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Four persons measure the length of a rod as \( 20.00 \, \text{cm} \), \( 19.75 \, \text{cm} \), \( 17.01 \, \text{cm} \) and \( 18.25 \, \text{cm} \). The relative error in the measurement of average length of the rod is:

In a screw gauge, the zero of the circular scale lies \( 3 \) divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is \( 1 \, \text{mm} \) and the circular scale reading is \( 51 \) then the correct thickness of the sheet is ______ \( \text{mm} \). [Assume least count is \( 0.01 \, \text{mm} \)]

Two-point charges \( 2q \) and \( q \) are placed at vertex \( A \) and centre of face \( CDEF \) of the cube as shown in figure. The electric flux passing through the cube is:

A thin prism with angle \( 5^\circ \) of refractive index \( 1.72 \) is combined with another prism of refractive index \( 1.9 \) to produce dispersion without deviation. The angle of second prism is ______.

Two small balls with masses \( m \) and \( 2m \) are attached to both ends of a rigid rod of length \( d \) and negligible mass. If angular momentum of this system is \( L \) about an axis \( (A) \) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about \( A \) is:

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius \( R \) and length \( L \) \( (R < L) \) about an axis passing through the mid points of opposite sides, is: [Take the mass of the entire loop as \( M \)]

The de-Broglie wavelength of an oxygen molecule at \( 27^\circ \text{C} \) is \( x \times 10^{-12} \, \text{m} \). The value of \( x \) is: [Take Planck’s constant \( = 6.63 \times 10^{-34} \, \text{J}\cdot\text{s} \), Boltzmann constant \( = 1.38 \times 10^{-23} \, \text{J K}^{-1} \), mass of oxygen molecule \( = 5.31 \times 10^{-26} \, \text{kg} \)]

A \( 20 \, \text{m} \) long uniform copper wire held horizontally is allowed to fall under the gravity \( (g = 10 \, \text{m/s}^{2}) \) through a uniform horizontal magnetic field of \( 0.5 \, \text{Gauss} \) perpendicular to the length of the wire. The induced EMF across the wire when it travels a vertical distance of \( 200 \, \text{m} \) is ______ mV.

A wire of uniform resistance \( \lambda \, \Omega / \text{m} \) is bent into a circle of radius \( r \) and another piece of wire with length \( 2r \) is connected between points A and B (AOB) as shown in figure. The equivalent resistance between points A and B is ______ \( \Omega \).

An object is projected with kinetic energy \( K \) from a point A at an angle \( 60^\circ \) with the horizontal. The ratio of the difference in kinetic energies at points B and C to that at point A (see figure), in the absence of air friction is:

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Consider a ferromagnetic material:
Assertion (A): The individual atoms in a ferromagnetic material possess a magnetic dipole moment and interact with one another in such a way that they spontaneously align themselves forming domains.
Reason (R): At high enough temperature, the domain structure of ferromagnetic material disintegrates. Thus, magnetization will disappear at high enough temperature known as Curie temperature.
In the light of the above statements, choose the correct answer from the options given below:

The following diagram shows a Zener diode as a voltage regulator. The Zener diode is rated at \( V_z = 5 \, \text{V} \) and the desired current in load is \( 5 \, \text{mA} \). The unregulated voltage source can supply upto \( 25 \, \text{V} \). Considering the Zener diode can withstand four times of the load current, the value of resistor \( R_s \) (shown in circuit) should be ______ \( \Omega \).

Match List-I with List-II.

Choose the correct answer from the options given below:

In hydrogen atom spectrum, (\( R \rightarrow \) Rydberg's constant)
A. the maximum wavelength of the radiation of Lyman series is \( \dfrac{4}{3R} \)
B. the Balmer series lies in the visible region of the spectrum
C. the minimum wavelength of the radiation of Paschen series is \( \dfrac{9}{R} \)
D. the minimum wavelength of Lyman series is \( \dfrac{5}{4R} \)
Choose the correct answer from the options given below:

Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media A and B are \( 2.4 \times 10^8 \, \text{m/s} \) and \( 2.7 \times 10^8 \, \text{m/s} \), respectively, then the value of critical angle is:

A simple pendulum of string length \( 30 \, \text{cm} \) performs \( 20 \) oscillations in \( 10 \, \text{s} \). The length of the string required for the pendulum to perform \( 40 \) oscillations in the same time duration is _______ \( \text{cm} \).
[Assume that the mass of the pendulum remains same]

Two blocks with masses \( 100 \, \text{g} \) and \( 200 \, \text{g} \) are attached to the ends of springs A and B as shown in figure. The energy stored in A is \( E \). The energy stored in B, when spring constants \( k_A \), \( k_B \) of A and B, respectively satisfy the relation \( 4k_A = 3k_B \), is:

The strain-stress plot for materials A, B, C and D is shown in the figure. Which material has the largest Young's modulus?

In a perfectly inelastic collision, two spheres made of the same material with masses \( 15 \, \text{kg} \) and \( 25 \, \text{kg} \), moving in opposite directions with speeds of \( 10 \, \text{m/s} \) and \( 30 \, \text{m/s} \), respectively, strike each other and stick together. The rise in temperature (in \( ^\circ \text{C} \)), if all the heat produced during the collision is retained by these spheres, is: [Specific heat of sphere material \( 31 \, \text{cal/(kg} \cdot ^\circ\text{C)} \) and \( 1 \, \text{cal} = 4.2 \, \text{J} \)]

The space between the plates of a parallel plate capacitor of capacitance \( C \) (without any dielectric) is now filled with three dielectric slabs of dielectric constants \( K_1 = 2 \), \( K_2 = 3 \) and \( K_3 = 5 \) (as shown in figure). If new capacitance is \( \frac{n}{3} C \) then the value of \( n \) is ______.

Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is \( 50 \, \text{mA} \) for \( V = 5 \sin(100 t) \). The values of \( L \) and \( R \) are shown in the figure. The capacitance of the capacitor \( (C) \) used is ______ \( \mu\text{F} \).

In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are \( 2:1 \) and \( 1:2 \) respectively. The corresponding ratio of the distances between the slits and the respective screens \( (D1 / D2) \) is ______.

The equation of the electric field of an electromagnetic wave propagating through free space is given by:
\( E = \sqrt{377} \sin(6.27 \times 10^{3} t - 2.09 \times 10^{-5} x) \, \text{N}/\text{C} \)
The average power of the electromagnetic wave is \( \left(\frac{1}{\alpha}\right) \, \text{W}/\text{m}^{2} \). The value of \( \alpha \) is ______.
[Take \( \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}} = 377 \) in SI units]

A simple pendulum made of mass \( 10 \, \text{g} \) and a metallic wire of length \( 10 \, \text{cm} \) is suspended vertically in a uniform magnetic field of \( 2 \, \text{T} \). The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of \( 60^\circ \) with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ______ \( \text{mV} \). [Take \( g = 10 \, \text{m}/\text{s}^{2} \)]