Question:
If \(x^3 + y^3 = (100010111)_2\) and \(x + y = (11111)_2\), then what is \((x – y)^2 + xy\) equal to?
यदि \(x^3 + y^3 = (100010111)_2\) और \(x + y = (11111)_2\) हैं, तो \((x – y)^2 + xy\) किसके बराबर है?
Options:
- \((1101)_2\)
- \((1011)_2\)
- \((1001)_2\)
- \((1111)_2\)
- \((1101)_2\)
- \((1011)_2\)
- \((1001)_2\)
- \((1111)_2\)
Correct Answer: c)
Explanation:
Note that \((x-y)^2 + xy = x^2 – 2xy + y^2 + xy = x^2 – xy + y^2\).
Since \(x^3 + y^3 = (x+y)(x^2 – xy + y^2)\), we have \((x-y)^2 + xy = \dfrac{x^3+y^3}{x+y}\).
\(x^3+y^3 = (100010111)_2 = 256 + 16 + 4 + 2 + 1 = 279\).
\(x+y = (11111)_2 = 16 + 8 + 4 + 2 + 1 = 31\).
\(\dfrac{279}{31} = 9 = (1001)_2\).
\((x-y)^2 + xy = x^2 – xy + y^2 = \dfrac{x^3+y^3}{x+y}\)।
\(x^3+y^3 = 279\) तथा \(x+y = 31\)।
\(\dfrac{279}{31} = 9 = (1001)_2\)।