Question

Assertion (A): The function \( f(x) = x^2 - x + 1 \) is strictly increasing on \((-1, 1)\). Reason (R): If \( f(x) \) is continuous on \([a, b]\) and derivable on \((a, b)\), then \( f(x) \) is strictly increasing on \([a, b]\) if \( f'(x)>0 \) for all \( x \in (a, b) \).

Answer 1

Step 1: Calculate the derivative of \( f(x) = x^2 - x + 1 \). The derivative is \( f'(x) = 2x - 1 \).
Step 2: Examine the sign of \( f'(x) \) on the interval \((-1, 1)\).
- When \( x = \frac{1}{2} \), \( f'(x) = 0 \).
- When \( x<\frac{1}{2} \), \( f'(x)<0 \), indicating \( f(x) \) is decreasing.
- When \( x>\frac{1}{2} \), \( f'(x)>0 \), indicating \( f(x) \) is increasing.
Step 3: Assertion (A) is false because \( f(x) \) is not strictly increasing over the entire interval \((-1,1)\).
Step 4: Reason (R) is true as it states a valid mathematical theorem. Therefore, Assertion (A) is false, and Reason (R) is true.

Answer 2

Step 1: The mean for a binomial distribution is calculated as \( \mu = np = 200 \times 0.04 = 8 \). Given that a Poisson approximation is used, the mean of the Poisson distribution is also 8. Therefore, Assertion (A) is true.
Step 2: The probability mass function for a Poisson distribution is defined as: \[ P(X = k) = \frac{e^{-\mu} \cdot \mu^k}{k!} \] . With \( \mu = 8 \) and \( k = 4 \), the calculation is: \[ P(X = 4) = \frac{e^{-8} \cdot 8^4}{4!} = \frac{512}{3e^8} \] . The provided expression aligns with this calculation, confirming that Reason (R) is true. 
Step 3: However, Reason (R) does not provide the direct explanation for why the mean of the Poisson distribution is 8. The mean of a Poisson distribution, when approximated from a binomial distribution, is determined by \( \lambda = np \), not by the calculation of \( P(X = 4) \). Consequently, both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation for Assertion (A).