NCERT Solutions Class 12

NCERT SolutionsMathematics
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 1)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 2)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 3)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 4)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 5)
 Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 1)
 Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 2)
 Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 3)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 1)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 2)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 3)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 4)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 5)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 1)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 2)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 3)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 4)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 5)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 6)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 7)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 1)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 2)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 3)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 4)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 5)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 6)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 7)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 8)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 9)

NCERT SolutionsChemistry
 Aldehydes, Ketones and Carboxylic Acids : NCERT Solutions – Class 12 Chemistry
 Alcohols, Phenols and Ethers : NCERT Solutions – Class 12 Chemistry
 Amines : NCERT Solutions – Class 12 Chemistry
 Biomolecules : NCERT Solutions – Class 12 Chemistry
 Chemical Kinetics : NCERT Solutions – Class 12 Chemistry
 Chemistry in Everyday Life : NCERT Solutions – Class 12 Chemistry
 Coordination Compounds : NCERT Solutions – Class 12 Chemistry
 Electrochemistry : NCERT Solutions – Class 12 Chemistry
 General Principles and Processes of Isolation of Elements : NCERT Solutions – Class 12 Chemistry
 Haloalkanes and Haloarenes : NCERT Solutions – Class 12 Chemistry
 Polymers : NCERT Solutions – Class 12 Chemistry
 Surface Chemistry : NCERT Solutions – Class 12 Chemistry
 The dand fBlock Elements : NCERT Solutions – Class 12 Chemistry
 The pBlock Elements : NCERT Solutions – Class 12 Chemistry
 The Solid State : NCERT Solutions – Class 12 Chemistry
 Solutions : NCERT Solutions – Class 12 Chemistry

