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: Compute the derivative of \( f(x) = x^2 - x + 1 \): \[ f'(x) = 2x - 1. \] 
Step 2: Analyze the sign of \( f'(x) \) on \((-1, 1)\): - At \( x = \frac{1}{2} \), \( f'(x) = 0 \). - For \( x<\frac{1}{2} \), \( f'(x)<0 \), meaning \( f(x) \) is decreasing. - For \( x>\frac{1}{2} \), \( f'(x)>0 \), meaning \( f(x) \) is increasing. 
Step 3: Since \( f(x) \) is not strictly increasing throughout \((-1,1)\), Assertion (A) is false. 
Step 4: Reason (R) states a correct mathematical theorem, so it is true. Thus, the correct answer is that Assertion (A) is false, but Reason (R) is true.

Answer 2

Step 1: For a binomial distribution, the mean is given by: \[ \mu = np = 200 \times 0.04 = 8. \] Since Poisson approximation is used, the mean of the Poisson distribution is also 8. Thus, Assertion (A) is true. 
Step 2: The probability mass function of a Poisson distribution is: \[ P(X = k) = \frac{e^{-\mu} \cdot \mu^k}{k!}. \] Substituting \( \mu = 8 \) and \( k = 4 \): \[ P(X = 4) = \frac{e^{-8} \cdot 8^4}{4!} = \frac{512}{3e^8}. \] Since the given expression matches this calculation, Reason (R) is also true. 
Step 3: However, Reason (R) does not directly explain why the mean of the Poisson distribution is 8. The mean of a Poisson distribution is derived from the binomial approximation (\( \lambda = np \)), not from the probability calculation of \( P(X = 4) \). Thus, Assertion (A) and Reason (R) are both true, but Reason (R) is not the correct explanation of Assertion (A).