Which of the following expressions are exactly equal in value ?
1. `( 3x - y) ^(2) - ( 5x ^(2) - 2xy)`
2. `(2x - y ) ^(2)`
3. `(2x pm y ) ^(2) - 2 xy`
4. `(2x +3y ) ^(2) - 2 xy`

Options

A. 1 and 2 only
B. 1, 2 and 3 only
C. 2 and 4 only
D. 1,2 and 4 only

Answer

To determine which of the given expressions are exactly equal in value, we will simplify each expression step by step.

### Step 1: Simplify the first expression
**Expression 1:** \((3x - y)^2 - (5x^2 - 2xy)\)

1. Expand \((3x - y)^2\):
\[ (3x - y)^2 = (3x)^2 - 2(3x)(y) + y^2 = 9x^2 - 6xy + y^2 \]

2. Substitute back into the expression:
\[ 9x^2 - 6xy + y^2 - (5x^2 - 2xy) \]

3. Distribute the negative sign:
\[ 9x^2 - 6xy + y^2 - 5x^2 + 2xy \]

4. Combine like terms:
\[ (9x^2 - 5x^2) + (-6xy + 2xy) + y^2 = 4x^2 - 4xy + y^2 \]

So, the first expression simplifies to:
\[ 4x^2 - 4xy + y^2 \]

### Step 2: Simplify the second expression
**Expression 2:** \((2x - y)^2\)

1. Expand \((2x - y)^2\):
\[ (2x - y)^2 = (2x)^2 - 2(2x)(y) + y^2 = 4x^2 - 4xy + y^2 \]

So, the second expression simplifies to:
\[ 4x^2 - 4xy + y^2 \]

### Step 3: Simplify the third expression
**Expression 3:** \((2x \pm y)^2 - 2xy\)

1. Consider both cases: \((2x + y)^2\) and \((2x - y)^2\).

**Case 1:** \((2x + y)^2\)
\[ (2x + y)^2 = 4x^2 + 4xy + y^2 \]
Then:
\[ 4x^2 + 4xy + y^2 - 2xy = 4x^2 + 2xy + y^2 \]

**Case 2:** \((2x - y)^2\)
\[ (2x - y)^2 = 4x^2 - 4xy + y^2 \]
Then:
\[ 4x^2 - 4xy + y^2 - 2xy = 4x^2 - 6xy + y^2 \]

Thus, the third expression can yield two different results depending on the sign.

### Step 4: Simplify the fourth expression
**Expression 4:** \((2x + 3y)^2 - 2xy\)

1. Expand \((2x + 3y)^2\):
\[ (2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 = 4x^2 + 12xy + 9y^2 \]

2. Substitute back into the expression:
\[ 4x^2 + 12xy + 9y^2 - 2xy = 4x^2 + 10xy + 9y^2 \]

### Conclusion
Now we compare the simplified expressions:

1. Expression 1: \(4x^2 - 4xy + y^2\)
2. Expression 2: \(4x^2 - 4xy + y^2\)
3. Expression 3:
- Case 1: \(4x^2 + 2xy + y^2\)
- Case 2: \(4x^2 - 6xy + y^2\)
4. Expression 4: \(4x^2 + 10xy + 9y^2\)

From the simplifications, we see that:
- Expression 1 and Expression 2 are equal: \(4x^2 - 4xy + y^2\).
- Expression 3 and Expression 4 do not match either of the first two expressions.

### Final Answer
The expressions that are exactly equal in value are **Expression 1 and Expression 2**.