JEE (Advanced) 2022 Paper-2

Let $$\bar{z}$$ denote the complex conjugate of a complex number $$z$$. If $$z$$ is a non-zero complex number for which both real and imaginary parts of $$(\bar{z})^2 + \dfrac{1}{z^2}$$ are integers, then which of the following is/are possible value(s) of $$|z|$$?

Let G be a circle of radius R > 0. Let G$$_1$$, G$$_2$$, ..., G$$_n$$ be $$n$$ circles of equal radius $$r > 0$$. Suppose each of the $$n$$ circles G$$_1$$, G$$_2$$, ..., G$$_n$$ touches the circle G externally. Also, for $$i = 1, 2, ..., n-1$$, the circle G$$_i$$ touches G$$_{i+1}$$ externally, and G$$_n$$ touches G$$_1$$ externally. Then, which of the following statements is/are TRUE?

Let $$\hat{i}$$, $$\hat{j}$$ and $$\hat{k}$$ be the unit vectors along the three positive coordinate axes. Let

$$\vec{a} = 3\hat{i} + \hat{j} - \hat{k}$$,

$$\vec{b} = \hat{i} + b_2\hat{j} + b_3\hat{k}$$,      $$b_2, b_3 \in \mathbb{R}$$,

$$\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$$,      $$c_1, c_2, c_3 \in \mathbb{R}$$

be three vectors such that $$b_2 b_3 > 0$$, $$\vec{a} \cdot \vec{b} = 0$$ and $$\begin{pmatrix} 0 & -c_3 & c_2 \\ c_3 & 0 & -c_1 \\ -c_2 & c_1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{pmatrix} 3 - c_1 \\ 1 - c_2 \\ -1 - c_3 \end{pmatrix}$$.

Then, which of the following is/are TRUE?

For $$x \in \mathbb{R}$$, let the function $$y(x)$$ be the solution of the differential equation $$\dfrac{dy}{dx} + 12y = \cos\left(\dfrac{\pi}{12}x\right)$$, $$y(0) = 0$$.

Then, which of the following statements is/are TRUE?

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?

If $$M = \begin{pmatrix} \dfrac{5}{2} & \dfrac{3}{2} \\ -\dfrac{3}{2} & -\dfrac{1}{2} \end{pmatrix}$$, then which of the following matrices is equal to $$M^{2022}$$?

Suppose that

Box-I contains 8 red, 3 blue and 5 green balls,

Box-II contains 24 red, 9 blue and 15 green balls,

Box-III contains 1 blue, 12 green and 3 yellow balls,

Box-IV contains 10 green, 16 orange and 6 white balls.

A ball is chosen randomly from Box-I; call this ball $$b$$. If $$b$$ is red then a ball is chosen randomly from Box-II, if $$b$$ is blue then a ball is chosen randomly from Box-III, and if $$b$$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

For positive integer $$n$$, define

$$f(n) = n + \dfrac{16 + 5n - 3n^2}{4n + 3n^2} + \dfrac{32 + n - 3n^2}{8n + 3n^2} + \dfrac{48 - 3n - 3n^2}{12n + 3n^2} + \cdots + \dfrac{25n - 7n^2}{7n^2}$$.

Then the value of $$\displaystyle\lim_{n \to \infty} f(n)$$ is equal to

A particle of mass 1 kg is subjected to a force which depends on the position as $$\vec{F} = -k(x\hat{i} + y\hat{j})$$ kg m s$$^{-2}$$ with $$k = 1$$ kg s$$^{-2}$$. At time $$t = 0$$, the particle's position $$\vec{r} = \left(\dfrac{1}{\sqrt{2}}\hat{i} + \sqrt{2}\hat{j}\right)$$ m and its velocity $$\vec{v} = \left(-\sqrt{2}\hat{i} + \sqrt{2}\hat{j} + \dfrac{2}{\pi}\hat{k}\right)$$ m s$$^{-1}$$. Let $$v_x$$ and $$v_y$$ denote the x and y components of the particle's velocity, respectively. Ignore gravity. When $$z = 0.5$$ m, the value of $$(x v_y - y v_x)$$ is _______ m$$^2$$ s$$^{-1}$$.

In a radioactive decay chain reaction, $$^{230}_{90}$$Th nucleus decays into $$^{214}_{84}$$Po nucleus. The ratio of the number of $$\alpha$$ to number of $$\beta^-$$ particles emitted in this process is _______.