Number Representations And Computer Arithmetic in Digital Logic : Practice MCQ Questions & Answers with Explanations

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

(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.

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.

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.