  Instructions

For the following questions answer them individually

Question 41

# Let$$f(x) = \frac{\sin \pi x}{x^2}, x > 0.$$Let $$x_1 < x_2 < x_3 < ... < x_n < ...$$ be all the points of local maximum offand $$y_1 < y_2 < y_3 < ... < y_n < ...$$ be all the points of local minimum off.Then which of the following options is/are correct? Question 42

# For $$a \in R, |a| > 1,$$ let $$\lim_{n \rightarrow \infty} \left(\frac{1 + \sqrt{2} + ... + \sqrt{n}}{n^{\frac{7}{3} \left(\frac{1}{\left(an + 1\right)^2} + \frac{1}{\left(an + 2\right)^2} + ... + \frac{1}{\left(an + n\right)^2} \right)}}\right) = 54.$$Then the possible value(s) of a is/are Question 43

# Let $$f : R \rightarrow R$$ be given by $$f(x) = (x - 1)(x - 2)(x - 5).$$ Define $$F(x) = \int_{0}^{x} f(t) dt, x > 0.$$Then which of the following options is/are correct? Question 44

# Three lines $$L_1 : \overrightarrow{r} = \lambda \widehat{i}, \lambda \in R$$$$L_2: \overrightarrow{r} = \widehat{k} + \mu \widehat{j}, \mu \in R$$ and $$L_3: \overrightarrow{r} = \widehat{i} + \widehat{j} + v \widehat{k}, v \in R$$are given. For which point(s) Q on $$L_2$$ can wefind a point P on $$L_1$$ and a point R on $$L_3$$ so that P,Q and R are collinear? Question 45

# Suppose $$det\begin{bmatrix} \sum_{k = 0}^nk & \sum_{k = 0}^n {^nC_k k^2} \\\sum_{k = 0}^n {^n C_k k} & \sum_{k = 0}^n {^n C_k 3^k} \end{bmatrix} = 0$$holds for somepositive integer n. Then $$\sum_{k = 0}^n \frac {^nC_k}{k + 1}$$ Equals __________

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# Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is__

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# Let |X| denote the number of elements in a set X. Let S = {1, 2,3, 4,5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that $$1 \leq |B| < |A|$$, equals ________

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# The value of $$\sec^{-1} \left(\frac{1}{4}\sum_{k = 0}^{10} \sec \left(\frac{7\pi}{12} + \frac{k\pi}{2}\right) \sec \left(\frac{7\pi}{12} + \frac{\left(k + 1\right)\pi}{2}\right) \right)$$in the interval $$\begin{bmatrix}-\frac {\pi}{4}, & \frac{3\pi}{4} \end{bmatrix}$$ equals _______

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# The value of the integral$$\int_{0}^{\pi/2} \frac{3 \sqrt{\cos \theta}}{\left(\sqrt{\cos \theta} + \sqrt{\sin \theta}\right)^5} d \theta$$equals ______

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# Let $$\overrightarrow{a} = 2\widehat{i} + \widehat{j} - \widehat{k}$$ and $$\overrightarrow{b} = \widehat{i} + 2\widehat{j} + \widehat{k}$$ be two vectors. Consider a vector $$\overrightarrow{c} = \alpha \overrightarrow{a} + \beta \overrightarrow{b}, \alpha, \beta \in R.$$ If the projection of $$\overrightarrow{c}$$ on the vector $$\left(\overrightarrow{a} + \overrightarrow{b}\right)$$ is $$3\sqrt{2},$$ then the minimum value of $$\left(\overrightarrow{c} - \left(\overrightarrow{a} \times \overrightarrow{b}\right)\right) . \overrightarrow{c}$$ equals _________ .

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