Given below are two statements: Statement I: (167)10 = (10100111)2 Statement…
2023
Given below are two statements:
Statement I: (167)10 = (10100111)2
Statement II: (11010110)2 = (214)10
In the light of the above statements, choose the correct answer from the options given below
Answer: A. Both Statement I and Statement II are true — ConceptIn a positional numeral system, each digit contributes its value multiplied by the base raised to that digit’s position, counted from zero at the…
- A.
Both Statement I and Statement II are true
- B.
Both Statement I and Statement II are false
- C.
Statement I is true but Statement II is false
- D.
Statement I is false but Statement II is true
Attempted by 12 students.
Show answer & explanation
Correct answer: A
Concept
In a positional numeral system, each digit contributes its value multiplied by the base raised to that digit’s position, counted from zero at the right.
Therefore, a binary numeral is converted to decimal by summing every bit multiplied by the corresponding power of 2.
Application
For Statement I, expand 10100111 in base 2: 1×27 + 0×26 + 1×25 + 0×24 + 0×23 + 1×22 + 1×21 + 1×20 = 128 + 32 + 4 + 2 + 1 = 167. Thus Statement I holds.
For Statement II, expand 11010110 in base 2: 1×27 + 1×26 + 0×25 + 1×24 + 0×23 + 1×22 + 1×21 + 0×20 = 128 + 64 + 16 + 4 + 2 = 214. Thus Statement II holds.
Cross-check
Repeated division gives 167 ÷ 2 remainders 1, 1, 1, 0, 0, 1, 0, 1, which read upward as 10100111; similarly, 214 gives 11010110. Both reverse conversions agree.
Result
Both Statement I and Statement II are true.