Instructions

For the following questions answer them individually

Question 51

Question 52

Question 53

Question 54

Question 55

Question 56

# Let $$f'(x) = \frac{192x^3}{2 + \sin^4 \pi x}$$ for all $$x \in R$$ with $$f\left(\frac{1}{2}\right) = 0$$. If $$m \leq \int_{\frac{1}{2}}^{1} f(x) dx \leq M,$$ then the possible values of m and M are

Instructions

Let $$n_1$$ and $$n_2$$ be the number of red and black balls, respectively, in box I. Let $$n_3$$ and $$n_4$$ be the number of red and black balls, respectively, in box II.

Question 57

Question 58

# A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $$\frac{1}{3},$$ then the correct option(s) with the possible values of $$n_1 and n_2$$ is(are)

Instructions

Let $$F : R \rightarrow R$$ be a thrice differentiable function. Suppose that F(1) = 0, F(3) = -4 and $$F'(x) < 0$$ for all x \in $$(\frac{1}{2} , 3)$$. Let $$f(x) = xF(x)$$ for all $$x \in R$$.

Question 59

Question 60

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