JEE (Advanced) 2022 Paper-1

Let $$P_1$$ and $$P_2$$ be two planes given by

$$P_1: 10x + 15y + 12z - 60 = 0,$$

$$P_2: -2x + 5y + 4z - 20 = 0.$$

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $$P_1$$ and $$P_2$$?

Let $$S$$ be the reflection of a point $$Q$$ with respect to the plane given by

$$\vec{r} = -(t+p)\hat{i} + t\hat{j} + (1+p)\hat{k}$$

where $$t$$, $$p$$ are real parameters and $$\hat{i}$$, $$\hat{j}$$, $$\hat{k}$$ are the unit vectors along the three positive coordinate axes. If the position vectors of $$Q$$ and $$S$$ are $$10\hat{i} + 15\hat{j} + 20\hat{k}$$ and $$\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$$ respectively, then which of the following is/are TRUE?

Consider the parabola $$y^2 = 4x$$. Let $$S$$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $$P = (-2, 1)$$ meet the parabola at $$P_1$$ and $$P_2$$. Let $$Q_1$$ and $$Q_2$$ be points on the lines $$SP_1$$ and $$SP_2$$ respectively such that $$PQ_1$$ is perpendicular to $$SP_1$$ and $$PQ_2$$ is perpendicular to $$SP_2$$. Then, which of the following is/are TRUE?

Let $$|M|$$ denote the determinant of a square matrix $$M$$. Let $$g: \left[0, \frac{\pi}{2}\right] \to \mathbb{R}$$ be the function defined by

$$g(\theta) = \sqrt{f(\theta) - 1} + \sqrt{f\left(\frac{\pi}{2} - \theta\right) - 1}$$

where

$$f(\theta) = \frac{1}{2} \begin{vmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{vmatrix} + \begin{vmatrix} \sin\pi & \cos\left(\theta + \frac{\pi}{4}\right) & \tan\left(\theta - \frac{\pi}{4}\right) \\ \sin\left(\theta - \frac{\pi}{4}\right) & -\cos\frac{\pi}{2} & \log_e\left(\frac{4}{\pi}\right) \\ \cot\left(\theta + \frac{\pi}{4}\right) & \log_e\left(\frac{\pi}{4}\right) & \tan\pi \end{vmatrix}.$$

Let $$p(x)$$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $$g(\theta)$$, and $$p(2) = 2 - \sqrt{2}$$. Then, which of the following is/are TRUE?

Consider the following lists:

The correct option is:

Two players, $$P_1$$ and $$P_2$$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $$x$$ and $$y$$ denote the readings on the die rolled by $$P_1$$ and $$P_2$$, respectively. If $$x > y$$, then $$P_1$$ scores 5 points and $$P_2$$ scores 0 point. If $$x = y$$, then each player scores 2 points. If $$x < y$$, then $$P_1$$ scores 0 point and $$P_2$$ scores 5 points. Let $$X_i$$ and $$Y_i$$ be the total scores of $$P_1$$ and $$P_2$$, respectively, after playing the $$i^{th}$$ round.

The correct option is:

Let $$p$$, $$q$$, $$r$$ be nonzero real numbers that are, respectively, the $$10^{th}$$, $$100^{th}$$ and $$1000^{th}$$ terms of a harmonic progression. Consider the system of linear equations

$$x + y + z = 1$$

$$10x + 100y + 1000z = 0$$

$$qr \, x + pr \, y + pq \, z = 0.$$

The correct option is:

Consider the ellipse

$$\frac{x^2}{4} + \frac{y^2}{3} = 1.$$

Let $$H(\alpha, 0)$$, $$0 < \alpha < 2$$, be a point. A straight line drawn through $$H$$ parallel to the $$y$$-axis crosses the ellipse and its auxiliary circle at points $$E$$ and $$F$$ respectively, in the first quadrant. The tangent to the ellipse at the point $$E$$ intersects the positive $$x$$-axis at a point $$G$$. Suppose the straight line joining $$F$$ and the origin makes an angle $$\phi$$ with the positive $$x$$-axis.

The correct option is:

Two spherical stars $$A$$ and $$B$$ have densities $$\rho_A$$ and $$\rho_B$$, respectively. $$A$$ and $$B$$ have the same radius, and their masses $$M_A$$ and $$M_B$$ are related by $$M_B = 2M_A$$. Due to an interaction process, star $$A$$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $$\rho_A$$. The entire mass lost by $$A$$ is deposited as a thick spherical shell on $$B$$ with the density of the shell being $$\rho_A$$. If $$v_A$$ and $$v_B$$ are the escape velocities from $$A$$ and $$B$$ after the interaction process, the ratio $$\frac{v_B}{v_A} = \sqrt{\frac{10n}{15^{1/3}}}$$. The value of $$n$$ is ______.

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction $$^{16}_7N + ^4_2He \to ^1_1H + ^{19}_8O$$ in a laboratory frame is $$n$$ (in MeV). Assume that $$^{16}_7N$$ is at rest in the laboratory frame. The masses of $$^{16}_7N$$, $$^4_2He$$, $$^1_1H$$ and $$^{19}_8O$$ can be taken to be 16.006 $$u$$, 4.003 $$u$$, 1.008 $$u$$ and 19.003 $$u$$, respectively, where 1 $$u$$ = 930 MeV$$c^{-2}$$. The value of $$n$$ is ______.