1
JEE Advanced 2022 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1

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:

A
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (T); (IV) $$\rightarrow(S)$$
B
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (T); (IV) $$\rightarrow$$ (T)
C
(I) $$\rightarrow$$ (P); (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{S})$$
D
(I) $$\rightarrow$$ (P); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (Q); (IV) $$\rightarrow$$ (T)
2
JEE Advanced 2022 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1

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: A (I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (T) B (I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (R) C (I) $$\rightarrow(\mathrm{Q})$$; (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{P})$$; (IV) $$\rightarrow$$ (R) D (I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (P); (IV) $$\rightarrow$$ (T) 3 JEE Advanced 2022 Paper 1 Online MCQ (Single Correct Answer) +3 -1 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:

A
$(\mathrm{I}) \rightarrow(\mathrm{R}) ;(\mathrm{II}) \rightarrow(\mathrm{S}) ;(\mathrm{III}) \rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$
B
(I) $\rightarrow$ (R); (II) $\rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$
C
(I) $\rightarrow(\mathrm{Q}) ;(\mathrm{II}) \rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$
D
(I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (P)
4
JEE Advanced 2022 Paper 1 Online
Numerical
+3
-0

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 __________ .

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