Read the following statements carefully and select the correct option.
Statement-1 : The perimeter of a triangle whose sides are `2p^(2)+3p+1`, `p^(2)+7` and `3p^(2)-2p+3` is `6p^(2)-p+11`.
Statement-2 : The value of the expression `x^(5)-y^(4)+y^(3)-x^(2)+1`, when x=2 and y=1 is 29.

Options

A. Statement-1 is true but Statement-2 is false
B. Statement-1 is false but Statement-2 is true
C. Both Statement-1 and Statement-2 are true
D. Both Statement-1 and Statement-2 are false

Answer

To solve the problem, we need to analyze both statements provided in the question.

**Step 1: Analyze Statement 1**
We are given the sides of a triangle:
1. Side 1: \(2p^2 + 3p + 1\)
2. Side 2: \(p^2 + 7\)
3. Side 3: \(3p^2 - 2p + 3\)

To find the perimeter of the triangle, we need to sum these three sides.

**Calculation of Perimeter:**
\[ \text{Perimeter} = (2p^2 + 3p + 1) + (p^2 + 7) + (3p^2 - 2p + 3) \]

Now, let's combine like terms:
- Combine \(p^2\) terms:
\[ 2p^2 + p^2 + 3p^2 = 6p^2 \]
- Combine \(p\) terms:
\[ 3p - 2p = p \]
- Combine constant terms:
\[ 1 + 7 + 3 = 11 \]

Putting it all together, we have:
\[ \text{Perimeter} = 6p^2 + p + 11 \]

**Conclusion for Statement 1:**
The perimeter calculated is \(6p^2 + p + 11\), which is different from the statement given in the question \(6p^2 - p + 11\). Therefore, **Statement 1 is false**.

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**Step 2: Analyze Statement 2**
We need to evaluate the expression:
\[ x^5 - y^4 + y^3 - x^2 + 1 \]
for \(x = 2\) and \(y = 1\).

**Substituting the values:**
\[ = (2^5) - (1^4) + (1^3) - (2^2) + 1 \]

Now, calculate each term:
- \(2^5 = 32\)
- \(1^4 = 1\)
- \(1^3 = 1\)
- \(2^2 = 4\)

Now substitute these values back into the expression:
\[ = 32 - 1 + 1 - 4 + 1 \]

**Perform the arithmetic:**
\[ = 32 - 1 = 31 \]
\[ 31 + 1 = 32 \]
\[ 32 - 4 = 28 \]
\[ 28 + 1 = 29 \]

**Conclusion for Statement 2:**
The value of the expression when \(x = 2\) and \(y = 1\) is indeed 29. Therefore, **Statement 2 is true**.

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**Final Conclusion:**
- Statement 1 is false.
- Statement 2 is true.

Thus, the correct option is that Statement 1 is false and Statement 2 is true.

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