The value of $\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e x\right)^2+1\right)^{-1}}}\right) d x$ is

1

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)

2

$ 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)

3

The value of $(\sin 70^\circ)\,\big(\cot 10^\circ \cot 70^\circ - 1\big)$ is:

Topic: JEE Main 2025 (23 January Morning Shift)

4

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)

5

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)

6

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…

Topic: JEE Main 2025 (23 January Morning Shift)

7

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)

8

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)

9

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…

Topic: JEE Main 2025 (23 January Morning Shift)

10

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e 2,\,k)$. If the curve satisfies the differential equa…

Topic: JEE Main 2025 (23 January Morning Shift)

1

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)

2

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)

3

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)

4

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)

5

If the first term of an A.P. is $3$ and the sum of its first four terms is equal to one-fifth of the sum of the next fo…

Topic: JEE Main 2025 (23 January Morning Shift)

6

Let the arc $AC$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $AC$ divides the arc …

Topic: JEE Main 2025 (23 January Morning Shift)

7

Let the area of $\triangle PQR$ with vertices $P(5,4),\ Q(-2,4)$ and $R(a,b)$ be $35$ square units. If its orthocenter …

Topic: JEE Main 2025 (23 January Morning Shift)

8

If $A,B$ and $\big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\big)$ are non-singular matrices of the same or…

Topic: JEE Main 2025 (23 January Morning Shift)

9

Let $R=\{(1,2),(2,3),(3,3)\}$ be a relation on the set $\{1,2,3,4\}$. The minimum number of ordered pairs that must be …

Topic: JEE Main 2025 (23 January Morning Shift)