Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above. The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is
The number of minutes spent by two students, X and Y, exercising every day in a given week are shown in the bar chart above. The number of days in the given week in which one of the students spent a minimum of 10\% more than the other student, on a given day, is
The least number of squares that must be added so that the line P-Q becomes the line of symmetry is
The current population of a city is 11,02,500. If it has been increasing at the rate of 5% per annum, what was its population 2 years ago?
In a high school having equal number of boy students and girl students, 75% of the students study Science and the remaining 25% students study Commerce. Commerce students are two times more likely to be a boy than are Science students. The amount of information gained in knowing that a randomly selected girl student studies Commerce (rounded off to three decimal places) is _________ bits.
The circuit in the figure contains a current source driving a load having an inductor and a resistor in series, with a shunt capacitor across the load. The ammeter is assumed to have zero resistance. The switch is closed at time \( t = 0 \). \[ \text{Initially, when the switch is open, the capacitor is discharged and the ammeter reads zero ampere. After the switch is closed, the ammeter reading keeps fluctuating for some time till it settles to a final steady value. The maximum ammeter reading that one will observe after the switch is closed (rounded off to two decimal places) is A.} \]
A unity feedback system that uses proportional-integral (PI) control is shown in the figure.
The propagation delay of the exclusive-OR (XOR) gate in the circuit in the figure is 3 ns. The propagation delay of all the flip-flops is assumed to be zero. The clock (Clk) frequency provided to the circuit is 500 MHz.
A circuit with an ideal OPAMP is shown in the figure. A pulse \( V_{\text{IN}} \) of 20 ms duration is applied to the input. The capacitors are initially uncharged.
For the transistor M1 in the circuit shown in the figure, \( \mu_n C_{\text{ox}} = 100 \, \mu \text{A/V}^2 \) and \( \frac{W}{L} = 10 \), where \( \mu_n \) is the mobility of electrons, \( C_{\text{ox}} \) is the oxide capacitance per unit area, \( W \) is the width, and \( L \) is the length. The channel length modulation coefficient is ignored. If the gate-to-source voltage \( V_{GS} \) is 1 V to keep the transistor at the edge of saturation, then the threshold voltage of the transistor (rounded off to one decimal place) is _________ V.
An asymmetrical periodic pulse train \( v_{\text{in}} \) of 10 V amplitude with on-time \( T_{\text{on}} = 1 \, \text{ms} \) and off-time \( T_{\text{off}} = 1 \, \mu \text{s} \) is applied to the circuit shown in the figure. The diode \( D_1 \) is ideal. \[ \text{The difference between the maximum voltage and minimum voltage of the output waveform \( v_o \) (in integer) is V.} \]
A silicon P-N junction is shown in the figure. The doping in the P region is \( 5 \times 10^{16} \, \text{cm}^{-3} \) and doping in the N region is \( 10 \times 10^{16} \, \text{cm}^{-3} \). The parameters given are: Built-in voltage \( \Phi_{bi} = 0.8 \, \text{V} \) Electron charge \( q = 1.6 \times 10^{-19} \, \text{C} \) Vacuum permittivity \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \) Relative permittivity of silicon \( \epsilon_{si} = 12 \) The magnitude of reverse bias voltage that would completely deplete one of the two regions (P or N) prior to the other (rounded off to one decimal place) is V.
The exponential Fourier series representation of a continuous-time periodic signal \( x(t) \) is defined as \[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j k \omega_0 t} \] where \( \omega_0 \) is the fundamental angular frequency of \( x(t) \) and the coefficients of the series are \( a_k \). The following information is given about \( x(t) \) and \( a_k \): \( x(t) \) is real and even, having a fundamental period of 6. The average value of The average power of the signal \( x(t) \) (rounded off to one decimal place) is _________.
