Digital Logic : Practice Questions and Answers

$\begin{aligned} & \cos ^{-1} \mathrm{a}-\cos ^{-1} \mathrm{~b}=\cos ^{-1}\left(a b+\sqrt{1-\mathrm{a}^{2}} \cdot \sqrt{1-\mathrm{b}^{2}}\right) \\\ \therefore & \cos ^{-1} x-\cos ^{-1} \frac{y}{3} \\\ = & \cos ^{-1}\left(\frac{x y}{3}+\sqrt{1-x^{2}} \cdot \sqrt{1-\frac{y^{2}}{9}}\right)=a \\\ \therefore & \frac{x y}{3}+\frac{\sqrt{1-x^{2}} \cdot \sqrt{9-y^{2}}}{3}=\cos \alpha \\\ & x y+\sqrt{1-x^{2}} \cdot \sqrt{9-y^{2}}=3 \cos \alpha \\\ & x y-3 \cos \alpha=-\sqrt{1-x^{2}} \cdot \sqrt{9-y^{2}} \end{aligned}$ squaring on both sides, we get $x^{2} y^{2}-6 x y \cos \alpha+9 \cos ^{2} \alpha=\left(1-x^{2}\right)\left(9-y^{2}\right)$ $x^{2} y^{2}-6 x y \cos \alpha+9 \cos ^{2} \alpha=9-y^{2}-9 x^{2}+x^{2} y^{2}$ i.e., $9 x^{2}-6 x y \cos \alpha+y^{2}=9 \sin ^{2} \alpha$

Explanation: The JK Flip-Flop Definition: A JK flip-flop is a type of digital storage element (or bistable multivibrator) that can operate in various modes, such as set, reset, toggle, or no change, depending on the states of its inputs (J and K) and the clock pulse. It is similar to the SR (Set-Reset) flip-flop but resolves the ambiguity when both inputs are high. The JK flip-flop is widely used in digital circuits due to its versatility and ability to handle the "race condition" present in SR flip-flops.

Explanation: Demultiplexer (Demux) Definition: A demultiplexer (Demux) is a combinational logic circuit that takes a single input signal and channels it into one of several output lines. The specific output line that is selected for the input signal is determined by a set of select lines. It essentially performs the reverse operation of a multiplexer (Mux).

(A + B')(C' + D). Given Boolean function: A'B + CD' + A'B + CD' First simplify the given expression: A'B + A'B = A'B (redundant term) CD' + CD' = CD' (redundant term) So, the simplified function = A'B + CD' Now, we are asked to find the complement of this function: Let F = A'B + CD' Then the complement is: F' = (A'B + CD')' Apply De Morgans Law: (A'B + CD')' = (A'B)' · (CD')' (A'B)' = A + B' (CD')' = C' + D Therefore, F' = (A + B')(C' + D) Hence, the correct answer is:Option 3) (A + B')(C' + D)

correct matching of Boolean Algebra Laws with their corresponding Axioms is as follows: Absorption Law: a + (a b) = a Bounded Law: a + 1 = 1 Identity Law: a + 0 = a Distributive Law: a (b + c) = (a b) + (a c) Based on this, the correct matching is: A - IV (Absorption Law - a + (a b) = a) B - I (Bounded Law - a + 1 = 1) C - II (Identity Law - a + 0 = a) D - III (Distributive Law - a (b + c) = (a b) + (a c)) Additional Information Understanding Boolean Algebra is essential in digital electronics and computer science for designing and analyzing digital circuits. Boolean Algebra allows for the simplification of logic expressions, making it easier to implement and optimize digital systems. These fundamental laws and axioms provide the foundation for more complex theorems and applications in Boolean logic and digital design.

Key Points Frequency-division multiplexing: In telecommunications, frequency-division multiplexing (FDM) is a technique by which the total bandwidth available in a communication medium is divided into a series of non-overlapping frequency bands, each of which is used to carry a separate signal. For example, the coaxial cable used by cable television systems has a bandwidth of about 1000 MHz, but the passband of each television channel is only 6 MHz wide, so there is room for many channels on the cable. Time-division multiplexing: Time-division multiplexing (TDM) is a method of transmitting and receiving independent signals over a common signal path by means of synchronized switches at each end of the transmission line so that each signal appears on the line only a fraction of time in an alternating pattern.