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NCERT SolutionsPhysics
 Electrostatic Potential And Capacitance : NCERT Solutions – Class 12 Physics
 Electric Charges And Fields : NCERT Solutions – Class 12 Physics
 Semiconductor Electronics: Materials, Devices And Simple Circuits : NCERT Solutions – Class 12 Physics
 Ray Optics And Optical Instruments : NCERT Solutions – Class 12 Physics
 Nuclei : NCERT Solutions – Class 12 Physics
 Moving Charges And Magnetism : NCERT Solutions – Class 12 Physics
 Magnetism And Matter : NCERT Solutions – Class 12 Physics
 Electromagnetic Induction : NCERT Solutions – Class 12 Physics
 Dual Nature Of Radiation And Matter : NCERT Solutions – Class 12 Physics
 Current Electricity : NCERT Solutions – Class 12 Physics
 Communication Systems : NCERT Solutions – Class 12 Physics
 Atoms : NCERT Solutions – Class 12 Physics
 Alternating Current : NCERT Solutions – Class 12 Physics
Semiconductor Electronics: Materials, Devices And Simple Circuits : NCERT Solutions – Class 12 Physics
14.1. In an ntype silicon, which of the following statement is true:
(a) Electrons are majority carriers and trivalent atoms are the dopants.
(b) Electrons are minority carriers and pentavalent atoms are the dopants.
(c) Holes are minority carriers and pentavalent atoms are the dopants.
(d) Holes are majority carriers and trivalent atoms are the dopants.
Answer
The correct statement is (c).
In an ntype silicon, the electrons are the majority carriers, while the holes are the minority carriers. An ntype semiconductor is obtained when pentavalent atoms, such as phosphorus, are doped in silicon atoms.
14.2. Which of the statements given in Exercise 14.1 is true for ptype semiconductors.
Answer
The correct statement is (d).
In a ptype semiconductor, the holes are the majority carriers, while the electrons are the minority carriers. A ptype semiconductor is obtained when trivalent atoms, such as aluminium, are doped in silicon atoms.
14.3. Carbon, silicon and germanium have four valence electrons each. These are characterised by valence and conduction bands separated by energy band gap respectively equal to (E_{g})_{C}, (E_{g})_{Si }and (E_{g})_{Ge}. Which of the following statements is true?
(a) (E_{g})_{Si }< (E_{g})_{Ge} < (E_{g})_{C}
(b) (E_{g})_{C} < (E_{g})_{Ge} > (E_{g})_{Si}
(c) (E_{g})_{C} > (E_{g})_{Si} > (E_{g})_{Ge}
(d) (E_{g})_{C} = (E_{g})_{Si }= (E_{g})_{Ge}
ANSWER:
The correct statement is (c).
Of the three given elements, the energy band gap of carbon is the maximum and that of germanium is the least.
The energy band gap of these elements are related as: (E_{g})_{C} > (E_{g})_{Si} > (E_{g})_{Ge}
14.4. In an unbiased pn junction, holes diffuse from the pregion to nregion because
(a) free electrons in the nregion attract them.
(b) they move across the junction by the potential difference.
(c) hole concentration in pregion is more as compared to nregion.
(d) All the above.
Answer
The correct statement is (c).
The diffusion of charge carriers across a junction takes place from the region of higher concentration to the region of lower concentration. In this case, the pregion has greater concentration of holes than the nregion. Hence, in an unbiased pn junction, holes diffuse from the pregion to the nregion.
14.5. When a forward bias is applied to a pn junction, it
(a) raises the potential barrier.
(b) reduces the majority carrier current to zero.
(c) lowers the potential barrier.
(d) None of the above.
Answer
The correct statement is (c).
When a forward bias is applied to a pn junction, it lowers the value of potential barrier. In the case of a forward bias, the potential barrier opposes the applied voltage. Hence, the potential barrier across the junction gets reduced.
14.6. For transistor action, which of the following statements are correct:
(a) Base, emitter and collector regions should have similar size and doping concentrations.
(b) The base region must be very thin and lightly doped.
(c) The emitter junction is forward biased and collector junction is reverse biased.
(d) Both the emitter junction as well as the collector junction are forward biased.
Answer
The correct statement is (b), (c).
For a transistor action, the junction must be lightly doped so that the base region is very thin. Also, the emitter junction must be forwardbiased and collector junction should be reversebiased.
14.7. For a transistor amplifier, the voltage gain
(a) remains constant for all frequencies.
(b) is high at high and low frequencies and constant in the middle frequency range.
(c) is low at high and low frequencies and constant at mid frequencies.
(d) None of the above.
Answer
The correct statement is (c).
The voltage gain of a transistor amplifier is constant at mid frequency range only. It is low at high and low frequencies.
14.8. In halfwave rectification, what is the output frequency if the input frequency is 50 Hz. What is the output frequency of a fullwave rectifier for the same input frequency.
Answer
Input frequency = 50 Hz
For a halfwave rectifier, the output frequency is equal to the input frequency.
∴Output frequency = 50 Hz
For a fullwave rectifier, the output frequency is twice the input frequency.
∴Output frequency = 2 × 50 = 100 Hz
14.9. For a CEtransistor amplifier, the audio signal voltage across the collected resistance of 2 kΩ is 2 V. Suppose the current amplification factor of the transistor is 100, find the input signal voltage and base current, if the base resistance is 1 kΩ.
Answer
Collector resistance, R_{C} = 2 kΩ = 2000 Ω
Audio signal voltage across the collector resistance, V = 2 V
Current amplification factor of the transistor, β = 100
Base resistance, R_{B} = 1 kΩ = 1000 Ω
Input signal voltage = V_{i}
Base current = I_{B}
We have the amplification relation as:
Voltage amplification
Therefore, the input signal voltage of the amplifier is 0.01 V.
