1
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $$S = \left\{ {x \in \mathbb{Z}:x(66 - x) \ge {5 \over 9}M} \right\}$$ and the event $$\mathrm{A = \{ x \in S:x\,is\,a\,multiple\,of\,3\}}$$. Then P(A) is equal to :

A
$$\frac{1}{3}$$
B
$$\frac{1}{5}$$
C
$$\frac{7}{22}$$
D
$$\frac{15}{44}$$
2
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations

$$x + y + z = 1$$

$$2x + \mathrm{N}y + 2z = 2$$

$$3x + 3y + \mathrm{N}z = 3$$

has unique solution is $${k \over 6}$$, then the sum of value of k and all possible values of N is :

A
18
B
21
C
20
D
19
3
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

Let $$\Omega$$ be the sample space and $$\mathrm{A \subseteq \Omega}$$ be an event.

Given below are two statements :

(S1) : If P(A) = 0, then A = $$\phi$$

(S2) : If P(A) = 1, then A = $$\Omega$$

Then :

A
both (S1) and (S2) are true
B
both (S1) and (S2) are false
C
only (S2) is true
D
only (S1) is true
4
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :

A
$$\frac{4}{9}$$
B
$$\frac{5}{18}$$
C
$$\frac{1}{6}$$
D
$$\frac{3}{10}$$
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