Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x$ Solution: Given $I=\int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x$ Integral is of form $\int \frac{p x+q}{\sqrt{a x^{2}+b x+c}} d x$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$ $\Rightarrow p x+q=\lambda(2 a x+b)+\mu$ $\Rightarrow 5 x+3=\lambda(2 x+4)+\mu$ $\therefore \lambda=5 / 2$ and $\mu=-7$ Let $5 x...
Read More →Write the truth table for the circuit shown
Question: Write the truth table for the circuit shown in the figure. Name the gate that the circuit resembles. Solution: The given circuit is AND and the truth table is:...
Read More →Explain why elemental semiconductor
Question: Explain why elemental semiconductor cannot be used to make visible LEDs. Solution: Elemental semiconductor cannot be used to make visible LEDs because the bandgap of elemental semiconductor is in the infrared region and not in the visible region. The energy gap for Si is 1.1eV, wavelength is 1129 nm and the energy gap for Ge is 0.7eV, wavelength is 1725 nm....
Read More →How would you set up a circuit to obtain
Question: How would you set up a circuit to obtain NOT gate using a transistor? Solution: (a) It has only one input and only one output (b) Using the Boolean expression (c) Realization of NOT gate (d) Using truth table...
Read More →Two car garages have a common gate
Question: Two car garages have a common gate which needs to open automatically when a car enters either of the garages or cars enter both. Devise a circuit that resembles this situation using diodes for this situation. Solution: OR gate is used for explaining the situation:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{2 x+3}{\sqrt{x^{2}+4 x+5}} d x$ Solution: Given I $=\int \frac{2 x+3}{\sqrt{x^{2}+4 x+5}} d x$ Integral is of form $\int \frac{p x+q}{\sqrt{a x^{2}+b x+c}} d x$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$ $\Rightarrow p x+q=\lambda(2 a x+b)+\mu$ $\Rightarrow 2 x+3=\lambda(2 x+4)+\mu$ $\therefore \lambda=1 / 2$ and $\mu=-1$ Let $2 x+...
Read More →Let A and B be two events associated with a random experiment
Question: Let A and B be two events associated with a random experiment for which P(A) = 0.4, P(B) = 0.5 and P(A or B) = 0.6. Find P(A and B). Solution: Given : P(A) = 0.4, P(A or B) = 0.6 and P(B) = 0.5 To find : P(A and B) Formula used : P(A or B) = P(A) + P(B) - P(A and B) Substituting in the above formula we get $0.6=0.4+0.5-P(A$ and $B)$ $0.6=0.9-\mathrm{P}(\mathrm{A}$ and $\mathrm{B})$ $P(A$ and $B)=0.9-0.6$ $P(A$ and $B)=0.3$ $\mathrm{P}(\mathrm{A}$ and $\mathrm{B})=0.3$...
Read More →If the resistance R1 is increased,
Question: If the resistance R1 is increased, how will the readings of the ammeter and voltmeter change? Solution: IbR1 + Vbe = Vbb Base current = Ib = Vbb Vbe/R1 Ib is proportional to 1/R1 Therefore, when R1 is increased, Ib decreases....
Read More →Three photodiodes D1, D2 and D3
Question: Three photodiodes D1, D2 and D3 are made of semiconductors having band gaps of 2.5eV, 2eV and 3eV, respectively. Which ones will be able to detect light of wavelength 6000 Ao? Solution: Wavelength = = 6000 Ao= 6000 10-10m Energy of the light photon = E = hc/ = 2.06 eV...
Read More →(i) Name the type of a diode whose characteristics
Question: (i) Name the type of a diode whose characteristics are shown in the figure (a) and in the figure (b) (ii) What does the point P in the figure (a) represent? (iii) What does the points P and Q in the figure (b) represent? Solution: (i) Figure a) represents the characteristics of Zener diode and figure b) represents the solar cell. (ii) Figure a) point P is the representation of Zener breakdown voltage. (iii) Figure b) point Q is the representation of zero voltage and negative current....
Read More →If A and B are two events associated with a random experiment for
Question: If A and B are two events associated with a random experiment for which P(A) = 0.60, P(A or B) = 0.85 and P(A and B) = 0.42, find P(B). Solution: Given : P(A) = 0.60, P(A or B) = 0.85 and P(A and B) = 0.42 To find : P(B) Formula used : $P(A$ or $B)=P(A)+P(B)-P(A$ and $B)$ Substituting in the above formula we get, $0.85=0.60+P(B)-0.42$ $0.85=0.18+P(B)$ $0.85-0.18=P(B)$ $0.67=P(B)$ $P(B)=0.67$...
Read More →In a CE transistor amplifier
Question: In a CE transistor amplifier there is a current and voltage gain associated with the circuit. In other words, there is a power gain. Considering power a measure of energy, does the circuit violate conservation of energy? Solution: The circuit wont violate the conservation of energy as the power gain in the CE transistor is very high. Also, extra power is required for the amplification....
Read More →The amplifiers X, Y and Z are connected in series.
Question: The amplifiers X, Y and Z are connected in series. If the voltage gains of X, Y and Z are 10, 20 and 30, respectively and the input signal is 1 mV peak value, then what is the output signal voltage (peak value) (i) if dc supply voltage is 10V? (ii) if dc supply voltage is 5V? Solution: Voltage gain in X = vx = 10 Voltage gain in Y = vy = 20 Voltage gain in Z = vz = 30 Total voltage amplification = vx.vy.vz = 6V (i) If dc supply voltage is 10V then the output voltage is 6V (ii) If dc su...
