Assertion (A ) : the equation whose roots are exceed by 2 then those of ` 2x^3 +3x^2 -4x +5=0` is ` 2x^3 -9x^2 +8x+9=0`
Reason (R ) : the equation whose roots are exceed by h than those of ` f(x) =0` is ` f(x-h) =0`

Options

A. both A and R are true R is the correct explanation of A
B. both A and R are true and R is not correct explanation of A
C. A is true and R is false
D. A is false and R is true

Answer

To solve the problem, we need to determine if the assertion (A) and reason (R) provided are true.

### Step-by-Step Solution:

1. **Identify the Original Equation**:
The original equation given is:
\[ 2x^3 + 3x^2 - 4x + 5 = 0 \]
Let's denote this function as \( f(x) \).

2. **Understand the Transformation**:
The assertion states that the roots of the new equation exceed those of the original equation by 2. According to the reason provided, if the roots of \( f(x) = 0 \) are increased by \( h \), then the new equation can be expressed as:
\[ f(x - h) = 0 \]
Here, \( h = 2 \).

3. **Substitute \( x - 2 \) into \( f(x) \)**:
We need to find \( f(x - 2) \):
\[ f(x - 2) = 2(x - 2)^3 + 3(x - 2)^2 - 4(x - 2) + 5 \]

4. **Expand \( f(x - 2) \)**:
- First, calculate \( (x - 2)^3 \):
\[ (x - 2)^3 = x^3 - 6x^2 + 12x - 8 \]
- Now, calculate \( (x - 2)^2 \):
\[ (x - 2)^2 = x^2 - 4x + 4 \]
- Substitute these into \( f(x - 2) \):
\[ f(x - 2) = 2(x^3 - 6x^2 + 12x - 8) + 3(x^2 - 4x + 4) - 4(x - 2) + 5 \]

5. **Distribute and Combine Like Terms**:
- Expanding each term:
\[ = 2x^3 - 12x^2 + 24x - 16 + 3x^2 - 12x + 12 - 4x + 8 + 5 \]
- Combine like terms:
- \( 2x^3 \)
- \( -12x^2 + 3x^2 = -9x^2 \)
- \( 24x - 12x - 4x = 8x \)
- \( -16 + 12 + 8 + 5 = 9 \)

6. **Final Expression**:
Thus, we have:
\[ f(x - 2) = 2x^3 - 9x^2 + 8x + 9 = 0 \]

7. **Conclusion**:
The new equation whose roots exceed those of the original equation by 2 is:
\[ 2x^3 - 9x^2 + 8x + 9 = 0 \]
This confirms that assertion (A) is true.

8. **Verify the Reason**:
The reason (R) states that the equation whose roots exceed by \( h \) is given by \( f(x - h) = 0 \). Since we used \( h = 2 \) and derived the correct equation, the reason is also true.

### Final Answer:
Both the assertion (A) and the reason (R) are true, and R is the correct explanation of A.

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