Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be $\hat{i}+2\hat{j}+\hat{k}$, $\hat{i}+3\hat{j}-2\hat{k}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median through $A$ of $\triangle ABC$ at the point $E$. If the length of $AD$ is $\dfrac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\dfrac{\sqrt{805}}{6\sqrt{2}}$, then the position vector of $E$ is:
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Let $f(x)=\log_e x$ and $g(x)=\dfrac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$. Then the domain of $f\circ g$ is:
Topic: JEE Main 2025 (23 January Morning Shift)
The number of words that can be formed using all the letters of the word "DAUGHTER" such that all the vowels never come…
Topic: JEE Main 2025 (23 January Morning Shift)
Let $P$ be the foot of the perpendicular from the point $Q(10,-3,-1)$ on the line $\dfrac{x-3}{7}=\dfrac{y-2}{-1}=\dfra…
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Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e 2,\,k)$. If the curve satisfies the differential equa…
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If $\dfrac{\pi}{2}\le x\le \dfrac{3\pi}{4}$, then $\cos^{-1}\!\left(\dfrac{12}{13}\cos x+\dfrac{5}{13}\sin x\right)$ is…
Topic: JEE Main 2025 (23 January Morning Shift)
One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face m…
Topic: JEE Main 2025 (23 January Morning Shift)
The value of $(\sin 70^\circ)\,\big(\cot 10^\circ \cot 70^\circ - 1\big)$ is:
Topic: JEE Main 2025 (23 January Morning Shift)
$
f(x)=\left\{\begin{array}{l}
\frac{2}{x}\left\{\sin \left(k_1+1\right) x+\sin \left(k_2-1\right) x\right\}, \quad x0
…
Topic: JEE Main 2025 (23 January Morning Shift)
If the system of equations
$(\lambda-1)x+(\lambda-4)y+\lambda z=5$
$\lambda x+(\lambda-1)y+(\lambda-4)z=7$
$(\lambd…
Topic: JEE Main 2025 (23 January Morning Shift)
The value of $\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(…
Topic: JEE Main 2025 (23 January Morning Shift)
Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. T…
Topic: JEE Main 2025 (23 January Morning Shift)
Let $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}, z \in C$, be the equation of a circle with center at $C$. …
Topic: JEE Main 2025 (23 January Morning Shift)
Let $\mathrm{I}(x)=\int \frac{d x}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\f…
Topic: JEE Main 2025 (23 January Morning Shift)
If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the…
Topic: JEE Main 2025 (23 January Morning Shift)
