scales of least count of 1 cm and 1 mm, respectively. The values of \(\lambda\) and \(m\) are known precisely to be 600 nm and 3, respectively. The maximum absolute error (in \(\mu \mathrm{m}\)) in the value of \(b\) estimated using the diffraction maximum that occurs for \(m=3\) with \(d=5 \mathrm{~mm}\) and \(D=1 \mathrm{~m}\) is ______.

The dimensions of a cone are measured using a scale with a least count of \(2 \mathrm{~mm}\). The diameter of the base and the height are both measured to be \(20.0 \mathrm{~cm}\). The maximum percentage error in the determination of the volume is _______ .

List I contains four combinations of two lenses ( 1 and 2 ) whose focal lengths (in cm ) are indicated in the figures. In all cases, the object is placed 20 cm from the first lens on the left, and the distance between the two lenses is 5 cm . List II contains the positions of the final images.

	List - I		List - II
(I)		(P)	Final image is formed at 7.5 cm on the right side of lens 2.
(II)		(Q)	Final image is formed at 60.0 cm on the right side of lens 2.
(III)		(R)	Final image is formed at 30.0 cm on the left side of lens 2.
(IV)		(S)	Final image is formed at 6.0 cm on the right side of lens 2.
		(T)	Final image is formed at 30.0 cm on the right side of lens 2.

Which one of the following options is correct?

A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\overrightarrow{v}=\alpha \left(y\stackrel{\wedge }{x}+2x\stackrel{\wedge }{y}\right)$ where $\alpha$ is a non-zero constant. What is the force $\overrightarrow{F}$ acting on the particle?

A thin uniform rod of length \(L\) and certain mass is kept on a frictionless horizontal table with a massless string of length \(L\) fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point \(\mathrm{O}\). If a horizontal impulse \(P\) is imparted to the rod at a distance \(x=L / n\) from the mid-point of the rod (see figure), then the rod and string revolve together around the point \(\mathrm{O}\), with the rod remaining aligned with the string. In such a case, the value of \(n\) is _______ .

Paragraph :
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 \(\mu\).
Question :
The net external force acting on the disk when its centre of mass is at displacement \(x\) with respect to its equilibrium position is

A student skates up a ramp that makes an angle $30°$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity)

The circle passing through the point \((-1,0)\) and touching the \(Y\)-axis at \((0,2)\), also passes through the point

The total number of basic groups in the following form of lysine is

The graph between \(1 / \lambda\) and stopping potential \((V)\) of three metals having work function \(\phi_1, \phi_2\) and \(\phi_3\) in an experiment of photoelectric effect is plotted as shown in the figure. Which of the following statement(s) is/are correct? [Here \(\lambda\) is the wavelength of the incident ray].

Paragraph:
The reaction of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with freshly prepared \(\mathrm{FeSO}_{4}\) solution produces a dark blue precipitate called Turnbull's blue. Reaction of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with the \(\mathrm{FeSO}_{4}\) solution in complete absence of air produces a white precipitate \(\mathbf{X}\), which turns blue in air. Mixing the \(\mathrm{FeSO}_{4}\) solution with \(\mathrm{NaNO}_{3}\), followed by a slow addition of concentrated \(\mathrm{H}_{2} \mathrm{SO}_{4}\) through the side of the test tube produces a brown ring.
Question:
Precipitate \(\mathbf{X}\) is

Paragraph II: Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=0\), the value of \(v_{\mathrm{cm}} /(a \omega)\) will be ________ .

The shape of \(\mathrm{XeO}_{2} \mathrm{~F}_{2}\) molecule is