1
JEE Main 2025 (Online) 23rd January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the point A divide the line segment joining the points $\mathrm{P}(-1,-1,2)$ and $\mathrm{Q}(5,5,10)$ internally in the ratio $r: 1(r>0)$. If O is the origin and $(\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}})-\frac{1}{5}|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}|^2=10$, then the value of r is :

A
$\sqrt7$
B
14
C
7
D
3
2
JEE Main 2025 (Online) 23rd January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

A board has 16 squares as shown in the figure :

JEE Main 2025 (Online) 23rd January Evening Shift Mathematics - Probability Question 14 English

Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

A
$\frac{3}{5}$
B
$\frac{4}{5}$
C
$\frac{23}{30}$
D
$\frac{7}{10}$
3
JEE Main 2025 (Online) 23rd January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m$, $n$ are coprime numbers, then $m+n$ is equal to :

A
14
B
6
C
21
D
9
4
JEE Main 2025 (Online) 23rd January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $\mathrm{X}=\mathbf{R} \times \mathbf{R}$. Define a relation R on X as :

$$\left(a_1, b_1\right) R\left(a_2, b_2\right) \Leftrightarrow b_1=b_2$$

Statement I: $\quad \mathrm{R}$ is an equivalence relation.

Statement II : For some $(\mathrm{a}, \mathrm{b}) \in \mathrm{X}$, the $\operatorname{set} \mathrm{S}=\{(x, y) \in \mathrm{X}:(x, y) \mathrm{R}(\mathrm{a}, \mathrm{b})\}$ represents a line parallel to $y=x$.

In the light of the above statements, choose the correct answer from the options given below :

A
Both Statement I and Statement II are true
B
Statement I is true but Statement II is false
C
Both Statement I and Statement II are false
D
Statement I is false but Statement II is true
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