An OPAMP is connected in a circuit with a Zener diode as shown in the figure. The value of resistance \( R \) in kΩ for obtaining a regulated output \( V_0 = 9 \, \text{V} \) is:
A sphere of radius \( R \) has a uniform charge density \( \rho \). A sphere of smaller radius \( \frac{R}{2} \) is cut out from the original sphere, as shown in the figure below. The center of the cut-out sphere lies at \( z = \frac{R}{2} \). After the smaller sphere has been cut out, the magnitude of the electric field at \( z = - \frac{R}{2} \) is \( \frac{\rho R}{2 \epsilon_0} \). The value of the integer \( n \) is:
In a coaxial cable, the radius of the inner conductor is 2 mm and that of the outer one is 5 mm. The inner conductor is at a potential of 10 V, while the outer conductor is grounded. The value of the potential at a distance of 3.5 mm from the axis is:
A rectangular loop of dimension \( L \) and width \( w \) moves with a constant velocity \( v \) away from an infinitely long straight wire carrying a current \( I \) in the plane of the loop as shown in the figure below. Let \( R \) be the resistance of the loop. What is the current in the loop at the instant the near-side is at a distance \( r \) from the wire?
An n-p-n transistor is connected in a circuit as shown in the figure. If \( I_C = 1 \, \text{mA}, \, \beta = 50, \, V_{BE} = 0.7 \, \text{V}, \) and the current through \( R_2 \) is \( 10 I_B \), where \( I_B \) is the base current. Then the ratio \( \frac{R_1}{R_2} \) is:
For the Fourier series of the following function of period \( 2\pi \): The ratio (to the nearest integer) of the Fourier coefficients of the first and the third harmonic is:
Consider two coherent point sources \( S_1 \) and \( S_2 \) separated by a small distance along a vertical line and two screens \( P_1 \) and \( P_2 \) placed as shown in the figure. Which one of the choices represents the shapes of the interference fringes at the central regions on the screens?
Consider a thin long insulator coated conducting wire carrying current \( I \). It is now wound once around an insulating thin disc of radius \( R \) to bring the wire back on the same side, as shown in the figure. The magnetic field at the center of the disc is equal to:
Which one of the following graphs represents the derivative \( f'(x) = \frac{df}{dx} \) of the function \( f(x) = \frac{1}{1+x^2} \) most closely (graphs are schematic and not drawn to scale)?
To demonstrate Bernoulli's principle, an instructor arranges two circular horizontal plates of radii \( b \) each with distance \( d \) (\( d \ll b \)) between them (see figure). The upper plate has a hole of radius \( a \) in the middle. On blowing air at a speed \( v_0 \) through the hole so that the flow rate of air is \( \pi a^2 v_0 \), it is seen that the lower plate does not fall. If the density of air is \( \rho \), the upward force on the lower plate is well approximated by the formula (assume that the region with \( r<a \) does not contribute to the upward force and the speed of air at the edges is negligible):
A pendulum is made of a massless string of length \( L \) and a small bob of negligible size and mass \( m \). It is released making an angle \( \theta_0 \) (<< 1 rad) from the vertical. When passing through the vertical, the string slips a bit from the pivot so that its length increases by a small amount \( \delta \) (\( \delta \ll L \)) in negligible time. If it swings up to angle \( \theta_1 \) on the other side before starting to swing back, then to a good approximation, which of the following expressions is correct?
A uniform rigid meter-scale is held horizontally with one of its end at the edge of a table and the other supported by hand. Some coins of negligible mass are kept on the meter scale as shown in the figure. As the hand supporting the scale is removed, the scale starts rotating about its edge on the table and the coins start moving. If a photograph of the rotating scale is taken soon after, it will look closest to:
Consider a uniform thin circular disk of radius \( R \) and mass \( M \). A concentric square of side \( R/2 \) is cut out from the disk (see figure). What is the moment of inertia of the resultant disk about an axis passing through the centre of the disk and perpendicular to it?
Consider an inertial frame \( S' \) moving at speed \( c/2 \) away from another inertial frame \( S \) along the common x-axis, where \( c \) is the speed of light. As observed from \( S' \), a particle is moving with speed \( c/2 \) in the \( y' \) direction, as shown in the figure. The speed of the particle as seen from \( S \) is:
Which one of the following schematic curves best represents the variation of conductivity \( \sigma \) of a metal with temperature \( T \)?
Shown in the figure is a combination of logic gates. The output values at P and Q are correctly represented by which of the following?
Consider two, single turn, co-planar, concentric coils of radii \( R_1 \) and \( R_2 \) with \( R_1 \gg R_2 \). The mutual inductance between the two coils is proportional to:
A current \( I = 10A \) flows in an infinitely long wire along the axis of a hemisphere. The value of \( \int (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{s} \) over the hemispherical surface as shown in the figure is:
Consider the following circuit with two identical Si diodes. The input ac voltage waveform has the peak voltage \( V_P = 2V \), as shown. The voltage waveform across PQ will be represented by:
