If $z_1,z_2,z_3\in\mathbb{C}$ are the vertices of an equilateral triangle whose centroid is $z_0$, then $\displaystyle \sum_{k=1}^{3}(z_k-z_0)^2$ is equal to

1

Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$- and $z$-axes, respectively, is h…

Topic: JEE Main 2025 (3 April Evening Shift)

2

Line $L_1$ of slope $2$ and line $L_2$ of slope $\dfrac{1}{2}$ intersect at the origin $O$. In the first quadrant, $P_1…

Topic: JEE Main 2025 (3 April Evening Shift)

3

The area of the region ${(x,y): |x-y|\le y \le 4\sqrt{x}}$ is

Topic: JEE Main 2025 (3 April Evening Shift)

1

Let $C$ be the circle of minimum area enclosing the ellipse $E:\ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ with eccentricity…

Topic: JEE Main 2025 (3 April Evening Shift)

2

The shortest distance between the curves $y^2=8x$ and $x^2+y^2+12y+35=0$ is:

Topic: JEE Main 2025 (3 April Evening Shift)

3

Consider the lines $x(3\lambda+1)+y(7\lambda+2)=17\lambda+5$, $\lambda$ being a parameter, all passing through a point …

Topic: JEE Main 2025 (3 April Evening Shift)

4

Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}+3\tan^2 x,y+3y=\sec^2 x$, $y(0)=\dfrac{1}{3}+…

Topic: JEE Main 2025 (3 April Evening Shift)

5

Let $f$ be a function such that $f(x)+3f\left(\dfrac{24}{x}\right)=4x,; x\ne0$. Then $f(3)+f(8)$ is equal to

Topic: JEE Main 2025 (3 April Evening Shift)

6

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k,\dfrac{k}{2}\right)$ from t…

Topic: JEE Main 2025 (3 April Evening Shift)

7

Let $A={-2,-1,0,1,2,3}$. Let $R$ be a relation on $A$ defined by $xRy$ iff $y=\max{x,1}$. Let $l$ be the number of elem…

Topic: JEE Main 2025 (3 April Evening Shift)

8

Let the mean and variance of five observations $x_1=1,\ x_2=3,\ x_3=a,\ x_4=7,\ x_5=b,\ a>b$ be $5$ and $10$ respective…

Topic: JEE Main 2025 (3 April Evening Shift)

9

The integral $\displaystyle \int_{0}^{\pi}\frac{8x,dx}{4\cos^{2}x+\sin^{2}x}$ is equal to

Topic: JEE Main 2025 (3 April Evening Shift)

10

The distance of the point $(7,10,11)$ from the line $\dfrac{x-4}{1}=\dfrac{y-4}{0}=\dfrac{z-2}{3}$ along the line $\dfr…

Topic: JEE Main 2025 (3 April Evening Shift)