JEE (Advanced) 2019 Paper-2

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?

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

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?

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?

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 __________

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__

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 ________

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 _______

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 ______

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 _________ .