Key Points The combinational circuit is a circuit in which we combine the different gates in the circuit, for example, encoder, decoder, multiplexer and demultiplexer. Some of the characteristics of combinational circuits are following − The output of a combinational circuit at any instant of time depends only on thelevels present at input terminals. The combinational circuit does not use any memory.

Key Points The number of multiplexers required to construct the bus is equal ton, where n is thenumber of bits in each register. The size of each multiplexer must bek: 1 since it multiplexes k data lines, where k is the number of registers. Therefore If there are 8 registers of 16 bits each, then we need 16 multiplexers, each of which is an 8: 1 multiplexer to create the 16-bit data bus.

The correct option is (1) Concept:- In the number system, we know hexadecimal is base 16 and octal is base-8 Calculation: To convert hexadecimal D.C to octal, First, convert D.C16 into decimal = D16 = D × 160 + C × 16-1 = 13.7510 Now, we have to convert 13.7510 to octal 13/8 = 1 with remainder 5 1/8 = 0 with remainder 1 13 = 15 ----(1) For converting decimal fraction 0.75 to octal number, 0.75 × 8 = 6 + 0 0.75 = 0.6 ---(2) ∴ 13.7510 = 15.68 ------ [ from (1) and (2) Hence, hexadecimal number D.C converted to octal is equals: 15.6

The correct answer is: option 4 Key Points Explanation: Convert the binary number and the hexadecimal number to decimal: 11012 = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13 A516 = 10×16¹ + 5×16⁰ = 160 + 5 = 165 Sum: 13 + 165 = 178 Therefore the correct decimal value of the sum is 178,

The correct answer is ASCII Code. Key Points ASCII (American Standard Code for Information Interchange) is a character encoding standard used to represent text in computers, telecommunications, and other devices. Each character, including letters, numbers, and symbols, is assigned a unique numeric value that is stored as a binary number in memory. For example, the character 'A' is represented by the numeric value 65 in ASCII, which is stored in binary as 01000001.

Calculations: Following choices are: 46, 51, 60, 42 1) 46 - The sum of digits is 10, not 6, so it can't be this. 2) 51 - The tens digit isn't double the ones digit, so it can't be this. 3) 60 - This exceeds the upper limit of less than 6 tens, so it can't be this.

Concept used: The successor of any number is the value that comes immediately after the original number. Calculation: The successor of 10999999 is (10999999 + 1) = 11000000 ∴ The correct answer is 11000000

Concept: The closure property of the multiplication of natural numbers states that the multiplication of two or more natural numbers always result in a natural number. Explanation: Hence, two or more natural numbers multiplied results in a natural number. For example, 5 × 2= 10, is a natural number.

Given: While calculating the value of 95 × 103, a student writes 95 × 103 = 10000 + 10a - 15. Concept used: ⇒ (a - b) × (c + d) = (a × c) + (a × d) - (b × c) - (b × d) Calculation: We can write 95 × 103 = (100 - 5) × (100 + 3) ⇒ (100 ×100) + (100 × 3) - (5 × 100) - (5 × 3) ⇒ 10000 + 300 - 500 - 15 ⇒ 10000 - 200 - 15 We can write it as, ⇒ 10000 - 20 × 10 - 15 Compare with given statement, 10000 + 10a - 15. ⇒ a = - 20 ∴ Option 3 is correct.

The correct answer is One bit of data. Key Points A flip-flop is a basic digital memory circuit, also known as a bistable multivibrator. It has two stable states, which can be used to store binary information. Each flip-flop can store exactly one bit of data, representing either a 0 or a 1.

Concept used: XL represents 40 in Roman numerals, LX represents 60, and LXXIV represents 74. Calculations: To find the value of (XL + LX - LXXIV), we first perform the addition and subtraction inside the parentheses, as follows: XL + LX - LXXIV = 40 + 60 - 74 Simplifying the addition and subtraction gives: = 100 - 74 = 26 = XXVI

Given: Distance = 2,896,819,200,000 metres Concept used: The standard form of a number is expressed in the powers of 10. Solution: Distance = 2,896,819,200,000 metres = 2.8968192 x 1012 m Hence, the correct option is 1.