Statement I : The lines `x(a+2b) +y(a+3b)=a+b ` are concurrent at the point `(2,-1)`
Statement II : The lines `x+y -1 =0 and 2x + 3y -1 = 0 ` intersect at the point `(2,-1)`

Options

A. Statement I is true ,statement II is true , statement II is a correct explanation for statement I
B. Statement I is true ,statement II is true statement II is not a correct explanation for statement I
C. Statement I is true ,statement II is false
D. Statement I is false ,statement II is true

Answer

To solve the problem, we need to analyze both statements and verify their validity step by step.

### Statement I:
The lines given by the equation \( x(a + 2b) + y(a + 3b) = a + b \) are concurrent at the point \( (2, -1) \).

**Step 1:** Substitute \( x = 2 \) and \( y = -1 \) into the equation.

\[ 2(a + 2b) + (-1)(a + 3b) = a + b \]

**Step 2:** Simplify the left-hand side.

\[ 2(a + 2b) - (a + 3b) = 2a + 4b - a - 3b \]

**Step 3:** Combine like terms.

\[ (2a - a) + (4b - 3b) = a + b \]

This simplifies to:

\[ a + b = a + b \]

**Step 4:** Since both sides of the equation are equal, Statement I is true.

### Statement II:
The lines given by the equations \( x + y - 1 = 0 \) and \( 2x + 3y - 1 = 0 \) intersect at the point \( (2, -1) \).

**Step 1:** Substitute \( x = 2 \) and \( y = -1 \) into the first equation.

\[ 2 + (-1) - 1 = 0 \]

**Step 2:** Simplify the left-hand side.

\[ 2 - 1 - 1 = 0 \implies 0 = 0 \]

This confirms that the point satisfies the first equation.

**Step 3:** Now substitute \( x = 2 \) and \( y = -1 \) into the second equation.

\[ 2(2) + 3(-1) - 1 = 0 \]

**Step 4:** Simplify the left-hand side.

\[ 4 - 3 - 1 = 0 \implies 0 = 0 \]

This confirms that the point satisfies the second equation as well.

### Conclusion:
Both statements are true. However, we need to determine if Statement II serves as a complete explanation for Statement I.

**Step 1:** Analyze the relationship between the two statements. The lines in Statement I are parameterized by \( a \) and \( b \), while Statement II deals with specific lines.

**Step 2:** Set \( a = 0 \) and \( b = 1 \) in Statement I's equation:

\[ x(0 + 2 \cdot 1) + y(0 + 3 \cdot 1) = 0 + 1 \implies 2x + 3y = 1 \]

**Step 3:** Set \( a = 1 \) and \( b = 0 \):

\[ x(1 + 2 \cdot 0) + y(1 + 3 \cdot 0) = 1 + 0 \implies x + y = 1 \]

**Step 4:** The two lines derived from Statement I are indeed the lines in Statement II. However, Statement II does not explain the concurrency of the lines for all values of \( a \) and \( b \).

Thus, Statement II is not a complete explanation for Statement I.

### Final Answer:
Both statements are true, but Statement II does not completely explain Statement I.