← 2019 Paper 2

UPSC 2019 Maths Optional Paper 2 Q6b — Step-by-Step Solution

15 marks · Section B

Number systems · Numerical Analysis · asked 7× in 13 yrs · Read the full method →

Question

Find the equivalent numbers given in a specified number to the system mentioned against them:

• (i) Integer $524$ in binary system.
• (ii) $101010110101\cdot101101011$ to octal system.
• (iii) decimal number $5280$ to hexadecimal system.
• (iv) Find the unknown number $(1101\cdot101)_8\to(?)_{10}$.

Technique

Repeated division for integer→base, positional expansion for base→decimal, and 3-bit grouping for binary↔octal (since $8=2^3$).

Solution

(i) $524_{10}\to$ binary (repeated division by 2)

Read remainders bottom-up:

Check: $512+8+4=524$. ✓

(ii) $101010110101\cdot101101011_2\to$ octal (group in 3 bits about the point)

Integer part, group from the point leftward (pad left): $101\,010\,110\,101$

Fractional part, group from the point rightward (pad right): $101\,101\,011$

(iii) $5280_{10}\to$ hexadecimal (repeated division by 16)

Bottom-up: $1,4,A,0$:

Check: $1\cdot4096+4\cdot256+10\cdot16+0=4096+1024+160=5280$. ✓

(iv) $(1101.101)_8\to$ decimal (positional expansion, base 8)

Answer