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 $$i^{\text {th }}$$ round.
| List-I | List-II |
|---|---|
| (I) Probability of $$\left(X_{2} \geq Y_{2}\right)$$ is | (P) $$\frac{3}{8}$$ |
| (II) Probability of $$\left(X_{2}>Y_{2}\right)$$ is | (Q) $$\frac{11}{16}$$ |
| (III) Probability of $$\left(X_{3}=Y_{3}\right)$$ is | (R) $$\frac{5}{16}$$ |
| (IV) Probability of $$\left(X_{3}>Y_{3}\right)$$ is | (S) $$\frac{355}{864}$$ |
| (T) $$\frac{77}{432}$$ |
The correct option is:
Let $$p, q, r$$ be nonzero real numbers that are, respectively, the $$10^{\text {th }}, 100^{\text {th }}$$ and $$1000^{\text {th }}$$ terms of a harmonic progression. Consider the system of linear equations
$$$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} $$$
| List-I | List-II |
|---|---|
| (I) If $$\frac{q}{r}=10$$, then the system of linear equations has | (P) $$x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$$ as a solution |
| (II) If $$\frac{p}{r} \neq 100$$, then the system of linear equations has | (Q) $$x=\frac{10}{9}, y=-\frac{1}{9}, z=0$$ as a solution |
| (III) If $$\frac{p}{q} \neq 10$$, then the system of linear equations has | (R) infinitely many solutions |
| (IV) If $$\frac{p}{q}=10$$, then the system of linear equations has | (S) no solution |
| (T) at least one solution |
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.
| List-I | List-II |
|---|---|
| (I) If $\phi=\frac{\pi}{4}$, then the area of the triangle $F G H$ is | (P) $\frac{(\sqrt{3}-1)^{4}}{8}$ |
| (II) If $\phi=\frac{\pi}{3}$, then the area of the triangle $F G H$ is | (Q) 1 |
| (III) If $\phi=\frac{\pi}{6}$, then the area of the triangle $F G H$ is | (R) $\frac{3}{4}$ |
| (IV) If $\phi=\frac{\pi}{12}$, then the area of the triangle $F G H$ is | (S) $\frac{1}{2 \sqrt{3}}$ |
| (T) $\frac{3 \sqrt{3}}{2}$ |
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 $M_{B}=2 M_{A}$. 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 $\frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}}$. The value of $n$ is __________ .