JEE (Advanced) 2019 Paper-2

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 off
and $$y_1 < y_2 < y_3 < ... < y_n < ...$$ be all the points of local minimum off.
Then which of the following options is/are correct?

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Question 42

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

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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?

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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?

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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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Question 46

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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Question 47

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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Question 48

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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Question 49

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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Question 50

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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