Solving the Binary Number Problem

The problem asks us to find the value of $(x-y)^2 + xy$ given two equations involving $x$ and $y$ with binary numbers. To solve this, we will first convert the binary numbers to their decimal equivalents and then use algebraic identities.

Converting Binary to Decimal

Let's convert the given binary numbers to decimal:

• $x^3 + y^3 = (100010111)_2$
• $x + y = (11111)_2$

Conversion of $(100010111)_2$ to decimal:

$(100010111)_2 = 1 \times 2^8 + 0 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$

$= 1 \times 256 + 0 \times 128 + 0 \times 64 + 0 \times 32 + 1 \times 16 + 0 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times 1$

$= 256 + 0 + 0 + 0 + 16 + 0 + 4 + 2 + 1$

$= 279$

So, $x^3 + y^3 = 279$ in decimal.

Conversion of $(11111)_2$ to decimal:

$(11111)_2 = 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$

$= 1 \times 16 + 1 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times 1$

$= 16 + 8 + 4 + 2 + 1$

$= 31$

So, $x + y = 31$ in decimal.

Using Algebraic Identities

We are asked to find the value of $(x-y)^2 + xy$. Let's simplify this expression using algebraic identities:

$(x-y)^2 + xy = (x^2 - 2xy + y^2) + xy$

$(x-y)^2 + xy = x^2 - 2xy + y^2 + xy$

$(x-y)^2 + xy = x^2 - xy + y^2$

Now, consider the algebraic identity for the sum of cubes:

$x^3 + y^3 = (x+y)(x^2 - xy + y^2)$

Notice that the expression we need to find, $x^2 - xy + y^2$, is a factor in the sum of cubes identity. We can rearrange the identity to solve for $x^2 - xy + y^2$:

$x^2 - xy + y^2 = \frac{x^3 + y^3}{x+y}$

Calculating the Value

Now we can substitute the decimal values we found:

$x^2 - xy + y^2 = \frac{279}{31}$

Performing the division:

$279 \div 31 = 9$

So, the value of $(x-y)^2 + xy$ in decimal is 9.

Converting Decimal Back to Binary

The options are given in binary form, so we need to convert the decimal value 9 back to binary.

To convert a decimal number to binary, we repeatedly divide the decimal number by 2 and record the remainders. The binary representation is formed by reading the remainders from bottom to top.

Reading the remainders from bottom up, we get $(1001)_2$.

Therefore, $(x-y)^2 + xy = (1001)_2$.

Final Answer Verification

The calculated value $(1001)_2$ matches one of the given options.

Revision Table: Key Concepts

Additional Information: Number Systems and Identities

This problem combines concepts from different areas of mathematics: number systems (binary and decimal) and algebra (identities). Understanding how to convert between different number bases is crucial in computer science and digital electronics. Binary is the base-2 number system, using only digits 0 and 1. Decimal is the base-10 system, which is the standard system we use daily.

Algebraic identities are equations that are true for all possible values of the variables. They are powerful tools for simplifying expressions and solving equations. The identity for the sum of cubes, $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$, and the square of a difference, $(a-b)^2 = a^2 - 2ab + b^2$, are examples of fundamental identities used here.

Being comfortable with both number base conversions and algebraic manipulation is key to solving such problems efficiently in competitive exams.