1
JEE Advanced 2023 Paper 1 Online
+3
-1
Let $X=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: \frac{x^2}{8}+\frac{y^2}{20}<1\right.$ and $\left.y^2<5 x\right\}$. Three distinct points $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is :
A
$\frac{71}{220}$
B
$\frac{73}{220}$
C
$\frac{79}{220}$
D
$\frac{83}{220}$
2
JEE Advanced 2022 Paper 2 Online
+3
-1
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

A
$\frac{15}{256}$
B
$\frac{3}{16}$
C
$\frac{5}{52}$
D
$\frac{1}{8}$
3
JEE Advanced 2022 Paper 1 Online
+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)
4
JEE Advanced 2021 Paper 1 Online
+3
-1
Consider three sets E1 = {1, 2, 3}, F1 = {1, 3, 4} and G1 = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E1, and let S1 denote the set of these chosen elements. Let E2 = E1 $$-$$ S1 and F2 = F1 $$\cup$$ S1. Now two elements are chosen at random, without replacement, from the set F2 and let S2 denote the set of these chosen elements.

Let G2 = G1 $$\cup$$ S2. Finally, two elements are chosen at random, without replacement, from the set G2 and let S3 denote the set of these chosen elements.

Let E3 = E2 $$\cup$$ S3. Given that E1 = E3, let p be the conditional probability of the event S1 = {1, 2}. Then the value of p is
A
$${1 \over 5}$$
B
$${3 \over 5}$$
C
$${1 \over 2}$$
D
$${2 \over 5}$$
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