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

Let the arc $A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac{\text { length of } \operatorname{arc} A B}{\text { length of } \operatorname{arc} B C}=\frac{1}{5}$, and $\overrightarrow{O C}=\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}$, then $\alpha+\sqrt{2}(\sqrt{3}-1) \beta$ is equal to

A
$2 \sqrt{3}$
B
$5 \sqrt{3}$
C
$2+\sqrt{3}$
D
$2-\sqrt{3}$
2
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the area of a $\triangle P Q R$ with vertices $P(5,4), Q(-2,4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $c+2 d$ is equal to

A
$3$
B
$\frac{7}{3}$
C
$2$
D
$\frac{8}{3}$
3
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $\mathrm{A}, \mathrm{B}, \operatorname{and}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of $A\left(\operatorname{adj}\left(A^{-1}\right)+\operatorname{adj}\left(B^{-1}\right)\right)^{-1} B$, is equal to

A
$\frac{A B^{-1}}{|A|}+\frac{B A^{-1}}{|B|}$
B
$\operatorname{adj}\left(\mathrm{B}^{-1}\right)+\operatorname{adj}\left(\mathrm{A}^{-1}\right)$
C
$\mathrm{AB}^{-1}+\mathrm{A}^{-1} \mathrm{~B}$
D
$\frac{1}{|A B|}(\operatorname{adj}(B)+\operatorname{adj}(A))$
4
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $\mathrm{R}=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

A
9
B
8
C
7
D
10
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