Base resistance is given by the relation:
Therefore, the base current of the amplifier is 10 μA.
14.10. Two amplifiers are connected one after the other in series (cascaded). The first amplifier has a voltage gain of 10 and the second has a voltage gain of 20. If the input signal is 0.01 volt, calculate the output ac signal.
Answer
Voltage gain of the first amplifier, V_{1} = 10
Voltage gain of the second amplifier, V_{2} = 20
Input signal voltage, V_{i} = 0.01 V
Output AC signal voltage = V_{o}
The total voltage gain of a twostage cascaded amplifier is given by the product of voltage gains of both the stages, i.e.,
V = V_{1} × V_{2}
= 10 × 20 = 200
We have the relation:
V_{0} = V × V_{i}
= 200 × 0.01 = 2 V
Therefore, the output AC signal of the given amplifier is 2 V.
14.11. A pn photodiode is fabricated from a semiconductor with band gap of 2.8 eV. Can it detect a wavelength of 6000 nm?
Answer
Energy band gap of the given photodiode, E_{g} = 2.8 eV
Wavelength, λ = 6000 nm = 6000 × 10^{−9} m
The energy of a signal is given by the relation:
E =
Where,
h = Planck’s constant
= 6.626 × 10^{−34} Js
c = Speed of light
= 3 × 10^{8} m/s
E
= 3.313 × 10^{−20} J
But 1.6 × 10^{−19} J = 1 eV
∴E = 3.313 × 10^{−20} J
The energy of a signal of wavelength 6000 nm is 0.207 eV, which is less than 2.8 eV − the energy band gap of a photodiode. Hence, the photodiode cannot detect the signal.
14.12. The number of silicon atoms per m^{3} is 5 × 10^{28}. This is doped simultaneously with 5 × 10^{22} atoms per m^{3} of Arsenic and 5 × 10^{20} per m^{3} atoms of Indium. Calculate the number of electrons and holes. Given that n_{i}= 1.5 × 10^{16} m^{−3}. Is the material ntype or ptype?
Answer
Number of silicon atoms, N = 5 × 10^{28} atoms/m^{3}
Number of arsenic atoms, n_{As} = 5 × 10^{22} atoms/m^{3}
Number of indium atoms, n_{In} = 5 × 10^{20} atoms/m^{3}
Number of thermallygenerated electrons, n_{i} = 1.5 × 10^{16} electrons/m^{3}
Number of electrons, n_{e} = 5 × 10^{22} − 1.5 × 10^{16} ≈ 4.99 × 10^{22}
Number of holes = n_{h}
In thermal equilibrium, the concentrations of electrons and holes in a semiconductor are related as:
n_{e}n_{h} = n_{i}^{2}
Therefore, the number of electrons is approximately 4.99 × 10^{22} and the number of holes is about 4.51 × 10^{9}. Since the number of electrons is more than the number of holes, the material is an ntype semiconductor.
14.13. In an intrinsic semiconductor the energy gap E_{g}is 1.2 eV. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at 600K and that at 300K? Assume that the temperature dependence of intrinsic carrier concentration n_{i}is given by
where n_{0 }is a constant.
Answer
Energy gap of the given intrinsic semiconductor, E_{g} = 1.2 eV
The temperature dependence of the intrinsic carrierconcentration is written as:
Where,
k_{B} = Boltzmann constant = 8.62 × 10^{−5} eV/K
T = Temperature
n_{0} = Constant
Initial temperature, T_{1} = 300 K
The intrinsic carrierconcentration at this temperature can be written as:
… (1)
Final temperature, T_{2} = 600 K
The intrinsic carrierconcentration at this temperature can be written as:
… (2)
The ratio between the conductivities at 600 K and at 300 K is equal to the ratio between the respective intrinsic carrierconcentrations at these temperatures.
Therefore, the ratio between the conductivities is 1.09 × 10^{5}.
14.14. In a pn junction diode, the current I can be expressed as
where I_{0} is called the reverse saturation current, V is the voltage across the diode and is positive for forward bias and negative for reverse bias, and I is the current through the diode, k_{B}is the Boltzmann constant (8.6×10^{−5} eV/K) and T is the absolute temperature. If for a given diode I_{0} = 5 × 10^{−12 }A and T = 300 K, then
(a) What will be the forward current at a forward voltage of 0.6 V?
(b) What will be the increase in the current if the voltage across the diode is increased to 0.7 V?
(c) What is the dynamic resistance?
(d) What will be the current if reverse bias voltage changes from 1 V to 2 V?
Answer
In a pn junction diode, the expression for current is given as:
Where,
I_{0} = Reverse saturation current = 5 × 10^{−12} A
T = Absolute temperature = 300 K
k_{B} = Boltzmann constant = 8.6 × 10^{−5} eV/K = 1.376 × 10^{−23} J K^{−1}
V = Voltage across the diode
(a) Forward voltage, V = 0.6 V
∴Current, I
Therefore, the forward current is about 0.0256 A.
(b) For forward voltage, V^{’} = 0.7 V, we can write:
Hence, the increase in current, ΔI = I^{‘} − I
= 1.257 − 0.0256 = 1.23 A
(c) Dynamic resistance
(d) If the reverse bias voltage changes from 1 V to 2 V, then the current (I) will almost remain equal to I_{0} in both cases. Therefore, the dynamic resistance in the reverse bias will be infinite.
14.15. You are given the two circuits as shown in Fig. 14.44. Show that circuit (a) acts as OR gate while the circuit (b) acts as AND gate.
Answer
(a) A and B are the inputs and Y is the output of the given circuit. The left half of the given figure acts as the NOR Gate, while the right half acts as the NOT Gate. This is shown in the following figure.
Hence, the output of the NOR Gate =
This will be the input for the NOT Gate. Its output will be = A + B
∴Y = A + B
Hence, this circuit functions as an OR Gate.
(b) A and B are the inputs and Y is the output of the given circuit. It can be observed from the following figure that the inputs of the right half NOR Gate are the outputs of the two NOT Gates.
Hence, the output of the given circuit can be written as:
Hence, this circuit functions as an AND Gate.
14.16. Write the truth table for a NAND gate connected as given in Fig. 14.45.
Hence identify the exact logic operation carried out by this circuit.
Answer
A acts as the two inputs of the NAND gate and Y is the output, as shown in the following figure.
Hence, the output can be written as:
The truth table for equation (i) can be drawn as:

