Question:

Let 𝑓: ℝ→ ℝ be a function defined by 𝑓(𝑥)=𝑥²−𝑥, 𝑥 ∈ ℝ. Let 𝑔: ℝ→ℝ be a twice differentiable function such that 𝑔(𝑥)=0 has exactly three distinct roots in the open interval (0, 1). Let ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥), 𝑥 ∈ ℝ, and ℎ′′ be the second order derivative of the function ℎ. If 𝑛 is the number of roots of ℎ′′(𝑥)=0 in (0, 1), then the minimum possible value of 𝑛 equals ________

IIT JAM MS - 2023
IIT JAM MS

Updated On: Oct 1, 2024

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Correct Answer: 3

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The correct answer is: 3

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Suppose that the weights (in kgs) of six months old babies, monitored at a healthcare facility, have $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$ are unknown parameters. Let $X_1, X_2, \ldots, X_9$ be a random sample of the weights of such babies. Let $\overline{X} = \frac{1}{9} \sum_{i=1}^{9} X_i$ , $S = \sqrt{\frac{1}{8} \sum_{i=1}^{9} (X_i - \overline{X})^2}$ and let a 95% confidence interval for $\mu$ based on $t$ -distribution be of the form $(\overline{X} - h(S), \overline{X} + h(S))$ , for an appropriate function $h$ of random variable $S$ . If the observed values of $\overline{X}$ and $S^2$ are 9 and 9.5, respectively, then the width of the confidence interval is equal to __________ (round off to 2 decimal places) (You may use $t_{9,0.025} = 2.262, t_{8,0.025} = 2.306, t_{9,0.05} = 1.833, t_{8,0.05} = 1.86$ ).
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- IIT JAM MS - 2024
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- IIT JAM MS - 2024
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