The number of integers that satisfy the equality \((x^2-5x+7)^{x+1} = 1\) is
- 2
- 3
- 5
- 4
The Correct Option is B
Solution and Explanation
For \( a^b = 1 \), the following cases hold:
- \( a = 1 \): valid for any \( b \)
- \( a = -1 \): valid only if \( b \) is even
- \( b = 0 \): valid for any \( a \neq 0 \)
Step-by-step Evaluation:
Case 1: \( x + 1 = 0 \Rightarrow x = -1 \)
Any non-zero base to the power 0 equals 1. Check the base: \[ (x^2 - 5x + 7) = (-1)^2 + 5 + 7 = 1 + 5 + 7 = 13 \neq 0 \] So, this is valid. ✅ \( x = -1 \) is a solution.
Case 2: \( x^2 - 5x + 7 = 1 \)
Solve: \[ x^2 - 5x + 6 = 0 \Rightarrow (x - 2)(x - 3) = 0 \Rightarrow x = 2, \; x = 3 \] Both are valid integer solutions.
✅ \( x = 2 \) and \( x = 3 \) are solutions.
Case 3: \( x^2 - 5x + 7 = -1 \)
\[ x^2 - 5x + 8 = 0 \Rightarrow \text{Discriminant} = 25 - 32 = -7 < 0 \] No real roots. ❌ No integer solution.
Final Answer:
The total number of integer solutions is: \[ \boxed{3} \] Correct Option: (B)
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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
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