A
Y
0
1
1
0
This circuit functions as a NOT gate. The symbol for this logic circuit is shown as:
14.17. You are given two circuits as shown in Fig. 14.46, which consist of NAND gates. Identify the logic operation carried out by the two circuits.
Answer
In both the given circuits, A and B are the inputs and Y is the output.
(a) The output of the left NAND gate will be, as shown in the following figure.
Hence, the output of the combination of the two NAND gates is given as:
Hence, this circuit functions as an AND gate.
(b) is the output of the upper left of the NAND gate and is the output of the lower half of the NAND gate, as shown in the following figure.
Hence, the output of the combination of the NAND gates will be given as:
Hence, this circuit functions as an OR gate.
14.18. Write the truth table for circuit given in Fig. 14.47 below consisting of NOR gates and identify the logic operation (OR, AND, NOT) which this circuit is performing.
(Hint: A = 0, B = 1 then A and B inputs of second NOR gate will be 0 and hence Y=1. Similarly work out the values of Y for other combinations of A and B. Compare with the truth table of OR, AND, NOT gates and find the correct one.)
Answer
A and B are the inputs of the given circuit. The output of the first NOR gate is. It can be observed from the following figure that the inputs of the second NOR gate become the out put of the first one.
Hence, the output of the combination is given as:
The truth table for this operation is given as:

A
B
Y (=A + B)
0
0
0
0
1
1
1
0
1
1
1
1
This is the truth table of an OR gate. Hence, this circuit functions as an OR gate.
14.19. Write the truth table for the circuits given in Fig. 14.48 consisting of NOR gates only. Identify the logic operations (OR, AND, NOT) performed by the two circuits.
Answer
(a) A acts as the two inputs of the NOR gate and Y is the output, as shown in the following figure. Hence, the output of the circuit is.
The truth table for the same is given as:

A
Y
0
1
1
0
This is the truth table of a NOT gate. Hence, this circuit functions as a NOT gate.
(b) A and B are the inputs and Y is the output of the given circuit. By using the result obtained in solution (a), we can infer that the outputs of the first two NOR gates areas shown in the following figure.
are the inputs for the last NOR gate. Hence, the output for the circuit can be written as:
The truth table for the same can be written as:

A
B
Y (=A⋅B)
0
0
0
0
1
0
1
0
0
1
1
1
This is the truth table of an AND gate. Hence, this circuit functions as an AND gate.