Expand the following:
(i) (a + 3b)2
(ii) (2p – 3q)2
(iii) (2a + 1 / 2a)2
(iv) (x – 3y – 2z)2
(i) (a + 3b)2
(a + 3b)2 = a2 + 2(a) (3b) + (3b)2 [Using (x + y)2 = x2 + 2xy + y2]
We get,
= a2 + 6ab + 9b2
(ii) (2p – 3q)2
= (2p)2 – 2 (2p) (3q) + (3q)2
We get,
= 4p2 – 12pq + 9q2
(iii) (2a + 1 / 2a)2
= (2a)2 + 2 (2a) (1 / 2a) + (1 / 2a)2
We get,
= {4a2 + 2 + (1 / 4a2)}
(iv) (x – 3y – 2z)2
= x2 + (3y)2 + (2z)2 + 2 (x) (-3y) + 2 (-3y) (-2z) + 2 (x) (-2z)
[Using (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac]
We get,
= x2 + 9y2 + 4z2 – 6xy + 12yz – 4xz
Expand the following:
(i) (3x + 4) (2x – 1)
(ii) (2x – 5) (2x + 5) (2x – 3)
(i) (3x + 4) (2x – 1)
= 6x2 – 3x + 8x – 4
We get,
= 6x2 + 5x – 4
(ii) (2x – 5) (2x + 5) (2x – 3)
= (4x2 – 25) (2x – 3)
We get,
= 8x3 – 12x2 – 50x + 75
Expand the following:
(i) (a + 4) (a + 7)
(ii) (m + 8) (m – 7)
(iii) (x – 5) (x – 4)
(i) (a + 4) (a + 7)
= a2 + 4a + 7a + 28
We get,
= a2 + 11a + 28
(ii) (m + 8) (m – 7)
= m2 + 8m – 7m – 56
We get,
= m2 + m – 56
(iii) (x – 5) (x – 4)
= x2 – 5x – 4x + 20
We get,
= x2 – 9x + 20
Write the truth table for the circuits given in figure consisting of NOR gates only. Identify the logic operations (OR, AND, NOT) performed by the two circuits.
Boolean expression for logic circuit (a)
Here the given NOR gate with short circuit input is acting as NOT gate. Its truth table is
Boolean expression for logic circuit (b)
Hence, the logic circuit acts like AND gate. Its truth table is
Write the truth table for circuit given in figure 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 Yfor other combinations of A and B. Compare with the truth table of OR, AND, NOT gates and find the correct one.)
Let us find the Boolean expression for the logic circuit.
Hence, the output of given logic circuit shows that logic circuit act as OR gate. Truth Table
You are given two circuits as shown in figure which consist of NAND gates. Identify the logic operation carried out by the two circuits.
(a) Let us find Boolean expression for given logic circuit
(b) By De-Morgan’s theorem
so, the given logic circuit acts as OR gate.
Write the truth table for a NAND gate connected as given in following figure.
Hence identify the exact logic operation carried out by this circuit.
The Boolean expression for NAND gate will be
YOU are given the two circuits as shown in figure. Show that circuit (a) acts as OR gate while the circuit (b) acts as AND gate.
Let us first find the Boolean expression for logic circuit (a)
By De-Morgan’s theorem we know
so, above logic circuit provides output as AND gate
In a p-n junction diode, the current I can be
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, kB is the Boltzmann constant (8.6 × 10-5 eV/K) and T is the absolute temperature. If for a given diode I0 = 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?
The current I through a junction diode is given as
(a) When V = 0.6 V
(b) When V = 0.7 V,
(c)
(d) For both the voltages, the current I will be almost equal to I0, showing almost infinite dynamic resistance in the reverse bias. I=-I0= – 5 × 10-12 A.
In an intrinsic semiconductor the energy gap Eg is 1.2 eV. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at 600 K and that at 300 K? Assume that the temperature dependence of intrinsic carrier concentration n, is given by
Conductivity is given by s = e (ne µe +nh µh) For intrinsic semiconductor ne = nh = ni Also mobility of holes (p,,) « mobility of electrons (pj So, conductivity a = enepe „ Temperature dependence of intrinsic carrier concentration
Conductivity increases rapidly with the rise of temperature.
Science (7568)
Maths (4467)
Technology (239)
English (170)
Grades