Read More →In a single throw of three dice, find the probability of getting
Question: In a single throw of three dice, find the probability of getting (i) a total of 5 (ii) a total of at most 5 Solution: Total no.of cases will be 6 x 6 x 6 = 216(because each die can have values from 1 to 6) Desired outcomes are those whose sum up to $5 .$ Desired outcomes are $(1,1,3),(1,3,$, $1),(1,2,2),(2,1,2),(2,2,1),(3,1,1)$ i.e. total of 6 cases As we know, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Therefo...
Read More →Can the potential barrier across a p-n junction
Question: Can the potential barrier across a p-n junction be measured by simply connecting a voltmeter across the junction? Solution: No, the potential barrier across a p-n junction cannot be measured by simply connecting a voltmeter across the junction because the resistance of the voltmeter is very high....
Read More →Sn, C, and Si, Ge are all group XIV elements.
Question: Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are semiconductors. Why? Solution: Sn is a conductor because the energy gap for Sn is 0 eV, C is an insulator because the energy gap is 5.4 eV while Si and Ge are semiconductors because the energy gaps are 1.1 eV and 0.7 eV respectively. These energy gaps are related to their atomic size and that makes one conductor, the other insulator and the rest semiconductors....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{2 x+1}{\sqrt{x^{2}+4 x+3}} d x$ Solution: Given $I=\int \frac{2 x+1}{\sqrt{x^{2}+4 x+3}} d x$ Integral is of form $\int \frac{p x+q}{\sqrt{a x^{2}+b x+c}} d x$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$ $\Rightarrow p x+q=\lambda(2 a x+b)+\mu$ $\Rightarrow 2 x+1=\lambda(2 x+4)+\mu$ $\therefore \lambda=1$ and $\mu=-3$ Let $2 x+1=2 x...
Read More →Why are elemental dopants for Silicon
Question: Why are elemental dopants for Silicon or Germanium usually chosen from group XIII or group XV? Solution: The elemental dopants for silicon or germanium is usually chosen from group XIII or group XV because the size of the dopant should be compatible with the semiconductor and they even form covalent bonds....
Read More →In a lottery, a person chooses six different numbers at random from 1 to 20.
Question: In a lottery, a person chooses six different numbers at random from 1 to 20. If these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? Solution: All numbers are different (given in question), this will be the same as picking r different objects from n objects which is ${ }^{n} \mathrm{C}_{r}$ Here, n= 20 and r = 6(as we have to pick 6 different objects from 20 objects) Now we shal...
Read More →The breakdown in a reverse-biased p–n junction
Question: The breakdown in a reverse-biased pn junction diode is more likely to occur due to (a) large velocity of the minority charge carriers if the doping concentration is small (b) large velocity of the minority charge carriers if the doping concentration is large (c) strong electric field in a depletion region if the doping concentration is small (d) strong electric field in the depletion region if the doping concentration is large Solution: (a) large velocity of the minority charge carrier...
Read More →To reduce the ripples in a rectifier circuit
Question: To reduce the ripples in a rectifier circuit with capacitor filter (a) RL should be increased (b) input frequency should be decreased (c) input frequency should be increased (d) capacitors with high capacitance should be used Solution: (a) RL should be increased (c) input frequency should be increased (d) capacitors with high capacitance should be used...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sqrt{\frac{1-x}{1+x}} d x$ Solution: Given $I=\int \frac{\sqrt{1-x}}{\sqrt{1+x}} d x$ Rationalizing the denominator, $\Rightarrow \int \sqrt{\frac{1-x}{1+x}} d x=\int \sqrt{\frac{1-x}{1+x} \times \frac{1-x}{1-x}} d x$ $=\int \frac{1-x}{\sqrt{1-x^{2}}} d x$ Integral is of form $\int \frac{p x+q}{\sqrt{a x^{2}+b x+c}} d x$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\ma...
Read More →What happens during regulation action of a Zener diode?
Question: What happens during regulation action of a Zener diode? (a) The current in and voltage across the Zenor remains fixed (b) The current through the series Resistance (Rs ) changes (c) The Zener resistance is constant (d) The resistance offered by the Zener changes. Solution: (b) The current through the series Resistance (Rs ) changes (d) The resistance offered by the Zener changes....
Read More →A combination lock on a suitcase has 3 wheels, each labeled with nine digits from 1 to 9.
Question: A combination lock on a suitcase has 3 wheels, each labeled with nine digits from 1 to 9. If an opening combination is a particular sequence of three digits with no repeats, what is the probability of a person guessing the right combination? Solution: As repetition is not allowed total no.of cases possible is $9 \times 8 \times 7$ (because if one of the numbers occupies a wheel, then the other wheel cannot be occupied by this number, i.e. next wheel have 1 less case than the previous w...
Read More →In the depletion region of a diode
Question: In the depletion region of a diode (a) there are no mobile charges (b) equal number of holes and electrons exist, making the region neutral (c) recombination of holes and electrons has taken place (d) immobile charged ions exist Solution: (a) there are no mobile charges (b) equal number of holes and electrons exist, making the region neutral (d) immobile charged ions exist...
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