From all the English alphabets, five letters are chosen and arranged in alphabetical order. The total number of ways in which the middle letter is M is:

1

The product of all solutions of the equation $e^{5(\log_e x)^2+3}=x^8,\ x>0$, is:

Topic: JEE Main 2025 (22 January Morning Shift)

2

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the…

Topic: JEE Main 2025 (22 January Morning Shift)

3

Let $A=\{1,2,3, \ldots, 10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m< n\right.$ and $\left.\operatorname{gcd}(m, n)=1…

Topic: JEE Main 2025 (22 January Morning Shift)

4

Let $L_1:\ \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $L_2:\ \dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$ be tw…

Topic: JEE Main 2025 (22 January Morning Shift)

5

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=-\frac{\pi}{4}$, $\arg(z_2)=0$ …

Topic: JEE Main 2025 (22 January Morning Shift)

6

Let the foci of a hyperbola be $(1,14)$ and $(1,-12)$. If it passes through the point $(1,6)$, then the length of its l…

Topic: JEE Main 2025 (22 January Morning Shift)

7

The area of the region, inside the circle $ (x - 2\sqrt{3})^2 + y^2 = 12 $ and outside the parabola $ y^2 = 2\sqrt{3}x …

Topic: JEE Main 2025 (22 January Morning Shift)

1

A circle $C$ of radius $2$ lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a…

Topic: JEE Main 2025 (22 January Morning Shift)

2

Let $x=x(y)$ be the solution of the differential equation $y^2\,dx+\left(x-\dfrac{1}{y}\right)dy=0$. If $x(1)=1$, then …

Topic: JEE Main 2025 (22 January Morning Shift)

3

A coin is tossed three times. Let X denote the number of times a tail follows a head. If \mu and \sigma^2 denote the me…

Topic: JEE Main 2025 (22 January Morning Shift)

4

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for …

Topic: JEE Main 2025 (22 January Morning Shift)

5

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the…

Topic: JEE Main 2025 (22 January Morning Shift)

6

Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive terms. If $a_1a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal t…

Topic: JEE Main 2025 (22 January Morning Shift)

7

The number of non-empty equivalence relations on the set {1,2,3} is:

Topic: JEE Main 2025 (22 January Morning Shift)

8

Let for $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x, \quad \mathrm{I}_1=\int_0^{\pi / 4} f(x) \mathrm{d} x$ a…

Topic: JEE Main 2025 (22 January Morning Shift)

9

Let the parabola $y=x^2+px-3$ meet the coordinate axes at the points $P,Q,R$. If the circle $C$ with centre at $(-1,-1)…

Topic: JEE Main 2025 (22 January Morning Shift)

10

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16…

Topic: JEE Main 2025 (22 January Morning Shift)