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

Let P be the foot of the perpendicular from the point $\mathrm{Q}(10,-3,-1)$ on the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}$. Then the area of the right angled triangle $P Q R$, where $R$ is the point $(3,-2,1)$, is

A
 $\sqrt{30}$
B
$9 \sqrt{15}$
C
$3 \sqrt{30}$
D
$8 \sqrt{15}$
2
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is :

A
34000
B
37000
C
35000
D
36000
3
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $f(x)=\log _{\mathrm{e}} x$ and $g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}$. Then the domain of $f \circ g$ is

A
$(0, \infty)$
B
$[1, \infty)$
C
$\mathbb{R}$
D
$[0, \infty)$
4
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the position vectors of the vertices $\mathrm{A}, \mathrm{B}$ and C of a tetrahedron ABCD be $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-2 \hat{k}$ and $2 \hat{i}+\hat{j}-\hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $A B C$ meets the median line segment through $A$ of the triangle $A B C$ at the point $E$. If the length of $A D$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6 \sqrt{2}}$, then the position vector of E is

A
$\frac{1}{6}(7 \hat{\mathrm{i}}+12 \hat{\mathrm{j}}+\hat{\mathrm{k}})$
B
$\frac{1}{12}(7 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$
C
$\frac{1}{6}(12 \hat{i}+12 \hat{j}+\hat{k})$
D
$\frac{1}{2}(\hat{i}+4 \hat{j}+7 \hat{k})$
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